xref: /libCEED/interface/ceed-preconditioning.c (revision 6574a04ff2135c3834f1b6ef9a4ec7566c4782db)

// Copyright (c) 2017-2022, Lawrence Livermore National Security, LLC and other CEED contributors.
// All Rights Reserved. See the top-level LICENSE and NOTICE files for details.
//
// SPDX-License-Identifier: BSD-2-Clause
//
// This file is part of CEED:  http://github.com/ceed

#include <assert.h>
#include <ceed-impl.h>
#include <ceed.h>
#include <ceed/backend.h>
#include <math.h>
#include <stdbool.h>
#include <stdio.h>
#include <string.h>

/// @file
/// Implementation of CeedOperator preconditioning interfaces

/// ----------------------------------------------------------------------------
/// CeedOperator Library Internal Preconditioning Functions
/// ----------------------------------------------------------------------------
/// @addtogroup CeedOperatorDeveloper
/// @{

/**
  @brief Duplicate a CeedQFunction with a reference Ceed to fallback for advanced CeedOperator functionality

  @param[in]  fallback_ceed Ceed on which to create fallback CeedQFunction
  @param[in]  qf            CeedQFunction to create fallback for
  @param[out] qf_fallback   fallback CeedQFunction

  @return An error code: 0 - success, otherwise - failure

  @ref Developer
**/
static int CeedQFunctionCreateFallback(Ceed fallback_ceed, CeedQFunction qf, CeedQFunction *qf_fallback) {
  // Check if NULL qf passed in
  if (!qf) return CEED_ERROR_SUCCESS;

  CeedDebug256(qf->ceed, 1, "---------- CeedOperator Fallback ----------\n");
  CeedDebug(qf->ceed, "Creating fallback CeedQFunction\n");

  char *source_path_with_name = "";
  if (qf->source_path) {
    size_t path_len = strlen(qf->source_path), name_len = strlen(qf->kernel_name);
    CeedCall(CeedCalloc(path_len + name_len + 2, &source_path_with_name));
    memcpy(source_path_with_name, qf->source_path, path_len);
    memcpy(&source_path_with_name[path_len], ":", 1);
    memcpy(&source_path_with_name[path_len + 1], qf->kernel_name, name_len);
  } else {
    CeedCall(CeedCalloc(1, &source_path_with_name));
  }

  CeedCall(CeedQFunctionCreateInterior(fallback_ceed, qf->vec_length, qf->function, source_path_with_name, qf_fallback));
  {
    CeedQFunctionContext ctx;

    CeedCall(CeedQFunctionGetContext(qf, &ctx));
    CeedCall(CeedQFunctionSetContext(*qf_fallback, ctx));
  }
  for (CeedInt i = 0; i < qf->num_input_fields; i++) {
    CeedCall(CeedQFunctionAddInput(*qf_fallback, qf->input_fields[i]->field_name, qf->input_fields[i]->size, qf->input_fields[i]->eval_mode));
  }
  for (CeedInt i = 0; i < qf->num_output_fields; i++) {
    CeedCall(CeedQFunctionAddOutput(*qf_fallback, qf->output_fields[i]->field_name, qf->output_fields[i]->size, qf->output_fields[i]->eval_mode));
  }
  CeedCall(CeedFree(&source_path_with_name));

  return CEED_ERROR_SUCCESS;
}

/**
  @brief Duplicate a CeedOperator with a reference Ceed to fallback for advanced CeedOperator functionality

  @param[in,out] op CeedOperator to create fallback for

  @return An error code: 0 - success, otherwise - failure

  @ref Developer
**/
static int CeedOperatorCreateFallback(CeedOperator op) {
  bool is_composite;
  Ceed ceed_fallback;

  // Check not already created
  if (op->op_fallback) return CEED_ERROR_SUCCESS;

  // Fallback Ceed
  CeedCall(CeedGetOperatorFallbackCeed(op->ceed, &ceed_fallback));
  if (!ceed_fallback) return CEED_ERROR_SUCCESS;

  CeedDebug256(op->ceed, 1, "---------- CeedOperator Fallback ----------\n");
  CeedDebug(op->ceed, "Creating fallback CeedOperator\n");

  // Clone Op
  CeedOperator op_fallback;
  CeedCall(CeedOperatorIsComposite(op, &is_composite));
  if (is_composite) {
    CeedInt       num_suboperators;
    CeedOperator *sub_operators;

    CeedCall(CeedCompositeOperatorCreate(ceed_fallback, &op_fallback));
    CeedCall(CeedCompositeOperatorGetNumSub(op, &num_suboperators));
    CeedCall(CeedCompositeOperatorGetSubList(op, &sub_operators));
    for (CeedInt i = 0; i < num_suboperators; i++) {
      CeedOperator op_sub_fallback;

      CeedCall(CeedOperatorGetFallback(sub_operators[i], &op_sub_fallback));
      CeedCall(CeedCompositeOperatorAddSub(op_fallback, op_sub_fallback));
    }
  } else {
    CeedQFunction qf_fallback = NULL, dqf_fallback = NULL, dqfT_fallback = NULL;
    CeedCall(CeedQFunctionCreateFallback(ceed_fallback, op->qf, &qf_fallback));
    CeedCall(CeedQFunctionCreateFallback(ceed_fallback, op->dqf, &dqf_fallback));
    CeedCall(CeedQFunctionCreateFallback(ceed_fallback, op->dqfT, &dqfT_fallback));
    CeedCall(CeedOperatorCreate(ceed_fallback, qf_fallback, dqf_fallback, dqfT_fallback, &op_fallback));
    for (CeedInt i = 0; i < op->qf->num_input_fields; i++) {
      CeedCall(CeedOperatorSetField(op_fallback, op->input_fields[i]->field_name, op->input_fields[i]->elem_rstr, op->input_fields[i]->basis,
                                    op->input_fields[i]->vec));
    }
    for (CeedInt i = 0; i < op->qf->num_output_fields; i++) {
      CeedCall(CeedOperatorSetField(op_fallback, op->output_fields[i]->field_name, op->output_fields[i]->elem_rstr, op->output_fields[i]->basis,
                                    op->output_fields[i]->vec));
    }
    CeedCall(CeedQFunctionAssemblyDataReferenceCopy(op->qf_assembled, &op_fallback->qf_assembled));
    if (op_fallback->num_qpts == 0) {
      CeedCall(CeedOperatorSetNumQuadraturePoints(op_fallback, op->num_qpts));
    }
    // Cleanup
    CeedCall(CeedQFunctionDestroy(&qf_fallback));
    CeedCall(CeedQFunctionDestroy(&dqf_fallback));
    CeedCall(CeedQFunctionDestroy(&dqfT_fallback));
  }
  CeedCall(CeedOperatorSetName(op_fallback, op->name));
  CeedCall(CeedOperatorCheckReady(op_fallback));
  op->op_fallback = op_fallback;

  return CEED_ERROR_SUCCESS;
}

/**
  @brief Retrieve fallback CeedOperator with a reference Ceed for advanced CeedOperator functionality

  @param[in]  op          CeedOperator to retrieve fallback for
  @param[out] op_fallback Fallback CeedOperator

  @return An error code: 0 - success, otherwise - failure

  @ref Developer
**/
int CeedOperatorGetFallback(CeedOperator op, CeedOperator *op_fallback) {
  // Create if needed
  if (!op->op_fallback) {
    CeedCall(CeedOperatorCreateFallback(op));
  }
  if (op->op_fallback) {
    bool is_debug;

    CeedCall(CeedIsDebug(op->ceed, &is_debug));
    if (is_debug) {
      Ceed        ceed, ceed_fallback;
      const char *resource, *resource_fallback;

      CeedCall(CeedOperatorGetCeed(op, &ceed));
      CeedCall(CeedGetOperatorFallbackCeed(ceed, &ceed_fallback));
      CeedCall(CeedGetResource(ceed, &resource));
      CeedCall(CeedGetResource(ceed_fallback, &resource_fallback));

      CeedDebug256(ceed, 1, "---------- CeedOperator Fallback ----------\n");
      CeedDebug(ceed, "Falling back from %s operator at address %ld to %s operator at address %ld\n", resource, op, resource_fallback,
                op->op_fallback);
    }
  }
  *op_fallback = op->op_fallback;

  return CEED_ERROR_SUCCESS;
}

/**
  @brief Select correct basis matrix pointer based on CeedEvalMode

  @param[in]  basis     CeedBasis from which to get the basis matrix
  @param[in]  eval_mode Current basis evaluation mode
  @param[in]  identity  Pointer to identity matrix
  @param[out] basis_ptr Basis pointer to set

  @ref Developer
**/
static inline int CeedOperatorGetBasisPointer(CeedBasis basis, CeedEvalMode eval_mode, const CeedScalar *identity, const CeedScalar **basis_ptr) {
  switch (eval_mode) {
    case CEED_EVAL_NONE:
      *basis_ptr = identity;
      break;
    case CEED_EVAL_INTERP:
      CeedCall(CeedBasisGetInterp(basis, basis_ptr));
      break;
    case CEED_EVAL_GRAD:
      CeedCall(CeedBasisGetGrad(basis, basis_ptr));
      break;
    case CEED_EVAL_DIV:
      CeedCall(CeedBasisGetDiv(basis, basis_ptr));
      break;
    case CEED_EVAL_CURL:
      CeedCall(CeedBasisGetCurl(basis, basis_ptr));
      break;
    case CEED_EVAL_WEIGHT:
      break;  // Caught by QF Assembly
  }
  assert(*basis_ptr != NULL);

  return CEED_ERROR_SUCCESS;
}

/**
  @brief Create point block restriction for active operator field

  @param[in]  rstr            Original CeedElemRestriction for active field
  @param[out] pointblock_rstr Address of the variable where the newly created CeedElemRestriction will be stored

  @return An error code: 0 - success, otherwise - failure

  @ref Developer
**/
static int CeedOperatorCreateActivePointBlockRestriction(CeedElemRestriction rstr, CeedElemRestriction *pointblock_rstr) {
  Ceed ceed;
  CeedCall(CeedElemRestrictionGetCeed(rstr, &ceed));
  const CeedInt *offsets;
  CeedCall(CeedElemRestrictionGetOffsets(rstr, CEED_MEM_HOST, &offsets));

  // Expand offsets
  CeedInt  num_elem, num_comp, elem_size, comp_stride, *pointblock_offsets;
  CeedSize l_size;
  CeedCall(CeedElemRestrictionGetNumElements(rstr, &num_elem));
  CeedCall(CeedElemRestrictionGetNumComponents(rstr, &num_comp));
  CeedCall(CeedElemRestrictionGetElementSize(rstr, &elem_size));
  CeedCall(CeedElemRestrictionGetCompStride(rstr, &comp_stride));
  CeedCall(CeedElemRestrictionGetLVectorSize(rstr, &l_size));
  CeedInt shift = num_comp;
  if (comp_stride != 1) shift *= num_comp;
  CeedCall(CeedCalloc(num_elem * elem_size, &pointblock_offsets));
  for (CeedInt i = 0; i < num_elem * elem_size; i++) {
    pointblock_offsets[i] = offsets[i] * shift;
  }

  // Create new restriction
  CeedCall(CeedElemRestrictionCreate(ceed, num_elem, elem_size, num_comp * num_comp, 1, l_size * num_comp, CEED_MEM_HOST, CEED_OWN_POINTER,
                                     pointblock_offsets, pointblock_rstr));

  // Cleanup
  CeedCall(CeedElemRestrictionRestoreOffsets(rstr, &offsets));

  return CEED_ERROR_SUCCESS;
}

/**
  @brief Core logic for assembling operator diagonal or point block diagonal

  @param[in]  op            CeedOperator to assemble point block diagonal
  @param[in]  request       Address of CeedRequest for non-blocking completion, else CEED_REQUEST_IMMEDIATE
  @param[in]  is_pointblock Boolean flag to assemble diagonal or point block diagonal
  @param[out] assembled     CeedVector to store assembled diagonal

  @return An error code: 0 - success, otherwise - failure

  @ref Developer
**/
static inline int CeedSingleOperatorAssembleAddDiagonal_Core(CeedOperator op, CeedRequest *request, const bool is_pointblock, CeedVector assembled) {
  Ceed ceed;
  CeedCall(CeedOperatorGetCeed(op, &ceed));

  // Assemble QFunction
  CeedQFunction       qf;
  const CeedScalar   *assembled_qf_array;
  CeedVector          assembled_qf;
  CeedElemRestriction assembled_elem_rstr;
  CeedInt             num_input_fields, num_output_fields;
  CeedInt             layout[3];

  CeedCall(CeedOperatorGetQFunction(op, &qf));
  CeedCall(CeedQFunctionGetNumArgs(qf, &num_input_fields, &num_output_fields));
  CeedCall(CeedOperatorLinearAssembleQFunctionBuildOrUpdate(op, &assembled_qf, &assembled_elem_rstr, request));
  CeedCall(CeedElemRestrictionGetELayout(assembled_elem_rstr, &layout));
  CeedCall(CeedElemRestrictionDestroy(&assembled_elem_rstr));
  CeedCall(CeedVectorGetArrayRead(assembled_qf, CEED_MEM_HOST, &assembled_qf_array));

  // Get assembly data
  CeedOperatorAssemblyData data;
  const CeedEvalMode     **eval_modes_in, **eval_modes_out;
  CeedInt                 *num_eval_modes_in, *num_eval_modes_out, num_active_bases;
  CeedSize               **eval_mode_offsets_in, **eval_mode_offsets_out, num_output_components;
  CeedBasis               *active_bases;
  CeedElemRestriction     *active_elem_rstrs;
  CeedCall(CeedOperatorGetOperatorAssemblyData(op, &data));
  CeedCall(CeedOperatorAssemblyDataGetEvalModes(data, &num_active_bases, &num_eval_modes_in, &eval_modes_in, &eval_mode_offsets_in,
                                                &num_eval_modes_out, &eval_modes_out, &eval_mode_offsets_out, &num_output_components));
  CeedCall(CeedOperatorAssemblyDataGetBases(data, NULL, &active_bases, NULL, NULL));
  CeedCall(CeedOperatorAssemblyDataGetElemRestrictions(data, NULL, &active_elem_rstrs));

  // Loop over all active bases
  for (CeedInt b = 0; b < num_active_bases; b++) {
    // Assemble point block diagonal restriction, if needed
    CeedElemRestriction diag_elem_rstr = active_elem_rstrs[b];

    if (is_pointblock) {
      CeedElemRestriction point_block_elem_rstr;

      CeedCall(CeedOperatorCreateActivePointBlockRestriction(diag_elem_rstr, &point_block_elem_rstr));
      diag_elem_rstr = point_block_elem_rstr;
    }

    // Create diagonal vector
    CeedVector elem_diag;
    CeedCall(CeedElemRestrictionCreateVector(diag_elem_rstr, NULL, &elem_diag));

    // Assemble element operator diagonals
    CeedScalar *elem_diag_array;
    CeedInt     num_elem, num_nodes, num_qpts, num_components;

    CeedCall(CeedVectorSetValue(elem_diag, 0.0));
    CeedCall(CeedVectorGetArray(elem_diag, CEED_MEM_HOST, &elem_diag_array));
    CeedCall(CeedElemRestrictionGetNumElements(diag_elem_rstr, &num_elem));
    CeedCall(CeedBasisGetNumNodes(active_bases[b], &num_nodes));
    CeedCall(CeedBasisGetNumComponents(active_bases[b], &num_components));
    CeedCall(CeedBasisGetNumQuadraturePoints(active_bases[b], &num_qpts));

    // Construct identity matrix for basis if required
    bool        has_eval_none = false;
    CeedScalar *identity      = NULL;
    for (CeedInt i = 0; i < num_eval_modes_in[b]; i++) {
      has_eval_none = has_eval_none || (eval_modes_in[b][i] == CEED_EVAL_NONE);
    }
    for (CeedInt i = 0; i < num_eval_modes_out[b]; i++) {
      has_eval_none = has_eval_none || (eval_modes_out[b][i] == CEED_EVAL_NONE);
    }
    if (has_eval_none) {
      CeedCall(CeedCalloc(num_qpts * num_nodes, &identity));
      for (CeedInt i = 0; i < (num_nodes < num_qpts ? num_nodes : num_qpts); i++) identity[i * num_nodes + i] = 1.0;
    }

    // Compute the diagonal of B^T D B
    // Each element
    for (CeedInt e = 0; e < num_elem; e++) {
      // Each basis eval mode pair
      CeedInt      d_out              = 0, q_comp_out;
      CeedEvalMode eval_mode_out_prev = CEED_EVAL_NONE;
      for (CeedInt e_out = 0; e_out < num_eval_modes_out[b]; e_out++) {
        const CeedScalar *B_t = NULL;
        CeedOperatorGetBasisPointer(active_bases[b], eval_modes_out[b][e_out], identity, &B_t);
        CeedCall(CeedBasisGetNumQuadratureComponents(active_bases[b], eval_modes_out[b][e_out], &q_comp_out));
        if (q_comp_out > 1) {
          if (e_out == 0 || eval_modes_out[b][e_out] != eval_mode_out_prev) d_out = 0;
          else B_t = &B_t[(++d_out) * num_qpts * num_nodes];
        }
        eval_mode_out_prev = eval_modes_out[b][e_out];

        CeedInt      d_in              = 0, q_comp_in;
        CeedEvalMode eval_mode_in_prev = CEED_EVAL_NONE;
        for (CeedInt e_in = 0; e_in < num_eval_modes_in[b]; e_in++) {
          const CeedScalar *B = NULL;
          CeedOperatorGetBasisPointer(active_bases[b], eval_modes_in[b][e_in], identity, &B);
          CeedCall(CeedBasisGetNumQuadratureComponents(active_bases[b], eval_modes_in[b][e_in], &q_comp_in));
          if (q_comp_in > 1) {
            if (e_in == 0 || eval_modes_in[b][e_in] != eval_mode_in_prev) d_in = 0;
            else B = &B[(++d_in) * num_qpts * num_nodes];
          }
          eval_mode_in_prev = eval_modes_in[b][e_in];

          // Each component
          for (CeedInt c_out = 0; c_out < num_components; c_out++) {
            // Each qpt/node pair
            for (CeedInt q = 0; q < num_qpts; q++) {
              if (is_pointblock) {
                // Point Block Diagonal
                for (CeedInt c_in = 0; c_in < num_components; c_in++) {
                  const CeedInt c_offset = (eval_mode_offsets_in[b][e_in] + c_in) * num_output_components + eval_mode_offsets_out[b][e_out] + c_out;
                  const CeedScalar qf_value = assembled_qf_array[q * layout[0] + c_offset * layout[1] + e * layout[2]];
                  for (CeedInt n = 0; n < num_nodes; n++) {
                    elem_diag_array[((e * num_components + c_out) * num_components + c_in) * num_nodes + n] +=
                        B_t[q * num_nodes + n] * qf_value * B[q * num_nodes + n];
                  }
                }
              } else {
                // Diagonal Only
                const CeedInt    c_offset = (eval_mode_offsets_in[b][e_in] + c_out) * num_output_components + eval_mode_offsets_out[b][e_out] + c_out;
                const CeedScalar qf_value = assembled_qf_array[q * layout[0] + c_offset * layout[1] + e * layout[2]];
                for (CeedInt n = 0; n < num_nodes; n++) {
                  elem_diag_array[(e * num_components + c_out) * num_nodes + n] += B_t[q * num_nodes + n] * qf_value * B[q * num_nodes + n];
                }
              }
            }
          }
        }
      }
    }
    CeedCall(CeedVectorRestoreArray(elem_diag, &elem_diag_array));

    // Assemble local operator diagonal
    CeedCall(CeedElemRestrictionApply(diag_elem_rstr, CEED_TRANSPOSE, elem_diag, assembled, request));

    // Cleanup
    if (is_pointblock) CeedCall(CeedElemRestrictionDestroy(&diag_elem_rstr));
    CeedCall(CeedVectorDestroy(&elem_diag));
    CeedCall(CeedFree(&identity));
  }
  CeedCall(CeedVectorRestoreArrayRead(assembled_qf, &assembled_qf_array));
  CeedCall(CeedVectorDestroy(&assembled_qf));

  return CEED_ERROR_SUCCESS;
}

/**
  @brief Core logic for assembling composite operator diagonal

  @param[in]  op            CeedOperator to assemble point block diagonal
  @param[in]  request       Address of CeedRequest for non-blocking completion, else CEED_REQUEST_IMMEDIATE
  @param[in]  is_pointblock Boolean flag to assemble diagonal or point block diagonal
  @param[out] assembled     CeedVector to store assembled diagonal

  @return An error code: 0 - success, otherwise - failure

  @ref Developer
**/
static inline int CeedCompositeOperatorLinearAssembleAddDiagonal(CeedOperator op, CeedRequest *request, const bool is_pointblock,
                                                                 CeedVector assembled) {
  CeedInt       num_sub;
  CeedOperator *suboperators;
  CeedCall(CeedCompositeOperatorGetNumSub(op, &num_sub));
  CeedCall(CeedCompositeOperatorGetSubList(op, &suboperators));
  for (CeedInt i = 0; i < num_sub; i++) {
    if (is_pointblock) {
      CeedCall(CeedOperatorLinearAssembleAddPointBlockDiagonal(suboperators[i], assembled, request));
    } else {
      CeedCall(CeedOperatorLinearAssembleAddDiagonal(suboperators[i], assembled, request));
    }
  }
  return CEED_ERROR_SUCCESS;
}

/**
  @brief Build nonzero pattern for non-composite operator

  Users should generally use CeedOperatorLinearAssembleSymbolic()

  @param[in]  op     CeedOperator to assemble nonzero pattern
  @param[in]  offset Offset for number of entries
  @param[out] rows   Row number for each entry
  @param[out] cols   Column number for each entry

  @return An error code: 0 - success, otherwise - failure

  @ref Developer
**/
static int CeedSingleOperatorAssembleSymbolic(CeedOperator op, CeedInt offset, CeedInt *rows, CeedInt *cols) {
  Ceed ceed;
  bool is_composite;
  CeedCall(CeedOperatorGetCeed(op, &ceed));
  CeedCall(CeedOperatorIsComposite(op, &is_composite));

  CeedCheck(!is_composite, ceed, CEED_ERROR_UNSUPPORTED, "Composite operator not supported");

  CeedSize num_nodes;
  CeedCall(CeedOperatorGetActiveVectorLengths(op, &num_nodes, NULL));
  CeedElemRestriction rstr_in;
  CeedCall(CeedOperatorGetActiveElemRestriction(op, &rstr_in));
  CeedInt num_elem, elem_size, num_comp;
  CeedCall(CeedElemRestrictionGetNumElements(rstr_in, &num_elem));
  CeedCall(CeedElemRestrictionGetElementSize(rstr_in, &elem_size));
  CeedCall(CeedElemRestrictionGetNumComponents(rstr_in, &num_comp));
  CeedInt layout_er[3];
  CeedCall(CeedElemRestrictionGetELayout(rstr_in, &layout_er));

  CeedInt local_num_entries = elem_size * num_comp * elem_size * num_comp * num_elem;

  // Determine elem_dof relation
  CeedVector index_vec;
  CeedCall(CeedVectorCreate(ceed, num_nodes, &index_vec));
  CeedScalar *array;
  CeedCall(CeedVectorGetArrayWrite(index_vec, CEED_MEM_HOST, &array));
  for (CeedInt i = 0; i < num_nodes; i++) array[i] = i;
  CeedCall(CeedVectorRestoreArray(index_vec, &array));
  CeedVector elem_dof;
  CeedCall(CeedVectorCreate(ceed, num_elem * elem_size * num_comp, &elem_dof));
  CeedCall(CeedVectorSetValue(elem_dof, 0.0));
  CeedCall(CeedElemRestrictionApply(rstr_in, CEED_NOTRANSPOSE, index_vec, elem_dof, CEED_REQUEST_IMMEDIATE));
  const CeedScalar *elem_dof_a;
  CeedCall(CeedVectorGetArrayRead(elem_dof, CEED_MEM_HOST, &elem_dof_a));
  CeedCall(CeedVectorDestroy(&index_vec));

  // Determine i, j locations for element matrices
  CeedInt count = 0;
  for (CeedInt e = 0; e < num_elem; e++) {
    for (CeedInt comp_in = 0; comp_in < num_comp; comp_in++) {
      for (CeedInt comp_out = 0; comp_out < num_comp; comp_out++) {
        for (CeedInt i = 0; i < elem_size; i++) {
          for (CeedInt j = 0; j < elem_size; j++) {
            const CeedInt elem_dof_index_row = i * layout_er[0] + (comp_out)*layout_er[1] + e * layout_er[2];
            const CeedInt elem_dof_index_col = j * layout_er[0] + comp_in * layout_er[1] + e * layout_er[2];

            const CeedInt row = elem_dof_a[elem_dof_index_row];
            const CeedInt col = elem_dof_a[elem_dof_index_col];

            rows[offset + count] = row;
            cols[offset + count] = col;
            count++;
          }
        }
      }
    }
  }
  CeedCheck(count == local_num_entries, ceed, CEED_ERROR_MAJOR, "Error computing assembled entries");
  CeedCall(CeedVectorRestoreArrayRead(elem_dof, &elem_dof_a));
  CeedCall(CeedVectorDestroy(&elem_dof));

  return CEED_ERROR_SUCCESS;
}

/**
  @brief Assemble nonzero entries for non-composite operator

  Users should generally use CeedOperatorLinearAssemble()

  @param[in]  op     CeedOperator to assemble
  @param[in]  offset Offset for number of entries
  @param[out] values Values to assemble into matrix

  @return An error code: 0 - success, otherwise - failure

  @ref Developer
**/
static int CeedSingleOperatorAssemble(CeedOperator op, CeedInt offset, CeedVector values) {
  Ceed ceed;
  bool is_composite;
  CeedCall(CeedOperatorGetCeed(op, &ceed));
  CeedCall(CeedOperatorIsComposite(op, &is_composite));

  CeedCheck(!is_composite, ceed, CEED_ERROR_UNSUPPORTED, "Composite operator not supported");

  // Early exit for empty operator
  {
    CeedInt num_elem = 0;

    CeedCall(CeedOperatorGetNumElements(op, &num_elem));
    if (num_elem == 0) return CEED_ERROR_SUCCESS;
  }

  if (op->LinearAssembleSingle) {
    // Backend version
    CeedCall(op->LinearAssembleSingle(op, offset, values));
    return CEED_ERROR_SUCCESS;
  } else {
    // Operator fallback
    CeedOperator op_fallback;

    CeedCall(CeedOperatorGetFallback(op, &op_fallback));
    if (op_fallback) {
      CeedCall(CeedSingleOperatorAssemble(op_fallback, offset, values));
      return CEED_ERROR_SUCCESS;
    }
  }

  // Assemble QFunction
  CeedQFunction qf;
  CeedCall(CeedOperatorGetQFunction(op, &qf));
  CeedVector          assembled_qf;
  CeedElemRestriction rstr_q;
  CeedCall(CeedOperatorLinearAssembleQFunctionBuildOrUpdate(op, &assembled_qf, &rstr_q, CEED_REQUEST_IMMEDIATE));
  CeedSize qf_length;
  CeedCall(CeedVectorGetLength(assembled_qf, &qf_length));

  CeedInt            num_input_fields, num_output_fields;
  CeedOperatorField *input_fields;
  CeedOperatorField *output_fields;
  CeedCall(CeedOperatorGetFields(op, &num_input_fields, &input_fields, &num_output_fields, &output_fields));

  // Get assembly data
  CeedOperatorAssemblyData data;
  CeedCall(CeedOperatorGetOperatorAssemblyData(op, &data));
  const CeedEvalMode **eval_modes_in, **eval_modes_out;
  CeedInt             *num_eval_modes_in, *num_eval_modes_out, num_active_bases;
  CeedCall(CeedOperatorAssemblyDataGetEvalModes(data, &num_active_bases, &num_eval_modes_in, &eval_modes_in, NULL, &num_eval_modes_out,
                                                &eval_modes_out, NULL, NULL));
  CeedBasis *bases;
  CeedCall(CeedOperatorAssemblyDataGetBases(data, NULL, &bases, NULL, NULL));
  CeedBasis basis_in = bases[0];

  CeedCheck(num_active_bases == 1, ceed, CEED_ERROR_UNSUPPORTED, "Cannot assemble operator with multiple active bases");
  CeedCheck(num_eval_modes_in[0] > 0 && num_eval_modes_out[0] > 0, ceed, CEED_ERROR_UNSUPPORTED, "Cannot assemble operator with out inputs/outputs");

  CeedElemRestriction active_rstr;
  CeedInt             num_elem, elem_size, num_qpts, num_comp;
  CeedCall(CeedOperatorGetActiveElemRestriction(op, &active_rstr));
  CeedCall(CeedElemRestrictionGetNumElements(active_rstr, &num_elem));
  CeedCall(CeedElemRestrictionGetElementSize(active_rstr, &elem_size));
  CeedCall(CeedElemRestrictionGetNumComponents(active_rstr, &num_comp));
  CeedCall(CeedBasisGetNumQuadraturePoints(basis_in, &num_qpts));

  CeedInt local_num_entries = elem_size * num_comp * elem_size * num_comp * num_elem;

  // loop over elements and put in data structure
  const CeedScalar *assembled_qf_array;
  CeedCall(CeedVectorGetArrayRead(assembled_qf, CEED_MEM_HOST, &assembled_qf_array));

  CeedInt layout_qf[3];
  CeedCall(CeedElemRestrictionGetELayout(rstr_q, &layout_qf));
  CeedCall(CeedElemRestrictionDestroy(&rstr_q));

  // we store B_mat_in, B_mat_out, BTD, elem_mat in row-major order
  const CeedScalar **B_mats_in, **B_mats_out;
  CeedCall(CeedOperatorAssemblyDataGetBases(data, NULL, NULL, &B_mats_in, &B_mats_out));
  const CeedScalar *B_mat_in = B_mats_in[0], *B_mat_out = B_mats_out[0];
  CeedScalar        BTD_mat[elem_size * num_qpts * num_eval_modes_in[0]];
  CeedScalar        elem_mat[elem_size * elem_size];
  CeedInt           count = 0;
  CeedScalar       *vals;
  CeedCall(CeedVectorGetArray(values, CEED_MEM_HOST, &vals));
  for (CeedInt e = 0; e < num_elem; e++) {
    for (CeedInt comp_in = 0; comp_in < num_comp; comp_in++) {
      for (CeedInt comp_out = 0; comp_out < num_comp; comp_out++) {
        // Compute B^T*D
        for (CeedInt n = 0; n < elem_size; n++) {
          for (CeedInt q = 0; q < num_qpts; q++) {
            for (CeedInt e_in = 0; e_in < num_eval_modes_in[0]; e_in++) {
              const CeedInt btd_index = n * (num_qpts * num_eval_modes_in[0]) + (num_eval_modes_in[0] * q + e_in);
              CeedScalar    sum       = 0.0;
              for (CeedInt e_out = 0; e_out < num_eval_modes_out[0]; e_out++) {
                const CeedInt b_out_index     = (num_eval_modes_out[0] * q + e_out) * elem_size + n;
                const CeedInt eval_mode_index = ((e_in * num_comp + comp_in) * num_eval_modes_out[0] + e_out) * num_comp + comp_out;
                const CeedInt qf_index        = q * layout_qf[0] + eval_mode_index * layout_qf[1] + e * layout_qf[2];
                sum += B_mat_out[b_out_index] * assembled_qf_array[qf_index];
              }
              BTD_mat[btd_index] = sum;
            }
          }
        }
        // form element matrix itself (for each block component)
        CeedCall(CeedMatrixMatrixMultiply(ceed, BTD_mat, B_mat_in, elem_mat, elem_size, elem_size, num_qpts * num_eval_modes_in[0]));

        // put element matrix in coordinate data structure
        for (CeedInt i = 0; i < elem_size; i++) {
          for (CeedInt j = 0; j < elem_size; j++) {
            vals[offset + count] = elem_mat[i * elem_size + j];
            count++;
          }
        }
      }
    }
  }
  CeedCheck(count == local_num_entries, ceed, CEED_ERROR_MAJOR, "Error computing entries");
  CeedCall(CeedVectorRestoreArray(values, &vals));

  CeedCall(CeedVectorRestoreArrayRead(assembled_qf, &assembled_qf_array));
  CeedCall(CeedVectorDestroy(&assembled_qf));

  return CEED_ERROR_SUCCESS;
}

/**
  @brief Count number of entries for assembled CeedOperator

  @param[in]  op          CeedOperator to assemble
  @param[out] num_entries Number of entries in assembled representation

  @return An error code: 0 - success, otherwise - failure

  @ref Utility
**/
static int CeedSingleOperatorAssemblyCountEntries(CeedOperator op, CeedInt *num_entries) {
  bool                is_composite;
  CeedElemRestriction rstr;
  CeedInt             num_elem, elem_size, num_comp;

  CeedCall(CeedOperatorIsComposite(op, &is_composite));
  CeedCheck(!is_composite, op->ceed, CEED_ERROR_UNSUPPORTED, "Composite operator not supported");
  CeedCall(CeedOperatorGetActiveElemRestriction(op, &rstr));
  CeedCall(CeedElemRestrictionGetNumElements(rstr, &num_elem));
  CeedCall(CeedElemRestrictionGetElementSize(rstr, &elem_size));
  CeedCall(CeedElemRestrictionGetNumComponents(rstr, &num_comp));
  *num_entries = elem_size * num_comp * elem_size * num_comp * num_elem;

  return CEED_ERROR_SUCCESS;
}

/**
  @brief Common code for creating a multigrid coarse operator and level transfer operators for a CeedOperator

  @param[in]  op_fine      Fine grid operator
  @param[in]  p_mult_fine  L-vector multiplicity in parallel gather/scatter, or NULL if not creating prolongation/restriction operators
  @param[in]  rstr_coarse  Coarse grid restriction
  @param[in]  basis_coarse Coarse grid active vector basis
  @param[in]  basis_c_to_f Basis for coarse to fine interpolation, or NULL if not creating prolongation/restriction operators
  @param[out] op_coarse    Coarse grid operator
  @param[out] op_prolong   Coarse to fine operator, or NULL
  @param[out] op_restrict  Fine to coarse operator, or NULL

  @return An error code: 0 - success, otherwise - failure

  @ref Developer
**/
static int CeedSingleOperatorMultigridLevel(CeedOperator op_fine, CeedVector p_mult_fine, CeedElemRestriction rstr_coarse, CeedBasis basis_coarse,
                                            CeedBasis basis_c_to_f, CeedOperator *op_coarse, CeedOperator *op_prolong, CeedOperator *op_restrict) {
  Ceed       ceed;
  CeedVector mult_vec = NULL;
  CeedCall(CeedOperatorGetCeed(op_fine, &ceed));

  // Check for composite operator
  bool is_composite;
  CeedCall(CeedOperatorIsComposite(op_fine, &is_composite));
  CeedCheck(!is_composite, ceed, CEED_ERROR_UNSUPPORTED, "Automatic multigrid setup for composite operators not supported");

  // Coarse Grid
  CeedCall(CeedOperatorCreate(ceed, op_fine->qf, op_fine->dqf, op_fine->dqfT, op_coarse));
  CeedElemRestriction rstr_fine = NULL;
  // -- Clone input fields
  for (CeedInt i = 0; i < op_fine->qf->num_input_fields; i++) {
    if (op_fine->input_fields[i]->vec == CEED_VECTOR_ACTIVE) {
      rstr_fine = op_fine->input_fields[i]->elem_rstr;
      CeedCall(CeedOperatorSetField(*op_coarse, op_fine->input_fields[i]->field_name, rstr_coarse, basis_coarse, CEED_VECTOR_ACTIVE));
    } else {
      CeedCall(CeedOperatorSetField(*op_coarse, op_fine->input_fields[i]->field_name, op_fine->input_fields[i]->elem_rstr,
                                    op_fine->input_fields[i]->basis, op_fine->input_fields[i]->vec));
    }
  }
  // -- Clone output fields
  for (CeedInt i = 0; i < op_fine->qf->num_output_fields; i++) {
    if (op_fine->output_fields[i]->vec == CEED_VECTOR_ACTIVE) {
      CeedCall(CeedOperatorSetField(*op_coarse, op_fine->output_fields[i]->field_name, rstr_coarse, basis_coarse, CEED_VECTOR_ACTIVE));
    } else {
      CeedCall(CeedOperatorSetField(*op_coarse, op_fine->output_fields[i]->field_name, op_fine->output_fields[i]->elem_rstr,
                                    op_fine->output_fields[i]->basis, op_fine->output_fields[i]->vec));
    }
  }
  // -- Clone QFunctionAssemblyData
  CeedCall(CeedQFunctionAssemblyDataReferenceCopy(op_fine->qf_assembled, &(*op_coarse)->qf_assembled));

  // Multiplicity vector
  if (op_restrict || op_prolong) {
    CeedVector mult_e_vec;

    CeedCheck(p_mult_fine, ceed, CEED_ERROR_INCOMPATIBLE, "Prolongation or restriction operator creation requires fine grid multiplicity vector");
    CeedCall(CeedElemRestrictionCreateVector(rstr_fine, &mult_vec, &mult_e_vec));
    CeedCall(CeedVectorSetValue(mult_e_vec, 0.0));
    CeedCall(CeedElemRestrictionApply(rstr_fine, CEED_NOTRANSPOSE, p_mult_fine, mult_e_vec, CEED_REQUEST_IMMEDIATE));
    CeedCall(CeedVectorSetValue(mult_vec, 0.0));
    CeedCall(CeedElemRestrictionApply(rstr_fine, CEED_TRANSPOSE, mult_e_vec, mult_vec, CEED_REQUEST_IMMEDIATE));
    CeedCall(CeedVectorDestroy(&mult_e_vec));
    CeedCall(CeedVectorReciprocal(mult_vec));
  }

  // Clone name
  bool   has_name = op_fine->name;
  size_t name_len = op_fine->name ? strlen(op_fine->name) : 0;
  CeedCall(CeedOperatorSetName(*op_coarse, op_fine->name));

  // Check that coarse to fine basis is provided if prolong/restrict operators are requested
  CeedCheck(basis_c_to_f || (!op_restrict && !op_prolong), ceed, CEED_ERROR_INCOMPATIBLE,
            "Prolongation or restriction operator creation requires coarse-to-fine basis");

  // Restriction/Prolongation Operators
  CeedInt num_comp;
  CeedCall(CeedBasisGetNumComponents(basis_coarse, &num_comp));

  // Restriction
  if (op_restrict) {
    CeedInt             *num_comp_r_data;
    CeedQFunction        qf_restrict;
    CeedQFunctionContext ctx_r;

    CeedCall(CeedQFunctionCreateInteriorByName(ceed, "Scale", &qf_restrict));
    CeedCall(CeedCalloc(1, &num_comp_r_data));
    num_comp_r_data[0] = num_comp;
    CeedCall(CeedQFunctionContextCreate(ceed, &ctx_r));
    CeedCall(CeedQFunctionContextSetData(ctx_r, CEED_MEM_HOST, CEED_OWN_POINTER, sizeof(*num_comp_r_data), num_comp_r_data));
    CeedCall(CeedQFunctionSetContext(qf_restrict, ctx_r));
    CeedCall(CeedQFunctionContextDestroy(&ctx_r));
    CeedCall(CeedQFunctionAddInput(qf_restrict, "input", num_comp, CEED_EVAL_NONE));
    CeedCall(CeedQFunctionAddInput(qf_restrict, "scale", num_comp, CEED_EVAL_NONE));
    CeedCall(CeedQFunctionAddOutput(qf_restrict, "output", num_comp, CEED_EVAL_INTERP));
    CeedCall(CeedQFunctionSetUserFlopsEstimate(qf_restrict, num_comp));

    CeedCall(CeedOperatorCreate(ceed, qf_restrict, CEED_QFUNCTION_NONE, CEED_QFUNCTION_NONE, op_restrict));
    CeedCall(CeedOperatorSetField(*op_restrict, "input", rstr_fine, CEED_BASIS_COLLOCATED, CEED_VECTOR_ACTIVE));
    CeedCall(CeedOperatorSetField(*op_restrict, "scale", rstr_fine, CEED_BASIS_COLLOCATED, mult_vec));
    CeedCall(CeedOperatorSetField(*op_restrict, "output", rstr_coarse, basis_c_to_f, CEED_VECTOR_ACTIVE));

    // Set name
    char *restriction_name;
    CeedCall(CeedCalloc(17 + name_len, &restriction_name));
    sprintf(restriction_name, "restriction%s%s", has_name ? " for " : "", has_name ? op_fine->name : "");
    CeedCall(CeedOperatorSetName(*op_restrict, restriction_name));
    CeedCall(CeedFree(&restriction_name));

    // Check
    CeedCall(CeedOperatorCheckReady(*op_restrict));

    // Cleanup
    CeedCall(CeedQFunctionDestroy(&qf_restrict));
  }

  // Prolongation
  if (op_prolong) {
    CeedInt             *num_comp_p_data;
    CeedQFunction        qf_prolong;
    CeedQFunctionContext ctx_p;

    CeedCall(CeedQFunctionCreateInteriorByName(ceed, "Scale", &qf_prolong));
    CeedCall(CeedCalloc(1, &num_comp_p_data));
    num_comp_p_data[0] = num_comp;
    CeedCall(CeedQFunctionContextCreate(ceed, &ctx_p));
    CeedCall(CeedQFunctionContextSetData(ctx_p, CEED_MEM_HOST, CEED_OWN_POINTER, sizeof(*num_comp_p_data), num_comp_p_data));
    CeedCall(CeedQFunctionSetContext(qf_prolong, ctx_p));
    CeedCall(CeedQFunctionContextDestroy(&ctx_p));
    CeedCall(CeedQFunctionAddInput(qf_prolong, "input", num_comp, CEED_EVAL_INTERP));
    CeedCall(CeedQFunctionAddInput(qf_prolong, "scale", num_comp, CEED_EVAL_NONE));
    CeedCall(CeedQFunctionAddOutput(qf_prolong, "output", num_comp, CEED_EVAL_NONE));
    CeedCall(CeedQFunctionSetUserFlopsEstimate(qf_prolong, num_comp));

    CeedCall(CeedOperatorCreate(ceed, qf_prolong, CEED_QFUNCTION_NONE, CEED_QFUNCTION_NONE, op_prolong));
    CeedCall(CeedOperatorSetField(*op_prolong, "input", rstr_coarse, basis_c_to_f, CEED_VECTOR_ACTIVE));
    CeedCall(CeedOperatorSetField(*op_prolong, "scale", rstr_fine, CEED_BASIS_COLLOCATED, mult_vec));
    CeedCall(CeedOperatorSetField(*op_prolong, "output", rstr_fine, CEED_BASIS_COLLOCATED, CEED_VECTOR_ACTIVE));

    // Set name
    char *prolongation_name;
    CeedCall(CeedCalloc(18 + name_len, &prolongation_name));
    sprintf(prolongation_name, "prolongation%s%s", has_name ? " for " : "", has_name ? op_fine->name : "");
    CeedCall(CeedOperatorSetName(*op_prolong, prolongation_name));
    CeedCall(CeedFree(&prolongation_name));

    // Check
    CeedCall(CeedOperatorCheckReady(*op_prolong));

    // Cleanup
    CeedCall(CeedQFunctionDestroy(&qf_prolong));
  }

  // Check
  CeedCall(CeedOperatorCheckReady(*op_coarse));

  // Cleanup
  CeedCall(CeedVectorDestroy(&mult_vec));
  CeedCall(CeedBasisDestroy(&basis_c_to_f));

  return CEED_ERROR_SUCCESS;
}

/**
  @brief Build 1D mass matrix and Laplacian with perturbation

  @param[in]  interp_1d   Interpolation matrix in one dimension
  @param[in]  grad_1d     Gradient matrix in one dimension
  @param[in]  q_weight_1d Quadrature weights in one dimension
  @param[in]  P_1d        Number of basis nodes in one dimension
  @param[in]  Q_1d        Number of quadrature points in one dimension
  @param[in]  dim         Dimension of basis
  @param[out] mass        Assembled mass matrix in one dimension
  @param[out] laplace     Assembled perturbed Laplacian in one dimension

  @return An error code: 0 - success, otherwise - failure

  @ref Developer
**/
CeedPragmaOptimizeOff
static int CeedBuildMassLaplace(const CeedScalar *interp_1d, const CeedScalar *grad_1d, const CeedScalar *q_weight_1d, CeedInt P_1d, CeedInt Q_1d,
                                CeedInt dim, CeedScalar *mass, CeedScalar *laplace) {
  for (CeedInt i = 0; i < P_1d; i++) {
    for (CeedInt j = 0; j < P_1d; j++) {
      CeedScalar sum = 0.0;
      for (CeedInt k = 0; k < Q_1d; k++) sum += interp_1d[k * P_1d + i] * q_weight_1d[k] * interp_1d[k * P_1d + j];
      mass[i + j * P_1d] = sum;
    }
  }
  // -- Laplacian
  for (CeedInt i = 0; i < P_1d; i++) {
    for (CeedInt j = 0; j < P_1d; j++) {
      CeedScalar sum = 0.0;
      for (CeedInt k = 0; k < Q_1d; k++) sum += grad_1d[k * P_1d + i] * q_weight_1d[k] * grad_1d[k * P_1d + j];
      laplace[i + j * P_1d] = sum;
    }
  }
  CeedScalar perturbation = dim > 2 ? 1e-6 : 1e-4;
  for (CeedInt i = 0; i < P_1d; i++) laplace[i + P_1d * i] += perturbation;
  return CEED_ERROR_SUCCESS;
}
CeedPragmaOptimizeOn

/// @}

/// ----------------------------------------------------------------------------
/// CeedOperator Backend API
/// ----------------------------------------------------------------------------
/// @addtogroup CeedOperatorBackend
/// @{

/**
  @brief Create object holding CeedQFunction assembly data for CeedOperator

  @param[in]  ceed A Ceed object where the CeedQFunctionAssemblyData will be created
  @param[out] data Address of the variable where the newly created CeedQFunctionAssemblyData will be stored

  @return An error code: 0 - success, otherwise - failure

  @ref Backend
**/
int CeedQFunctionAssemblyDataCreate(Ceed ceed, CeedQFunctionAssemblyData *data) {
  CeedCall(CeedCalloc(1, data));
  (*data)->ref_count = 1;
  (*data)->ceed      = ceed;
  CeedCall(CeedReference(ceed));

  return CEED_ERROR_SUCCESS;
}

/**
  @brief Increment the reference counter for a CeedQFunctionAssemblyData

  @param[in,out] data CeedQFunctionAssemblyData to increment the reference counter

  @return An error code: 0 - success, otherwise - failure

  @ref Backend
**/
int CeedQFunctionAssemblyDataReference(CeedQFunctionAssemblyData data) {
  data->ref_count++;
  return CEED_ERROR_SUCCESS;
}

/**
  @brief Set re-use of CeedQFunctionAssemblyData

  @param[in,out] data       CeedQFunctionAssemblyData to mark for reuse
  @param[in]     reuse_data Boolean flag indicating data re-use

  @return An error code: 0 - success, otherwise - failure

  @ref Backend
**/
int CeedQFunctionAssemblyDataSetReuse(CeedQFunctionAssemblyData data, bool reuse_data) {
  data->reuse_data        = reuse_data;
  data->needs_data_update = true;
  return CEED_ERROR_SUCCESS;
}

/**
  @brief Mark QFunctionAssemblyData as stale

  @param[in,out] data              CeedQFunctionAssemblyData to mark as stale
  @param[in]     needs_data_update Boolean flag indicating if update is needed or completed

  @return An error code: 0 - success, otherwise - failure

  @ref Backend
**/
int CeedQFunctionAssemblyDataSetUpdateNeeded(CeedQFunctionAssemblyData data, bool needs_data_update) {
  data->needs_data_update = needs_data_update;
  return CEED_ERROR_SUCCESS;
}

/**
  @brief Determine if QFunctionAssemblyData needs update

  @param[in]  data             CeedQFunctionAssemblyData to mark as stale
  @param[out] is_update_needed Boolean flag indicating if re-assembly is required

  @return An error code: 0 - success, otherwise - failure

  @ref Backend
**/
int CeedQFunctionAssemblyDataIsUpdateNeeded(CeedQFunctionAssemblyData data, bool *is_update_needed) {
  *is_update_needed = !data->reuse_data || data->needs_data_update;
  return CEED_ERROR_SUCCESS;
}

/**
  @brief Copy the pointer to a CeedQFunctionAssemblyData.
           Both pointers should be destroyed with `CeedCeedQFunctionAssemblyDataDestroy()`.

           Note: If the value of `data_copy` passed to this function is non-NULL, then it is assumed that `*data_copy` is a pointer to a
             CeedQFunctionAssemblyData. This CeedQFunctionAssemblyData will be destroyed if `data_copy` is the only reference to this
             CeedQFunctionAssemblyData.

  @param[in]     data      CeedQFunctionAssemblyData to copy reference to
  @param[in,out] data_copy Variable to store copied reference

  @return An error code: 0 - success, otherwise - failure

  @ref Backend
**/
int CeedQFunctionAssemblyDataReferenceCopy(CeedQFunctionAssemblyData data, CeedQFunctionAssemblyData *data_copy) {
  CeedCall(CeedQFunctionAssemblyDataReference(data));
  CeedCall(CeedQFunctionAssemblyDataDestroy(data_copy));
  *data_copy = data;
  return CEED_ERROR_SUCCESS;
}

/**
  @brief Get setup status for internal objects for CeedQFunctionAssemblyData

  @param[in]  data     CeedQFunctionAssemblyData to retrieve status
  @param[out] is_setup Boolean flag for setup status

  @return An error code: 0 - success, otherwise - failure

  @ref Backend
**/
int CeedQFunctionAssemblyDataIsSetup(CeedQFunctionAssemblyData data, bool *is_setup) {
  *is_setup = data->is_setup;
  return CEED_ERROR_SUCCESS;
}

/**
  @brief Set internal objects for CeedQFunctionAssemblyData

  @param[in,out] data CeedQFunctionAssemblyData to set objects
  @param[in]     vec  CeedVector to store assembled CeedQFunction at quadrature points
  @param[in]     rstr CeedElemRestriction for CeedVector containing assembled CeedQFunction

  @return An error code: 0 - success, otherwise - failure

  @ref Backend
**/
int CeedQFunctionAssemblyDataSetObjects(CeedQFunctionAssemblyData data, CeedVector vec, CeedElemRestriction rstr) {
  CeedCall(CeedVectorReferenceCopy(vec, &data->vec));
  CeedCall(CeedElemRestrictionReferenceCopy(rstr, &data->rstr));

  data->is_setup = true;
  return CEED_ERROR_SUCCESS;
}

int CeedQFunctionAssemblyDataGetObjects(CeedQFunctionAssemblyData data, CeedVector *vec, CeedElemRestriction *rstr) {
  CeedCheck(data->is_setup, data->ceed, CEED_ERROR_INCOMPLETE, "Internal objects not set; must call CeedQFunctionAssemblyDataSetObjects first.");

  CeedCall(CeedVectorReferenceCopy(data->vec, vec));
  CeedCall(CeedElemRestrictionReferenceCopy(data->rstr, rstr));

  return CEED_ERROR_SUCCESS;
}

/**
  @brief Destroy CeedQFunctionAssemblyData

  @param[in,out] data  CeedQFunctionAssemblyData to destroy

  @return An error code: 0 - success, otherwise - failure

  @ref Backend
**/
int CeedQFunctionAssemblyDataDestroy(CeedQFunctionAssemblyData *data) {
  if (!*data || --(*data)->ref_count > 0) {
    *data = NULL;
    return CEED_ERROR_SUCCESS;
  }
  CeedCall(CeedDestroy(&(*data)->ceed));
  CeedCall(CeedVectorDestroy(&(*data)->vec));
  CeedCall(CeedElemRestrictionDestroy(&(*data)->rstr));

  CeedCall(CeedFree(data));
  return CEED_ERROR_SUCCESS;
}

/**
  @brief Get CeedOperatorAssemblyData

  @param[in]  op   CeedOperator to assemble
  @param[out] data CeedQFunctionAssemblyData

  @return An error code: 0 - success, otherwise - failure

  @ref Backend
**/
int CeedOperatorGetOperatorAssemblyData(CeedOperator op, CeedOperatorAssemblyData *data) {
  if (!op->op_assembled) {
    CeedOperatorAssemblyData data;

    CeedCall(CeedOperatorAssemblyDataCreate(op->ceed, op, &data));
    op->op_assembled = data;
  }
  *data = op->op_assembled;

  return CEED_ERROR_SUCCESS;
}

/**
  @brief Create object holding CeedOperator assembly data.

    The CeedOperatorAssemblyData holds an array with references to every active CeedBasis used in the CeedOperator.
    An array with references to the corresponding active CeedElemRestrictions is also stored.
    For each active CeedBasis, the CeedOperatorAssemblyData holds an array of all input and output CeedEvalModes for this CeedBasis.
    The CeedOperatorAssemblyData holds an array of offsets for indexing into the assembled CeedQFunction arrays to the row representing each
      CeedEvalMode.
    The number of input columns across all active bases for the assembled CeedQFunction is also stored.
    Lastly, the CeedOperatorAssembly data holds assembled matrices representing the full action of the CeedBasis for all CeedEvalModes.

  @param[in]  ceed Ceed object where the CeedOperatorAssemblyData will be created
  @param[in]  op   CeedOperator to be assembled
  @param[out] data Address of the variable where the newly created CeedOperatorAssemblyData will be stored

  @return An error code: 0 - success, otherwise - failure

  @ref Backend
**/
int CeedOperatorAssemblyDataCreate(Ceed ceed, CeedOperator op, CeedOperatorAssemblyData *data) {
  CeedInt num_active_bases = 0;

  // Allocate
  CeedCall(CeedCalloc(1, data));
  (*data)->ceed = ceed;
  CeedCall(CeedReference(ceed));

  // Build OperatorAssembly data
  CeedQFunction       qf;
  CeedQFunctionField *qf_fields;
  CeedOperatorField  *op_fields;
  CeedInt             num_input_fields;
  CeedCall(CeedOperatorGetQFunction(op, &qf));
  CeedCall(CeedQFunctionGetFields(qf, &num_input_fields, &qf_fields, NULL, NULL));
  CeedCall(CeedOperatorGetFields(op, NULL, &op_fields, NULL, NULL));

  // Determine active input basis
  CeedInt       *num_eval_modes_in = NULL, *num_eval_modes_out = NULL, offset = 0;
  CeedEvalMode **eval_modes_in = NULL, **eval_modes_out = NULL;
  CeedSize     **eval_mode_offsets_in = NULL, **eval_mode_offsets_out = NULL;
  for (CeedInt i = 0; i < num_input_fields; i++) {
    CeedVector vec;
    CeedCall(CeedOperatorFieldGetVector(op_fields[i], &vec));
    if (vec == CEED_VECTOR_ACTIVE) {
      CeedBasis    basis_in = NULL;
      CeedEvalMode eval_mode;
      CeedInt      index = -1, dim, num_comp, q_comp;
      CeedCall(CeedOperatorFieldGetBasis(op_fields[i], &basis_in));
      CeedCall(CeedQFunctionFieldGetEvalMode(qf_fields[i], &eval_mode));
      CeedCall(CeedBasisGetDimension(basis_in, &dim));
      CeedCall(CeedBasisGetNumComponents(basis_in, &num_comp));
      CeedCall(CeedBasisGetNumQuadratureComponents(basis_in, eval_mode, &q_comp));
      for (CeedInt i = 0; i < num_active_bases; i++) {
        if ((*data)->active_bases[i] == basis_in) index = i;
      }
      if (index == -1) {
        CeedElemRestriction elem_rstr_in;
        index = num_active_bases;
        CeedCall(CeedRealloc(num_active_bases + 1, &(*data)->active_bases));
        (*data)->active_bases[num_active_bases] = NULL;
        CeedCall(CeedBasisReferenceCopy(basis_in, &(*data)->active_bases[num_active_bases]));
        CeedCall(CeedRealloc(num_active_bases + 1, &(*data)->active_elem_rstrs));
        (*data)->active_elem_rstrs[num_active_bases] = NULL;
        CeedCall(CeedOperatorFieldGetElemRestriction(op_fields[i], &elem_rstr_in));
        CeedCall(CeedElemRestrictionReferenceCopy(elem_rstr_in, &(*data)->active_elem_rstrs[num_active_bases]));
        CeedCall(CeedRealloc(num_active_bases + 1, &num_eval_modes_in));
        CeedCall(CeedRealloc(num_active_bases + 1, &num_eval_modes_out));
        num_eval_modes_in[index]  = 0;
        num_eval_modes_out[index] = 0;
        CeedCall(CeedRealloc(num_active_bases + 1, &eval_modes_in));
        CeedCall(CeedRealloc(num_active_bases + 1, &eval_modes_out));
        eval_modes_in[index]  = NULL;
        eval_modes_out[index] = NULL;
        CeedCall(CeedRealloc(num_active_bases + 1, &eval_mode_offsets_in));
        CeedCall(CeedRealloc(num_active_bases + 1, &eval_mode_offsets_out));
        eval_mode_offsets_in[index]  = NULL;
        eval_mode_offsets_out[index] = NULL;
        CeedCall(CeedRealloc(num_active_bases + 1, &(*data)->assembled_bases_in));
        CeedCall(CeedRealloc(num_active_bases + 1, &(*data)->assembled_bases_out));
        (*data)->assembled_bases_in[index]  = NULL;
        (*data)->assembled_bases_out[index] = NULL;
        num_active_bases++;
      }
      if (eval_mode != CEED_EVAL_WEIGHT) {
        // q_comp = 1 if CEED_EVAL_NONE, CEED_EVAL_WEIGHT caught by QF Assembly
        CeedCall(CeedRealloc(num_eval_modes_in[index] + q_comp, &eval_modes_in[index]));
        CeedCall(CeedRealloc(num_eval_modes_in[index] + q_comp, &eval_mode_offsets_in[index]));
        for (CeedInt d = 0; d < q_comp; d++) {
          eval_modes_in[index][num_eval_modes_in[index] + d]        = eval_mode;
          eval_mode_offsets_in[index][num_eval_modes_in[index] + d] = offset;
          offset += num_comp;
        }
        num_eval_modes_in[index] += q_comp;
      }
    }
  }
  (*data)->num_eval_modes_in    = num_eval_modes_in;
  (*data)->eval_modes_in        = eval_modes_in;
  (*data)->eval_mode_offsets_in = eval_mode_offsets_in;

  // Determine active output basis
  CeedInt num_output_fields;
  CeedCall(CeedQFunctionGetFields(qf, NULL, NULL, &num_output_fields, &qf_fields));
  CeedCall(CeedOperatorGetFields(op, NULL, NULL, NULL, &op_fields));
  offset = 0;
  for (CeedInt i = 0; i < num_output_fields; i++) {
    CeedVector vec;
    CeedCall(CeedOperatorFieldGetVector(op_fields[i], &vec));
    if (vec == CEED_VECTOR_ACTIVE) {
      CeedBasis    basis_out = NULL;
      CeedEvalMode eval_mode;
      CeedInt      index = -1, dim, num_comp, q_comp;
      CeedCall(CeedOperatorFieldGetBasis(op_fields[i], &basis_out));
      CeedCall(CeedQFunctionFieldGetEvalMode(qf_fields[i], &eval_mode));
      CeedCall(CeedBasisGetDimension(basis_out, &dim));
      CeedCall(CeedBasisGetNumComponents(basis_out, &num_comp));
      CeedCall(CeedBasisGetNumQuadratureComponents(basis_out, eval_mode, &q_comp));
      for (CeedInt i = 0; i < num_active_bases; i++) {
        if ((*data)->active_bases[i] == basis_out) index = i;
      }
      if (index == -1) {
        CeedElemRestriction elem_rstr_out;

        index = num_active_bases;
        CeedCall(CeedRealloc(num_active_bases + 1, &(*data)->active_bases));
        (*data)->active_bases[num_active_bases] = NULL;
        CeedCall(CeedBasisReferenceCopy(basis_out, &(*data)->active_bases[num_active_bases]));
        CeedCall(CeedRealloc(num_active_bases + 1, &(*data)->active_elem_rstrs));
        (*data)->active_elem_rstrs[num_active_bases] = NULL;
        CeedCall(CeedOperatorFieldGetElemRestriction(op_fields[i], &elem_rstr_out));
        CeedCall(CeedElemRestrictionReferenceCopy(elem_rstr_out, &(*data)->active_elem_rstrs[num_active_bases]));
        CeedCall(CeedRealloc(num_active_bases + 1, &num_eval_modes_in));
        CeedCall(CeedRealloc(num_active_bases + 1, &num_eval_modes_out));
        num_eval_modes_in[index]  = 0;
        num_eval_modes_out[index] = 0;
        CeedCall(CeedRealloc(num_active_bases + 1, &eval_modes_in));
        CeedCall(CeedRealloc(num_active_bases + 1, &eval_modes_out));
        eval_modes_in[index]  = NULL;
        eval_modes_out[index] = NULL;
        CeedCall(CeedRealloc(num_active_bases + 1, &eval_mode_offsets_in));
        CeedCall(CeedRealloc(num_active_bases + 1, &eval_mode_offsets_out));
        eval_mode_offsets_in[index]  = NULL;
        eval_mode_offsets_out[index] = NULL;
        CeedCall(CeedRealloc(num_active_bases + 1, &(*data)->assembled_bases_in));
        CeedCall(CeedRealloc(num_active_bases + 1, &(*data)->assembled_bases_out));
        (*data)->assembled_bases_in[index]  = NULL;
        (*data)->assembled_bases_out[index] = NULL;
        num_active_bases++;
      }
      if (eval_mode != CEED_EVAL_WEIGHT) {
        // q_comp = 1 if CEED_EVAL_NONE, CEED_EVAL_WEIGHT caught by QF Assembly
        CeedCall(CeedRealloc(num_eval_modes_out[index] + q_comp, &eval_modes_out[index]));
        CeedCall(CeedRealloc(num_eval_modes_out[index] + q_comp, &eval_mode_offsets_out[index]));
        for (CeedInt d = 0; d < q_comp; d++) {
          eval_modes_out[index][num_eval_modes_out[index] + d]        = eval_mode;
          eval_mode_offsets_out[index][num_eval_modes_out[index] + d] = offset;
          offset += num_comp;
        }
        num_eval_modes_out[index] += q_comp;
      }
    }
  }
  (*data)->num_output_components = offset;
  (*data)->num_eval_modes_out    = num_eval_modes_out;
  (*data)->eval_modes_out        = eval_modes_out;
  (*data)->eval_mode_offsets_out = eval_mode_offsets_out;
  (*data)->num_active_bases      = num_active_bases;

  return CEED_ERROR_SUCCESS;
}

/**
  @brief Get CeedOperator CeedEvalModes for assembly.

    Note: See CeedOperatorAssemblyDataCreate for a full description of the data stored in this object.

  @param[in]  data                  CeedOperatorAssemblyData
  @param[out] num_active_bases      Total number of active bases
  @param[out] num_eval_modes_in     Pointer to hold array of numbers of input CeedEvalModes, or NULL.
                                      `eval_modes_in[0]` holds an array of eval modes for the first active basis.
  @param[out] eval_modes_in         Pointer to hold arrays of input CeedEvalModes, or NULL.
  @param[out] eval_mode_offsets_in  Pointer to hold arrays of input offsets at each quadrature point.
  @param[out] num_eval_modes_out    Pointer to hold array of numbers of output CeedEvalModes, or NULL
  @param[out] eval_modes_out        Pointer to hold arrays of output CeedEvalModes, or NULL.
  @param[out] eval_mode_offsets_out Pointer to hold arrays of output offsets at each quadrature point
  @param[out] num_output_components The number of columns in the assembled CeedQFunction matrix for each quadrature point,
                                      including contributions of all active bases

  @return An error code: 0 - success, otherwise - failure


  @ref Backend
**/
int CeedOperatorAssemblyDataGetEvalModes(CeedOperatorAssemblyData data, CeedInt *num_active_bases, CeedInt **num_eval_modes_in,
                                         const CeedEvalMode ***eval_modes_in, CeedSize ***eval_mode_offsets_in, CeedInt **num_eval_modes_out,
                                         const CeedEvalMode ***eval_modes_out, CeedSize ***eval_mode_offsets_out, CeedSize *num_output_components) {
  if (num_active_bases) *num_active_bases = data->num_active_bases;
  if (num_eval_modes_in) *num_eval_modes_in = data->num_eval_modes_in;
  if (eval_modes_in) *eval_modes_in = (const CeedEvalMode **)data->eval_modes_in;
  if (eval_mode_offsets_in) *eval_mode_offsets_in = data->eval_mode_offsets_in;
  if (num_eval_modes_out) *num_eval_modes_out = data->num_eval_modes_out;
  if (eval_modes_out) *eval_modes_out = (const CeedEvalMode **)data->eval_modes_out;
  if (eval_mode_offsets_out) *eval_mode_offsets_out = data->eval_mode_offsets_out;
  if (num_output_components) *num_output_components = data->num_output_components;

  return CEED_ERROR_SUCCESS;
}

/**
  @brief Get CeedOperator CeedBasis data for assembly.

    Note: See CeedOperatorAssemblyDataCreate for a full description of the data stored in this object.

  @param[in]  data                CeedOperatorAssemblyData
  @param[out] num_active_bases    Number of active bases, or NULL
  @param[out] active_bases        Pointer to hold active CeedBasis, or NULL
  @param[out] assembled_bases_in  Pointer to hold assembled active input B, or NULL
  @param[out] assembled_bases_out Pointer to hold assembled active output B, or NULL

  @return An error code: 0 - success, otherwise - failure

  @ref Backend
**/
int CeedOperatorAssemblyDataGetBases(CeedOperatorAssemblyData data, CeedInt *num_active_bases, CeedBasis **active_bases,
                                     const CeedScalar ***assembled_bases_in, const CeedScalar ***assembled_bases_out) {
  // Assemble B_in, B_out if needed
  if (assembled_bases_in && !data->assembled_bases_in[0]) {
    CeedInt num_qpts;

    CeedCall(CeedBasisGetNumQuadraturePoints(data->active_bases[0], &num_qpts));
    for (CeedInt b = 0; b < data->num_active_bases; b++) {
      CeedInt     num_nodes;
      CeedScalar *B_in = NULL, *identity = NULL;
      bool        has_eval_none = false;

      CeedCall(CeedBasisGetNumNodes(data->active_bases[b], &num_nodes));
      CeedCall(CeedCalloc(num_qpts * num_nodes * data->num_eval_modes_in[b], &B_in));

      for (CeedInt i = 0; i < data->num_eval_modes_in[b]; i++) {
        has_eval_none = has_eval_none || (data->eval_modes_in[b][i] == CEED_EVAL_NONE);
      }
      if (has_eval_none) {
        CeedCall(CeedCalloc(num_qpts * num_nodes, &identity));
        for (CeedInt i = 0; i < (num_nodes < num_qpts ? num_nodes : num_qpts); i++) {
          identity[i * num_nodes + i] = 1.0;
        }
      }

      for (CeedInt q = 0; q < num_qpts; q++) {
        for (CeedInt n = 0; n < num_nodes; n++) {
          CeedInt      d_in              = 0, q_comp_in;
          CeedEvalMode eval_mode_in_prev = CEED_EVAL_NONE;
          for (CeedInt e_in = 0; e_in < data->num_eval_modes_in[b]; e_in++) {
            const CeedInt     qq = data->num_eval_modes_in[b] * q;
            const CeedScalar *B  = NULL;
            CeedOperatorGetBasisPointer(data->active_bases[b], data->eval_modes_in[b][e_in], identity, &B);
            CeedCall(CeedBasisGetNumQuadratureComponents(data->active_bases[b], data->eval_modes_in[b][e_in], &q_comp_in));
            if (q_comp_in > 1) {
              if (e_in == 0 || data->eval_modes_in[b][e_in] != eval_mode_in_prev) d_in = 0;
              else B = &B[(++d_in) * num_qpts * num_nodes];
            }
            eval_mode_in_prev                 = data->eval_modes_in[b][e_in];
            B_in[(qq + e_in) * num_nodes + n] = B[q * num_nodes + n];
          }
        }
      }
      if (identity) CeedCall(CeedFree(identity));
      data->assembled_bases_in[b] = B_in;
    }
  }

  if (assembled_bases_out && !data->assembled_bases_out[0]) {
    CeedInt num_qpts;

    CeedCall(CeedBasisGetNumQuadraturePoints(data->active_bases[0], &num_qpts));
    for (CeedInt b = 0; b < data->num_active_bases; b++) {
      CeedInt     num_nodes;
      bool        has_eval_none = false;
      CeedScalar *B_out = NULL, *identity = NULL;

      CeedCall(CeedBasisGetNumNodes(data->active_bases[b], &num_nodes));
      CeedCall(CeedCalloc(num_qpts * num_nodes * data->num_eval_modes_out[b], &B_out));

      for (CeedInt i = 0; i < data->num_eval_modes_out[b]; i++) {
        has_eval_none = has_eval_none || (data->eval_modes_out[b][i] == CEED_EVAL_NONE);
      }
      if (has_eval_none) {
        CeedCall(CeedCalloc(num_qpts * num_nodes, &identity));
        for (CeedInt i = 0; i < (num_nodes < num_qpts ? num_nodes : num_qpts); i++) {
          identity[i * num_nodes + i] = 1.0;
        }
      }

      for (CeedInt q = 0; q < num_qpts; q++) {
        for (CeedInt n = 0; n < num_nodes; n++) {
          CeedInt      d_out              = 0, q_comp_out;
          CeedEvalMode eval_mode_out_prev = CEED_EVAL_NONE;
          for (CeedInt e_out = 0; e_out < data->num_eval_modes_out[b]; e_out++) {
            const CeedInt     qq = data->num_eval_modes_out[b] * q;
            const CeedScalar *B  = NULL;
            CeedOperatorGetBasisPointer(data->active_bases[b], data->eval_modes_out[b][e_out], identity, &B);
            CeedCall(CeedBasisGetNumQuadratureComponents(data->active_bases[b], data->eval_modes_out[b][e_out], &q_comp_out));
            if (q_comp_out > 1) {
              if (e_out == 0 || data->eval_modes_out[b][e_out] != eval_mode_out_prev) d_out = 0;
              else B = &B[(++d_out) * num_qpts * num_nodes];
            }
            eval_mode_out_prev                  = data->eval_modes_out[b][e_out];
            B_out[(qq + e_out) * num_nodes + n] = B[q * num_nodes + n];
          }
        }
      }
      if (identity) CeedCall(CeedFree(identity));
      data->assembled_bases_out[b] = B_out;
    }
  }

  // Pass out assembled data
  if (active_bases) *active_bases = data->active_bases;
  if (assembled_bases_in) *assembled_bases_in = (const CeedScalar **)data->assembled_bases_in;
  if (assembled_bases_out) *assembled_bases_out = (const CeedScalar **)data->assembled_bases_out;

  return CEED_ERROR_SUCCESS;
}

/**
  @brief Get CeedOperator CeedBasis data for assembly.

  Note: See CeedOperatorAssemblyDataCreate for a full description of the data stored in this object.

  @param[in]  data                  CeedOperatorAssemblyData
  @param[out] num_active_elem_rstrs Number of active element restrictions, or NULL
  @param[out] active_elem_rstrs     Pointer to hold active CeedElemRestrictions, or NULL

  @return An error code: 0 - success, otherwise - failure

  @ref Backend
**/
int CeedOperatorAssemblyDataGetElemRestrictions(CeedOperatorAssemblyData data, CeedInt *num_active_elem_rstrs,
                                                CeedElemRestriction **active_elem_rstrs) {
  if (num_active_elem_rstrs) *num_active_elem_rstrs = data->num_active_bases;
  if (active_elem_rstrs) *active_elem_rstrs = data->active_elem_rstrs;

  return CEED_ERROR_SUCCESS;
}

/**
  @brief Destroy CeedOperatorAssemblyData

  @param[in,out] data CeedOperatorAssemblyData to destroy

  @return An error code: 0 - success, otherwise - failure

  @ref Backend
**/
int CeedOperatorAssemblyDataDestroy(CeedOperatorAssemblyData *data) {
  if (!*data) {
    *data = NULL;
    return CEED_ERROR_SUCCESS;
  }
  CeedCall(CeedDestroy(&(*data)->ceed));
  for (CeedInt b = 0; b < (*data)->num_active_bases; b++) {
    CeedCall(CeedBasisDestroy(&(*data)->active_bases[b]));
    CeedCall(CeedElemRestrictionDestroy(&(*data)->active_elem_rstrs[b]));
    CeedCall(CeedFree(&(*data)->eval_modes_in[b]));
    CeedCall(CeedFree(&(*data)->eval_modes_out[b]));
    CeedCall(CeedFree(&(*data)->eval_mode_offsets_in[b]));
    CeedCall(CeedFree(&(*data)->eval_mode_offsets_out[b]));
    CeedCall(CeedFree(&(*data)->assembled_bases_in[b]));
    CeedCall(CeedFree(&(*data)->assembled_bases_out[b]));
  }
  CeedCall(CeedFree(&(*data)->active_bases));
  CeedCall(CeedFree(&(*data)->active_elem_rstrs));
  CeedCall(CeedFree(&(*data)->num_eval_modes_in));
  CeedCall(CeedFree(&(*data)->num_eval_modes_out));
  CeedCall(CeedFree(&(*data)->eval_modes_in));
  CeedCall(CeedFree(&(*data)->eval_modes_out));
  CeedCall(CeedFree(&(*data)->eval_mode_offsets_in));
  CeedCall(CeedFree(&(*data)->eval_mode_offsets_out));
  CeedCall(CeedFree(&(*data)->assembled_bases_in));
  CeedCall(CeedFree(&(*data)->assembled_bases_out));

  CeedCall(CeedFree(data));
  return CEED_ERROR_SUCCESS;
}

/// @}

/// ----------------------------------------------------------------------------
/// CeedOperator Public API
/// ----------------------------------------------------------------------------
/// @addtogroup CeedOperatorUser
/// @{

/**
  @brief Assemble a linear CeedQFunction associated with a CeedOperator

  This returns a CeedVector containing a matrix at each quadrature point providing the action of the CeedQFunction associated with the CeedOperator.
    The vector `assembled` is of shape `[num_elements, num_input_fields, num_output_fields, num_quad_points]` and contains column-major matrices
representing the action of the CeedQFunction for a corresponding quadrature point on an element.

  Inputs and outputs are in the order provided by the
user when adding CeedOperator fields. For example, a CeedQFunction with inputs 'u' and 'gradu' and outputs 'gradv' and 'v', provided in that order,
would result in an assembled QFunction that consists of (1 + dim) x (dim + 1) matrices at each quadrature point acting on the input [u, du_0, du_1]
and producing the output [dv_0, dv_1, v].

  Note: Calling this function asserts that setup is complete and sets the CeedOperator as immutable.

  @param[in]  op        CeedOperator to assemble CeedQFunction
  @param[out] assembled CeedVector to store assembled CeedQFunction at quadrature points
  @param[out] rstr      CeedElemRestriction for CeedVector containing assembled CeedQFunction
  @param[in]  request   Address of CeedRequest for non-blocking completion, else @ref CEED_REQUEST_IMMEDIATE

  @return An error code: 0 - success, otherwise - failure

  @ref User
**/
int CeedOperatorLinearAssembleQFunction(CeedOperator op, CeedVector *assembled, CeedElemRestriction *rstr, CeedRequest *request) {
  CeedCall(CeedOperatorCheckReady(op));

  if (op->LinearAssembleQFunction) {
    // Backend version
    CeedCall(op->LinearAssembleQFunction(op, assembled, rstr, request));
  } else {
    // Operator fallback
    CeedOperator op_fallback;

    CeedCall(CeedOperatorGetFallback(op, &op_fallback));
    if (op_fallback) CeedCall(CeedOperatorLinearAssembleQFunction(op_fallback, assembled, rstr, request));
    else return CeedError(op->ceed, CEED_ERROR_UNSUPPORTED, "Backend does not support CeedOperatorLinearAssembleQFunction");
  }
  return CEED_ERROR_SUCCESS;
}

/**
  @brief Assemble CeedQFunction and store result internally.
           Return copied references of stored data to the caller.
           Caller is responsible for ownership and destruction of the copied references.
           See also @ref CeedOperatorLinearAssembleQFunction

  @param[in]  op        CeedOperator to assemble CeedQFunction
  @param[out] assembled CeedVector to store assembled CeedQFunction at quadrature points
  @param[out] rstr      CeedElemRestriction for CeedVector containing assembledCeedQFunction
  @param[in]  request   Address of CeedRequest for non-blocking completion, else @ref CEED_REQUEST_IMMEDIATE

  @return An error code: 0 - success, otherwise - failure

  @ref User
**/
int CeedOperatorLinearAssembleQFunctionBuildOrUpdate(CeedOperator op, CeedVector *assembled, CeedElemRestriction *rstr, CeedRequest *request) {
  CeedCall(CeedOperatorCheckReady(op));

  if (op->LinearAssembleQFunctionUpdate) {
    // Backend version
    bool                qf_assembled_is_setup;
    CeedVector          assembled_vec  = NULL;
    CeedElemRestriction assembled_rstr = NULL;

    CeedCall(CeedQFunctionAssemblyDataIsSetup(op->qf_assembled, &qf_assembled_is_setup));
    if (qf_assembled_is_setup) {
      bool update_needed;

      CeedCall(CeedQFunctionAssemblyDataGetObjects(op->qf_assembled, &assembled_vec, &assembled_rstr));
      CeedCall(CeedQFunctionAssemblyDataIsUpdateNeeded(op->qf_assembled, &update_needed));
      if (update_needed) {
        CeedCall(op->LinearAssembleQFunctionUpdate(op, assembled_vec, assembled_rstr, request));
      }
    } else {
      CeedCall(op->LinearAssembleQFunction(op, &assembled_vec, &assembled_rstr, request));
      CeedCall(CeedQFunctionAssemblyDataSetObjects(op->qf_assembled, assembled_vec, assembled_rstr));
    }
    CeedCall(CeedQFunctionAssemblyDataSetUpdateNeeded(op->qf_assembled, false));

    // Copy reference from internally held copy
    *assembled = NULL;
    *rstr      = NULL;
    CeedCall(CeedVectorReferenceCopy(assembled_vec, assembled));
    CeedCall(CeedVectorDestroy(&assembled_vec));
    CeedCall(CeedElemRestrictionReferenceCopy(assembled_rstr, rstr));
    CeedCall(CeedElemRestrictionDestroy(&assembled_rstr));
  } else {
    // Operator fallback
    CeedOperator op_fallback;

    CeedCall(CeedOperatorGetFallback(op, &op_fallback));
    if (op_fallback) CeedCall(CeedOperatorLinearAssembleQFunctionBuildOrUpdate(op_fallback, assembled, rstr, request));
    else return CeedError(op->ceed, CEED_ERROR_UNSUPPORTED, "Backend does not support CeedOperatorLinearAssembleQFunctionUpdate");
  }

  return CEED_ERROR_SUCCESS;
}

/**
  @brief Assemble the diagonal of a square linear CeedOperator

  This overwrites a CeedVector with the diagonal of a linear CeedOperator.

  Note: Currently only non-composite CeedOperators with a single field and composite CeedOperators with single field sub-operators are supported.

  Note: Calling this function asserts that setup is complete and sets the CeedOperator as immutable.

  @param[in]  op        CeedOperator to assemble CeedQFunction
  @param[out] assembled CeedVector to store assembled CeedOperator diagonal
  @param[in]  request   Address of CeedRequest for non-blocking completion, else @ref CEED_REQUEST_IMMEDIATE

  @return An error code: 0 - success, otherwise - failure

  @ref User
**/
int CeedOperatorLinearAssembleDiagonal(CeedOperator op, CeedVector assembled, CeedRequest *request) {
  bool is_composite;
  CeedCall(CeedOperatorCheckReady(op));
  CeedCall(CeedOperatorIsComposite(op, &is_composite));

  CeedSize input_size = 0, output_size = 0;
  CeedCall(CeedOperatorGetActiveVectorLengths(op, &input_size, &output_size));
  CeedCheck(input_size == output_size, op->ceed, CEED_ERROR_DIMENSION, "Operator must be square");

  // Early exit for empty operator
  if (!is_composite) {
    CeedInt num_elem = 0;

    CeedCall(CeedOperatorGetNumElements(op, &num_elem));
    if (num_elem == 0) return CEED_ERROR_SUCCESS;
  }

  if (op->LinearAssembleDiagonal) {
    // Backend version
    CeedCall(op->LinearAssembleDiagonal(op, assembled, request));
    return CEED_ERROR_SUCCESS;
  } else if (op->LinearAssembleAddDiagonal) {
    // Backend version with zeroing first
    CeedCall(CeedVectorSetValue(assembled, 0.0));
    CeedCall(op->LinearAssembleAddDiagonal(op, assembled, request));
    return CEED_ERROR_SUCCESS;
  } else {
    // Operator fallback
    CeedOperator op_fallback;

    CeedCall(CeedOperatorGetFallback(op, &op_fallback));
    if (op_fallback) {
      CeedCall(CeedOperatorLinearAssembleDiagonal(op_fallback, assembled, request));
      return CEED_ERROR_SUCCESS;
    }
  }
  // Default interface implementation
  CeedCall(CeedVectorSetValue(assembled, 0.0));
  CeedCall(CeedOperatorLinearAssembleAddDiagonal(op, assembled, request));

  return CEED_ERROR_SUCCESS;
}

/**
  @brief Assemble the diagonal of a square linear CeedOperator

  This sums into a CeedVector the diagonal of a linear CeedOperator.

  Note: Currently only non-composite CeedOperators with a single field and composite CeedOperators with single field sub-operators are supported.

  Note: Calling this function asserts that setup is complete and sets the CeedOperator as immutable.

  @param[in]  op        CeedOperator to assemble CeedQFunction
  @param[out] assembled CeedVector to store assembled CeedOperator diagonal
  @param[in]  request   Address of CeedRequest for non-blocking completion, else @ref CEED_REQUEST_IMMEDIATE

  @return An error code: 0 - success, otherwise - failure

  @ref User
**/
int CeedOperatorLinearAssembleAddDiagonal(CeedOperator op, CeedVector assembled, CeedRequest *request) {
  bool is_composite;
  CeedCall(CeedOperatorCheckReady(op));
  CeedCall(CeedOperatorIsComposite(op, &is_composite));

  CeedSize input_size = 0, output_size = 0;
  CeedCall(CeedOperatorGetActiveVectorLengths(op, &input_size, &output_size));
  CeedCheck(input_size == output_size, op->ceed, CEED_ERROR_DIMENSION, "Operator must be square");

  // Early exit for empty operator
  if (!is_composite) {
    CeedInt num_elem = 0;

    CeedCall(CeedOperatorGetNumElements(op, &num_elem));
    if (num_elem == 0) return CEED_ERROR_SUCCESS;
  }

  if (op->LinearAssembleAddDiagonal) {
    // Backend version
    CeedCall(op->LinearAssembleAddDiagonal(op, assembled, request));
    return CEED_ERROR_SUCCESS;
  } else {
    // Operator fallback
    CeedOperator op_fallback;

    CeedCall(CeedOperatorGetFallback(op, &op_fallback));
    if (op_fallback) {
      CeedCall(CeedOperatorLinearAssembleAddDiagonal(op_fallback, assembled, request));
      return CEED_ERROR_SUCCESS;
    }
  }
  // Default interface implementation
  if (is_composite) {
    CeedCall(CeedCompositeOperatorLinearAssembleAddDiagonal(op, request, false, assembled));
  } else {
    CeedCall(CeedSingleOperatorAssembleAddDiagonal_Core(op, request, false, assembled));
  }

  return CEED_ERROR_SUCCESS;
}

/**
  @brief Assemble the point block diagonal of a square linear CeedOperator

  This overwrites a CeedVector with the point block diagonal of a linear CeedOperator.

  Note: Currently only non-composite CeedOperators with a single field and composite CeedOperators with single field sub-operators are supported.

  Note: Calling this function asserts that setup is complete and sets the CeedOperator as immutable.

  @param[in]  op        CeedOperator to assemble CeedQFunction
  @param[out] assembled CeedVector to store assembled CeedOperator point block diagonal, provided in row-major form with an @a num_comp * @a num_comp
block at each node. The dimensions of this vector are derived from the active vector for the CeedOperator. The array has shape [nodes, component out,
component in].
  @param[in]  request   Address of CeedRequest for non-blocking completion, else @ref CEED_REQUEST_IMMEDIATE

  @return An error code: 0 - success, otherwise - failure

  @ref User
**/
int CeedOperatorLinearAssemblePointBlockDiagonal(CeedOperator op, CeedVector assembled, CeedRequest *request) {
  bool is_composite;
  CeedCall(CeedOperatorCheckReady(op));
  CeedCall(CeedOperatorIsComposite(op, &is_composite));

  CeedSize input_size = 0, output_size = 0;
  CeedCall(CeedOperatorGetActiveVectorLengths(op, &input_size, &output_size));
  CeedCheck(input_size == output_size, op->ceed, CEED_ERROR_DIMENSION, "Operator must be square");

  // Early exit for empty operator
  if (!is_composite) {
    CeedInt num_elem = 0;

    CeedCall(CeedOperatorGetNumElements(op, &num_elem));
    if (num_elem == 0) return CEED_ERROR_SUCCESS;
  }

  if (op->LinearAssemblePointBlockDiagonal) {
    // Backend version
    CeedCall(op->LinearAssemblePointBlockDiagonal(op, assembled, request));
    return CEED_ERROR_SUCCESS;
  } else if (op->LinearAssembleAddPointBlockDiagonal) {
    // Backend version with zeroing first
    CeedCall(CeedVectorSetValue(assembled, 0.0));
    CeedCall(CeedOperatorLinearAssembleAddPointBlockDiagonal(op, assembled, request));
    return CEED_ERROR_SUCCESS;
  } else {
    // Operator fallback
    CeedOperator op_fallback;

    CeedCall(CeedOperatorGetFallback(op, &op_fallback));
    if (op_fallback) {
      CeedCall(CeedOperatorLinearAssemblePointBlockDiagonal(op_fallback, assembled, request));
      return CEED_ERROR_SUCCESS;
    }
  }
  // Default interface implementation
  CeedCall(CeedVectorSetValue(assembled, 0.0));
  CeedCall(CeedOperatorLinearAssembleAddPointBlockDiagonal(op, assembled, request));

  return CEED_ERROR_SUCCESS;
}

/**
  @brief Assemble the point block diagonal of a square linear CeedOperator

  This sums into a CeedVector with the point block diagonal of a linear CeedOperator.

  Note: Currently only non-composite CeedOperators with a single field and composite CeedOperators with single field sub-operators are supported.

  Note: Calling this function asserts that setup is complete and sets the CeedOperator as immutable.

  @param[in]  op        CeedOperator to assemble CeedQFunction
  @param[out] assembled CeedVector to store assembled CeedOperator point block diagonal, provided in row-major form with an @a num_comp * @a num_comp
block at each node. The dimensions of this vector are derived from the active vector for the CeedOperator. The array has shape [nodes, component out,
component in].
  @param[in]  request Address of CeedRequest for non-blocking completion, else @ref CEED_REQUEST_IMMEDIATE

  @return An error code: 0 - success, otherwise - failure

  @ref User
**/
int CeedOperatorLinearAssembleAddPointBlockDiagonal(CeedOperator op, CeedVector assembled, CeedRequest *request) {
  bool is_composite;
  CeedCall(CeedOperatorCheckReady(op));
  CeedCall(CeedOperatorIsComposite(op, &is_composite));

  CeedSize input_size = 0, output_size = 0;
  CeedCall(CeedOperatorGetActiveVectorLengths(op, &input_size, &output_size));
  CeedCheck(input_size == output_size, op->ceed, CEED_ERROR_DIMENSION, "Operator must be square");

  // Early exit for empty operator
  if (!is_composite) {
    CeedInt num_elem = 0;

    CeedCall(CeedOperatorGetNumElements(op, &num_elem));
    if (num_elem == 0) return CEED_ERROR_SUCCESS;
  }

  if (op->LinearAssembleAddPointBlockDiagonal) {
    // Backend version
    CeedCall(op->LinearAssembleAddPointBlockDiagonal(op, assembled, request));
    return CEED_ERROR_SUCCESS;
  } else {
    // Operator fallback
    CeedOperator op_fallback;

    CeedCall(CeedOperatorGetFallback(op, &op_fallback));
    if (op_fallback) {
      CeedCall(CeedOperatorLinearAssembleAddPointBlockDiagonal(op_fallback, assembled, request));
      return CEED_ERROR_SUCCESS;
    }
  }
  // Default interface implementation
  if (is_composite) {
    CeedCall(CeedCompositeOperatorLinearAssembleAddDiagonal(op, request, true, assembled));
  } else {
    CeedCall(CeedSingleOperatorAssembleAddDiagonal_Core(op, request, true, assembled));
  }

  return CEED_ERROR_SUCCESS;
}

/**
   @brief Fully assemble the nonzero pattern of a linear operator.

   Expected to be used in conjunction with CeedOperatorLinearAssemble().

   The assembly routines use coordinate format, with num_entries tuples of the form (i, j, value) which indicate that value should be added to the
matrix in entry (i, j). Note that the (i, j) pairs are not unique and may repeat. This function returns the number of entries and their (i, j)
locations, while CeedOperatorLinearAssemble() provides the values in the same ordering.

   This will generally be slow unless your operator is low-order.

   Note: Calling this function asserts that setup is complete and sets the CeedOperator as immutable.

   @param[in]  op          CeedOperator to assemble
   @param[out] num_entries Number of entries in coordinate nonzero pattern
   @param[out] rows        Row number for each entry
   @param[out] cols        Column number for each entry

   @ref User
**/
int CeedOperatorLinearAssembleSymbolic(CeedOperator op, CeedSize *num_entries, CeedInt **rows, CeedInt **cols) {
  CeedInt       num_suboperators, single_entries;
  CeedOperator *sub_operators;
  bool          is_composite;
  CeedCall(CeedOperatorCheckReady(op));
  CeedCall(CeedOperatorIsComposite(op, &is_composite));

  if (op->LinearAssembleSymbolic) {
    // Backend version
    CeedCall(op->LinearAssembleSymbolic(op, num_entries, rows, cols));
    return CEED_ERROR_SUCCESS;
  } else {
    // Operator fallback
    CeedOperator op_fallback;

    CeedCall(CeedOperatorGetFallback(op, &op_fallback));
    if (op_fallback) {
      CeedCall(CeedOperatorLinearAssembleSymbolic(op_fallback, num_entries, rows, cols));
      return CEED_ERROR_SUCCESS;
    }
  }

  // Default interface implementation

  // count entries and allocate rows, cols arrays
  *num_entries = 0;
  if (is_composite) {
    CeedCall(CeedCompositeOperatorGetNumSub(op, &num_suboperators));
    CeedCall(CeedCompositeOperatorGetSubList(op, &sub_operators));
    for (CeedInt k = 0; k < num_suboperators; ++k) {
      CeedCall(CeedSingleOperatorAssemblyCountEntries(sub_operators[k], &single_entries));
      *num_entries += single_entries;
    }
  } else {
    CeedCall(CeedSingleOperatorAssemblyCountEntries(op, &single_entries));
    *num_entries += single_entries;
  }
  CeedCall(CeedCalloc(*num_entries, rows));
  CeedCall(CeedCalloc(*num_entries, cols));

  // assemble nonzero locations
  CeedInt offset = 0;
  if (is_composite) {
    CeedCall(CeedCompositeOperatorGetNumSub(op, &num_suboperators));
    CeedCall(CeedCompositeOperatorGetSubList(op, &sub_operators));
    for (CeedInt k = 0; k < num_suboperators; ++k) {
      CeedCall(CeedSingleOperatorAssembleSymbolic(sub_operators[k], offset, *rows, *cols));
      CeedCall(CeedSingleOperatorAssemblyCountEntries(sub_operators[k], &single_entries));
      offset += single_entries;
    }
  } else {
    CeedCall(CeedSingleOperatorAssembleSymbolic(op, offset, *rows, *cols));
  }

  return CEED_ERROR_SUCCESS;
}

/**
   @brief Fully assemble the nonzero entries of a linear operator.

   Expected to be used in conjunction with CeedOperatorLinearAssembleSymbolic().

   The assembly routines use coordinate format, with num_entries tuples of the form (i, j, value) which indicate that value should be added to the
matrix in entry (i, j). Note that the (i, j) pairs are not unique and may repeat. This function returns the values of the nonzero entries to be added,
their (i, j) locations are provided by CeedOperatorLinearAssembleSymbolic()

   This will generally be slow unless your operator is low-order.

   Note: Calling this function asserts that setup is complete and sets the CeedOperator as immutable.

   @param[in]  op     CeedOperator to assemble
   @param[out] values Values to assemble into matrix

   @ref User
**/
int CeedOperatorLinearAssemble(CeedOperator op, CeedVector values) {
  CeedInt       num_suboperators, single_entries = 0;
  CeedOperator *sub_operators;
  bool          is_composite;
  CeedCall(CeedOperatorCheckReady(op));
  CeedCall(CeedOperatorIsComposite(op, &is_composite));

  // Early exit for empty operator
  if (!is_composite) {
    CeedInt num_elem = 0;

    CeedCall(CeedOperatorGetNumElements(op, &num_elem));
    if (num_elem == 0) return CEED_ERROR_SUCCESS;
  }

  if (op->LinearAssemble) {
    // Backend version
    CeedCall(op->LinearAssemble(op, values));
    return CEED_ERROR_SUCCESS;
  } else {
    // Operator fallback
    CeedOperator op_fallback;

    CeedCall(CeedOperatorGetFallback(op, &op_fallback));
    if (op_fallback) {
      CeedCall(CeedOperatorLinearAssemble(op_fallback, values));
      return CEED_ERROR_SUCCESS;
    }
  }

  // Default interface implementation
  CeedInt offset = 0;
  CeedCall(CeedVectorSetValue(values, 0.0));
  if (is_composite) {
    CeedCall(CeedCompositeOperatorGetNumSub(op, &num_suboperators));
    CeedCall(CeedCompositeOperatorGetSubList(op, &sub_operators));
    for (CeedInt k = 0; k < num_suboperators; k++) {
      CeedCall(CeedSingleOperatorAssemble(sub_operators[k], offset, values));
      CeedCall(CeedSingleOperatorAssemblyCountEntries(sub_operators[k], &single_entries));
      offset += single_entries;
    }
  } else {
    CeedCall(CeedSingleOperatorAssemble(op, offset, values));
  }

  return CEED_ERROR_SUCCESS;
}

/**
  @brief Get the multiplicity of nodes across suboperators in a composite CeedOperator

  Note: Calling this function asserts that setup is complete and sets the CeedOperator as immutable.

  @param[in]  op               Composite CeedOperator
  @param[in]  num_skip_indices Number of suboperators to skip
  @param[in]  skip_indices     Array of indices of suboperators to skip
  @param[out] mult             Vector to store multiplicity (of size l_size)

  @return An error code: 0 - success, otherwise - failure

  @ref User
**/
int CeedCompositeOperatorGetMultiplicity(CeedOperator op, CeedInt num_skip_indices, CeedInt *skip_indices, CeedVector mult) {
  CeedCall(CeedOperatorCheckReady(op));

  Ceed                ceed;
  CeedInt             num_suboperators;
  CeedSize            l_vec_len;
  CeedScalar         *mult_array;
  CeedVector          ones_l_vec;
  CeedElemRestriction elem_rstr;
  CeedOperator       *sub_operators;

  CeedCall(CeedOperatorGetCeed(op, &ceed));

  // Zero mult vector
  CeedCall(CeedVectorSetValue(mult, 0.0));

  // Get suboperators
  CeedCall(CeedCompositeOperatorGetNumSub(op, &num_suboperators));
  CeedCall(CeedCompositeOperatorGetSubList(op, &sub_operators));
  if (num_suboperators == 0) return CEED_ERROR_SUCCESS;

  // Work vector
  CeedCall(CeedVectorGetLength(mult, &l_vec_len));
  CeedCall(CeedVectorCreate(ceed, l_vec_len, &ones_l_vec));
  CeedCall(CeedVectorSetValue(ones_l_vec, 1.0));
  CeedCall(CeedVectorGetArray(mult, CEED_MEM_HOST, &mult_array));

  // Compute multiplicity across suboperators
  for (CeedInt i = 0; i < num_suboperators; i++) {
    const CeedScalar *sub_mult_array;
    CeedVector        sub_mult_l_vec, ones_e_vec;

    // -- Check for suboperator to skip
    for (CeedInt j = 0; j < num_skip_indices; j++) {
      if (skip_indices[j] == i) continue;
    }

    // -- Sub operator multiplicity
    CeedCall(CeedOperatorGetActiveElemRestriction(sub_operators[i], &elem_rstr));
    CeedCall(CeedElemRestrictionCreateVector(elem_rstr, &sub_mult_l_vec, &ones_e_vec));
    CeedCall(CeedVectorSetValue(sub_mult_l_vec, 0.0));
    CeedCall(CeedElemRestrictionApply(elem_rstr, CEED_NOTRANSPOSE, ones_l_vec, ones_e_vec, CEED_REQUEST_IMMEDIATE));
    CeedCall(CeedElemRestrictionApply(elem_rstr, CEED_TRANSPOSE, ones_e_vec, sub_mult_l_vec, CEED_REQUEST_IMMEDIATE));
    CeedCall(CeedVectorGetArrayRead(sub_mult_l_vec, CEED_MEM_HOST, &sub_mult_array));
    // ---- Flag every node present in the current suboperator
    for (CeedInt j = 0; j < l_vec_len; j++) {
      if (sub_mult_array[j] > 0.0) mult_array[j] += 1.0;
    }
    CeedCall(CeedVectorRestoreArrayRead(sub_mult_l_vec, &sub_mult_array));
    CeedCall(CeedVectorDestroy(&sub_mult_l_vec));
    CeedCall(CeedVectorDestroy(&ones_e_vec));
  }
  CeedCall(CeedVectorRestoreArray(mult, &mult_array));
  CeedCall(CeedVectorDestroy(&ones_l_vec));

  return CEED_ERROR_SUCCESS;
}

/**
  @brief Create a multigrid coarse operator and level transfer operators for a CeedOperator, creating the prolongation basis from the fine and coarse
grid interpolation

  Note: Calling this function asserts that setup is complete and sets all four CeedOperators as immutable.

  @param[in]  op_fine      Fine grid operator
  @param[in]  p_mult_fine  L-vector multiplicity in parallel gather/scatter, or NULL if not creating prolongation/restriction operators
  @param[in]  rstr_coarse  Coarse grid restriction
  @param[in]  basis_coarse Coarse grid active vector basis
  @param[out] op_coarse    Coarse grid operator
  @param[out] op_prolong   Coarse to fine operator, or NULL
  @param[out] op_restrict  Fine to coarse operator, or NULL

  @return An error code: 0 - success, otherwise - failure

  @ref User
**/
int CeedOperatorMultigridLevelCreate(CeedOperator op_fine, CeedVector p_mult_fine, CeedElemRestriction rstr_coarse, CeedBasis basis_coarse,
                                     CeedOperator *op_coarse, CeedOperator *op_prolong, CeedOperator *op_restrict) {
  CeedCall(CeedOperatorCheckReady(op_fine));

  // Build prolongation matrix, if required
  CeedBasis basis_c_to_f = NULL;
  if (op_prolong || op_restrict) {
    CeedBasis basis_fine;
    CeedCall(CeedOperatorGetActiveBasis(op_fine, &basis_fine));
    CeedCall(CeedBasisCreateProjection(basis_coarse, basis_fine, &basis_c_to_f));
  }

  // Core code
  CeedCall(CeedSingleOperatorMultigridLevel(op_fine, p_mult_fine, rstr_coarse, basis_coarse, basis_c_to_f, op_coarse, op_prolong, op_restrict));

  return CEED_ERROR_SUCCESS;
}

/**
  @brief Create a multigrid coarse operator and level transfer operators for a CeedOperator with a tensor basis for the active basis

  Note: Calling this function asserts that setup is complete and sets all four CeedOperators as immutable.

  @param[in]  op_fine       Fine grid operator
  @param[in]  p_mult_fine   L-vector multiplicity in parallel gather/scatter, or NULL if not creating prolongation/restriction operators
  @param[in]  rstr_coarse   Coarse grid restriction
  @param[in]  basis_coarse  Coarse grid active vector basis
  @param[in]  interp_c_to_f Matrix for coarse to fine interpolation, or NULL if not creating prolongation/restriction operators
  @param[out] op_coarse     Coarse grid operator
  @param[out] op_prolong    Coarse to fine operator, or NULL
  @param[out] op_restrict   Fine to coarse operator, or NULL

  @return An error code: 0 - success, otherwise - failure

  @ref User
**/
int CeedOperatorMultigridLevelCreateTensorH1(CeedOperator op_fine, CeedVector p_mult_fine, CeedElemRestriction rstr_coarse, CeedBasis basis_coarse,
                                             const CeedScalar *interp_c_to_f, CeedOperator *op_coarse, CeedOperator *op_prolong,
                                             CeedOperator *op_restrict) {
  CeedCall(CeedOperatorCheckReady(op_fine));
  Ceed ceed;
  CeedCall(CeedOperatorGetCeed(op_fine, &ceed));

  // Check for compatible quadrature spaces
  CeedBasis basis_fine;
  CeedCall(CeedOperatorGetActiveBasis(op_fine, &basis_fine));
  CeedInt Q_f, Q_c;
  CeedCall(CeedBasisGetNumQuadraturePoints(basis_fine, &Q_f));
  CeedCall(CeedBasisGetNumQuadraturePoints(basis_coarse, &Q_c));
  CeedCheck(Q_f == Q_c, ceed, CEED_ERROR_DIMENSION, "Bases must have compatible quadrature spaces");

  // Create coarse to fine basis, if required
  CeedBasis basis_c_to_f = NULL;
  if (op_prolong || op_restrict) {
    // Check if interpolation matrix is provided
    CeedCheck(interp_c_to_f, ceed, CEED_ERROR_INCOMPATIBLE,
              "Prolongation or restriction operator creation requires coarse-to-fine interpolation matrix");
    CeedInt dim, num_comp, num_nodes_c, P_1d_f, P_1d_c;
    CeedCall(CeedBasisGetDimension(basis_fine, &dim));
    CeedCall(CeedBasisGetNumComponents(basis_fine, &num_comp));
    CeedCall(CeedBasisGetNumNodes1D(basis_fine, &P_1d_f));
    CeedCall(CeedElemRestrictionGetElementSize(rstr_coarse, &num_nodes_c));
    P_1d_c = dim == 1 ? num_nodes_c : dim == 2 ? sqrt(num_nodes_c) : cbrt(num_nodes_c);
    CeedScalar *q_ref, *q_weight, *grad;
    CeedCall(CeedCalloc(P_1d_f, &q_ref));
    CeedCall(CeedCalloc(P_1d_f, &q_weight));
    CeedCall(CeedCalloc(P_1d_f * P_1d_c * dim, &grad));
    CeedCall(CeedBasisCreateTensorH1(ceed, dim, num_comp, P_1d_c, P_1d_f, interp_c_to_f, grad, q_ref, q_weight, &basis_c_to_f));
    CeedCall(CeedFree(&q_ref));
    CeedCall(CeedFree(&q_weight));
    CeedCall(CeedFree(&grad));
  }

  // Core code
  CeedCall(CeedSingleOperatorMultigridLevel(op_fine, p_mult_fine, rstr_coarse, basis_coarse, basis_c_to_f, op_coarse, op_prolong, op_restrict));
  return CEED_ERROR_SUCCESS;
}

/**
  @brief Create a multigrid coarse operator and level transfer operators for a CeedOperator with a non-tensor basis for the active vector

  Note: Calling this function asserts that setup is complete and sets all four CeedOperators as immutable.

  @param[in]  op_fine       Fine grid operator
  @param[in]  p_mult_fine   L-vector multiplicity in parallel gather/scatter, or NULL if not creating prolongation/restriction operators
  @param[in]  rstr_coarse   Coarse grid restriction
  @param[in]  basis_coarse  Coarse grid active vector basis
  @param[in]  interp_c_to_f Matrix for coarse to fine interpolation, or NULL if not creating prolongation/restriction operators
  @param[out] op_coarse     Coarse grid operator
  @param[out] op_prolong    Coarse to fine operator, or NULL
  @param[out] op_restrict   Fine to coarse operator, or NULL

  @return An error code: 0 - success, otherwise - failure

  @ref User
**/
int CeedOperatorMultigridLevelCreateH1(CeedOperator op_fine, CeedVector p_mult_fine, CeedElemRestriction rstr_coarse, CeedBasis basis_coarse,
                                       const CeedScalar *interp_c_to_f, CeedOperator *op_coarse, CeedOperator *op_prolong,
                                       CeedOperator *op_restrict) {
  CeedCall(CeedOperatorCheckReady(op_fine));
  Ceed ceed;
  CeedCall(CeedOperatorGetCeed(op_fine, &ceed));

  // Check for compatible quadrature spaces
  CeedBasis basis_fine;
  CeedCall(CeedOperatorGetActiveBasis(op_fine, &basis_fine));
  CeedInt Q_f, Q_c;
  CeedCall(CeedBasisGetNumQuadraturePoints(basis_fine, &Q_f));
  CeedCall(CeedBasisGetNumQuadraturePoints(basis_coarse, &Q_c));
  CeedCheck(Q_f == Q_c, ceed, CEED_ERROR_DIMENSION, "Bases must have compatible quadrature spaces");

  // Coarse to fine basis
  CeedBasis basis_c_to_f = NULL;
  if (op_prolong || op_restrict) {
    // Check if interpolation matrix is provided
    CeedCheck(interp_c_to_f, ceed, CEED_ERROR_INCOMPATIBLE,
              "Prolongation or restriction operator creation requires coarse-to-fine interpolation matrix");
    CeedElemTopology topo;
    CeedCall(CeedBasisGetTopology(basis_fine, &topo));
    CeedInt dim, num_comp, num_nodes_c, num_nodes_f;
    CeedCall(CeedBasisGetDimension(basis_fine, &dim));
    CeedCall(CeedBasisGetNumComponents(basis_fine, &num_comp));
    CeedCall(CeedBasisGetNumNodes(basis_fine, &num_nodes_f));
    CeedCall(CeedElemRestrictionGetElementSize(rstr_coarse, &num_nodes_c));
    CeedScalar *q_ref, *q_weight, *grad;
    CeedCall(CeedCalloc(num_nodes_f * dim, &q_ref));
    CeedCall(CeedCalloc(num_nodes_f, &q_weight));
    CeedCall(CeedCalloc(num_nodes_f * num_nodes_c * dim, &grad));
    CeedCall(CeedBasisCreateH1(ceed, topo, num_comp, num_nodes_c, num_nodes_f, interp_c_to_f, grad, q_ref, q_weight, &basis_c_to_f));
    CeedCall(CeedFree(&q_ref));
    CeedCall(CeedFree(&q_weight));
    CeedCall(CeedFree(&grad));
  }

  // Core code
  CeedCall(CeedSingleOperatorMultigridLevel(op_fine, p_mult_fine, rstr_coarse, basis_coarse, basis_c_to_f, op_coarse, op_prolong, op_restrict));
  return CEED_ERROR_SUCCESS;
}

/**
  @brief Build a FDM based approximate inverse for each element for a CeedOperator

  This returns a CeedOperator and CeedVector to apply a Fast Diagonalization Method based approximate inverse.
    This function obtains the simultaneous diagonalization for the 1D mass and Laplacian operators, \f$M = V^T V, K = V^T S V\f$.
    The assembled QFunction is used to modify the eigenvalues from simultaneous diagonalization and obtain an approximate inverse of the form \f$V^T
\hat S V\f$. The CeedOperator must be linear and non-composite. The associated CeedQFunction must therefore also be linear.

  Note: Calling this function asserts that setup is complete and sets the CeedOperator as immutable.

  @param[in]  op      CeedOperator to create element inverses
  @param[out] fdm_inv CeedOperator to apply the action of a FDM based inverse for each element
  @param[in]  request Address of CeedRequest for non-blocking completion, else @ref CEED_REQUEST_IMMEDIATE

  @return An error code: 0 - success, otherwise - failure

  @ref User
**/
int CeedOperatorCreateFDMElementInverse(CeedOperator op, CeedOperator *fdm_inv, CeedRequest *request) {
  CeedCall(CeedOperatorCheckReady(op));

  if (op->CreateFDMElementInverse) {
    // Backend version
    CeedCall(op->CreateFDMElementInverse(op, fdm_inv, request));
    return CEED_ERROR_SUCCESS;
  } else {
    // Operator fallback
    CeedOperator op_fallback;

    CeedCall(CeedOperatorGetFallback(op, &op_fallback));
    if (op_fallback) {
      CeedCall(CeedOperatorCreateFDMElementInverse(op_fallback, fdm_inv, request));
      return CEED_ERROR_SUCCESS;
    }
  }

  // Default interface implementation
  Ceed ceed, ceed_parent;
  CeedCall(CeedOperatorGetCeed(op, &ceed));
  CeedCall(CeedGetOperatorFallbackParentCeed(ceed, &ceed_parent));
  ceed_parent = ceed_parent ? ceed_parent : ceed;
  CeedQFunction qf;
  CeedCall(CeedOperatorGetQFunction(op, &qf));

  // Determine active input basis
  bool                interp = false, grad = false;
  CeedBasis           basis = NULL;
  CeedElemRestriction rstr  = NULL;
  CeedOperatorField  *op_fields;
  CeedQFunctionField *qf_fields;
  CeedInt             num_input_fields;
  CeedCall(CeedOperatorGetFields(op, &num_input_fields, &op_fields, NULL, NULL));
  CeedCall(CeedQFunctionGetFields(qf, NULL, &qf_fields, NULL, NULL));
  for (CeedInt i = 0; i < num_input_fields; i++) {
    CeedVector vec;
    CeedCall(CeedOperatorFieldGetVector(op_fields[i], &vec));
    if (vec == CEED_VECTOR_ACTIVE) {
      CeedEvalMode eval_mode;
      CeedCall(CeedQFunctionFieldGetEvalMode(qf_fields[i], &eval_mode));
      interp = interp || eval_mode == CEED_EVAL_INTERP;
      grad   = grad || eval_mode == CEED_EVAL_GRAD;
      CeedCall(CeedOperatorFieldGetBasis(op_fields[i], &basis));
      CeedCall(CeedOperatorFieldGetElemRestriction(op_fields[i], &rstr));
    }
  }
  CeedCheck(basis, ceed, CEED_ERROR_BACKEND, "No active field set");
  CeedSize l_size = 1;
  CeedInt  P_1d, Q_1d, num_nodes, num_qpts, dim, num_comp = 1, num_elem = 1;
  CeedCall(CeedBasisGetNumNodes1D(basis, &P_1d));
  CeedCall(CeedBasisGetNumNodes(basis, &num_nodes));
  CeedCall(CeedBasisGetNumQuadraturePoints1D(basis, &Q_1d));
  CeedCall(CeedBasisGetNumQuadraturePoints(basis, &num_qpts));
  CeedCall(CeedBasisGetDimension(basis, &dim));
  CeedCall(CeedBasisGetNumComponents(basis, &num_comp));
  CeedCall(CeedElemRestrictionGetNumElements(rstr, &num_elem));
  CeedCall(CeedElemRestrictionGetLVectorSize(rstr, &l_size));

  // Build and diagonalize 1D Mass and Laplacian
  bool is_tensor_basis;
  CeedCall(CeedBasisIsTensor(basis, &is_tensor_basis));
  CeedCheck(is_tensor_basis, ceed, CEED_ERROR_BACKEND, "FDMElementInverse only supported for tensor bases");
  CeedScalar *mass, *laplace, *x, *fdm_interp, *lambda;
  CeedCall(CeedCalloc(P_1d * P_1d, &mass));
  CeedCall(CeedCalloc(P_1d * P_1d, &laplace));
  CeedCall(CeedCalloc(P_1d * P_1d, &x));
  CeedCall(CeedCalloc(P_1d * P_1d, &fdm_interp));
  CeedCall(CeedCalloc(P_1d, &lambda));
  // -- Build matrices
  const CeedScalar *interp_1d, *grad_1d, *q_weight_1d;
  CeedCall(CeedBasisGetInterp1D(basis, &interp_1d));
  CeedCall(CeedBasisGetGrad1D(basis, &grad_1d));
  CeedCall(CeedBasisGetQWeights(basis, &q_weight_1d));
  CeedCall(CeedBuildMassLaplace(interp_1d, grad_1d, q_weight_1d, P_1d, Q_1d, dim, mass, laplace));

  // -- Diagonalize
  CeedCall(CeedSimultaneousDiagonalization(ceed, laplace, mass, x, lambda, P_1d));
  CeedCall(CeedFree(&mass));
  CeedCall(CeedFree(&laplace));
  for (CeedInt i = 0; i < P_1d; i++) {
    for (CeedInt j = 0; j < P_1d; j++) fdm_interp[i + j * P_1d] = x[j + i * P_1d];
  }
  CeedCall(CeedFree(&x));

  // Assemble QFunction
  CeedVector          assembled;
  CeedElemRestriction rstr_qf;
  CeedCall(CeedOperatorLinearAssembleQFunctionBuildOrUpdate(op, &assembled, &rstr_qf, request));
  CeedInt layout[3];
  CeedCall(CeedElemRestrictionGetELayout(rstr_qf, &layout));
  CeedCall(CeedElemRestrictionDestroy(&rstr_qf));
  CeedScalar max_norm = 0;
  CeedCall(CeedVectorNorm(assembled, CEED_NORM_MAX, &max_norm));

  // Calculate element averages
  CeedInt           num_modes = (interp ? 1 : 0) + (grad ? dim : 0);
  CeedScalar       *elem_avg;
  const CeedScalar *assembled_array, *q_weight_array;
  CeedVector        q_weight;
  CeedCall(CeedVectorCreate(ceed_parent, num_qpts, &q_weight));
  CeedCall(CeedBasisApply(basis, 1, CEED_NOTRANSPOSE, CEED_EVAL_WEIGHT, CEED_VECTOR_NONE, q_weight));
  CeedCall(CeedVectorGetArrayRead(assembled, CEED_MEM_HOST, &assembled_array));
  CeedCall(CeedVectorGetArrayRead(q_weight, CEED_MEM_HOST, &q_weight_array));
  CeedCall(CeedCalloc(num_elem, &elem_avg));
  const CeedScalar qf_value_bound = max_norm * 100 * CEED_EPSILON;
  for (CeedInt e = 0; e < num_elem; e++) {
    CeedInt count = 0;
    for (CeedInt q = 0; q < num_qpts; q++) {
      for (CeedInt i = 0; i < num_comp * num_comp * num_modes * num_modes; i++) {
        if (fabs(assembled_array[q * layout[0] + i * layout[1] + e * layout[2]]) > qf_value_bound) {
          elem_avg[e] += assembled_array[q * layout[0] + i * layout[1] + e * layout[2]] / q_weight_array[q];
          count++;
        }
      }
    }
    if (count) {
      elem_avg[e] /= count;
    } else {
      elem_avg[e] = 1.0;
    }
  }
  CeedCall(CeedVectorRestoreArrayRead(assembled, &assembled_array));
  CeedCall(CeedVectorDestroy(&assembled));
  CeedCall(CeedVectorRestoreArrayRead(q_weight, &q_weight_array));
  CeedCall(CeedVectorDestroy(&q_weight));

  // Build FDM diagonal
  CeedVector  q_data;
  CeedScalar *q_data_array, *fdm_diagonal;
  CeedCall(CeedCalloc(num_comp * num_nodes, &fdm_diagonal));
  const CeedScalar fdm_diagonal_bound = num_nodes * CEED_EPSILON;
  for (CeedInt c = 0; c < num_comp; c++) {
    for (CeedInt n = 0; n < num_nodes; n++) {
      if (interp) fdm_diagonal[c * num_nodes + n] = 1.0;
      if (grad) {
        for (CeedInt d = 0; d < dim; d++) {
          CeedInt i = (n / CeedIntPow(P_1d, d)) % P_1d;
          fdm_diagonal[c * num_nodes + n] += lambda[i];
        }
      }
      if (fabs(fdm_diagonal[c * num_nodes + n]) < fdm_diagonal_bound) fdm_diagonal[c * num_nodes + n] = fdm_diagonal_bound;
    }
  }
  CeedCall(CeedVectorCreate(ceed_parent, num_elem * num_comp * num_nodes, &q_data));
  CeedCall(CeedVectorSetValue(q_data, 0.0));
  CeedCall(CeedVectorGetArrayWrite(q_data, CEED_MEM_HOST, &q_data_array));
  for (CeedInt e = 0; e < num_elem; e++) {
    for (CeedInt c = 0; c < num_comp; c++) {
      for (CeedInt n = 0; n < num_nodes; n++) q_data_array[(e * num_comp + c) * num_nodes + n] = 1. / (elem_avg[e] * fdm_diagonal[c * num_nodes + n]);
    }
  }
  CeedCall(CeedFree(&elem_avg));
  CeedCall(CeedFree(&fdm_diagonal));
  CeedCall(CeedVectorRestoreArray(q_data, &q_data_array));

  // Setup FDM operator
  // -- Basis
  CeedBasis   fdm_basis;
  CeedScalar *grad_dummy, *q_ref_dummy, *q_weight_dummy;
  CeedCall(CeedCalloc(P_1d * P_1d, &grad_dummy));
  CeedCall(CeedCalloc(P_1d, &q_ref_dummy));
  CeedCall(CeedCalloc(P_1d, &q_weight_dummy));
  CeedCall(CeedBasisCreateTensorH1(ceed_parent, dim, num_comp, P_1d, P_1d, fdm_interp, grad_dummy, q_ref_dummy, q_weight_dummy, &fdm_basis));
  CeedCall(CeedFree(&fdm_interp));
  CeedCall(CeedFree(&grad_dummy));
  CeedCall(CeedFree(&q_ref_dummy));
  CeedCall(CeedFree(&q_weight_dummy));
  CeedCall(CeedFree(&lambda));

  // -- Restriction
  CeedElemRestriction rstr_qd_i;
  CeedInt             strides[3] = {1, num_nodes, num_nodes * num_comp};
  CeedCall(CeedElemRestrictionCreateStrided(ceed_parent, num_elem, num_nodes, num_comp, num_elem * num_comp * num_nodes, strides, &rstr_qd_i));
  // -- QFunction
  CeedQFunction qf_fdm;
  CeedCall(CeedQFunctionCreateInteriorByName(ceed_parent, "Scale", &qf_fdm));
  CeedCall(CeedQFunctionAddInput(qf_fdm, "input", num_comp, CEED_EVAL_INTERP));
  CeedCall(CeedQFunctionAddInput(qf_fdm, "scale", num_comp, CEED_EVAL_NONE));
  CeedCall(CeedQFunctionAddOutput(qf_fdm, "output", num_comp, CEED_EVAL_INTERP));
  CeedCall(CeedQFunctionSetUserFlopsEstimate(qf_fdm, num_comp));
  // -- QFunction context
  CeedInt *num_comp_data;
  CeedCall(CeedCalloc(1, &num_comp_data));
  num_comp_data[0] = num_comp;
  CeedQFunctionContext ctx_fdm;
  CeedCall(CeedQFunctionContextCreate(ceed, &ctx_fdm));
  CeedCall(CeedQFunctionContextSetData(ctx_fdm, CEED_MEM_HOST, CEED_OWN_POINTER, sizeof(*num_comp_data), num_comp_data));
  CeedCall(CeedQFunctionSetContext(qf_fdm, ctx_fdm));
  CeedCall(CeedQFunctionContextDestroy(&ctx_fdm));
  // -- Operator
  CeedCall(CeedOperatorCreate(ceed_parent, qf_fdm, NULL, NULL, fdm_inv));
  CeedCall(CeedOperatorSetField(*fdm_inv, "input", rstr, fdm_basis, CEED_VECTOR_ACTIVE));
  CeedCall(CeedOperatorSetField(*fdm_inv, "scale", rstr_qd_i, CEED_BASIS_COLLOCATED, q_data));
  CeedCall(CeedOperatorSetField(*fdm_inv, "output", rstr, fdm_basis, CEED_VECTOR_ACTIVE));

  // Cleanup
  CeedCall(CeedVectorDestroy(&q_data));
  CeedCall(CeedBasisDestroy(&fdm_basis));
  CeedCall(CeedElemRestrictionDestroy(&rstr_qd_i));
  CeedCall(CeedQFunctionDestroy(&qf_fdm));

  return CEED_ERROR_SUCCESS;
}

/// @}