impls/ntrdc/ntrdc.c

#include <../src/snes/impls/ntrdc/ntrdcimpl.h> /*I   "petscsnes.h"   I*/

typedef struct {
  SNES snes;
  /*  Information on the regular SNES convergence test; which may have been user provided
      Copied from tr.c (maybe able to disposed, but this is a private function) - Heeho
      Same with SNESTR_KSPConverged_Private, SNESTR_KSPConverged_Destroy, and SNESTR_Converged_Private
 */

  PetscErrorCode (*convtest)(KSP, PetscInt, PetscReal, KSPConvergedReason *, void *);
  PetscErrorCode (*convdestroy)(void *);
  void *convctx;
} SNES_TRDC_KSPConverged_Ctx;

static PetscErrorCode SNESTRDC_KSPConverged_Private(KSP ksp, PetscInt n, PetscReal rnorm, KSPConvergedReason *reason, void *cctx)
{
  SNES_TRDC_KSPConverged_Ctx *ctx  = (SNES_TRDC_KSPConverged_Ctx *)cctx;
  SNES                        snes = ctx->snes;
  SNES_NEWTONTRDC            *neP  = (SNES_NEWTONTRDC *)snes->data;
  Vec                         x;
  PetscReal                   nrm;

  PetscFunctionBegin;
  PetscCall((*ctx->convtest)(ksp, n, rnorm, reason, ctx->convctx));
  if (*reason) PetscCall(PetscInfo(snes, "Default or user provided convergence test KSP iterations=%" PetscInt_FMT ", rnorm=%g\n", n, (double)rnorm));
  /* Determine norm of solution */
  PetscCall(KSPBuildSolution(ksp, NULL, &x));
  PetscCall(VecNorm(x, NORM_2, &nrm));
  if (nrm >= neP->delta) {
    PetscCall(PetscInfo(snes, "Ending linear iteration early, delta=%g, length=%g\n", (double)neP->delta, (double)nrm));
    *reason = KSP_CONVERGED_STEP_LENGTH;
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode SNESTRDC_KSPConverged_Destroy(void *cctx)
{
  SNES_TRDC_KSPConverged_Ctx *ctx = (SNES_TRDC_KSPConverged_Ctx *)cctx;

  PetscFunctionBegin;
  PetscCall((*ctx->convdestroy)(ctx->convctx));
  PetscCall(PetscFree(ctx));

  PetscFunctionReturn(PETSC_SUCCESS);
}

/*
   SNESTRDC_Converged_Private -test convergence JUST for
   the trust region tolerance.

*/
static PetscErrorCode SNESTRDC_Converged_Private(SNES snes, PetscInt it, PetscReal xnorm, PetscReal pnorm, PetscReal fnorm, SNESConvergedReason *reason, void *dummy)
{
  SNES_NEWTONTRDC *neP = (SNES_NEWTONTRDC *)snes->data;

  PetscFunctionBegin;
  *reason = SNES_CONVERGED_ITERATING;
  if (neP->delta < xnorm * snes->deltatol) {
    PetscCall(PetscInfo(snes, "Diverged due to too small a trust region %g<%g*%g\n", (double)neP->delta, (double)xnorm, (double)snes->deltatol));
    *reason = SNES_DIVERGED_TR_DELTA;
  } else if (snes->nfuncs >= snes->max_funcs && snes->max_funcs >= 0) {
    PetscCall(PetscInfo(snes, "Exceeded maximum number of function evaluations: %" PetscInt_FMT "\n", snes->max_funcs));
    *reason = SNES_DIVERGED_FUNCTION_COUNT;
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  SNESNewtonTRDCGetRhoFlag - Get whether the current solution update is within the trust-region.

  Logically Collective

  Input Parameter:
. snes - the nonlinear solver object

  Output Parameter:
. rho_flag - `PETSC_FALSE`

  Level: developer

.seealso: `SNESNEWTONTRDC`, `SNESNewtonTRDCPreCheck()`, `SNESNewtonTRDCGetPreCheck()`, , `SNESNewtonTRDCSetPreCheck()`,
          `SNESNewtonTRDCSetPostCheck()`, `SNESNewtonTRDCGetPostCheck()`
@*/
PetscErrorCode SNESNewtonTRDCGetRhoFlag(SNES snes, PetscBool *rho_flag)
{
  SNES_NEWTONTRDC *tr = (SNES_NEWTONTRDC *)snes->data;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(snes, SNES_CLASSID, 1);
  PetscAssertPointer(rho_flag, 2);
  *rho_flag = tr->rho_satisfied;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  SNESNewtonTRDCSetPreCheck - Sets a user function that is called before the search step has been determined.
  Allows the user a chance to change or override the trust region decision.

  Logically Collective

  Input Parameters:
+ snes - the nonlinear solver object
. func - [optional] function evaluation routine, for the calling sequence see `SNESNewtonTRDCPreCheck()`
- ctx  - [optional] user-defined context for private data for the function evaluation routine (may be `NULL`)

  Level: intermediate

  Note:
  This function is called BEFORE the function evaluation within the `SNESNEWTONTRDC` solver.

.seealso: `SNESNEWTONTRDC`, `SNESNewtonTRDCPreCheck()`, `SNESNewtonTRDCGetPreCheck()`, `SNESNewtonTRDCSetPostCheck()`, `SNESNewtonTRDCGetPostCheck()`,
          `SNESNewtonTRDCGetRhoFlag()`
@*/
PetscErrorCode SNESNewtonTRDCSetPreCheck(SNES snes, PetscErrorCode (*func)(SNES, Vec, Vec, PetscBool *, void *), void *ctx)
{
  SNES_NEWTONTRDC *tr = (SNES_NEWTONTRDC *)snes->data;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(snes, SNES_CLASSID, 1);
  if (func) tr->precheck = func;
  if (ctx) tr->precheckctx = ctx;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  SNESNewtonTRDCGetPreCheck - Gets the pre-check function

  Not Collective

  Input Parameter:
. snes - the nonlinear solver context

  Output Parameters:
+ func - [optional] function evaluation routine, for the calling sequence see `SNESNewtonTRDCPreCheck()`
- ctx  - [optional] user-defined context for private data for the function evaluation routine (may be `NULL`)

  Level: intermediate

.seealso: `SNESNEWTONTRDC`, `SNESNewtonTRDCSetPreCheck()`, `SNESNewtonTRDCPreCheck()`
@*/
PetscErrorCode SNESNewtonTRDCGetPreCheck(SNES snes, PetscErrorCode (**func)(SNES, Vec, Vec, PetscBool *, void *), void **ctx)
{
  SNES_NEWTONTRDC *tr = (SNES_NEWTONTRDC *)snes->data;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(snes, SNES_CLASSID, 1);
  if (func) *func = tr->precheck;
  if (ctx) *ctx = tr->precheckctx;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  SNESNewtonTRDCSetPostCheck - Sets a user function that is called after the search step has been determined but before the next
  function evaluation. Allows the user a chance to change or override the decision of the line search routine

  Logically Collective

  Input Parameters:
+ snes - the nonlinear solver object
. func - [optional] function evaluation routine, for the calling sequence see `SNESNewtonTRDCPostCheck()`
- ctx  - [optional] user-defined context for private data for the function evaluation routine (may be `NULL`)

  Level: intermediate

  Note:
  This function is called BEFORE the function evaluation within the `SNESNEWTONTRDC` solver while the function set in
  `SNESLineSearchSetPostCheck()` is called AFTER the function evaluation.

.seealso: `SNESNEWTONTRDC`, `SNESNewtonTRDCPostCheck()`, `SNESNewtonTRDCGetPostCheck()`, `SNESNewtonTRDCSetPreCheck()`, `SNESNewtonTRDCGetPreCheck()`
@*/
PetscErrorCode SNESNewtonTRDCSetPostCheck(SNES snes, PetscErrorCode (*func)(SNES, Vec, Vec, Vec, PetscBool *, PetscBool *, void *), void *ctx)
{
  SNES_NEWTONTRDC *tr = (SNES_NEWTONTRDC *)snes->data;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(snes, SNES_CLASSID, 1);
  if (func) tr->postcheck = func;
  if (ctx) tr->postcheckctx = ctx;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  SNESNewtonTRDCGetPostCheck - Gets the post-check function

  Not Collective

  Input Parameter:
. snes - the nonlinear solver context

  Output Parameters:
+ func - [optional] function evaluation routine, for the calling sequence see `SNESNewtonTRDCPostCheck()`
- ctx  - [optional] user-defined context for private data for the function evaluation routine (may be `NULL`)

  Level: intermediate

.seealso: `SNESNEWTONTRDC`, `SNESNewtonTRDCSetPostCheck()`, `SNESNewtonTRDCPostCheck()`
@*/
PetscErrorCode SNESNewtonTRDCGetPostCheck(SNES snes, PetscErrorCode (**func)(SNES, Vec, Vec, Vec, PetscBool *, PetscBool *, void *), void **ctx)
{
  SNES_NEWTONTRDC *tr = (SNES_NEWTONTRDC *)snes->data;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(snes, SNES_CLASSID, 1);
  if (func) *func = tr->postcheck;
  if (ctx) *ctx = tr->postcheckctx;
  PetscFunctionReturn(PETSC_SUCCESS);
}

// PetscClangLinter pragma disable: -fdoc-internal-linkage
/*@C
   SNESNewtonTRDCPreCheck - Called before the step has been determined in `SNESNEWTONTRDC`

   Logically Collective

   Input Parameters:
+  snes - the solver
.  X - The last solution
-  Y - The step direction

   Output Parameter:
.  changed_Y - Indicator that the step direction `Y` has been changed.

   Level: developer

.seealso: `SNESNEWTONTRDC`, `SNESNewtonTRDCSetPreCheck()`, `SNESNewtonTRDCGetPreCheck()`, `SNESNewtonTRDCPostCheck()`
@*/
static PetscErrorCode SNESNewtonTRDCPreCheck(SNES snes, Vec X, Vec Y, PetscBool *changed_Y)
{
  SNES_NEWTONTRDC *tr = (SNES_NEWTONTRDC *)snes->data;

  PetscFunctionBegin;
  *changed_Y = PETSC_FALSE;
  if (tr->precheck) {
    PetscCall((*tr->precheck)(snes, X, Y, changed_Y, tr->precheckctx));
    PetscValidLogicalCollectiveBool(snes, *changed_Y, 4);
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

// PetscClangLinter pragma disable: -fdoc-internal-linkage
/*@C
   SNESNewtonTRDCPostCheck - Called after the step has been determined in `SNESNEWTONTRDC` but before the function evaluation at that step

   Logically Collective

   Input Parameters:
+  snes - the solver
.  X - The last solution
.  Y - The full step direction
-  W - The updated solution, W = X - Y

   Output Parameters:
+  changed_Y - indicator if step has been changed
-  changed_W - Indicator if the new candidate solution `W` has been changed.

   Level: developer

   Note:
     If `Y` is changed then `W` is recomputed as `X` - `Y`

.seealso: `SNESNEWTONTRDC`, `SNESNEWTONTRDC`, `SNESNewtonTRDCSetPostCheck()`, `SNESNewtonTRDCGetPostCheck()`, `SNESNewtonTRDCPreCheck()
@*/
static PetscErrorCode SNESNewtonTRDCPostCheck(SNES snes, Vec X, Vec Y, Vec W, PetscBool *changed_Y, PetscBool *changed_W)
{
  SNES_NEWTONTRDC *tr = (SNES_NEWTONTRDC *)snes->data;

  PetscFunctionBegin;
  *changed_Y = PETSC_FALSE;
  *changed_W = PETSC_FALSE;
  if (tr->postcheck) {
    PetscCall((*tr->postcheck)(snes, X, Y, W, changed_Y, changed_W, tr->postcheckctx));
    PetscValidLogicalCollectiveBool(snes, *changed_Y, 5);
    PetscValidLogicalCollectiveBool(snes, *changed_W, 6);
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*
   SNESSolve_NEWTONTRDC - Implements Newton's Method with trust-region subproblem and adds dogleg Cauchy
   (Steepest Descent direction) step and direction if the trust region is not satisfied for solving system of
   nonlinear equations

*/
static PetscErrorCode SNESSolve_NEWTONTRDC(SNES snes)
{
  SNES_NEWTONTRDC            *neP = (SNES_NEWTONTRDC *)snes->data;
  Vec                         X, F, Y, G, W, GradF, YNtmp;
  Vec                         YCtmp;
  Mat                         jac;
  PetscInt                    maxits, i, j, lits, inner_count, bs;
  PetscReal                   rho, fnorm, gnorm, xnorm = 0, delta, ynorm, temp_xnorm, temp_ynorm; /* TRDC inner iteration */
  PetscReal                   inorms[99];                                                         /* need to make it dynamic eventually, fixed max block size of 99 for now */
  PetscReal                   deltaM, ynnorm, f0, mp, gTy, g, yTHy;                               /* rho calculation */
  PetscReal                   auk, gfnorm, ycnorm, c0, c1, c2, tau, tau_pos, tau_neg, gTBg;       /* Cauchy Point */
  KSP                         ksp;
  SNESConvergedReason         reason   = SNES_CONVERGED_ITERATING;
  PetscBool                   breakout = PETSC_FALSE;
  SNES_TRDC_KSPConverged_Ctx *ctx;
  PetscErrorCode (*convtest)(KSP, PetscInt, PetscReal, KSPConvergedReason *, void *), (*convdestroy)(void *);
  void *convctx;

  PetscFunctionBegin;
  maxits = snes->max_its;  /* maximum number of iterations */
  X      = snes->vec_sol;  /* solution vector */
  F      = snes->vec_func; /* residual vector */
  Y      = snes->work[0];  /* update vector */
  G      = snes->work[1];  /* updated residual */
  W      = snes->work[2];  /* temporary vector */
  GradF  = snes->work[3];  /* grad f = J^T F */
  YNtmp  = snes->work[4];  /* Newton solution */
  YCtmp  = snes->work[5];  /* Cauchy solution */

  PetscCheck(!snes->xl && !snes->xu && !snes->ops->computevariablebounds, PetscObjectComm((PetscObject)snes), PETSC_ERR_ARG_WRONGSTATE, "SNES solver %s does not support bounds", ((PetscObject)snes)->type_name);

  PetscCall(VecGetBlockSize(YNtmp, &bs));

  PetscCall(PetscObjectSAWsTakeAccess((PetscObject)snes));
  snes->iter = 0;
  PetscCall(PetscObjectSAWsGrantAccess((PetscObject)snes));

  /* Set the linear stopping criteria to use the More' trick. From tr.c */
  PetscCall(SNESGetKSP(snes, &ksp));
  PetscCall(KSPGetConvergenceTest(ksp, &convtest, &convctx, &convdestroy));
  if (convtest != SNESTRDC_KSPConverged_Private) {
    PetscCall(PetscNew(&ctx));
    ctx->snes = snes;
    PetscCall(KSPGetAndClearConvergenceTest(ksp, &ctx->convtest, &ctx->convctx, &ctx->convdestroy));
    PetscCall(KSPSetConvergenceTest(ksp, SNESTRDC_KSPConverged_Private, ctx, SNESTRDC_KSPConverged_Destroy));
    PetscCall(PetscInfo(snes, "Using Krylov convergence test SNESTRDC_KSPConverged_Private\n"));
  }

  if (!snes->vec_func_init_set) {
    PetscCall(SNESComputeFunction(snes, X, F)); /* F(X) */
  } else snes->vec_func_init_set = PETSC_FALSE;

  PetscCall(VecNorm(F, NORM_2, &fnorm)); /* fnorm <- || F || */
  SNESCheckFunctionNorm(snes, fnorm);
  PetscCall(VecNorm(X, NORM_2, &xnorm)); /* xnorm <- || X || */

  PetscCall(PetscObjectSAWsTakeAccess((PetscObject)snes));
  snes->norm = fnorm;
  PetscCall(PetscObjectSAWsGrantAccess((PetscObject)snes));
  delta      = xnorm ? neP->delta0 * xnorm : neP->delta0; /* initial trust region size scaled by xnorm */
  deltaM     = xnorm ? neP->deltaM * xnorm : neP->deltaM; /* maximum trust region size scaled by xnorm */
  neP->delta = delta;
  PetscCall(SNESLogConvergenceHistory(snes, fnorm, 0));
  PetscCall(SNESMonitor(snes, 0, fnorm));

  neP->rho_satisfied = PETSC_FALSE;

  /* test convergence */
  PetscUseTypeMethod(snes, converged, snes->iter, 0.0, 0.0, fnorm, &snes->reason, snes->cnvP);
  if (snes->reason) PetscFunctionReturn(PETSC_SUCCESS);

  for (i = 0; i < maxits; i++) {
    PetscBool changed_y;
    PetscBool changed_w;

    /* dogleg method */
    PetscCall(SNESComputeJacobian(snes, X, snes->jacobian, snes->jacobian_pre));
    SNESCheckJacobianDomainerror(snes);
    PetscCall(KSPSetOperators(snes->ksp, snes->jacobian, snes->jacobian));
    PetscCall(KSPSolve(snes->ksp, F, YNtmp)); /* Quasi Newton Solution */
    SNESCheckKSPSolve(snes);                  /* this is necessary but old tr.c did not have it*/
    PetscCall(KSPGetIterationNumber(snes->ksp, &lits));
    PetscCall(SNESGetJacobian(snes, &jac, NULL, NULL, NULL));

    /* rescale Jacobian, Newton solution update, and re-calculate delta for multiphase (multivariable)
       for inner iteration and Cauchy direction calculation
    */
    if (bs > 1 && neP->auto_scale_multiphase) {
      PetscCall(VecStrideNormAll(YNtmp, NORM_INFINITY, inorms));
      for (j = 0; j < bs; j++) {
        if (neP->auto_scale_max > 1.0) {
          if (inorms[j] < 1.0 / neP->auto_scale_max) inorms[j] = 1.0 / neP->auto_scale_max;
        }
        PetscCall(VecStrideSet(W, j, inorms[j]));
        PetscCall(VecStrideScale(YNtmp, j, 1.0 / inorms[j]));
        PetscCall(VecStrideScale(X, j, 1.0 / inorms[j]));
      }
      PetscCall(VecNorm(X, NORM_2, &xnorm));
      if (i == 0) {
        delta = neP->delta0 * xnorm;
      } else {
        delta = neP->delta * xnorm;
      }
      deltaM = neP->deltaM * xnorm;
      PetscCall(MatDiagonalScale(jac, NULL, W));
    }

    /* calculating GradF of minimization function */
    PetscCall(MatMultTranspose(jac, F, GradF)); /* grad f = J^T F */
    PetscCall(VecNorm(YNtmp, NORM_2, &ynnorm)); /* ynnorm <- || Y_newton || */

    inner_count        = 0;
    neP->rho_satisfied = PETSC_FALSE;
    while (1) {
      if (ynnorm <= delta) { /* see if the Newton solution is within the trust region */
        PetscCall(VecCopy(YNtmp, Y));
      } else if (neP->use_cauchy) { /* use Cauchy direction if enabled */
        PetscCall(MatMult(jac, GradF, W));
        PetscCall(VecDotRealPart(W, W, &gTBg));     /* completes GradF^T J^T J GradF */
        PetscCall(VecNorm(GradF, NORM_2, &gfnorm)); /* grad f norm <- || grad f || */
        if (gTBg <= 0.0) {
          auk = PETSC_MAX_REAL;
        } else {
          auk = PetscSqr(gfnorm) / gTBg;
        }
        auk = PetscMin(delta / gfnorm, auk);
        PetscCall(VecCopy(GradF, YCtmp));           /* this could be improved */
        PetscCall(VecScale(YCtmp, auk));            /* YCtmp, Cauchy solution*/
        PetscCall(VecNorm(YCtmp, NORM_2, &ycnorm)); /* ycnorm <- || Y_cauchy || */
        if (ycnorm >= delta) {                      /* see if the Cauchy solution meets the criteria */
          PetscCall(VecCopy(YCtmp, Y));
          PetscCall(PetscInfo(snes, "DL evaluated. delta: %8.4e, ynnorm: %8.4e, ycnorm: %8.4e\n", (double)delta, (double)ynnorm, (double)ycnorm));
        } else {                                  /* take ratio, tau, of Cauchy and Newton direction and step */
          PetscCall(VecAXPY(YNtmp, -1.0, YCtmp)); /* YCtmp = A, YNtmp = B */
          PetscCall(VecNorm(YNtmp, NORM_2, &c0)); /* this could be improved */
          c0 = PetscSqr(c0);
          PetscCall(VecDotRealPart(YCtmp, YNtmp, &c1));
          c1 = 2.0 * c1;
          PetscCall(VecNorm(YCtmp, NORM_2, &c2)); /* this could be improved */
          c2      = PetscSqr(c2) - PetscSqr(delta);
          tau_pos = (c1 + PetscSqrtReal(PetscSqr(c1) - 4. * c0 * c2)) / (2. * c0); /* quadratic formula */
          tau_neg = (c1 - PetscSqrtReal(PetscSqr(c1) - 4. * c0 * c2)) / (2. * c0);
          tau     = PetscMax(tau_pos, tau_neg); /* can tau_neg > tau_pos? I don't think so, but just in case. */
          PetscCall(PetscInfo(snes, "DL evaluated. tau: %8.4e, ynnorm: %8.4e, ycnorm: %8.4e\n", (double)tau, (double)ynnorm, (double)ycnorm));
          PetscCall(VecWAXPY(W, tau, YNtmp, YCtmp));
          PetscCall(VecAXPY(W, -tau, YCtmp));
          PetscCall(VecCopy(W, Y)); /* this could be improved */
        }
      } else {
        /* if Cauchy is disabled, only use Newton direction */
        auk = delta / ynnorm;
        PetscCall(VecScale(YNtmp, auk));
        PetscCall(VecCopy(YNtmp, Y)); /* this could be improved (many VecCopy, VecNorm)*/
      }

      PetscCall(VecNorm(Y, NORM_2, &ynorm)); /* compute the final ynorm  */
      f0 = 0.5 * PetscSqr(fnorm);            /* minimizing function f(X) */
      PetscCall(MatMult(jac, Y, W));
      PetscCall(VecDotRealPart(W, W, &yTHy)); /* completes GradY^T J^T J GradY */
      PetscCall(VecDotRealPart(GradF, Y, &gTy));
      mp = f0 - gTy + 0.5 * yTHy; /* quadratic model to satisfy, -gTy because our update is X-Y*/

      /* scale back solution update */
      if (bs > 1 && neP->auto_scale_multiphase) {
        for (j = 0; j < bs; j++) {
          PetscCall(VecStrideScale(Y, j, inorms[j]));
          if (inner_count == 0) {
            /* TRDC inner algorithm does not need scaled X after calculating delta in the outer iteration */
            /* need to scale back X to match Y and provide proper update to the external code */
            PetscCall(VecStrideScale(X, j, inorms[j]));
          }
        }
        if (inner_count == 0) PetscCall(VecNorm(X, NORM_2, &temp_xnorm)); /* only in the first iteration */
        PetscCall(VecNorm(Y, NORM_2, &temp_ynorm));
      } else {
        temp_xnorm = xnorm;
        temp_ynorm = ynorm;
      }
      inner_count++;

      /* Evaluate the solution to meet the improvement ratio criteria */
      PetscCall(SNESNewtonTRDCPreCheck(snes, X, Y, &changed_y));
      PetscCall(VecWAXPY(W, -1.0, Y, X));
      PetscCall(SNESNewtonTRDCPostCheck(snes, X, Y, W, &changed_y, &changed_w));
      if (changed_y) PetscCall(VecWAXPY(W, -1.0, Y, X));
      PetscCall(VecCopy(Y, snes->vec_sol_update));
      PetscCall(SNESComputeFunction(snes, W, G)); /*  F(X-Y) = G */
      PetscCall(VecNorm(G, NORM_2, &gnorm));      /* gnorm <- || g || */
      SNESCheckFunctionNorm(snes, gnorm);
      g = 0.5 * PetscSqr(gnorm); /* minimizing function g(W) */
      if (f0 == mp) rho = 0.0;
      else rho = (f0 - g) / (f0 - mp); /* actual improvement over predicted improvement */

      if (rho < neP->eta2) {
        delta *= neP->t1; /* shrink the region */
      } else if (rho > neP->eta3) {
        delta = PetscMin(neP->t2 * delta, deltaM); /* expand the region, but not greater than deltaM */
      }

      neP->delta = delta;
      if (rho >= neP->eta1) {
        /* unscale delta and xnorm before going to the next outer iteration */
        if (bs > 1 && neP->auto_scale_multiphase) {
          neP->delta = delta / xnorm;
          xnorm      = temp_xnorm;
          ynorm      = temp_ynorm;
        }
        neP->rho_satisfied = PETSC_TRUE;
        break; /* the improvement ratio is satisfactory */
      }
      PetscCall(PetscInfo(snes, "Trying again in smaller region\n"));

      /* check to see if progress is hopeless */
      neP->itflag = PETSC_FALSE;
      /* both delta, ynorm, and xnorm are either scaled or unscaled */
      PetscCall(SNESTRDC_Converged_Private(snes, snes->iter, xnorm, ynorm, fnorm, &reason, snes->cnvP));
      /* if multiphase state changes, break out inner iteration */
      if (reason == SNES_BREAKOUT_INNER_ITER) {
        if (bs > 1 && neP->auto_scale_multiphase) {
          /* unscale delta and xnorm before going to the next outer iteration */
          neP->delta = delta / xnorm;
          xnorm      = temp_xnorm;
          ynorm      = temp_ynorm;
        }
        reason = SNES_CONVERGED_ITERATING;
        break;
      }
      if (reason == SNES_CONVERGED_SNORM_RELATIVE) reason = SNES_DIVERGED_INNER;
      if (reason) {
        if (reason < 0) {
          /* We're not progressing, so return with the current iterate */
          PetscCall(SNESMonitor(snes, i + 1, fnorm));
          breakout = PETSC_TRUE;
          break;
        } else if (reason > 0) {
          /* We're converged, so return with the current iterate and update solution */
          PetscCall(SNESMonitor(snes, i + 1, fnorm));
          breakout = PETSC_FALSE;
          break;
        }
      }
      snes->numFailures++;
    }
    if (!breakout) {
      /* Update function and solution vectors */
      fnorm = gnorm;
      PetscCall(VecCopy(G, F));
      PetscCall(VecCopy(W, X));
      /* Monitor convergence */
      PetscCall(PetscObjectSAWsTakeAccess((PetscObject)snes));
      snes->iter  = i + 1;
      snes->norm  = fnorm;
      snes->xnorm = xnorm;
      snes->ynorm = ynorm;
      PetscCall(PetscObjectSAWsGrantAccess((PetscObject)snes));
      PetscCall(SNESLogConvergenceHistory(snes, snes->norm, lits));
      PetscCall(SNESMonitor(snes, snes->iter, snes->norm));
      /* Test for convergence, xnorm = || X || */
      neP->itflag = PETSC_TRUE;
      if (snes->ops->converged != SNESConvergedSkip) PetscCall(VecNorm(X, NORM_2, &xnorm));
      PetscUseTypeMethod(snes, converged, snes->iter, xnorm, ynorm, fnorm, &reason, snes->cnvP);
      if (reason) break;
    } else break;
  }

  /* PetscCall(PetscFree(inorms)); */
  if (i == maxits) {
    PetscCall(PetscInfo(snes, "Maximum number of iterations has been reached: %" PetscInt_FMT "\n", maxits));
    if (!reason) reason = SNES_DIVERGED_MAX_IT;
  }
  PetscCall(PetscObjectSAWsTakeAccess((PetscObject)snes));
  snes->reason = reason;
  PetscCall(PetscObjectSAWsGrantAccess((PetscObject)snes));
  if (convtest != SNESTRDC_KSPConverged_Private) {
    PetscCall(KSPGetAndClearConvergenceTest(ksp, &ctx->convtest, &ctx->convctx, &ctx->convdestroy));
    PetscCall(PetscFree(ctx));
    PetscCall(KSPSetConvergenceTest(ksp, convtest, convctx, convdestroy));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode SNESSetUp_NEWTONTRDC(SNES snes)
{
  PetscFunctionBegin;
  PetscCall(SNESSetWorkVecs(snes, 6));
  PetscCall(SNESSetUpMatrices(snes));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode SNESReset_NEWTONTRDC(SNES snes)
{
  PetscFunctionBegin;
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode SNESDestroy_NEWTONTRDC(SNES snes)
{
  PetscFunctionBegin;
  PetscCall(SNESReset_NEWTONTRDC(snes));
  PetscCall(PetscFree(snes->data));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode SNESSetFromOptions_NEWTONTRDC(SNES snes, PetscOptionItems *PetscOptionsObject)
{
  SNES_NEWTONTRDC *ctx = (SNES_NEWTONTRDC *)snes->data;

  PetscFunctionBegin;
  PetscOptionsHeadBegin(PetscOptionsObject, "SNES trust region options for nonlinear equations");
  PetscCall(PetscOptionsReal("-snes_trdc_tol", "Trust region tolerance", "SNESSetTrustRegionTolerance", snes->deltatol, &snes->deltatol, NULL));
  PetscCall(PetscOptionsReal("-snes_trdc_eta1", "eta1", "None", ctx->eta1, &ctx->eta1, NULL));
  PetscCall(PetscOptionsReal("-snes_trdc_eta2", "eta2", "None", ctx->eta2, &ctx->eta2, NULL));
  PetscCall(PetscOptionsReal("-snes_trdc_eta3", "eta3", "None", ctx->eta3, &ctx->eta3, NULL));
  PetscCall(PetscOptionsReal("-snes_trdc_t1", "t1", "None", ctx->t1, &ctx->t1, NULL));
  PetscCall(PetscOptionsReal("-snes_trdc_t2", "t2", "None", ctx->t2, &ctx->t2, NULL));
  PetscCall(PetscOptionsReal("-snes_trdc_deltaM", "deltaM", "None", ctx->deltaM, &ctx->deltaM, NULL));
  PetscCall(PetscOptionsReal("-snes_trdc_delta0", "delta0", "None", ctx->delta0, &ctx->delta0, NULL));
  PetscCall(PetscOptionsReal("-snes_trdc_auto_scale_max", "auto_scale_max", "None", ctx->auto_scale_max, &ctx->auto_scale_max, NULL));
  PetscCall(PetscOptionsBool("-snes_trdc_use_cauchy", "use_cauchy", "use Cauchy step and direction", ctx->use_cauchy, &ctx->use_cauchy, NULL));
  PetscCall(PetscOptionsBool("-snes_trdc_auto_scale_multiphase", "auto_scale_multiphase", "Auto scaling for proper cauchy direction", ctx->auto_scale_multiphase, &ctx->auto_scale_multiphase, NULL));
  PetscOptionsHeadEnd();
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode SNESView_NEWTONTRDC(SNES snes, PetscViewer viewer)
{
  SNES_NEWTONTRDC *tr = (SNES_NEWTONTRDC *)snes->data;
  PetscBool        iascii;

  PetscFunctionBegin;
  PetscCall(PetscObjectTypeCompare((PetscObject)viewer, PETSCVIEWERASCII, &iascii));
  if (iascii) {
    PetscCall(PetscViewerASCIIPrintf(viewer, "  Trust region tolerance %g (-snes_trtol)\n", (double)snes->deltatol));
    PetscCall(PetscViewerASCIIPrintf(viewer, "  eta1=%g, eta2=%g, eta3=%g\n", (double)tr->eta1, (double)tr->eta2, (double)tr->eta3));
    PetscCall(PetscViewerASCIIPrintf(viewer, "  delta0=%g, t1=%g, t2=%g, deltaM=%g\n", (double)tr->delta0, (double)tr->t1, (double)tr->t2, (double)tr->deltaM));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*MC
      SNESNEWTONTRDC - Newton based nonlinear solver that uses trust-region dogleg method with Cauchy direction

   Options Database Keys:
+   -snes_trdc_tol <tol> - trust region tolerance
.   -snes_trdc_eta1 <eta1> - trust region parameter 0.0 <= eta1 <= eta2, rho >= eta1 breaks out of the inner iteration (default: eta1=0.001)
.   -snes_trdc_eta2 <eta2> - trust region parameter 0.0 <= eta1 <= eta2, rho <= eta2 shrinks the trust region (default: eta2=0.25)
.   -snes_trdc_eta3 <eta3> - trust region parameter eta3 > eta2, rho >= eta3 expands the trust region (default: eta3=0.75)
.   -snes_trdc_t1 <t1> - trust region parameter, shrinking factor of trust region (default: 0.25)
.   -snes_trdc_t2 <t2> - trust region parameter, expanding factor of trust region (default: 2.0)
.   -snes_trdc_deltaM <deltaM> - trust region parameter, max size of trust region, deltaM*norm2(x) (default: 0.5)
.   -snes_trdc_delta0 <delta0> - trust region parameter, initial size of trust region, delta0*norm2(x) (default: 0.1)
.   -snes_trdc_auto_scale_max <auto_scale_max> - used with auto_scale_multiphase, caps the maximum auto-scaling factor
.   -snes_trdc_use_cauchy <use_cauchy> - True uses dogleg Cauchy (Steepest Descent direction) step & direction in the trust region algorithm
-   -snes_trdc_auto_scale_multiphase <auto_scale_multiphase> - True turns on auto-scaling for multivariable block matrix for Cauchy and trust region

   Level: intermediate

    Reference:
.   - * "Linear and Nonlinear Solvers for Simulating Multiphase Flow
    within Large-Scale Engineered Subsurface Systems" by Heeho D. Park, Glenn E. Hammond,
    Albert J. Valocchi, Tara LaForce.

.seealso: `SNESCreate()`, `SNES`, `SNESSetType()`, `SNESNEWTONLS`, `SNESSetTrustRegionTolerance()`,
          `SNESNewtonTRDCPreCheck()`, `SNESNewtonTRDCGetPreCheck()`, `SNESNewtonTRDCSetPostCheck()`, `SNESNewtonTRDCGetPostCheck()`,
          `SNESNewtonTRDCGetRhoFlag()`, `SNESNewtonTRDCSetPreCheck()`
M*/
PETSC_EXTERN PetscErrorCode SNESCreate_NEWTONTRDC(SNES snes)
{
  SNES_NEWTONTRDC *neP;

  PetscFunctionBegin;
  snes->ops->setup          = SNESSetUp_NEWTONTRDC;
  snes->ops->solve          = SNESSolve_NEWTONTRDC;
  snes->ops->destroy        = SNESDestroy_NEWTONTRDC;
  snes->ops->setfromoptions = SNESSetFromOptions_NEWTONTRDC;
  snes->ops->view           = SNESView_NEWTONTRDC;
  snes->ops->reset          = SNESReset_NEWTONTRDC;

  snes->usesksp = PETSC_TRUE;
  snes->usesnpc = PETSC_FALSE;

  snes->alwayscomputesfinalresidual = PETSC_TRUE;

  PetscCall(PetscNew(&neP));
  snes->data                 = (void *)neP;
  neP->delta                 = 0.0;
  neP->delta0                = 0.1;
  neP->eta1                  = 0.001;
  neP->eta2                  = 0.25;
  neP->eta3                  = 0.75;
  neP->t1                    = 0.25;
  neP->t2                    = 2.0;
  neP->deltaM                = 0.5;
  neP->sigma                 = 0.0001;
  neP->itflag                = PETSC_FALSE;
  neP->rnorm0                = 0.0;
  neP->ttol                  = 0.0;
  neP->use_cauchy            = PETSC_TRUE;
  neP->auto_scale_multiphase = PETSC_FALSE;
  neP->auto_scale_max        = -1.0;
  neP->rho_satisfied         = PETSC_FALSE;
  snes->deltatol             = 1.e-12;

  /* for multiphase (multivariable) scaling */
  /* may be used for dynamic allocation of inorms, but it fails snes_tutorials-ex3_13
     on test forced DIVERGED_JACOBIAN_DOMAIN test. I will use static array for now.
  PetscCall(VecGetBlockSize(snes->work[0],&neP->bs));
  PetscCall(PetscCalloc1(neP->bs,&neP->inorms));
  */

  PetscFunctionReturn(PETSC_SUCCESS);
}