static char help[] = "A toy example for testing forward and adjoint sensitivity analysis of an implicit ODE with a paramerized mass matrice.\n";

/*
  This example solves the simple ODE
    c x' = b x, x(0) = a,
  whose analytical solution is x(T)=a*exp(b/c*T), and calculates the derivative of x(T) w.r.t. c (by default) or w.r.t. b (can be enabled with command line option -der 2).

*/

#include <petscts.h>

typedef struct _n_User *User;
struct _n_User {
  PetscReal a;
  PetscReal b;
  PetscReal c;
  /* Sensitivity analysis support */
  PetscInt  steps;
  PetscReal ftime;
  Mat       Jac;  /* Jacobian matrix */
  Mat       Jacp; /* JacobianP matrix */
  Vec       x;
  Mat       sp;        /* forward sensitivity variables */
  Vec       lambda[1]; /* adjoint sensitivity variables */
  Vec       mup[1];    /* adjoint sensitivity variables */
  PetscInt  der;
};

static PetscErrorCode IFunction(TS ts, PetscReal t, Vec X, Vec Xdot, Vec F, void *ctx)
{
  User               user = (User)ctx;
  const PetscScalar *x, *xdot;
  PetscScalar       *f;

  PetscFunctionBeginUser;
  PetscCall(VecGetArrayRead(X, &x));
  PetscCall(VecGetArrayRead(Xdot, &xdot));
  PetscCall(VecGetArrayWrite(F, &f));
  f[0] = user->c * xdot[0] - user->b * x[0];
  PetscCall(VecRestoreArrayRead(X, &x));
  PetscCall(VecRestoreArrayRead(Xdot, &xdot));
  PetscCall(VecRestoreArrayWrite(F, &f));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode IJacobian(TS ts, PetscReal t, Vec X, Vec Xdot, PetscReal a, Mat A, Mat B, void *ctx)
{
  User               user     = (User)ctx;
  PetscInt           rowcol[] = {0};
  PetscScalar        J[1][1];
  const PetscScalar *x;

  PetscFunctionBeginUser;
  PetscCall(VecGetArrayRead(X, &x));
  J[0][0] = user->c * a - user->b * 1.0;
  PetscCall(MatSetValues(B, 1, rowcol, 1, rowcol, &J[0][0], INSERT_VALUES));
  PetscCall(VecRestoreArrayRead(X, &x));

  PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));
  if (A != B) {
    PetscCall(MatAssemblyBegin(B, MAT_FINAL_ASSEMBLY));
    PetscCall(MatAssemblyEnd(B, MAT_FINAL_ASSEMBLY));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode IJacobianP(TS ts, PetscReal t, Vec X, Vec Xdot, PetscReal shift, Mat A, void *ctx)
{
  User               user  = (User)ctx;
  PetscInt           row[] = {0}, col[] = {0};
  PetscScalar        J[1][1];
  const PetscScalar *x, *xdot;
  PetscReal          dt;

  PetscFunctionBeginUser;
  PetscCall(VecGetArrayRead(X, &x));
  PetscCall(VecGetArrayRead(Xdot, &xdot));
  PetscCall(TSGetTimeStep(ts, &dt));
  if (user->der == 1) J[0][0] = xdot[0];
  if (user->der == 2) J[0][0] = -x[0];
  PetscCall(MatSetValues(A, 1, row, 1, col, &J[0][0], INSERT_VALUES));
  PetscCall(VecRestoreArrayRead(X, &x));

  PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));
  PetscFunctionReturn(PETSC_SUCCESS);
}

int main(int argc, char **argv)
{
  TS             ts;
  PetscScalar   *x_ptr;
  PetscMPIInt    size;
  struct _n_User user;
  PetscInt       rows, cols;

  PetscFunctionBeginUser;
  PetscCall(PetscInitialize(&argc, &argv, NULL, help));

  PetscCallMPI(MPI_Comm_size(PETSC_COMM_WORLD, &size));
  PetscCheck(size == 1, PETSC_COMM_WORLD, PETSC_ERR_WRONG_MPI_SIZE, "This is a uniprocessor example only!");

  user.a     = 2.0;
  user.b     = 4.0;
  user.c     = 3.0;
  user.steps = 0;
  user.ftime = 1.0;
  user.der   = 1;
  PetscCall(PetscOptionsGetInt(NULL, NULL, "-der", &user.der, NULL));

  rows = 1;
  cols = 1;
  PetscCall(MatCreate(PETSC_COMM_WORLD, &user.Jac));
  PetscCall(MatSetSizes(user.Jac, PETSC_DECIDE, PETSC_DECIDE, 1, 1));
  PetscCall(MatSetFromOptions(user.Jac));
  PetscCall(MatSetUp(user.Jac));
  PetscCall(MatCreateVecs(user.Jac, &user.x, NULL));

  PetscCall(TSCreate(PETSC_COMM_WORLD, &ts));
  PetscCall(TSSetType(ts, TSBEULER));
  PetscCall(TSSetIFunction(ts, NULL, IFunction, &user));
  PetscCall(TSSetIJacobian(ts, user.Jac, user.Jac, IJacobian, &user));
  PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_MATCHSTEP));
  PetscCall(TSSetMaxTime(ts, user.ftime));

  PetscCall(VecGetArrayWrite(user.x, &x_ptr));
  x_ptr[0] = user.a;
  PetscCall(VecRestoreArrayWrite(user.x, &x_ptr));
  PetscCall(TSSetTimeStep(ts, 0.001));

  /* Set up forward sensitivity */
  PetscCall(MatCreate(PETSC_COMM_WORLD, &user.Jacp));
  PetscCall(MatSetSizes(user.Jacp, PETSC_DECIDE, PETSC_DECIDE, rows, cols));
  PetscCall(MatSetFromOptions(user.Jacp));
  PetscCall(MatSetUp(user.Jacp));
  PetscCall(MatCreateDense(PETSC_COMM_WORLD, PETSC_DECIDE, PETSC_DECIDE, rows, cols, NULL, &user.sp));
  PetscCall(MatZeroEntries(user.sp));
  PetscCall(TSForwardSetSensitivities(ts, cols, user.sp));
  PetscCall(TSSetIJacobianP(ts, user.Jacp, IJacobianP, &user));

  PetscCall(TSSetSaveTrajectory(ts));
  PetscCall(TSSetFromOptions(ts));

  PetscCall(TSSolve(ts, user.x));
  PetscCall(TSGetSolveTime(ts, &user.ftime));
  PetscCall(TSGetStepNumber(ts, &user.steps));
  PetscCall(VecGetArray(user.x, &x_ptr));
  PetscCall(PetscPrintf(PETSC_COMM_WORLD, "\n ode solution %g\n", (double)PetscRealPart(x_ptr[0])));
  PetscCall(VecRestoreArray(user.x, &x_ptr));
  PetscCall(PetscPrintf(PETSC_COMM_WORLD, "\n analytical solution %g\n", (double)(user.a * PetscExpReal(user.b / user.c * user.ftime))));

  if (user.der == 1) PetscCall(PetscPrintf(PETSC_COMM_WORLD, "\n analytical derivative w.r.t. c %g\n", (double)(-user.a * user.ftime * user.b / (user.c * user.c) * PetscExpReal(user.b / user.c * user.ftime))));
  if (user.der == 2) PetscCall(PetscPrintf(PETSC_COMM_WORLD, "\n analytical derivative w.r.t. b %g\n", (double)(user.a * user.ftime / user.c * PetscExpReal(user.b / user.c * user.ftime))));
  PetscCall(PetscPrintf(PETSC_COMM_WORLD, "\n forward sensitivity:\n"));
  PetscCall(MatView(user.sp, PETSC_VIEWER_STDOUT_WORLD));

  PetscCall(MatCreateVecs(user.Jac, &user.lambda[0], NULL));
  /* Set initial conditions for the adjoint integration */
  PetscCall(VecGetArrayWrite(user.lambda[0], &x_ptr));
  x_ptr[0] = 1.0;
  PetscCall(VecRestoreArrayWrite(user.lambda[0], &x_ptr));
  PetscCall(MatCreateVecs(user.Jacp, &user.mup[0], NULL));
  PetscCall(VecGetArrayWrite(user.mup[0], &x_ptr));
  x_ptr[0] = 0.0;
  PetscCall(VecRestoreArrayWrite(user.mup[0], &x_ptr));

  PetscCall(TSSetCostGradients(ts, 1, user.lambda, user.mup));
  PetscCall(TSAdjointSolve(ts));

  PetscCall(PetscPrintf(PETSC_COMM_WORLD, "\n adjoint sensitivity:\n"));
  PetscCall(VecView(user.mup[0], PETSC_VIEWER_STDOUT_WORLD));

  PetscCall(MatDestroy(&user.Jac));
  PetscCall(MatDestroy(&user.sp));
  PetscCall(MatDestroy(&user.Jacp));
  PetscCall(VecDestroy(&user.x));
  PetscCall(VecDestroy(&user.lambda[0]));
  PetscCall(VecDestroy(&user.mup[0]));
  PetscCall(TSDestroy(&ts));

  PetscCall(PetscFinalize());
  return 0;
}

/*TEST

    test:
      args: -ts_type beuler

    test:
      suffix: 2
      args: -ts_type cn
      output_file: output/ex23fwdadj_1.out

TEST*/