xref: /petsc/src/tao/pde_constrained/tutorials/parabolic.c (revision 48a46eb9bd028bec07ec0f396b1a3abb43f14558)

#include <petsc/private/taoimpl.h>

typedef struct {
  PetscInt  n;            /*  Number of variables */
  PetscInt  m;            /*  Number of constraints per time step */
  PetscInt  mx;           /*  grid points in each direction */
  PetscInt  nt;           /*  Number of time steps; as of now, must be divisible by 8 */
  PetscInt  ndata;        /*  Number of data points per sample */
  PetscInt  ns;           /*  Number of samples */
  PetscInt *sample_times; /*  Times of samples */
  IS        s_is;
  IS        d_is;

  VecScatter  state_scatter;
  VecScatter  design_scatter;
  VecScatter *yi_scatter;
  VecScatter *di_scatter;

  Mat       Js, Jd, JsBlockPrec, JsInv, JsBlock;
  PetscBool jformed, dsg_formed;

  PetscReal alpha; /*  Regularization parameter */
  PetscReal beta;  /*  Weight attributed to ||u||^2 in regularization functional */
  PetscReal noise; /*  Amount of noise to add to data */
  PetscReal ht;    /*  Time step */

  Mat Qblock, QblockT;
  Mat L, LT;
  Mat Div, Divwork;
  Mat Grad;
  Mat Av, Avwork, AvT;
  Mat DSG;
  Vec q;
  Vec ur; /*  reference */

  Vec  d;
  Vec  dwork;
  Vec *di;

  Vec y; /*  state variables */
  Vec ywork;

  Vec  ytrue;
  Vec *yi, *yiwork;

  Vec u; /*  design variables */
  Vec uwork;

  Vec utrue;
  Vec js_diag;
  Vec c; /*  constraint vector */
  Vec cwork;

  Vec lwork;
  Vec S;
  Vec Rwork, Swork, Twork;
  Vec Av_u;

  KSP solver;
  PC  prec;

  PetscInt ksp_its;
  PetscInt ksp_its_initial;
} AppCtx;

PetscErrorCode FormFunction(Tao, Vec, PetscReal *, void *);
PetscErrorCode FormGradient(Tao, Vec, Vec, void *);
PetscErrorCode FormFunctionGradient(Tao, Vec, PetscReal *, Vec, void *);
PetscErrorCode FormJacobianState(Tao, Vec, Mat, Mat, Mat, void *);
PetscErrorCode FormJacobianDesign(Tao, Vec, Mat, void *);
PetscErrorCode FormConstraints(Tao, Vec, Vec, void *);
PetscErrorCode FormHessian(Tao, Vec, Mat, Mat, void *);
PetscErrorCode Gather(Vec x, Vec state, VecScatter s_scat, Vec design, VecScatter d_scat);
PetscErrorCode Scatter(Vec x, Vec state, VecScatter s_scat, Vec design, VecScatter d_scat);
PetscErrorCode ParabolicInitialize(AppCtx *user);
PetscErrorCode ParabolicDestroy(AppCtx *user);
PetscErrorCode ParabolicMonitor(Tao, void *);

PetscErrorCode StateMatMult(Mat, Vec, Vec);
PetscErrorCode StateMatBlockMult(Mat, Vec, Vec);
PetscErrorCode StateMatMultTranspose(Mat, Vec, Vec);
PetscErrorCode StateMatGetDiagonal(Mat, Vec);
PetscErrorCode StateMatDuplicate(Mat, MatDuplicateOption, Mat *);
PetscErrorCode StateMatInvMult(Mat, Vec, Vec);
PetscErrorCode StateMatInvTransposeMult(Mat, Vec, Vec);
PetscErrorCode StateMatBlockPrecMult(PC, Vec, Vec);

PetscErrorCode DesignMatMult(Mat, Vec, Vec);
PetscErrorCode DesignMatMultTranspose(Mat, Vec, Vec);

PetscErrorCode Gather_i(Vec, Vec *, VecScatter *, PetscInt);
PetscErrorCode Scatter_i(Vec, Vec *, VecScatter *, PetscInt);

static char help[] = "";

int main(int argc, char **argv) {
  Vec      x, x0;
  Tao      tao;
  AppCtx   user;
  IS       is_allstate, is_alldesign;
  PetscInt lo, hi, hi2, lo2, ksp_old;
  PetscInt ntests = 1;
  PetscInt i;
#if defined(PETSC_USE_LOG)
  PetscLogStage stages[1];
#endif

  PetscFunctionBeginUser;
  PetscCall(PetscInitialize(&argc, &argv, (char *)0, help));
  user.mx = 8;
  PetscOptionsBegin(PETSC_COMM_WORLD, NULL, "parabolic example", NULL);
  PetscCall(PetscOptionsInt("-mx", "Number of grid points in each direction", "", user.mx, &user.mx, NULL));
  user.nt = 8;
  PetscCall(PetscOptionsInt("-nt", "Number of time steps", "", user.nt, &user.nt, NULL));
  user.ndata = 64;
  PetscCall(PetscOptionsInt("-ndata", "Numbers of data points per sample", "", user.ndata, &user.ndata, NULL));
  user.ns = 8;
  PetscCall(PetscOptionsInt("-ns", "Number of samples", "", user.ns, &user.ns, NULL));
  user.alpha = 1.0;
  PetscCall(PetscOptionsReal("-alpha", "Regularization parameter", "", user.alpha, &user.alpha, NULL));
  user.beta = 0.01;
  PetscCall(PetscOptionsReal("-beta", "Weight attributed to ||u||^2 in regularization functional", "", user.beta, &user.beta, NULL));
  user.noise = 0.01;
  PetscCall(PetscOptionsReal("-noise", "Amount of noise to add to data", "", user.noise, &user.noise, NULL));
  PetscCall(PetscOptionsInt("-ntests", "Number of times to repeat TaoSolve", "", ntests, &ntests, NULL));
  PetscOptionsEnd();

  user.m  = user.mx * user.mx * user.mx; /*  number of constraints per time step */
  user.n  = user.m * (user.nt + 1);      /*  number of variables */
  user.ht = (PetscReal)1 / user.nt;

  PetscCall(VecCreate(PETSC_COMM_WORLD, &user.u));
  PetscCall(VecCreate(PETSC_COMM_WORLD, &user.y));
  PetscCall(VecCreate(PETSC_COMM_WORLD, &user.c));
  PetscCall(VecSetSizes(user.u, PETSC_DECIDE, user.n - user.m * user.nt));
  PetscCall(VecSetSizes(user.y, PETSC_DECIDE, user.m * user.nt));
  PetscCall(VecSetSizes(user.c, PETSC_DECIDE, user.m * user.nt));
  PetscCall(VecSetFromOptions(user.u));
  PetscCall(VecSetFromOptions(user.y));
  PetscCall(VecSetFromOptions(user.c));

  /* Create scatters for reduced spaces.
     If the state vector y and design vector u are partitioned as
     [y_1; y_2; ...; y_np] and [u_1; u_2; ...; u_np] (with np = # of processors),
     then the solution vector x is organized as
     [y_1; u_1; y_2; u_2; ...; y_np; u_np].
     The index sets user.s_is and user.d_is correspond to the indices of the
     state and design variables owned by the current processor.
  */
  PetscCall(VecCreate(PETSC_COMM_WORLD, &x));

  PetscCall(VecGetOwnershipRange(user.y, &lo, &hi));
  PetscCall(VecGetOwnershipRange(user.u, &lo2, &hi2));

  PetscCall(ISCreateStride(PETSC_COMM_SELF, hi - lo, lo, 1, &is_allstate));
  PetscCall(ISCreateStride(PETSC_COMM_SELF, hi - lo, lo + lo2, 1, &user.s_is));

  PetscCall(ISCreateStride(PETSC_COMM_SELF, hi2 - lo2, lo2, 1, &is_alldesign));
  PetscCall(ISCreateStride(PETSC_COMM_SELF, hi2 - lo2, hi + lo2, 1, &user.d_is));

  PetscCall(VecSetSizes(x, hi - lo + hi2 - lo2, user.n));
  PetscCall(VecSetFromOptions(x));

  PetscCall(VecScatterCreate(x, user.s_is, user.y, is_allstate, &user.state_scatter));
  PetscCall(VecScatterCreate(x, user.d_is, user.u, is_alldesign, &user.design_scatter));
  PetscCall(ISDestroy(&is_alldesign));
  PetscCall(ISDestroy(&is_allstate));

  /* Create TAO solver and set desired solution method */
  PetscCall(TaoCreate(PETSC_COMM_WORLD, &tao));
  PetscCall(TaoSetType(tao, TAOLCL));

  /* Set up initial vectors and matrices */
  PetscCall(ParabolicInitialize(&user));

  PetscCall(Gather(x, user.y, user.state_scatter, user.u, user.design_scatter));
  PetscCall(VecDuplicate(x, &x0));
  PetscCall(VecCopy(x, x0));

  /* Set solution vector with an initial guess */
  PetscCall(TaoSetSolution(tao, x));
  PetscCall(TaoSetObjective(tao, FormFunction, &user));
  PetscCall(TaoSetGradient(tao, NULL, FormGradient, &user));
  PetscCall(TaoSetConstraintsRoutine(tao, user.c, FormConstraints, &user));

  PetscCall(TaoSetJacobianStateRoutine(tao, user.Js, user.JsBlockPrec, user.JsInv, FormJacobianState, &user));
  PetscCall(TaoSetJacobianDesignRoutine(tao, user.Jd, FormJacobianDesign, &user));

  PetscCall(TaoSetFromOptions(tao));
  PetscCall(TaoSetStateDesignIS(tao, user.s_is, user.d_is));

  /* SOLVE THE APPLICATION */
  PetscCall(PetscLogStageRegister("Trials", &stages[0]));
  PetscCall(PetscLogStagePush(stages[0]));
  user.ksp_its_initial = user.ksp_its;
  ksp_old              = user.ksp_its;
  for (i = 0; i < ntests; i++) {
    PetscCall(TaoSolve(tao));
    PetscCall(PetscPrintf(PETSC_COMM_WORLD, "KSP Iterations = %" PetscInt_FMT "\n", user.ksp_its - ksp_old));
    PetscCall(VecCopy(x0, x));
    PetscCall(TaoSetSolution(tao, x));
  }
  PetscCall(PetscLogStagePop());
  PetscCall(PetscBarrier((PetscObject)x));
  PetscCall(PetscPrintf(PETSC_COMM_WORLD, "KSP iterations within initialization: "));
  PetscCall(PetscPrintf(PETSC_COMM_WORLD, "%" PetscInt_FMT "\n", user.ksp_its_initial));
  PetscCall(PetscPrintf(PETSC_COMM_WORLD, "Total KSP iterations over %" PetscInt_FMT " trial(s): ", ntests));
  PetscCall(PetscPrintf(PETSC_COMM_WORLD, "%" PetscInt_FMT "\n", user.ksp_its));
  PetscCall(PetscPrintf(PETSC_COMM_WORLD, "KSP iterations per trial: "));
  PetscCall(PetscPrintf(PETSC_COMM_WORLD, "%" PetscInt_FMT "\n", (user.ksp_its - user.ksp_its_initial) / ntests));

  PetscCall(TaoDestroy(&tao));
  PetscCall(VecDestroy(&x));
  PetscCall(VecDestroy(&x0));
  PetscCall(ParabolicDestroy(&user));

  PetscCall(PetscFinalize());
  return 0;
}
/* ------------------------------------------------------------------- */
/*
   dwork = Qy - d
   lwork = L*(u-ur)
   f = 1/2 * (dwork.dork + alpha*lwork.lwork)
*/
PetscErrorCode FormFunction(Tao tao, Vec X, PetscReal *f, void *ptr) {
  PetscReal d1 = 0, d2 = 0;
  PetscInt  i, j;
  AppCtx   *user = (AppCtx *)ptr;

  PetscFunctionBegin;
  PetscCall(Scatter(X, user->y, user->state_scatter, user->u, user->design_scatter));
  PetscCall(Scatter_i(user->y, user->yi, user->yi_scatter, user->nt));
  for (j = 0; j < user->ns; j++) {
    i = user->sample_times[j];
    PetscCall(MatMult(user->Qblock, user->yi[i], user->di[j]));
  }
  PetscCall(Gather_i(user->dwork, user->di, user->di_scatter, user->ns));
  PetscCall(VecAXPY(user->dwork, -1.0, user->d));
  PetscCall(VecDot(user->dwork, user->dwork, &d1));

  PetscCall(VecWAXPY(user->uwork, -1.0, user->ur, user->u));
  PetscCall(MatMult(user->L, user->uwork, user->lwork));
  PetscCall(VecDot(user->lwork, user->lwork, &d2));

  *f = 0.5 * (d1 + user->alpha * d2);
  PetscFunctionReturn(0);
}

/* ------------------------------------------------------------------- */
/*
    state: g_s = Q' *(Qy - d)
    design: g_d = alpha*L'*L*(u-ur)
*/
PetscErrorCode FormGradient(Tao tao, Vec X, Vec G, void *ptr) {
  PetscInt i, j;
  AppCtx  *user = (AppCtx *)ptr;

  PetscFunctionBegin;
  PetscCall(Scatter(X, user->y, user->state_scatter, user->u, user->design_scatter));
  PetscCall(Scatter_i(user->y, user->yi, user->yi_scatter, user->nt));
  for (j = 0; j < user->ns; j++) {
    i = user->sample_times[j];
    PetscCall(MatMult(user->Qblock, user->yi[i], user->di[j]));
  }
  PetscCall(Gather_i(user->dwork, user->di, user->di_scatter, user->ns));
  PetscCall(VecAXPY(user->dwork, -1.0, user->d));
  PetscCall(Scatter_i(user->dwork, user->di, user->di_scatter, user->ns));
  PetscCall(VecSet(user->ywork, 0.0));
  PetscCall(Scatter_i(user->ywork, user->yiwork, user->yi_scatter, user->nt));
  for (j = 0; j < user->ns; j++) {
    i = user->sample_times[j];
    PetscCall(MatMult(user->QblockT, user->di[j], user->yiwork[i]));
  }
  PetscCall(Gather_i(user->ywork, user->yiwork, user->yi_scatter, user->nt));

  PetscCall(VecWAXPY(user->uwork, -1.0, user->ur, user->u));
  PetscCall(MatMult(user->L, user->uwork, user->lwork));
  PetscCall(MatMult(user->LT, user->lwork, user->uwork));
  PetscCall(VecScale(user->uwork, user->alpha));
  PetscCall(Gather(G, user->ywork, user->state_scatter, user->uwork, user->design_scatter));
  PetscFunctionReturn(0);
}

PetscErrorCode FormFunctionGradient(Tao tao, Vec X, PetscReal *f, Vec G, void *ptr) {
  PetscReal d1, d2;
  PetscInt  i, j;
  AppCtx   *user = (AppCtx *)ptr;

  PetscFunctionBegin;
  PetscCall(Scatter(X, user->y, user->state_scatter, user->u, user->design_scatter));
  PetscCall(Scatter_i(user->y, user->yi, user->yi_scatter, user->nt));
  for (j = 0; j < user->ns; j++) {
    i = user->sample_times[j];
    PetscCall(MatMult(user->Qblock, user->yi[i], user->di[j]));
  }
  PetscCall(Gather_i(user->dwork, user->di, user->di_scatter, user->ns));
  PetscCall(VecAXPY(user->dwork, -1.0, user->d));
  PetscCall(VecDot(user->dwork, user->dwork, &d1));
  PetscCall(Scatter_i(user->dwork, user->di, user->di_scatter, user->ns));
  PetscCall(VecSet(user->ywork, 0.0));
  PetscCall(Scatter_i(user->ywork, user->yiwork, user->yi_scatter, user->nt));
  for (j = 0; j < user->ns; j++) {
    i = user->sample_times[j];
    PetscCall(MatMult(user->QblockT, user->di[j], user->yiwork[i]));
  }
  PetscCall(Gather_i(user->ywork, user->yiwork, user->yi_scatter, user->nt));

  PetscCall(VecWAXPY(user->uwork, -1.0, user->ur, user->u));
  PetscCall(MatMult(user->L, user->uwork, user->lwork));
  PetscCall(VecDot(user->lwork, user->lwork, &d2));
  PetscCall(MatMult(user->LT, user->lwork, user->uwork));
  PetscCall(VecScale(user->uwork, user->alpha));
  *f = 0.5 * (d1 + user->alpha * d2);

  PetscCall(Gather(G, user->ywork, user->state_scatter, user->uwork, user->design_scatter));
  PetscFunctionReturn(0);
}

/* ------------------------------------------------------------------- */
/* A
MatShell object
*/
PetscErrorCode FormJacobianState(Tao tao, Vec X, Mat J, Mat JPre, Mat JInv, void *ptr) {
  AppCtx *user = (AppCtx *)ptr;

  PetscFunctionBegin;
  PetscCall(Scatter(X, user->y, user->state_scatter, user->u, user->design_scatter));
  PetscCall(VecSet(user->uwork, 0));
  PetscCall(VecAXPY(user->uwork, -1.0, user->u));
  PetscCall(VecExp(user->uwork));
  PetscCall(MatMult(user->Av, user->uwork, user->Av_u));
  PetscCall(VecCopy(user->Av_u, user->Swork));
  PetscCall(VecReciprocal(user->Swork));
  PetscCall(MatCopy(user->Div, user->Divwork, SAME_NONZERO_PATTERN));
  PetscCall(MatDiagonalScale(user->Divwork, NULL, user->Swork));
  if (user->dsg_formed) {
    PetscCall(MatProductNumeric(user->DSG));
  } else {
    PetscCall(MatMatMult(user->Divwork, user->Grad, MAT_INITIAL_MATRIX, PETSC_DEFAULT, &user->DSG));
    user->dsg_formed = PETSC_TRUE;
  }

  /* B = speye(nx^3) + ht*DSG; */
  PetscCall(MatScale(user->DSG, user->ht));
  PetscCall(MatShift(user->DSG, 1.0));
  PetscFunctionReturn(0);
}

/* ------------------------------------------------------------------- */
/* B */
PetscErrorCode FormJacobianDesign(Tao tao, Vec X, Mat J, void *ptr) {
  AppCtx *user = (AppCtx *)ptr;

  PetscFunctionBegin;
  PetscCall(Scatter(X, user->y, user->state_scatter, user->u, user->design_scatter));
  PetscFunctionReturn(0);
}

PetscErrorCode StateMatMult(Mat J_shell, Vec X, Vec Y) {
  PetscInt i;
  AppCtx  *user;

  PetscFunctionBegin;
  PetscCall(MatShellGetContext(J_shell, &user));
  PetscCall(Scatter_i(X, user->yi, user->yi_scatter, user->nt));
  PetscCall(MatMult(user->JsBlock, user->yi[0], user->yiwork[0]));
  for (i = 1; i < user->nt; i++) {
    PetscCall(MatMult(user->JsBlock, user->yi[i], user->yiwork[i]));
    PetscCall(VecAXPY(user->yiwork[i], -1.0, user->yi[i - 1]));
  }
  PetscCall(Gather_i(Y, user->yiwork, user->yi_scatter, user->nt));
  PetscFunctionReturn(0);
}

PetscErrorCode StateMatMultTranspose(Mat J_shell, Vec X, Vec Y) {
  PetscInt i;
  AppCtx  *user;

  PetscFunctionBegin;
  PetscCall(MatShellGetContext(J_shell, &user));
  PetscCall(Scatter_i(X, user->yi, user->yi_scatter, user->nt));
  for (i = 0; i < user->nt - 1; i++) {
    PetscCall(MatMult(user->JsBlock, user->yi[i], user->yiwork[i]));
    PetscCall(VecAXPY(user->yiwork[i], -1.0, user->yi[i + 1]));
  }
  i = user->nt - 1;
  PetscCall(MatMult(user->JsBlock, user->yi[i], user->yiwork[i]));
  PetscCall(Gather_i(Y, user->yiwork, user->yi_scatter, user->nt));
  PetscFunctionReturn(0);
}

PetscErrorCode StateMatBlockMult(Mat J_shell, Vec X, Vec Y) {
  AppCtx *user;

  PetscFunctionBegin;
  PetscCall(MatShellGetContext(J_shell, &user));
  PetscCall(MatMult(user->Grad, X, user->Swork));
  PetscCall(VecPointwiseDivide(user->Swork, user->Swork, user->Av_u));
  PetscCall(MatMult(user->Div, user->Swork, Y));
  PetscCall(VecAYPX(Y, user->ht, X));
  PetscFunctionReturn(0);
}

PetscErrorCode DesignMatMult(Mat J_shell, Vec X, Vec Y) {
  PetscInt i;
  AppCtx  *user;

  PetscFunctionBegin;
  PetscCall(MatShellGetContext(J_shell, &user));

  /* sdiag(1./v) */
  PetscCall(VecSet(user->uwork, 0));
  PetscCall(VecAXPY(user->uwork, -1.0, user->u));
  PetscCall(VecExp(user->uwork));

  /* sdiag(1./((Av*(1./v)).^2)) */
  PetscCall(MatMult(user->Av, user->uwork, user->Swork));
  PetscCall(VecPointwiseMult(user->Swork, user->Swork, user->Swork));
  PetscCall(VecReciprocal(user->Swork));

  /* (Av * (sdiag(1./v) * b)) */
  PetscCall(VecPointwiseMult(user->uwork, user->uwork, X));
  PetscCall(MatMult(user->Av, user->uwork, user->Twork));

  /* (sdiag(1./((Av*(1./v)).^2)) * (Av * (sdiag(1./v) * b))) */
  PetscCall(VecPointwiseMult(user->Swork, user->Twork, user->Swork));

  PetscCall(Scatter_i(user->y, user->yi, user->yi_scatter, user->nt));
  for (i = 0; i < user->nt; i++) {
    /* (sdiag(Grad*y(:,i)) */
    PetscCall(MatMult(user->Grad, user->yi[i], user->Twork));

    /* ht * Div * (sdiag(Grad*y(:,i)) * (sdiag(1./((Av*(1./v)).^2)) * (Av * (sdiag(1./v) * b)))) */
    PetscCall(VecPointwiseMult(user->Twork, user->Twork, user->Swork));
    PetscCall(MatMult(user->Div, user->Twork, user->yiwork[i]));
    PetscCall(VecScale(user->yiwork[i], user->ht));
  }
  PetscCall(Gather_i(Y, user->yiwork, user->yi_scatter, user->nt));

  PetscFunctionReturn(0);
}

PetscErrorCode DesignMatMultTranspose(Mat J_shell, Vec X, Vec Y) {
  PetscInt i;
  AppCtx  *user;

  PetscFunctionBegin;
  PetscCall(MatShellGetContext(J_shell, &user));

  /* sdiag(1./((Av*(1./v)).^2)) */
  PetscCall(VecSet(user->uwork, 0));
  PetscCall(VecAXPY(user->uwork, -1.0, user->u));
  PetscCall(VecExp(user->uwork));
  PetscCall(MatMult(user->Av, user->uwork, user->Rwork));
  PetscCall(VecPointwiseMult(user->Rwork, user->Rwork, user->Rwork));
  PetscCall(VecReciprocal(user->Rwork));

  PetscCall(VecSet(Y, 0.0));
  PetscCall(Scatter_i(user->y, user->yi, user->yi_scatter, user->nt));
  PetscCall(Scatter_i(X, user->yiwork, user->yi_scatter, user->nt));
  for (i = 0; i < user->nt; i++) {
    /* (Div' * b(:,i)) */
    PetscCall(MatMult(user->Grad, user->yiwork[i], user->Swork));

    /* sdiag(Grad*y(:,i)) */
    PetscCall(MatMult(user->Grad, user->yi[i], user->Twork));

    /* (sdiag(Grad*y(:,i)) * (Div' * b(:,i))) */
    PetscCall(VecPointwiseMult(user->Twork, user->Swork, user->Twork));

    /* (sdiag(1./((Av*(1./v)).^2)) * (sdiag(Grad*y(:,i)) * (Div' * b(:,i)))) */
    PetscCall(VecPointwiseMult(user->Twork, user->Rwork, user->Twork));

    /* (Av' * (sdiag(1./((Av*(1./v)).^2)) * (sdiag(Grad*y(:,i)) * (Div' * b(:,i))))) */
    PetscCall(MatMult(user->AvT, user->Twork, user->yiwork[i]));

    /* sdiag(1./v) * (Av' * (sdiag(1./((Av*(1./v)).^2)) * (sdiag(Grad*y(:,i)) * (Div' * b(:,i))))) */
    PetscCall(VecPointwiseMult(user->yiwork[i], user->uwork, user->yiwork[i]));
    PetscCall(VecAXPY(Y, user->ht, user->yiwork[i]));
  }
  PetscFunctionReturn(0);
}

PetscErrorCode StateMatBlockPrecMult(PC PC_shell, Vec X, Vec Y) {
  AppCtx *user;

  PetscFunctionBegin;
  PetscCall(PCShellGetContext(PC_shell, &user));

  if (user->dsg_formed) {
    PetscCall(MatSOR(user->DSG, X, 1.0, (MatSORType)(SOR_ZERO_INITIAL_GUESS | SOR_LOCAL_SYMMETRIC_SWEEP), 0.0, 1, 1, Y));
  } else SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "DSG not formed");
  PetscFunctionReturn(0);
}

PetscErrorCode StateMatInvMult(Mat J_shell, Vec X, Vec Y) {
  AppCtx  *user;
  PetscInt its, i;

  PetscFunctionBegin;
  PetscCall(MatShellGetContext(J_shell, &user));

  if (Y == user->ytrue) {
    /* First solve is done with true solution to set up problem */
    PetscCall(KSPSetTolerances(user->solver, 1e-8, PETSC_DEFAULT, PETSC_DEFAULT, PETSC_DEFAULT));
  } else {
    PetscCall(KSPSetTolerances(user->solver, PETSC_DEFAULT, PETSC_DEFAULT, PETSC_DEFAULT, PETSC_DEFAULT));
  }

  PetscCall(Scatter_i(X, user->yi, user->yi_scatter, user->nt));
  PetscCall(KSPSolve(user->solver, user->yi[0], user->yiwork[0]));
  PetscCall(KSPGetIterationNumber(user->solver, &its));
  user->ksp_its = user->ksp_its + its;

  for (i = 1; i < user->nt; i++) {
    PetscCall(VecAXPY(user->yi[i], 1.0, user->yiwork[i - 1]));
    PetscCall(KSPSolve(user->solver, user->yi[i], user->yiwork[i]));
    PetscCall(KSPGetIterationNumber(user->solver, &its));
    user->ksp_its = user->ksp_its + its;
  }
  PetscCall(Gather_i(Y, user->yiwork, user->yi_scatter, user->nt));
  PetscFunctionReturn(0);
}

PetscErrorCode StateMatInvTransposeMult(Mat J_shell, Vec X, Vec Y) {
  AppCtx  *user;
  PetscInt its, i;

  PetscFunctionBegin;
  PetscCall(MatShellGetContext(J_shell, &user));

  PetscCall(Scatter_i(X, user->yi, user->yi_scatter, user->nt));

  i = user->nt - 1;
  PetscCall(KSPSolve(user->solver, user->yi[i], user->yiwork[i]));

  PetscCall(KSPGetIterationNumber(user->solver, &its));
  user->ksp_its = user->ksp_its + its;

  for (i = user->nt - 2; i >= 0; i--) {
    PetscCall(VecAXPY(user->yi[i], 1.0, user->yiwork[i + 1]));
    PetscCall(KSPSolve(user->solver, user->yi[i], user->yiwork[i]));

    PetscCall(KSPGetIterationNumber(user->solver, &its));
    user->ksp_its = user->ksp_its + its;
  }

  PetscCall(Gather_i(Y, user->yiwork, user->yi_scatter, user->nt));
  PetscFunctionReturn(0);
}

PetscErrorCode StateMatDuplicate(Mat J_shell, MatDuplicateOption opt, Mat *new_shell) {
  AppCtx *user;

  PetscFunctionBegin;
  PetscCall(MatShellGetContext(J_shell, &user));

  PetscCall(MatCreateShell(PETSC_COMM_WORLD, PETSC_DETERMINE, PETSC_DETERMINE, user->m, user->m, user, new_shell));
  PetscCall(MatShellSetOperation(*new_shell, MATOP_MULT, (void (*)(void))StateMatMult));
  PetscCall(MatShellSetOperation(*new_shell, MATOP_DUPLICATE, (void (*)(void))StateMatDuplicate));
  PetscCall(MatShellSetOperation(*new_shell, MATOP_MULT_TRANSPOSE, (void (*)(void))StateMatMultTranspose));
  PetscCall(MatShellSetOperation(*new_shell, MATOP_GET_DIAGONAL, (void (*)(void))StateMatGetDiagonal));
  PetscFunctionReturn(0);
}

PetscErrorCode StateMatGetDiagonal(Mat J_shell, Vec X) {
  AppCtx *user;

  PetscFunctionBegin;
  PetscCall(MatShellGetContext(J_shell, &user));
  PetscCall(VecCopy(user->js_diag, X));
  PetscFunctionReturn(0);
}

PetscErrorCode FormConstraints(Tao tao, Vec X, Vec C, void *ptr) {
  /* con = Ay - q, A = [B  0  0 ... 0;
                       -I  B  0 ... 0;
                        0 -I  B ... 0;
                             ...     ;
                        0    ... -I B]
     B = ht * Div * Sigma * Grad + eye */
  PetscInt i;
  AppCtx  *user = (AppCtx *)ptr;

  PetscFunctionBegin;
  PetscCall(Scatter(X, user->y, user->state_scatter, user->u, user->design_scatter));
  PetscCall(Scatter_i(user->y, user->yi, user->yi_scatter, user->nt));
  PetscCall(MatMult(user->JsBlock, user->yi[0], user->yiwork[0]));
  for (i = 1; i < user->nt; i++) {
    PetscCall(MatMult(user->JsBlock, user->yi[i], user->yiwork[i]));
    PetscCall(VecAXPY(user->yiwork[i], -1.0, user->yi[i - 1]));
  }
  PetscCall(Gather_i(C, user->yiwork, user->yi_scatter, user->nt));
  PetscCall(VecAXPY(C, -1.0, user->q));
  PetscFunctionReturn(0);
}

PetscErrorCode Scatter(Vec x, Vec state, VecScatter s_scat, Vec design, VecScatter d_scat) {
  PetscFunctionBegin;
  PetscCall(VecScatterBegin(s_scat, x, state, INSERT_VALUES, SCATTER_FORWARD));
  PetscCall(VecScatterEnd(s_scat, x, state, INSERT_VALUES, SCATTER_FORWARD));
  PetscCall(VecScatterBegin(d_scat, x, design, INSERT_VALUES, SCATTER_FORWARD));
  PetscCall(VecScatterEnd(d_scat, x, design, INSERT_VALUES, SCATTER_FORWARD));
  PetscFunctionReturn(0);
}

PetscErrorCode Scatter_i(Vec y, Vec *yi, VecScatter *scat, PetscInt nt) {
  PetscInt i;

  PetscFunctionBegin;
  for (i = 0; i < nt; i++) {
    PetscCall(VecScatterBegin(scat[i], y, yi[i], INSERT_VALUES, SCATTER_FORWARD));
    PetscCall(VecScatterEnd(scat[i], y, yi[i], INSERT_VALUES, SCATTER_FORWARD));
  }
  PetscFunctionReturn(0);
}

PetscErrorCode Gather(Vec x, Vec state, VecScatter s_scat, Vec design, VecScatter d_scat) {
  PetscFunctionBegin;
  PetscCall(VecScatterBegin(s_scat, state, x, INSERT_VALUES, SCATTER_REVERSE));
  PetscCall(VecScatterEnd(s_scat, state, x, INSERT_VALUES, SCATTER_REVERSE));
  PetscCall(VecScatterBegin(d_scat, design, x, INSERT_VALUES, SCATTER_REVERSE));
  PetscCall(VecScatterEnd(d_scat, design, x, INSERT_VALUES, SCATTER_REVERSE));
  PetscFunctionReturn(0);
}

PetscErrorCode Gather_i(Vec y, Vec *yi, VecScatter *scat, PetscInt nt) {
  PetscInt i;

  PetscFunctionBegin;
  for (i = 0; i < nt; i++) {
    PetscCall(VecScatterBegin(scat[i], yi[i], y, INSERT_VALUES, SCATTER_REVERSE));
    PetscCall(VecScatterEnd(scat[i], yi[i], y, INSERT_VALUES, SCATTER_REVERSE));
  }
  PetscFunctionReturn(0);
}

PetscErrorCode ParabolicInitialize(AppCtx *user) {
  PetscInt    m, n, i, j, k, linear_index, istart, iend, iblock, lo, hi, lo2, hi2;
  Vec         XX, YY, ZZ, XXwork, YYwork, ZZwork, UTwork, yi, di, bc;
  PetscReal  *x, *y, *z;
  PetscReal   h, stime;
  PetscScalar hinv, neg_hinv, half = 0.5, sqrt_beta;
  PetscInt    im, indx1, indx2, indy1, indy2, indz1, indz2, nx, ny, nz;
  PetscReal   xri, yri, zri, xim, yim, zim, dx1, dx2, dy1, dy2, dz1, dz2, Dx, Dy, Dz;
  PetscScalar v, vx, vy, vz;
  IS          is_from_y, is_to_yi, is_from_d, is_to_di;
  /* Data locations */
  PetscScalar xr[64] = {0.4970, 0.8498, 0.7814, 0.6268, 0.7782, 0.6402, 0.3617, 0.3160, 0.3610, 0.5298, 0.6987, 0.3331, 0.7962, 0.5596, 0.3866, 0.6774, 0.5407, 0.4518, 0.6702, 0.6061, 0.7580, 0.8997,
                        0.5198, 0.8326, 0.2138, 0.9198, 0.3000, 0.2833, 0.8288, 0.7076, 0.1820, 0.0728, 0.8447, 0.2367, 0.3239, 0.6413, 0.3114, 0.4731, 0.1192, 0.9273, 0.5724, 0.4331, 0.5136, 0.3547,
                        0.4413, 0.2602, 0.5698, 0.7278, 0.5261, 0.6230, 0.2454, 0.3948, 0.7479, 0.6582, 0.4660, 0.5594, 0.7574, 0.1143, 0.5900, 0.1065, 0.4260, 0.3294, 0.8276, 0.0756};

  PetscScalar yr[64] = {0.7345, 0.9120, 0.9288, 0.7528, 0.4463, 0.4985, 0.2497, 0.6256, 0.3425, 0.9026, 0.6983, 0.4230, 0.7140, 0.2970, 0.4474, 0.8792, 0.6604, 0.2485, 0.7968, 0.6127, 0.1796, 0.2437,
                        0.5938, 0.6137, 0.3867, 0.5658, 0.4575, 0.1009, 0.0863, 0.3361, 0.0738, 0.3985, 0.6602, 0.1437, 0.0934, 0.5983, 0.5950, 0.0763, 0.0768, 0.2288, 0.5761, 0.1129, 0.3841, 0.6150,
                        0.6904, 0.6686, 0.1361, 0.4601, 0.4491, 0.3716, 0.1969, 0.6537, 0.6743, 0.6991, 0.4811, 0.5480, 0.1684, 0.4569, 0.6889, 0.8437, 0.3015, 0.2854, 0.8199, 0.2658};

  PetscScalar zr[64] = {0.7668, 0.8573, 0.2654, 0.2719, 0.1060, 0.1311, 0.6232, 0.2295, 0.8009, 0.2147, 0.2119, 0.9325, 0.4473, 0.3600, 0.3374, 0.3819, 0.4066, 0.5801, 0.1673, 0.0959, 0.4638, 0.8236,
                        0.8800, 0.2939, 0.2028, 0.8262, 0.2706, 0.6276, 0.9085, 0.6443, 0.8241, 0.0712, 0.1824, 0.7789, 0.4389, 0.8415, 0.7055, 0.6639, 0.3653, 0.2078, 0.1987, 0.2297, 0.4321, 0.8115,
                        0.4915, 0.7764, 0.4657, 0.4627, 0.4569, 0.4232, 0.8514, 0.0674, 0.3227, 0.1055, 0.6690, 0.6313, 0.9226, 0.5461, 0.4126, 0.2364, 0.6096, 0.7042, 0.3914, 0.0711};

  PetscFunctionBegin;
  PetscCall(PetscMalloc1(user->mx, &x));
  PetscCall(PetscMalloc1(user->mx, &y));
  PetscCall(PetscMalloc1(user->mx, &z));
  user->jformed    = PETSC_FALSE;
  user->dsg_formed = PETSC_FALSE;

  n         = user->mx * user->mx * user->mx;
  m         = 3 * user->mx * user->mx * (user->mx - 1);
  sqrt_beta = PetscSqrtScalar(user->beta);

  user->ksp_its         = 0;
  user->ksp_its_initial = 0;

  stime = (PetscReal)user->nt / user->ns;
  PetscCall(PetscMalloc1(user->ns, &user->sample_times));
  for (i = 0; i < user->ns; i++) { user->sample_times[i] = (PetscInt)(stime * ((PetscReal)i + 1.0) - 0.5); }

  PetscCall(VecCreate(PETSC_COMM_WORLD, &XX));
  PetscCall(VecCreate(PETSC_COMM_WORLD, &user->q));
  PetscCall(VecSetSizes(XX, PETSC_DECIDE, n));
  PetscCall(VecSetSizes(user->q, PETSC_DECIDE, n * user->nt));
  PetscCall(VecSetFromOptions(XX));
  PetscCall(VecSetFromOptions(user->q));

  PetscCall(VecDuplicate(XX, &YY));
  PetscCall(VecDuplicate(XX, &ZZ));
  PetscCall(VecDuplicate(XX, &XXwork));
  PetscCall(VecDuplicate(XX, &YYwork));
  PetscCall(VecDuplicate(XX, &ZZwork));
  PetscCall(VecDuplicate(XX, &UTwork));
  PetscCall(VecDuplicate(XX, &user->utrue));
  PetscCall(VecDuplicate(XX, &bc));

  /* Generate 3D grid, and collect ns (1<=ns<=8) right-hand-side vectors into user->q */
  h        = 1.0 / user->mx;
  hinv     = user->mx;
  neg_hinv = -hinv;

  PetscCall(VecGetOwnershipRange(XX, &istart, &iend));
  for (linear_index = istart; linear_index < iend; linear_index++) {
    i  = linear_index % user->mx;
    j  = ((linear_index - i) / user->mx) % user->mx;
    k  = ((linear_index - i) / user->mx - j) / user->mx;
    vx = h * (i + 0.5);
    vy = h * (j + 0.5);
    vz = h * (k + 0.5);
    PetscCall(VecSetValues(XX, 1, &linear_index, &vx, INSERT_VALUES));
    PetscCall(VecSetValues(YY, 1, &linear_index, &vy, INSERT_VALUES));
    PetscCall(VecSetValues(ZZ, 1, &linear_index, &vz, INSERT_VALUES));
    if ((vx < 0.6) && (vx > 0.4) && (vy < 0.6) && (vy > 0.4) && (vy < 0.6) && (vz < 0.6) && (vz > 0.4)) {
      v = 1000.0;
      PetscCall(VecSetValues(bc, 1, &linear_index, &v, INSERT_VALUES));
    }
  }

  PetscCall(VecAssemblyBegin(XX));
  PetscCall(VecAssemblyEnd(XX));
  PetscCall(VecAssemblyBegin(YY));
  PetscCall(VecAssemblyEnd(YY));
  PetscCall(VecAssemblyBegin(ZZ));
  PetscCall(VecAssemblyEnd(ZZ));
  PetscCall(VecAssemblyBegin(bc));
  PetscCall(VecAssemblyEnd(bc));

  /* Compute true parameter function
     ut = 0.5 + exp(-10*((x-0.5)^2+(y-0.5)^2+(z-0.5)^2)) */
  PetscCall(VecCopy(XX, XXwork));
  PetscCall(VecCopy(YY, YYwork));
  PetscCall(VecCopy(ZZ, ZZwork));

  PetscCall(VecShift(XXwork, -0.5));
  PetscCall(VecShift(YYwork, -0.5));
  PetscCall(VecShift(ZZwork, -0.5));

  PetscCall(VecPointwiseMult(XXwork, XXwork, XXwork));
  PetscCall(VecPointwiseMult(YYwork, YYwork, YYwork));
  PetscCall(VecPointwiseMult(ZZwork, ZZwork, ZZwork));

  PetscCall(VecCopy(XXwork, user->utrue));
  PetscCall(VecAXPY(user->utrue, 1.0, YYwork));
  PetscCall(VecAXPY(user->utrue, 1.0, ZZwork));
  PetscCall(VecScale(user->utrue, -10.0));
  PetscCall(VecExp(user->utrue));
  PetscCall(VecShift(user->utrue, 0.5));

  PetscCall(VecDestroy(&XX));
  PetscCall(VecDestroy(&YY));
  PetscCall(VecDestroy(&ZZ));
  PetscCall(VecDestroy(&XXwork));
  PetscCall(VecDestroy(&YYwork));
  PetscCall(VecDestroy(&ZZwork));
  PetscCall(VecDestroy(&UTwork));

  /* Initial guess and reference model */
  PetscCall(VecDuplicate(user->utrue, &user->ur));
  PetscCall(VecSet(user->ur, 0.0));

  /* Generate Grad matrix */
  PetscCall(MatCreate(PETSC_COMM_WORLD, &user->Grad));
  PetscCall(MatSetSizes(user->Grad, PETSC_DECIDE, PETSC_DECIDE, m, n));
  PetscCall(MatSetFromOptions(user->Grad));
  PetscCall(MatMPIAIJSetPreallocation(user->Grad, 2, NULL, 2, NULL));
  PetscCall(MatSeqAIJSetPreallocation(user->Grad, 2, NULL));
  PetscCall(MatGetOwnershipRange(user->Grad, &istart, &iend));

  for (i = istart; i < iend; i++) {
    if (i < m / 3) {
      iblock = i / (user->mx - 1);
      j      = iblock * user->mx + (i % (user->mx - 1));
      PetscCall(MatSetValues(user->Grad, 1, &i, 1, &j, &neg_hinv, INSERT_VALUES));
      j = j + 1;
      PetscCall(MatSetValues(user->Grad, 1, &i, 1, &j, &hinv, INSERT_VALUES));
    }
    if (i >= m / 3 && i < 2 * m / 3) {
      iblock = (i - m / 3) / (user->mx * (user->mx - 1));
      j      = iblock * user->mx * user->mx + ((i - m / 3) % (user->mx * (user->mx - 1)));
      PetscCall(MatSetValues(user->Grad, 1, &i, 1, &j, &neg_hinv, INSERT_VALUES));
      j = j + user->mx;
      PetscCall(MatSetValues(user->Grad, 1, &i, 1, &j, &hinv, INSERT_VALUES));
    }
    if (i >= 2 * m / 3) {
      j = i - 2 * m / 3;
      PetscCall(MatSetValues(user->Grad, 1, &i, 1, &j, &neg_hinv, INSERT_VALUES));
      j = j + user->mx * user->mx;
      PetscCall(MatSetValues(user->Grad, 1, &i, 1, &j, &hinv, INSERT_VALUES));
    }
  }

  PetscCall(MatAssemblyBegin(user->Grad, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(user->Grad, MAT_FINAL_ASSEMBLY));

  /* Generate arithmetic averaging matrix Av */
  PetscCall(MatCreate(PETSC_COMM_WORLD, &user->Av));
  PetscCall(MatSetSizes(user->Av, PETSC_DECIDE, PETSC_DECIDE, m, n));
  PetscCall(MatSetFromOptions(user->Av));
  PetscCall(MatMPIAIJSetPreallocation(user->Av, 2, NULL, 2, NULL));
  PetscCall(MatSeqAIJSetPreallocation(user->Av, 2, NULL));
  PetscCall(MatGetOwnershipRange(user->Av, &istart, &iend));

  for (i = istart; i < iend; i++) {
    if (i < m / 3) {
      iblock = i / (user->mx - 1);
      j      = iblock * user->mx + (i % (user->mx - 1));
      PetscCall(MatSetValues(user->Av, 1, &i, 1, &j, &half, INSERT_VALUES));
      j = j + 1;
      PetscCall(MatSetValues(user->Av, 1, &i, 1, &j, &half, INSERT_VALUES));
    }
    if (i >= m / 3 && i < 2 * m / 3) {
      iblock = (i - m / 3) / (user->mx * (user->mx - 1));
      j      = iblock * user->mx * user->mx + ((i - m / 3) % (user->mx * (user->mx - 1)));
      PetscCall(MatSetValues(user->Av, 1, &i, 1, &j, &half, INSERT_VALUES));
      j = j + user->mx;
      PetscCall(MatSetValues(user->Av, 1, &i, 1, &j, &half, INSERT_VALUES));
    }
    if (i >= 2 * m / 3) {
      j = i - 2 * m / 3;
      PetscCall(MatSetValues(user->Av, 1, &i, 1, &j, &half, INSERT_VALUES));
      j = j + user->mx * user->mx;
      PetscCall(MatSetValues(user->Av, 1, &i, 1, &j, &half, INSERT_VALUES));
    }
  }

  PetscCall(MatAssemblyBegin(user->Av, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(user->Av, MAT_FINAL_ASSEMBLY));

  /* Generate transpose of averaging matrix Av */
  PetscCall(MatTranspose(user->Av, MAT_INITIAL_MATRIX, &user->AvT));

  PetscCall(MatCreate(PETSC_COMM_WORLD, &user->L));
  PetscCall(MatSetSizes(user->L, PETSC_DECIDE, PETSC_DECIDE, m + n, n));
  PetscCall(MatSetFromOptions(user->L));
  PetscCall(MatMPIAIJSetPreallocation(user->L, 2, NULL, 2, NULL));
  PetscCall(MatSeqAIJSetPreallocation(user->L, 2, NULL));
  PetscCall(MatGetOwnershipRange(user->L, &istart, &iend));

  for (i = istart; i < iend; i++) {
    if (i < m / 3) {
      iblock = i / (user->mx - 1);
      j      = iblock * user->mx + (i % (user->mx - 1));
      PetscCall(MatSetValues(user->L, 1, &i, 1, &j, &neg_hinv, INSERT_VALUES));
      j = j + 1;
      PetscCall(MatSetValues(user->L, 1, &i, 1, &j, &hinv, INSERT_VALUES));
    }
    if (i >= m / 3 && i < 2 * m / 3) {
      iblock = (i - m / 3) / (user->mx * (user->mx - 1));
      j      = iblock * user->mx * user->mx + ((i - m / 3) % (user->mx * (user->mx - 1)));
      PetscCall(MatSetValues(user->L, 1, &i, 1, &j, &neg_hinv, INSERT_VALUES));
      j = j + user->mx;
      PetscCall(MatSetValues(user->L, 1, &i, 1, &j, &hinv, INSERT_VALUES));
    }
    if (i >= 2 * m / 3 && i < m) {
      j = i - 2 * m / 3;
      PetscCall(MatSetValues(user->L, 1, &i, 1, &j, &neg_hinv, INSERT_VALUES));
      j = j + user->mx * user->mx;
      PetscCall(MatSetValues(user->L, 1, &i, 1, &j, &hinv, INSERT_VALUES));
    }
    if (i >= m) {
      j = i - m;
      PetscCall(MatSetValues(user->L, 1, &i, 1, &j, &sqrt_beta, INSERT_VALUES));
    }
  }

  PetscCall(MatAssemblyBegin(user->L, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(user->L, MAT_FINAL_ASSEMBLY));

  PetscCall(MatScale(user->L, PetscPowScalar(h, 1.5)));

  /* Generate Div matrix */
  PetscCall(MatTranspose(user->Grad, MAT_INITIAL_MATRIX, &user->Div));

  /* Build work vectors and matrices */
  PetscCall(VecCreate(PETSC_COMM_WORLD, &user->S));
  PetscCall(VecSetSizes(user->S, PETSC_DECIDE, user->mx * user->mx * (user->mx - 1) * 3));
  PetscCall(VecSetFromOptions(user->S));

  PetscCall(VecCreate(PETSC_COMM_WORLD, &user->lwork));
  PetscCall(VecSetSizes(user->lwork, PETSC_DECIDE, m + user->mx * user->mx * user->mx));
  PetscCall(VecSetFromOptions(user->lwork));

  PetscCall(MatDuplicate(user->Div, MAT_SHARE_NONZERO_PATTERN, &user->Divwork));
  PetscCall(MatDuplicate(user->Av, MAT_SHARE_NONZERO_PATTERN, &user->Avwork));

  PetscCall(VecCreate(PETSC_COMM_WORLD, &user->d));
  PetscCall(VecSetSizes(user->d, PETSC_DECIDE, user->ndata * user->nt));
  PetscCall(VecSetFromOptions(user->d));

  PetscCall(VecDuplicate(user->S, &user->Swork));
  PetscCall(VecDuplicate(user->S, &user->Av_u));
  PetscCall(VecDuplicate(user->S, &user->Twork));
  PetscCall(VecDuplicate(user->S, &user->Rwork));
  PetscCall(VecDuplicate(user->y, &user->ywork));
  PetscCall(VecDuplicate(user->u, &user->uwork));
  PetscCall(VecDuplicate(user->u, &user->js_diag));
  PetscCall(VecDuplicate(user->c, &user->cwork));
  PetscCall(VecDuplicate(user->d, &user->dwork));

  /* Create matrix-free shell user->Js for computing A*x */
  PetscCall(MatCreateShell(PETSC_COMM_WORLD, PETSC_DETERMINE, PETSC_DETERMINE, user->m * user->nt, user->m * user->nt, user, &user->Js));
  PetscCall(MatShellSetOperation(user->Js, MATOP_MULT, (void (*)(void))StateMatMult));
  PetscCall(MatShellSetOperation(user->Js, MATOP_DUPLICATE, (void (*)(void))StateMatDuplicate));
  PetscCall(MatShellSetOperation(user->Js, MATOP_MULT_TRANSPOSE, (void (*)(void))StateMatMultTranspose));
  PetscCall(MatShellSetOperation(user->Js, MATOP_GET_DIAGONAL, (void (*)(void))StateMatGetDiagonal));

  /* Diagonal blocks of user->Js */
  PetscCall(MatCreateShell(PETSC_COMM_WORLD, PETSC_DETERMINE, PETSC_DETERMINE, user->m, user->m, user, &user->JsBlock));
  PetscCall(MatShellSetOperation(user->JsBlock, MATOP_MULT, (void (*)(void))StateMatBlockMult));
  /* Blocks are symmetric */
  PetscCall(MatShellSetOperation(user->JsBlock, MATOP_MULT_TRANSPOSE, (void (*)(void))StateMatBlockMult));

  /* Create a matrix-free shell user->JsBlockPrec for computing (U+D)\D*(L+D)\x, where JsBlock = L+D+U,
     D is diagonal, L is strictly lower triangular, and U is strictly upper triangular.
     This is an SSOR preconditioner for user->JsBlock. */
  PetscCall(MatCreateShell(PETSC_COMM_WORLD, PETSC_DETERMINE, PETSC_DETERMINE, user->m, user->m, user, &user->JsBlockPrec));
  PetscCall(MatShellSetOperation(user->JsBlockPrec, MATOP_MULT, (void (*)(void))StateMatBlockPrecMult));
  /* JsBlockPrec is symmetric */
  PetscCall(MatShellSetOperation(user->JsBlockPrec, MATOP_MULT_TRANSPOSE, (void (*)(void))StateMatBlockPrecMult));
  PetscCall(MatSetOption(user->JsBlockPrec, MAT_SYMMETRIC, PETSC_TRUE));
  PetscCall(MatSetOption(user->JsBlockPrec, MAT_SYMMETRY_ETERNAL, PETSC_TRUE));

  /* Create a matrix-free shell user->Jd for computing B*x */
  PetscCall(MatCreateShell(PETSC_COMM_WORLD, PETSC_DETERMINE, PETSC_DETERMINE, user->m * user->nt, user->m, user, &user->Jd));
  PetscCall(MatShellSetOperation(user->Jd, MATOP_MULT, (void (*)(void))DesignMatMult));
  PetscCall(MatShellSetOperation(user->Jd, MATOP_MULT_TRANSPOSE, (void (*)(void))DesignMatMultTranspose));

  /* User-defined routines for computing user->Js\x and user->Js^T\x*/
  PetscCall(MatCreateShell(PETSC_COMM_WORLD, PETSC_DETERMINE, PETSC_DETERMINE, user->m * user->nt, user->m * user->nt, user, &user->JsInv));
  PetscCall(MatShellSetOperation(user->JsInv, MATOP_MULT, (void (*)(void))StateMatInvMult));
  PetscCall(MatShellSetOperation(user->JsInv, MATOP_MULT_TRANSPOSE, (void (*)(void))StateMatInvTransposeMult));

  /* Solver options and tolerances */
  PetscCall(KSPCreate(PETSC_COMM_WORLD, &user->solver));
  PetscCall(KSPSetType(user->solver, KSPCG));
  PetscCall(KSPSetOperators(user->solver, user->JsBlock, user->JsBlockPrec));
  PetscCall(KSPSetInitialGuessNonzero(user->solver, PETSC_FALSE));
  PetscCall(KSPSetTolerances(user->solver, 1e-4, 1e-20, 1e3, 500));
  PetscCall(KSPSetFromOptions(user->solver));
  PetscCall(KSPGetPC(user->solver, &user->prec));
  PetscCall(PCSetType(user->prec, PCSHELL));

  PetscCall(PCShellSetApply(user->prec, StateMatBlockPrecMult));
  PetscCall(PCShellSetApplyTranspose(user->prec, StateMatBlockPrecMult));
  PetscCall(PCShellSetContext(user->prec, user));

  /* Create scatter from y to y_1,y_2,...,y_nt */
  PetscCall(PetscMalloc1(user->nt * user->m, &user->yi_scatter));
  PetscCall(VecCreate(PETSC_COMM_WORLD, &yi));
  PetscCall(VecSetSizes(yi, PETSC_DECIDE, user->mx * user->mx * user->mx));
  PetscCall(VecSetFromOptions(yi));
  PetscCall(VecDuplicateVecs(yi, user->nt, &user->yi));
  PetscCall(VecDuplicateVecs(yi, user->nt, &user->yiwork));

  PetscCall(VecGetOwnershipRange(user->y, &lo2, &hi2));
  istart = 0;
  for (i = 0; i < user->nt; i++) {
    PetscCall(VecGetOwnershipRange(user->yi[i], &lo, &hi));
    PetscCall(ISCreateStride(PETSC_COMM_SELF, hi - lo, lo, 1, &is_to_yi));
    PetscCall(ISCreateStride(PETSC_COMM_SELF, hi - lo, lo2 + istart, 1, &is_from_y));
    PetscCall(VecScatterCreate(user->y, is_from_y, user->yi[i], is_to_yi, &user->yi_scatter[i]));
    istart = istart + hi - lo;
    PetscCall(ISDestroy(&is_to_yi));
    PetscCall(ISDestroy(&is_from_y));
  }
  PetscCall(VecDestroy(&yi));

  /* Create scatter from d to d_1,d_2,...,d_ns */
  PetscCall(PetscMalloc1(user->ns * user->ndata, &user->di_scatter));
  PetscCall(VecCreate(PETSC_COMM_WORLD, &di));
  PetscCall(VecSetSizes(di, PETSC_DECIDE, user->ndata));
  PetscCall(VecSetFromOptions(di));
  PetscCall(VecDuplicateVecs(di, user->ns, &user->di));
  PetscCall(VecGetOwnershipRange(user->d, &lo2, &hi2));
  istart = 0;
  for (i = 0; i < user->ns; i++) {
    PetscCall(VecGetOwnershipRange(user->di[i], &lo, &hi));
    PetscCall(ISCreateStride(PETSC_COMM_SELF, hi - lo, lo, 1, &is_to_di));
    PetscCall(ISCreateStride(PETSC_COMM_SELF, hi - lo, lo2 + istart, 1, &is_from_d));
    PetscCall(VecScatterCreate(user->d, is_from_d, user->di[i], is_to_di, &user->di_scatter[i]));
    istart = istart + hi - lo;
    PetscCall(ISDestroy(&is_to_di));
    PetscCall(ISDestroy(&is_from_d));
  }
  PetscCall(VecDestroy(&di));

  /* Assemble RHS of forward problem */
  PetscCall(VecCopy(bc, user->yiwork[0]));
  for (i = 1; i < user->nt; i++) PetscCall(VecSet(user->yiwork[i], 0.0));
  PetscCall(Gather_i(user->q, user->yiwork, user->yi_scatter, user->nt));
  PetscCall(VecDestroy(&bc));

  /* Compute true state function yt given ut */
  PetscCall(VecCreate(PETSC_COMM_WORLD, &user->ytrue));
  PetscCall(VecSetSizes(user->ytrue, PETSC_DECIDE, n * user->nt));
  PetscCall(VecSetFromOptions(user->ytrue));

  /* First compute Av_u = Av*exp(-u) */
  PetscCall(VecSet(user->uwork, 0));
  PetscCall(VecAXPY(user->uwork, -1.0, user->utrue)); /*  Note: user->utrue */
  PetscCall(VecExp(user->uwork));
  PetscCall(MatMult(user->Av, user->uwork, user->Av_u));

  /* Symbolic DSG = Div * Grad */
  PetscCall(MatProductCreate(user->Div, user->Grad, NULL, &user->DSG));
  PetscCall(MatProductSetType(user->DSG, MATPRODUCT_AB));
  PetscCall(MatProductSetAlgorithm(user->DSG, "default"));
  PetscCall(MatProductSetFill(user->DSG, PETSC_DEFAULT));
  PetscCall(MatProductSetFromOptions(user->DSG));
  PetscCall(MatProductSymbolic(user->DSG));

  user->dsg_formed = PETSC_TRUE;

  /* Next form DSG = Div*Grad */
  PetscCall(MatCopy(user->Div, user->Divwork, SAME_NONZERO_PATTERN));
  PetscCall(MatDiagonalScale(user->Divwork, NULL, user->Av_u));
  if (user->dsg_formed) {
    PetscCall(MatProductNumeric(user->DSG));
  } else {
    PetscCall(MatMatMult(user->Div, user->Grad, MAT_INITIAL_MATRIX, PETSC_DEFAULT, &user->DSG));
    user->dsg_formed = PETSC_TRUE;
  }
  /* B = speye(nx^3) + ht*DSG; */
  PetscCall(MatScale(user->DSG, user->ht));
  PetscCall(MatShift(user->DSG, 1.0));

  /* Now solve for ytrue */
  PetscCall(StateMatInvMult(user->Js, user->q, user->ytrue));

  /* Initial guess y0 for state given u0 */

  /* First compute Av_u = Av*exp(-u) */
  PetscCall(VecSet(user->uwork, 0));
  PetscCall(VecAXPY(user->uwork, -1.0, user->u)); /*  Note: user->u */
  PetscCall(VecExp(user->uwork));
  PetscCall(MatMult(user->Av, user->uwork, user->Av_u));

  /* Next form DSG = Div*S*Grad */
  PetscCall(MatCopy(user->Div, user->Divwork, SAME_NONZERO_PATTERN));
  PetscCall(MatDiagonalScale(user->Divwork, NULL, user->Av_u));
  if (user->dsg_formed) {
    PetscCall(MatProductNumeric(user->DSG));
  } else {
    PetscCall(MatMatMult(user->Div, user->Grad, MAT_INITIAL_MATRIX, PETSC_DEFAULT, &user->DSG));

    user->dsg_formed = PETSC_TRUE;
  }
  /* B = speye(nx^3) + ht*DSG; */
  PetscCall(MatScale(user->DSG, user->ht));
  PetscCall(MatShift(user->DSG, 1.0));

  /* Now solve for y */
  PetscCall(StateMatInvMult(user->Js, user->q, user->y));

  /* Construct projection matrix Q, a block diagonal matrix consisting of nt copies of Qblock along the diagonal */
  PetscCall(MatCreate(PETSC_COMM_WORLD, &user->Qblock));
  PetscCall(MatSetSizes(user->Qblock, PETSC_DECIDE, PETSC_DECIDE, user->ndata, n));
  PetscCall(MatSetFromOptions(user->Qblock));
  PetscCall(MatMPIAIJSetPreallocation(user->Qblock, 8, NULL, 8, NULL));
  PetscCall(MatSeqAIJSetPreallocation(user->Qblock, 8, NULL));

  for (i = 0; i < user->mx; i++) {
    x[i] = h * (i + 0.5);
    y[i] = h * (i + 0.5);
    z[i] = h * (i + 0.5);
  }

  PetscCall(MatGetOwnershipRange(user->Qblock, &istart, &iend));
  nx = user->mx;
  ny = user->mx;
  nz = user->mx;
  for (i = istart; i < iend; i++) {
    xri = xr[i];
    im  = 0;
    xim = x[im];
    while (xri > xim && im < nx) {
      im  = im + 1;
      xim = x[im];
    }
    indx1 = im - 1;
    indx2 = im;
    dx1   = xri - x[indx1];
    dx2   = x[indx2] - xri;

    yri = yr[i];
    im  = 0;
    yim = y[im];
    while (yri > yim && im < ny) {
      im  = im + 1;
      yim = y[im];
    }
    indy1 = im - 1;
    indy2 = im;
    dy1   = yri - y[indy1];
    dy2   = y[indy2] - yri;

    zri = zr[i];
    im  = 0;
    zim = z[im];
    while (zri > zim && im < nz) {
      im  = im + 1;
      zim = z[im];
    }
    indz1 = im - 1;
    indz2 = im;
    dz1   = zri - z[indz1];
    dz2   = z[indz2] - zri;

    Dx = x[indx2] - x[indx1];
    Dy = y[indy2] - y[indy1];
    Dz = z[indz2] - z[indz1];

    j = indx1 + indy1 * nx + indz1 * nx * ny;
    v = (1 - dx1 / Dx) * (1 - dy1 / Dy) * (1 - dz1 / Dz);
    PetscCall(MatSetValues(user->Qblock, 1, &i, 1, &j, &v, INSERT_VALUES));

    j = indx1 + indy1 * nx + indz2 * nx * ny;
    v = (1 - dx1 / Dx) * (1 - dy1 / Dy) * (1 - dz2 / Dz);
    PetscCall(MatSetValues(user->Qblock, 1, &i, 1, &j, &v, INSERT_VALUES));

    j = indx1 + indy2 * nx + indz1 * nx * ny;
    v = (1 - dx1 / Dx) * (1 - dy2 / Dy) * (1 - dz1 / Dz);
    PetscCall(MatSetValues(user->Qblock, 1, &i, 1, &j, &v, INSERT_VALUES));

    j = indx1 + indy2 * nx + indz2 * nx * ny;
    v = (1 - dx1 / Dx) * (1 - dy2 / Dy) * (1 - dz2 / Dz);
    PetscCall(MatSetValues(user->Qblock, 1, &i, 1, &j, &v, INSERT_VALUES));

    j = indx2 + indy1 * nx + indz1 * nx * ny;
    v = (1 - dx2 / Dx) * (1 - dy1 / Dy) * (1 - dz1 / Dz);
    PetscCall(MatSetValues(user->Qblock, 1, &i, 1, &j, &v, INSERT_VALUES));

    j = indx2 + indy1 * nx + indz2 * nx * ny;
    v = (1 - dx2 / Dx) * (1 - dy1 / Dy) * (1 - dz2 / Dz);
    PetscCall(MatSetValues(user->Qblock, 1, &i, 1, &j, &v, INSERT_VALUES));

    j = indx2 + indy2 * nx + indz1 * nx * ny;
    v = (1 - dx2 / Dx) * (1 - dy2 / Dy) * (1 - dz1 / Dz);
    PetscCall(MatSetValues(user->Qblock, 1, &i, 1, &j, &v, INSERT_VALUES));

    j = indx2 + indy2 * nx + indz2 * nx * ny;
    v = (1 - dx2 / Dx) * (1 - dy2 / Dy) * (1 - dz2 / Dz);
    PetscCall(MatSetValues(user->Qblock, 1, &i, 1, &j, &v, INSERT_VALUES));
  }
  PetscCall(MatAssemblyBegin(user->Qblock, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(user->Qblock, MAT_FINAL_ASSEMBLY));

  PetscCall(MatTranspose(user->Qblock, MAT_INITIAL_MATRIX, &user->QblockT));
  PetscCall(MatTranspose(user->L, MAT_INITIAL_MATRIX, &user->LT));

  /* Add noise to the measurement data */
  PetscCall(VecSet(user->ywork, 1.0));
  PetscCall(VecAYPX(user->ywork, user->noise, user->ytrue));
  PetscCall(Scatter_i(user->ywork, user->yiwork, user->yi_scatter, user->nt));
  for (j = 0; j < user->ns; j++) {
    i = user->sample_times[j];
    PetscCall(MatMult(user->Qblock, user->yiwork[i], user->di[j]));
  }
  PetscCall(Gather_i(user->d, user->di, user->di_scatter, user->ns));

  /* Now that initial conditions have been set, let the user pass tolerance options to the KSP solver */
  PetscCall(KSPSetFromOptions(user->solver));
  PetscCall(PetscFree(x));
  PetscCall(PetscFree(y));
  PetscCall(PetscFree(z));
  PetscFunctionReturn(0);
}

PetscErrorCode ParabolicDestroy(AppCtx *user) {
  PetscInt i;

  PetscFunctionBegin;
  PetscCall(MatDestroy(&user->Qblock));
  PetscCall(MatDestroy(&user->QblockT));
  PetscCall(MatDestroy(&user->Div));
  PetscCall(MatDestroy(&user->Divwork));
  PetscCall(MatDestroy(&user->Grad));
  PetscCall(MatDestroy(&user->Av));
  PetscCall(MatDestroy(&user->Avwork));
  PetscCall(MatDestroy(&user->AvT));
  PetscCall(MatDestroy(&user->DSG));
  PetscCall(MatDestroy(&user->L));
  PetscCall(MatDestroy(&user->LT));
  PetscCall(KSPDestroy(&user->solver));
  PetscCall(MatDestroy(&user->Js));
  PetscCall(MatDestroy(&user->Jd));
  PetscCall(MatDestroy(&user->JsInv));
  PetscCall(MatDestroy(&user->JsBlock));
  PetscCall(MatDestroy(&user->JsBlockPrec));
  PetscCall(VecDestroy(&user->u));
  PetscCall(VecDestroy(&user->uwork));
  PetscCall(VecDestroy(&user->utrue));
  PetscCall(VecDestroy(&user->y));
  PetscCall(VecDestroy(&user->ywork));
  PetscCall(VecDestroy(&user->ytrue));
  PetscCall(VecDestroyVecs(user->nt, &user->yi));
  PetscCall(VecDestroyVecs(user->nt, &user->yiwork));
  PetscCall(VecDestroyVecs(user->ns, &user->di));
  PetscCall(PetscFree(user->yi));
  PetscCall(PetscFree(user->yiwork));
  PetscCall(PetscFree(user->di));
  PetscCall(VecDestroy(&user->c));
  PetscCall(VecDestroy(&user->cwork));
  PetscCall(VecDestroy(&user->ur));
  PetscCall(VecDestroy(&user->q));
  PetscCall(VecDestroy(&user->d));
  PetscCall(VecDestroy(&user->dwork));
  PetscCall(VecDestroy(&user->lwork));
  PetscCall(VecDestroy(&user->S));
  PetscCall(VecDestroy(&user->Swork));
  PetscCall(VecDestroy(&user->Av_u));
  PetscCall(VecDestroy(&user->Twork));
  PetscCall(VecDestroy(&user->Rwork));
  PetscCall(VecDestroy(&user->js_diag));
  PetscCall(ISDestroy(&user->s_is));
  PetscCall(ISDestroy(&user->d_is));
  PetscCall(VecScatterDestroy(&user->state_scatter));
  PetscCall(VecScatterDestroy(&user->design_scatter));
  for (i = 0; i < user->nt; i++) PetscCall(VecScatterDestroy(&user->yi_scatter[i]));
  for (i = 0; i < user->ns; i++) PetscCall(VecScatterDestroy(&user->di_scatter[i]));
  PetscCall(PetscFree(user->yi_scatter));
  PetscCall(PetscFree(user->di_scatter));
  PetscCall(PetscFree(user->sample_times));
  PetscFunctionReturn(0);
}

PetscErrorCode ParabolicMonitor(Tao tao, void *ptr) {
  Vec       X;
  PetscReal unorm, ynorm;
  AppCtx   *user = (AppCtx *)ptr;

  PetscFunctionBegin;
  PetscCall(TaoGetSolution(tao, &X));
  PetscCall(Scatter(X, user->ywork, user->state_scatter, user->uwork, user->design_scatter));
  PetscCall(VecAXPY(user->ywork, -1.0, user->ytrue));
  PetscCall(VecAXPY(user->uwork, -1.0, user->utrue));
  PetscCall(VecNorm(user->uwork, NORM_2, &unorm));
  PetscCall(VecNorm(user->ywork, NORM_2, &ynorm));
  PetscCall(PetscPrintf(MPI_COMM_WORLD, "||u-ut||=%g ||y-yt||=%g\n", (double)unorm, (double)ynorm));
  PetscFunctionReturn(0);
}

/*TEST

   build:
      requires: !complex

   test:
      args: -tao_cmonitor -tao_type lcl -ns 1 -tao_gatol 1.e-4 -ksp_max_it 30
      requires: !single

TEST*/