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

#include <petsctao.h>

typedef struct {
  PetscInt    n;     /*  Number of variables */
  PetscInt    m;     /*  Number of constraints */
  PetscInt    mx;    /*  grid points in each direction */
  PetscInt    nt;    /*  Number of time steps */
  PetscInt    ndata; /*  Number of data points per sample */
  IS          s_is;
  IS          d_is;
  VecScatter  state_scatter;
  VecScatter  design_scatter;
  VecScatter *uxi_scatter, *uyi_scatter, *ux_scatter, *uy_scatter, *ui_scatter;
  VecScatter *yi_scatter;

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

  PetscReal alpha; /*  Regularization parameter */
  PetscReal gamma;
  PetscReal ht; /*  Time step */
  PetscReal T;  /*  Final time */
  Mat       Q, QT;
  Mat       L, LT;
  Mat       Div, Divwork, Divxy[2];
  Mat       Grad, Gradxy[2];
  Mat       M;
  Mat      *C, *Cwork;
  /* Mat Hs,Hd,Hsd; */
  Vec       q;
  Vec       ur; /*  reference */

  Vec d;
  Vec dwork;

  Vec  y; /*  state variables */
  Vec  ywork;
  Vec  ytrue;
  Vec *yi, *yiwork, *ziwork;
  Vec *uxi, *uyi, *uxiwork, *uyiwork, *ui, *uiwork;

  Vec u; /*  design variables */
  Vec uwork, vwork;
  Vec utrue;

  Vec js_diag;

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

  Vec lwork;

  KSP      solver;
  PC       prec;
  PetscInt block_index;

  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 HyperbolicInitialize(AppCtx *user);
PetscErrorCode HyperbolicDestroy(AppCtx *user);
PetscErrorCode HyperbolicMonitor(Tao, void *);

PetscErrorCode StateMatMult(Mat, Vec, Vec);
PetscErrorCode StateMatBlockMult(Mat, Vec, Vec);
PetscErrorCode StateMatBlockMultTranspose(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 Scatter_yi(Vec, Vec *, VecScatter *, PetscInt); /*  y to y1,y2,...,y_nt */
PetscErrorCode Gather_yi(Vec, Vec *, VecScatter *, PetscInt);
PetscErrorCode Scatter_uxi_uyi(Vec, Vec *, VecScatter *, Vec *, VecScatter *, PetscInt); /*  u to ux_1,uy_1,ux_2,uy_2,...,u */
PetscErrorCode Gather_uxi_uyi(Vec, Vec *, VecScatter *, 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 = 32;
  PetscOptionsBegin(PETSC_COMM_WORLD, NULL, "hyperbolic example", NULL);
  PetscCall(PetscOptionsInt("-mx", "Number of grid points in each direction", "", user.mx, &user.mx, NULL));
  user.nt = 16;
  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.alpha = 10.0;
  PetscCall(PetscOptionsReal("-alpha", "Regularization parameter", "", user.alpha, &user.alpha, NULL));
  user.T = 1.0 / 32.0;
  PetscCall(PetscOptionsReal("-Tfinal", "Final time", "", user.T, &user.T, NULL));
  PetscCall(PetscOptionsInt("-ntests", "Number of times to repeat TaoSolve", "", ntests, &ntests, NULL));
  PetscOptionsEnd();

  user.m     = user.mx * user.mx * user.nt;     /*  number of constraints */
  user.n     = user.mx * user.mx * 3 * user.nt; /*  number of variables */
  user.ht    = user.T / user.nt;                /*  Time step */
  user.gamma = user.T * user.ht / (user.mx * user.mx);

  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));
  PetscCall(VecSetSizes(user.y, PETSC_DECIDE, user.m));
  PetscCall(VecSetSizes(user.c, PETSC_DECIDE, user.m));
  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(HyperbolicInitialize(&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.Js, 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(HyperbolicDestroy(&user));
  PetscCall(PetscFinalize());
  return 0;
}
/* ------------------------------------------------------------------- */
/*
   dwork = Qy - d
   lwork = L*(u-ur).^2
   f = 1/2 * (dwork.dork + alpha*y.lwork)
*/
PetscErrorCode FormFunction(Tao tao, Vec X, PetscReal *f, void *ptr) {
  PetscReal d1 = 0, d2 = 0;
  AppCtx   *user = (AppCtx *)ptr;

  PetscFunctionBegin;
  PetscCall(Scatter(X, user->y, user->state_scatter, user->u, user->design_scatter));
  PetscCall(MatMult(user->Q, user->y, user->dwork));
  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(VecPointwiseMult(user->uwork, user->uwork, user->uwork));
  PetscCall(MatMult(user->L, user->uwork, user->lwork));
  PetscCall(VecDot(user->y, user->lwork, &d2));
  *f = 0.5 * (d1 + user->alpha * d2);
  PetscFunctionReturn(0);
}

/* ------------------------------------------------------------------- */
/*
    state: g_s = Q' *(Qy - d) + 0.5*alpha*L*(u-ur).^2
    design: g_d = alpha*(L'y).*(u-ur)
*/
PetscErrorCode FormGradient(Tao tao, Vec X, Vec G, void *ptr) {
  AppCtx *user = (AppCtx *)ptr;

  PetscFunctionBegin;
  PetscCall(Scatter(X, user->y, user->state_scatter, user->u, user->design_scatter));
  PetscCall(MatMult(user->Q, user->y, user->dwork));
  PetscCall(VecAXPY(user->dwork, -1.0, user->d));

  PetscCall(MatMult(user->QT, user->dwork, user->ywork));

  PetscCall(MatMult(user->LT, user->y, user->uwork));
  PetscCall(VecWAXPY(user->vwork, -1.0, user->ur, user->u));
  PetscCall(VecPointwiseMult(user->uwork, user->vwork, user->uwork));
  PetscCall(VecScale(user->uwork, user->alpha));

  PetscCall(VecPointwiseMult(user->vwork, user->vwork, user->vwork));
  PetscCall(MatMult(user->L, user->vwork, user->lwork));
  PetscCall(VecAXPY(user->ywork, 0.5 * user->alpha, user->lwork));

  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;
  AppCtx   *user = (AppCtx *)ptr;

  PetscFunctionBegin;
  PetscCall(Scatter(X, user->y, user->state_scatter, user->u, user->design_scatter));
  PetscCall(MatMult(user->Q, user->y, user->dwork));
  PetscCall(VecAXPY(user->dwork, -1.0, user->d));

  PetscCall(MatMult(user->QT, user->dwork, user->ywork));

  PetscCall(VecDot(user->dwork, user->dwork, &d1));

  PetscCall(MatMult(user->LT, user->y, user->uwork));
  PetscCall(VecWAXPY(user->vwork, -1.0, user->ur, user->u));
  PetscCall(VecPointwiseMult(user->uwork, user->vwork, user->uwork));
  PetscCall(VecScale(user->uwork, user->alpha));

  PetscCall(VecPointwiseMult(user->vwork, user->vwork, user->vwork));
  PetscCall(MatMult(user->L, user->vwork, user->lwork));
  PetscCall(VecAXPY(user->ywork, 0.5 * user->alpha, user->lwork));

  PetscCall(VecDot(user->y, user->lwork, &d2));

  *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) {
  PetscInt i;
  AppCtx  *user = (AppCtx *)ptr;

  PetscFunctionBegin;
  PetscCall(Scatter(X, user->y, user->state_scatter, user->u, user->design_scatter));
  PetscCall(Scatter_yi(user->u, user->ui, user->ui_scatter, user->nt));
  PetscCall(Scatter_uxi_uyi(user->u, user->uxi, user->uxi_scatter, user->uyi, user->uyi_scatter, user->nt));
  for (i = 0; i < user->nt; i++) {
    PetscCall(MatCopy(user->Divxy[0], user->C[i], SUBSET_NONZERO_PATTERN));
    PetscCall(MatCopy(user->Divxy[1], user->Cwork[i], SAME_NONZERO_PATTERN));

    PetscCall(MatDiagonalScale(user->C[i], NULL, user->uxi[i]));
    PetscCall(MatDiagonalScale(user->Cwork[i], NULL, user->uyi[i]));
    PetscCall(MatAXPY(user->C[i], 1.0, user->Cwork[i], SUBSET_NONZERO_PATTERN));
    PetscCall(MatScale(user->C[i], user->ht));
    PetscCall(MatShift(user->C[i], 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_yi(X, user->yi, user->yi_scatter, user->nt));
  user->block_index = 0;
  PetscCall(MatMult(user->JsBlock, user->yi[0], user->yiwork[0]));

  for (i = 1; i < user->nt; i++) {
    user->block_index = i;
    PetscCall(MatMult(user->JsBlock, user->yi[i], user->yiwork[i]));
    PetscCall(MatMult(user->M, user->yi[i - 1], user->ziwork[i - 1]));
    PetscCall(VecAXPY(user->yiwork[i], -1.0, user->ziwork[i - 1]));
  }
  PetscCall(Gather_yi(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_yi(X, user->yi, user->yi_scatter, user->nt));

  for (i = 0; i < user->nt - 1; i++) {
    user->block_index = i;
    PetscCall(MatMultTranspose(user->JsBlock, user->yi[i], user->yiwork[i]));
    PetscCall(MatMult(user->M, user->yi[i + 1], user->ziwork[i + 1]));
    PetscCall(VecAXPY(user->yiwork[i], -1.0, user->ziwork[i + 1]));
  }

  i                 = user->nt - 1;
  user->block_index = i;
  PetscCall(MatMultTranspose(user->JsBlock, user->yi[i], user->yiwork[i]));
  PetscCall(Gather_yi(Y, user->yiwork, user->yi_scatter, user->nt));
  PetscFunctionReturn(0);
}

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

  PetscFunctionBegin;
  PetscCall(MatShellGetContext(J_shell, &user));
  i = user->block_index;
  PetscCall(VecPointwiseMult(user->uxiwork[i], X, user->uxi[i]));
  PetscCall(VecPointwiseMult(user->uyiwork[i], X, user->uyi[i]));
  PetscCall(Gather(user->uiwork[i], user->uxiwork[i], user->ux_scatter[i], user->uyiwork[i], user->uy_scatter[i]));
  PetscCall(MatMult(user->Div, user->uiwork[i], Y));
  PetscCall(VecAYPX(Y, user->ht, X));
  PetscFunctionReturn(0);
}

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

  PetscFunctionBegin;
  PetscCall(MatShellGetContext(J_shell, &user));
  i = user->block_index;
  PetscCall(MatMult(user->Grad, X, user->uiwork[i]));
  PetscCall(Scatter(user->uiwork[i], user->uxiwork[i], user->ux_scatter[i], user->uyiwork[i], user->uy_scatter[i]));
  PetscCall(VecPointwiseMult(user->uxiwork[i], user->uxi[i], user->uxiwork[i]));
  PetscCall(VecPointwiseMult(user->uyiwork[i], user->uyi[i], user->uyiwork[i]));
  PetscCall(VecWAXPY(Y, 1.0, user->uxiwork[i], user->uyiwork[i]));
  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));
  PetscCall(Scatter_yi(user->y, user->yi, user->yi_scatter, user->nt));
  PetscCall(Scatter_uxi_uyi(X, user->uxiwork, user->uxi_scatter, user->uyiwork, user->uyi_scatter, user->nt));
  for (i = 0; i < user->nt; i++) {
    PetscCall(VecPointwiseMult(user->uxiwork[i], user->yi[i], user->uxiwork[i]));
    PetscCall(VecPointwiseMult(user->uyiwork[i], user->yi[i], user->uyiwork[i]));
    PetscCall(Gather(user->uiwork[i], user->uxiwork[i], user->ux_scatter[i], user->uyiwork[i], user->uy_scatter[i]));
    PetscCall(MatMult(user->Div, user->uiwork[i], user->ziwork[i]));
    PetscCall(VecScale(user->ziwork[i], user->ht));
  }
  PetscCall(Gather_yi(Y, user->ziwork, 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));
  PetscCall(Scatter_yi(user->y, user->yi, user->yi_scatter, user->nt));
  PetscCall(Scatter_yi(X, user->yiwork, user->yi_scatter, user->nt));
  for (i = 0; i < user->nt; i++) {
    PetscCall(MatMult(user->Grad, user->yiwork[i], user->uiwork[i]));
    PetscCall(Scatter(user->uiwork[i], user->uxiwork[i], user->ux_scatter[i], user->uyiwork[i], user->uy_scatter[i]));
    PetscCall(VecPointwiseMult(user->uxiwork[i], user->yi[i], user->uxiwork[i]));
    PetscCall(VecPointwiseMult(user->uyiwork[i], user->yi[i], user->uyiwork[i]));
    PetscCall(Gather(user->uiwork[i], user->uxiwork[i], user->ux_scatter[i], user->uyiwork[i], user->uy_scatter[i]));
    PetscCall(VecScale(user->uiwork[i], user->ht));
  }
  PetscCall(Gather_yi(Y, user->uiwork, user->ui_scatter, user->nt));
  PetscFunctionReturn(0);
}

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

  PetscFunctionBegin;
  PetscCall(PCShellGetContext(PC_shell, &user));
  i = user->block_index;
  if (user->c_formed) {
    PetscCall(MatSOR(user->C[i], 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, "Not formed");
  PetscFunctionReturn(0);
}

PetscErrorCode StateMatBlockPrecMultTranspose(PC PC_shell, Vec X, Vec Y) {
  PetscInt i;
  AppCtx  *user;

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

  i = user->block_index;
  if (user->c_formed) {
    PetscCall(MatSOR(user->C[i], 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, "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 using true solution to set up problem */
    PetscCall(KSPSetTolerances(user->solver, 1e-4, 1e-20, PETSC_DEFAULT, PETSC_DEFAULT));
  } else {
    PetscCall(KSPSetTolerances(user->solver, PETSC_DEFAULT, PETSC_DEFAULT, PETSC_DEFAULT, PETSC_DEFAULT));
  }
  PetscCall(Scatter_yi(X, user->yi, user->yi_scatter, user->nt));
  PetscCall(Scatter_yi(Y, user->yiwork, user->yi_scatter, user->nt));
  PetscCall(Scatter_uxi_uyi(user->u, user->uxi, user->uxi_scatter, user->uyi, user->uyi_scatter, user->nt));

  user->block_index = 0;
  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(MatMult(user->M, user->yiwork[i - 1], user->ziwork[i - 1]));
    PetscCall(VecAXPY(user->yi[i], 1.0, user->ziwork[i - 1]));
    user->block_index = i;
    PetscCall(KSPSolve(user->solver, user->yi[i], user->yiwork[i]));

    PetscCall(KSPGetIterationNumber(user->solver, &its));
    user->ksp_its = user->ksp_its + its;
  }
  PetscCall(Gather_yi(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_yi(X, user->yi, user->yi_scatter, user->nt));
  PetscCall(Scatter_yi(Y, user->yiwork, user->yi_scatter, user->nt));
  PetscCall(Scatter_uxi_uyi(user->u, user->uxi, user->uxi_scatter, user->uyi, user->uyi_scatter, user->nt));

  i                 = user->nt - 1;
  user->block_index = i;
  PetscCall(KSPSolveTranspose(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(MatMult(user->M, user->yiwork[i + 1], user->ziwork[i + 1]));
    PetscCall(VecAXPY(user->yi[i], 1.0, user->ziwork[i + 1]));
    user->block_index = i;
    PetscCall(KSPSolveTranspose(user->solver, user->yi[i], user->yiwork[i]));

    PetscCall(KSPGetIterationNumber(user->solver, &its));
    user->ksp_its = user->ksp_its + its;
  }
  PetscCall(Gather_yi(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 = [C(u1)  0     0     ...   0;
                         -M  C(u2)   0     ...   0;
                          0   -M   C(u3)   ...   0;
                                      ...         ;
                          0    ...      -M C(u_nt)]
     C(u) = eye + ht*Div*[diag(u1); diag(u2)]       */
  PetscInt i;
  AppCtx  *user = (AppCtx *)ptr;

  PetscFunctionBegin;
  PetscCall(Scatter(X, user->y, user->state_scatter, user->u, user->design_scatter));
  PetscCall(Scatter_yi(user->y, user->yi, user->yi_scatter, user->nt));
  PetscCall(Scatter_uxi_uyi(user->u, user->uxi, user->uxi_scatter, user->uyi, user->uyi_scatter, user->nt));

  user->block_index = 0;
  PetscCall(MatMult(user->JsBlock, user->yi[0], user->yiwork[0]));

  for (i = 1; i < user->nt; i++) {
    user->block_index = i;
    PetscCall(MatMult(user->JsBlock, user->yi[i], user->yiwork[i]));
    PetscCall(MatMult(user->M, user->yi[i - 1], user->ziwork[i - 1]));
    PetscCall(VecAXPY(user->yiwork[i], -1.0, user->ziwork[i - 1]));
  }

  PetscCall(Gather_yi(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_uxi_uyi(Vec u, Vec *uxi, VecScatter *scatx, Vec *uyi, VecScatter *scaty, PetscInt nt) {
  PetscInt i;

  PetscFunctionBegin;
  for (i = 0; i < nt; i++) {
    PetscCall(VecScatterBegin(scatx[i], u, uxi[i], INSERT_VALUES, SCATTER_FORWARD));
    PetscCall(VecScatterEnd(scatx[i], u, uxi[i], INSERT_VALUES, SCATTER_FORWARD));
    PetscCall(VecScatterBegin(scaty[i], u, uyi[i], INSERT_VALUES, SCATTER_FORWARD));
    PetscCall(VecScatterEnd(scaty[i], u, uyi[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_uxi_uyi(Vec u, Vec *uxi, VecScatter *scatx, Vec *uyi, VecScatter *scaty, PetscInt nt) {
  PetscInt i;

  PetscFunctionBegin;
  for (i = 0; i < nt; i++) {
    PetscCall(VecScatterBegin(scatx[i], uxi[i], u, INSERT_VALUES, SCATTER_REVERSE));
    PetscCall(VecScatterEnd(scatx[i], uxi[i], u, INSERT_VALUES, SCATTER_REVERSE));
    PetscCall(VecScatterBegin(scaty[i], uyi[i], u, INSERT_VALUES, SCATTER_REVERSE));
    PetscCall(VecScatterEnd(scaty[i], uyi[i], u, INSERT_VALUES, SCATTER_REVERSE));
  }
  PetscFunctionReturn(0);
}

PetscErrorCode Scatter_yi(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_yi(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 HyperbolicInitialize(AppCtx *user) {
  PetscInt    n, i, j, linear_index, istart, iend, iblock, lo, hi;
  Vec         XX, YY, XXwork, YYwork, yi, uxi, ui, bc;
  PetscReal   h, sum;
  PetscScalar hinv, neg_hinv, quarter = 0.25, one = 1.0, half_hinv, neg_half_hinv;
  PetscScalar vx, vy, zero = 0.0;
  IS          is_from_y, is_to_yi, is_from_u, is_to_uxi, is_to_uyi;

  PetscFunctionBegin;
  user->jformed  = PETSC_FALSE;
  user->c_formed = PETSC_FALSE;

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

  n = user->mx * user->mx;

  h             = 1.0 / user->mx;
  hinv          = user->mx;
  neg_hinv      = -hinv;
  half_hinv     = hinv / 2.0;
  neg_half_hinv = neg_hinv / 2.0;

  /* Generate Grad matrix */
  PetscCall(MatCreate(PETSC_COMM_WORLD, &user->Grad));
  PetscCall(MatSetSizes(user->Grad, PETSC_DECIDE, PETSC_DECIDE, 2 * n, n));
  PetscCall(MatSetFromOptions(user->Grad));
  PetscCall(MatMPIAIJSetPreallocation(user->Grad, 3, NULL, 3, NULL));
  PetscCall(MatSeqAIJSetPreallocation(user->Grad, 3, NULL));
  PetscCall(MatGetOwnershipRange(user->Grad, &istart, &iend));

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

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

  PetscCall(MatCreate(PETSC_COMM_WORLD, &user->Gradxy[0]));
  PetscCall(MatSetSizes(user->Gradxy[0], PETSC_DECIDE, PETSC_DECIDE, n, n));
  PetscCall(MatSetFromOptions(user->Gradxy[0]));
  PetscCall(MatMPIAIJSetPreallocation(user->Gradxy[0], 3, NULL, 3, NULL));
  PetscCall(MatSeqAIJSetPreallocation(user->Gradxy[0], 3, NULL));
  PetscCall(MatGetOwnershipRange(user->Gradxy[0], &istart, &iend));

  for (i = istart; i < iend; i++) {
    iblock = i / user->mx;
    j      = iblock * user->mx + ((i + user->mx - 1) % user->mx);
    PetscCall(MatSetValues(user->Gradxy[0], 1, &i, 1, &j, &half_hinv, INSERT_VALUES));
    j = iblock * user->mx + ((i + 1) % user->mx);
    PetscCall(MatSetValues(user->Gradxy[0], 1, &i, 1, &j, &neg_half_hinv, INSERT_VALUES));
    PetscCall(MatSetValues(user->Gradxy[0], 1, &i, 1, &i, &zero, INSERT_VALUES));
  }
  PetscCall(MatAssemblyBegin(user->Gradxy[0], MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(user->Gradxy[0], MAT_FINAL_ASSEMBLY));

  PetscCall(MatCreate(PETSC_COMM_WORLD, &user->Gradxy[1]));
  PetscCall(MatSetSizes(user->Gradxy[1], PETSC_DECIDE, PETSC_DECIDE, n, n));
  PetscCall(MatSetFromOptions(user->Gradxy[1]));
  PetscCall(MatMPIAIJSetPreallocation(user->Gradxy[1], 3, NULL, 3, NULL));
  PetscCall(MatSeqAIJSetPreallocation(user->Gradxy[1], 3, NULL));
  PetscCall(MatGetOwnershipRange(user->Gradxy[1], &istart, &iend));

  for (i = istart; i < iend; i++) {
    j = (i + n - user->mx) % n;
    PetscCall(MatSetValues(user->Gradxy[1], 1, &i, 1, &j, &half_hinv, INSERT_VALUES));
    j = (j + 2 * user->mx) % n;
    PetscCall(MatSetValues(user->Gradxy[1], 1, &i, 1, &j, &neg_half_hinv, INSERT_VALUES));
    PetscCall(MatSetValues(user->Gradxy[1], 1, &i, 1, &i, &zero, INSERT_VALUES));
  }
  PetscCall(MatAssemblyBegin(user->Gradxy[1], MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(user->Gradxy[1], MAT_FINAL_ASSEMBLY));

  /* Generate Div matrix */
  PetscCall(MatTranspose(user->Grad, MAT_INITIAL_MATRIX, &user->Div));
  PetscCall(MatTranspose(user->Gradxy[0], MAT_INITIAL_MATRIX, &user->Divxy[0]));
  PetscCall(MatTranspose(user->Gradxy[1], MAT_INITIAL_MATRIX, &user->Divxy[1]));

  /* Off-diagonal averaging matrix */
  PetscCall(MatCreate(PETSC_COMM_WORLD, &user->M));
  PetscCall(MatSetSizes(user->M, PETSC_DECIDE, PETSC_DECIDE, n, n));
  PetscCall(MatSetFromOptions(user->M));
  PetscCall(MatMPIAIJSetPreallocation(user->M, 4, NULL, 4, NULL));
  PetscCall(MatSeqAIJSetPreallocation(user->M, 4, NULL));
  PetscCall(MatGetOwnershipRange(user->M, &istart, &iend));

  for (i = istart; i < iend; i++) {
    /* kron(Id,Av) */
    iblock = i / user->mx;
    j      = iblock * user->mx + ((i + user->mx - 1) % user->mx);
    PetscCall(MatSetValues(user->M, 1, &i, 1, &j, &quarter, INSERT_VALUES));
    j = iblock * user->mx + ((i + 1) % user->mx);
    PetscCall(MatSetValues(user->M, 1, &i, 1, &j, &quarter, INSERT_VALUES));

    /* kron(Av,Id) */
    j = (i + user->mx) % n;
    PetscCall(MatSetValues(user->M, 1, &i, 1, &j, &quarter, INSERT_VALUES));
    j = (i + n - user->mx) % n;
    PetscCall(MatSetValues(user->M, 1, &i, 1, &j, &quarter, INSERT_VALUES));
  }
  PetscCall(MatAssemblyBegin(user->M, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(user->M, MAT_FINAL_ASSEMBLY));

  /* Generate 2D grid */
  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, &XXwork));
  PetscCall(VecDuplicate(XX, &YYwork));
  PetscCall(VecDuplicate(XX, &user->d));
  PetscCall(VecDuplicate(XX, &user->dwork));

  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;
    vx = h * (i + 0.5);
    vy = h * (j + 0.5);
    PetscCall(VecSetValues(XX, 1, &linear_index, &vx, INSERT_VALUES));
    PetscCall(VecSetValues(YY, 1, &linear_index, &vy, INSERT_VALUES));
  }

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

  /* Compute final density function yT
     yT = 1.0 + exp(-30*((x-0.25)^2+(y-0.25)^2)) + exp(-30*((x-0.75)^2+(y-0.75)^2))
     yT = yT / (h^2*sum(yT)) */
  PetscCall(VecCopy(XX, XXwork));
  PetscCall(VecCopy(YY, YYwork));

  PetscCall(VecShift(XXwork, -0.25));
  PetscCall(VecShift(YYwork, -0.25));

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

  PetscCall(VecCopy(XXwork, user->dwork));
  PetscCall(VecAXPY(user->dwork, 1.0, YYwork));
  PetscCall(VecScale(user->dwork, -30.0));
  PetscCall(VecExp(user->dwork));
  PetscCall(VecCopy(user->dwork, user->d));

  PetscCall(VecCopy(XX, XXwork));
  PetscCall(VecCopy(YY, YYwork));

  PetscCall(VecShift(XXwork, -0.75));
  PetscCall(VecShift(YYwork, -0.75));

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

  PetscCall(VecCopy(XXwork, user->dwork));
  PetscCall(VecAXPY(user->dwork, 1.0, YYwork));
  PetscCall(VecScale(user->dwork, -30.0));
  PetscCall(VecExp(user->dwork));

  PetscCall(VecAXPY(user->d, 1.0, user->dwork));
  PetscCall(VecShift(user->d, 1.0));
  PetscCall(VecSum(user->d, &sum));
  PetscCall(VecScale(user->d, 1.0 / (h * h * sum)));

  /* Initial conditions of forward problem */
  PetscCall(VecDuplicate(XX, &bc));
  PetscCall(VecCopy(XX, XXwork));
  PetscCall(VecCopy(YY, YYwork));

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

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

  PetscCall(VecWAXPY(bc, 1.0, XXwork, YYwork));
  PetscCall(VecScale(bc, -50.0));
  PetscCall(VecExp(bc));
  PetscCall(VecShift(bc, 1.0));
  PetscCall(VecSum(bc, &sum));
  PetscCall(VecScale(bc, 1.0 / (h * h * sum)));

  /* Create scatter from y to y_1,y_2,...,y_nt */
  /*  TODO: Reorder for better parallelism. (This will require reordering Q and L as well.) */
  PetscCall(PetscMalloc1(user->nt * user->mx * user->mx, &user->yi_scatter));
  PetscCall(VecCreate(PETSC_COMM_WORLD, &yi));
  PetscCall(VecSetSizes(yi, PETSC_DECIDE, user->mx * user->mx));
  PetscCall(VecSetFromOptions(yi));
  PetscCall(VecDuplicateVecs(yi, user->nt, &user->yi));
  PetscCall(VecDuplicateVecs(yi, user->nt, &user->yiwork));
  PetscCall(VecDuplicateVecs(yi, user->nt, &user->ziwork));
  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, lo + i * user->mx * user->mx, 1, &is_from_y));
    PetscCall(VecScatterCreate(user->y, is_from_y, user->yi[i], is_to_yi, &user->yi_scatter[i]));
    PetscCall(ISDestroy(&is_to_yi));
    PetscCall(ISDestroy(&is_from_y));
  }

  /* Create scatter from u to ux_1,uy_1,ux_2,uy_2,...,ux_nt,uy_nt */
  /*  TODO: reorder for better parallelism */
  PetscCall(PetscMalloc1(user->nt * user->mx * user->mx, &user->uxi_scatter));
  PetscCall(PetscMalloc1(user->nt * user->mx * user->mx, &user->uyi_scatter));
  PetscCall(PetscMalloc1(user->nt * user->mx * user->mx, &user->ux_scatter));
  PetscCall(PetscMalloc1(user->nt * user->mx * user->mx, &user->uy_scatter));
  PetscCall(PetscMalloc1(2 * user->nt * user->mx * user->mx, &user->ui_scatter));
  PetscCall(VecCreate(PETSC_COMM_WORLD, &uxi));
  PetscCall(VecCreate(PETSC_COMM_WORLD, &ui));
  PetscCall(VecSetSizes(uxi, PETSC_DECIDE, user->mx * user->mx));
  PetscCall(VecSetSizes(ui, PETSC_DECIDE, 2 * user->mx * user->mx));
  PetscCall(VecSetFromOptions(uxi));
  PetscCall(VecSetFromOptions(ui));
  PetscCall(VecDuplicateVecs(uxi, user->nt, &user->uxi));
  PetscCall(VecDuplicateVecs(uxi, user->nt, &user->uyi));
  PetscCall(VecDuplicateVecs(uxi, user->nt, &user->uxiwork));
  PetscCall(VecDuplicateVecs(uxi, user->nt, &user->uyiwork));
  PetscCall(VecDuplicateVecs(ui, user->nt, &user->ui));
  PetscCall(VecDuplicateVecs(ui, user->nt, &user->uiwork));
  for (i = 0; i < user->nt; i++) {
    PetscCall(VecGetOwnershipRange(user->uxi[i], &lo, &hi));
    PetscCall(ISCreateStride(PETSC_COMM_SELF, hi - lo, lo, 1, &is_to_uxi));
    PetscCall(ISCreateStride(PETSC_COMM_SELF, hi - lo, lo + 2 * i * user->mx * user->mx, 1, &is_from_u));
    PetscCall(VecScatterCreate(user->u, is_from_u, user->uxi[i], is_to_uxi, &user->uxi_scatter[i]));

    PetscCall(ISDestroy(&is_to_uxi));
    PetscCall(ISDestroy(&is_from_u));

    PetscCall(VecGetOwnershipRange(user->uyi[i], &lo, &hi));
    PetscCall(ISCreateStride(PETSC_COMM_SELF, hi - lo, lo, 1, &is_to_uyi));
    PetscCall(ISCreateStride(PETSC_COMM_SELF, hi - lo, lo + (2 * i + 1) * user->mx * user->mx, 1, &is_from_u));
    PetscCall(VecScatterCreate(user->u, is_from_u, user->uyi[i], is_to_uyi, &user->uyi_scatter[i]));

    PetscCall(ISDestroy(&is_to_uyi));
    PetscCall(ISDestroy(&is_from_u));

    PetscCall(VecGetOwnershipRange(user->uxi[i], &lo, &hi));
    PetscCall(ISCreateStride(PETSC_COMM_SELF, hi - lo, lo, 1, &is_to_uxi));
    PetscCall(ISCreateStride(PETSC_COMM_SELF, hi - lo, lo, 1, &is_from_u));
    PetscCall(VecScatterCreate(user->ui[i], is_from_u, user->uxi[i], is_to_uxi, &user->ux_scatter[i]));

    PetscCall(ISDestroy(&is_to_uxi));
    PetscCall(ISDestroy(&is_from_u));

    PetscCall(VecGetOwnershipRange(user->uyi[i], &lo, &hi));
    PetscCall(ISCreateStride(PETSC_COMM_SELF, hi - lo, lo, 1, &is_to_uyi));
    PetscCall(ISCreateStride(PETSC_COMM_SELF, hi - lo, lo + user->mx * user->mx, 1, &is_from_u));
    PetscCall(VecScatterCreate(user->ui[i], is_from_u, user->uyi[i], is_to_uyi, &user->uy_scatter[i]));

    PetscCall(ISDestroy(&is_to_uyi));
    PetscCall(ISDestroy(&is_from_u));

    PetscCall(VecGetOwnershipRange(user->ui[i], &lo, &hi));
    PetscCall(ISCreateStride(PETSC_COMM_SELF, hi - lo, lo, 1, &is_to_uxi));
    PetscCall(ISCreateStride(PETSC_COMM_SELF, hi - lo, lo + 2 * i * user->mx * user->mx, 1, &is_from_u));
    PetscCall(VecScatterCreate(user->u, is_from_u, user->ui[i], is_to_uxi, &user->ui_scatter[i]));

    PetscCall(ISDestroy(&is_to_uxi));
    PetscCall(ISDestroy(&is_from_u));
  }

  /* RHS of forward problem */
  PetscCall(MatMult(user->M, bc, user->yiwork[0]));
  for (i = 1; i < user->nt; i++) PetscCall(VecSet(user->yiwork[i], 0.0));
  PetscCall(Gather_yi(user->q, user->yiwork, user->yi_scatter, user->nt));

  /* Compute true velocity field utrue */
  PetscCall(VecDuplicate(user->u, &user->utrue));
  for (i = 0; i < user->nt; i++) {
    PetscCall(VecCopy(YY, user->uxi[i]));
    PetscCall(VecScale(user->uxi[i], 150.0 * i * user->ht));
    PetscCall(VecCopy(XX, user->uyi[i]));
    PetscCall(VecShift(user->uyi[i], -10.0));
    PetscCall(VecScale(user->uyi[i], 15.0 * i * user->ht));
  }
  PetscCall(Gather_uxi_uyi(user->utrue, user->uxi, user->uxi_scatter, user->uyi, user->uyi_scatter, user->nt));

  /* Initial guess and reference model */
  PetscCall(VecDuplicate(user->utrue, &user->ur));
  for (i = 0; i < user->nt; i++) {
    PetscCall(VecCopy(XX, user->uxi[i]));
    PetscCall(VecShift(user->uxi[i], i * user->ht));
    PetscCall(VecCopy(YY, user->uyi[i]));
    PetscCall(VecShift(user->uyi[i], -i * user->ht));
  }
  PetscCall(Gather_uxi_uyi(user->ur, user->uxi, user->uxi_scatter, user->uyi, user->uyi_scatter, user->nt));

  /* Generate regularization matrix L */
  PetscCall(MatCreate(PETSC_COMM_WORLD, &user->LT));
  PetscCall(MatSetSizes(user->LT, PETSC_DECIDE, PETSC_DECIDE, 2 * n * user->nt, n * user->nt));
  PetscCall(MatSetFromOptions(user->LT));
  PetscCall(MatMPIAIJSetPreallocation(user->LT, 1, NULL, 1, NULL));
  PetscCall(MatSeqAIJSetPreallocation(user->LT, 1, NULL));
  PetscCall(MatGetOwnershipRange(user->LT, &istart, &iend));

  for (i = istart; i < iend; i++) {
    iblock = (i + n) / (2 * n);
    j      = i - iblock * n;
    PetscCall(MatSetValues(user->LT, 1, &i, 1, &j, &user->gamma, INSERT_VALUES));
  }

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

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

  /* Build work vectors and matrices */
  PetscCall(VecCreate(PETSC_COMM_WORLD, &user->lwork));
  PetscCall(VecSetType(user->lwork, VECMPI));
  PetscCall(VecSetSizes(user->lwork, PETSC_DECIDE, user->m));
  PetscCall(VecSetFromOptions(user->lwork));

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

  PetscCall(VecDuplicate(user->y, &user->ywork));
  PetscCall(VecDuplicate(user->u, &user->uwork));
  PetscCall(VecDuplicate(user->u, &user->vwork));
  PetscCall(VecDuplicate(user->u, &user->js_diag));
  PetscCall(VecDuplicate(user->c, &user->cwork));

  /* Create matrix-free shell user->Js for computing A*x */
  PetscCall(MatCreateShell(PETSC_COMM_WORLD, PETSC_DETERMINE, PETSC_DETERMINE, user->m, user->m, 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, n, n, user, &user->JsBlock));
  PetscCall(MatShellSetOperation(user->JsBlock, MATOP_MULT, (void (*)(void))StateMatBlockMult));
  PetscCall(MatShellSetOperation(user->JsBlock, MATOP_MULT_TRANSPOSE, (void (*)(void))StateMatBlockMultTranspose));

  /* 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 SOR preconditioner for user->JsBlock. */
  PetscCall(MatCreateShell(PETSC_COMM_WORLD, PETSC_DETERMINE, PETSC_DETERMINE, n, n, user, &user->JsBlockPrec));
  PetscCall(MatShellSetOperation(user->JsBlockPrec, MATOP_MULT, (void (*)(void))StateMatBlockPrecMult));
  PetscCall(MatShellSetOperation(user->JsBlockPrec, MATOP_MULT_TRANSPOSE, (void (*)(void))StateMatBlockPrecMultTranspose));

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

  /* Build matrices for SOR preconditioner */
  PetscCall(Scatter_uxi_uyi(user->u, user->uxi, user->uxi_scatter, user->uyi, user->uyi_scatter, user->nt));
  PetscCall(PetscMalloc1(5 * n, &user->C));
  PetscCall(PetscMalloc1(2 * n, &user->Cwork));
  for (i = 0; i < user->nt; i++) {
    PetscCall(MatDuplicate(user->Divxy[0], MAT_COPY_VALUES, &user->C[i]));
    PetscCall(MatDuplicate(user->Divxy[1], MAT_COPY_VALUES, &user->Cwork[i]));

    PetscCall(MatDiagonalScale(user->C[i], NULL, user->uxi[i]));
    PetscCall(MatDiagonalScale(user->Cwork[i], NULL, user->uyi[i]));
    PetscCall(MatAXPY(user->C[i], 1.0, user->Cwork[i], DIFFERENT_NONZERO_PATTERN));
    PetscCall(MatScale(user->C[i], user->ht));
    PetscCall(MatShift(user->C[i], 1.0));
  }

  /* Solver options and tolerances */
  PetscCall(KSPCreate(PETSC_COMM_WORLD, &user->solver));
  PetscCall(KSPSetType(user->solver, KSPGMRES));
  PetscCall(KSPSetOperators(user->solver, user->JsBlock, user->JsBlockPrec));
  PetscCall(KSPSetTolerances(user->solver, 1e-4, 1e-20, 1e3, 500));
  /* PetscCall(KSPSetTolerances(user->solver,1e-8,1e-16,1e3,500)); */
  PetscCall(KSPGetPC(user->solver, &user->prec));
  PetscCall(PCSetType(user->prec, PCSHELL));

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

  /* 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));
  user->c_formed = PETSC_TRUE;
  PetscCall(VecCopy(user->utrue, user->u)); /*  Set u=utrue temporarily for StateMatInv */
  PetscCall(VecSet(user->ytrue, 0.0));      /*  Initial guess */
  PetscCall(StateMatInvMult(user->Js, user->q, user->ytrue));
  PetscCall(VecCopy(user->ur, user->u)); /*  Reset u=ur */

  /* Initial guess y0 for state given u0 */
  PetscCall(StateMatInvMult(user->Js, user->q, user->y));

  /* Data discretization */
  PetscCall(MatCreate(PETSC_COMM_WORLD, &user->Q));
  PetscCall(MatSetSizes(user->Q, PETSC_DECIDE, PETSC_DECIDE, user->mx * user->mx, user->m));
  PetscCall(MatSetFromOptions(user->Q));
  PetscCall(MatMPIAIJSetPreallocation(user->Q, 0, NULL, 1, NULL));
  PetscCall(MatSeqAIJSetPreallocation(user->Q, 1, NULL));

  PetscCall(MatGetOwnershipRange(user->Q, &istart, &iend));

  for (i = istart; i < iend; i++) {
    j = i + user->m - user->mx * user->mx;
    PetscCall(MatSetValues(user->Q, 1, &i, 1, &j, &one, INSERT_VALUES));
  }

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

  PetscCall(MatTranspose(user->Q, MAT_INITIAL_MATRIX, &user->QT));

  PetscCall(VecDestroy(&XX));
  PetscCall(VecDestroy(&YY));
  PetscCall(VecDestroy(&XXwork));
  PetscCall(VecDestroy(&YYwork));
  PetscCall(VecDestroy(&yi));
  PetscCall(VecDestroy(&uxi));
  PetscCall(VecDestroy(&ui));
  PetscCall(VecDestroy(&bc));

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

PetscErrorCode HyperbolicDestroy(AppCtx *user) {
  PetscInt i;

  PetscFunctionBegin;
  PetscCall(MatDestroy(&user->Q));
  PetscCall(MatDestroy(&user->QT));
  PetscCall(MatDestroy(&user->Div));
  PetscCall(MatDestroy(&user->Divwork));
  PetscCall(MatDestroy(&user->Grad));
  PetscCall(MatDestroy(&user->L));
  PetscCall(MatDestroy(&user->LT));
  PetscCall(KSPDestroy(&user->solver));
  PetscCall(MatDestroy(&user->Js));
  PetscCall(MatDestroy(&user->Jd));
  PetscCall(MatDestroy(&user->JsBlockPrec));
  PetscCall(MatDestroy(&user->JsInv));
  PetscCall(MatDestroy(&user->JsBlock));
  PetscCall(MatDestroy(&user->Divxy[0]));
  PetscCall(MatDestroy(&user->Divxy[1]));
  PetscCall(MatDestroy(&user->Gradxy[0]));
  PetscCall(MatDestroy(&user->Gradxy[1]));
  PetscCall(MatDestroy(&user->M));
  for (i = 0; i < user->nt; i++) {
    PetscCall(MatDestroy(&user->C[i]));
    PetscCall(MatDestroy(&user->Cwork[i]));
  }
  PetscCall(PetscFree(user->C));
  PetscCall(PetscFree(user->Cwork));
  PetscCall(VecDestroy(&user->u));
  PetscCall(VecDestroy(&user->uwork));
  PetscCall(VecDestroy(&user->vwork));
  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->nt, &user->ziwork));
  PetscCall(VecDestroyVecs(user->nt, &user->uxi));
  PetscCall(VecDestroyVecs(user->nt, &user->uyi));
  PetscCall(VecDestroyVecs(user->nt, &user->uxiwork));
  PetscCall(VecDestroyVecs(user->nt, &user->uyiwork));
  PetscCall(VecDestroyVecs(user->nt, &user->ui));
  PetscCall(VecDestroyVecs(user->nt, &user->uiwork));
  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->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->uxi_scatter[i]));
    PetscCall(VecScatterDestroy(&user->uyi_scatter[i]));
    PetscCall(VecScatterDestroy(&user->ux_scatter[i]));
    PetscCall(VecScatterDestroy(&user->uy_scatter[i]));
    PetscCall(VecScatterDestroy(&user->ui_scatter[i]));
    PetscCall(VecScatterDestroy(&user->yi_scatter[i]));
  }
  PetscCall(PetscFree(user->uxi_scatter));
  PetscCall(PetscFree(user->uyi_scatter));
  PetscCall(PetscFree(user->ux_scatter));
  PetscCall(PetscFree(user->uy_scatter));
  PetscCall(PetscFree(user->ui_scatter));
  PetscCall(PetscFree(user->yi_scatter));
  PetscFunctionReturn(0);
}

PetscErrorCode HyperbolicMonitor(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:
      requires: !single
      args: -tao_cmonitor -tao_max_funcs 10 -tao_type lcl -tao_gatol 1.e-5

   test:
      suffix: guess_pod
      requires: !single
      args: -tao_cmonitor -tao_max_funcs 10 -tao_type lcl -ksp_guess_type pod -tao_gatol 1.e-5

TEST*/