Date: 2025-03-28T21:11:16+00:00

Actual source code: ex2.c

      1: static char help[] = "One-Shot Multigrid for Parameter Estimation Problem for the Poisson Equation.\n\
      2: Using the Interior Point Method.\n\n\n";

      4: /*F
      5:   We are solving the parameter estimation problem for the Laplacian. We will ask to minimize a Lagrangian
      6: function over $y$ and $u$, given by
      7: \begin{align}
      8:   L(u, a, \lambda) = \frac{1}{2} || Qu - d_A ||^2 || Qu - d_B ||^2 + \frac{\beta}{2} || L (a - a_r) ||^2 + \lambda F(u; a)
      9: \end{align}
     10: where $Q$ is a sampling operator, $L$ is a regularization operator, $F$ defines the PDE.

     12: Currently, we have perfect information, meaning $Q = I$, and then we need no regularization, $L = I$. We
     13: also give the null vector for the reference control $a_r$. Right now $\beta = 1$.

     15: The PDE will be the Laplace equation with homogeneous boundary conditions
     16: \begin{align}
     17:   -Delta u = a
     18: \end{align}

     20: F*/

     22: #include <petsc.h>
     23: #include <petscfe.h>

     25: typedef enum {
     26:   RUN_FULL,
     27:   RUN_TEST
     28: } RunType;

     30: typedef struct {
     31:   RunType   runType;        /* Whether to run tests, or solve the full problem */
     32:   PetscBool useDualPenalty; /* Penalize deviation from both goals */
     33: } AppCtx;

     35: static PetscErrorCode ProcessOptions(MPI_Comm comm, AppCtx *options)
     36: {
     37:   const char *runTypes[2] = {"full", "test"};
     38:   PetscInt    run;

     40:   PetscFunctionBeginUser;
     41:   options->runType        = RUN_FULL;
     42:   options->useDualPenalty = PETSC_FALSE;
     43:   PetscOptionsBegin(comm, "", "Inverse Problem Options", "DMPLEX");
     44:   run = options->runType;
     45:   PetscCall(PetscOptionsEList("-run_type", "The run type", "ex2.c", runTypes, 2, runTypes[options->runType], &run, NULL));
     46:   options->runType = (RunType)run;
     47:   PetscCall(PetscOptionsBool("-use_dual_penalty", "Penalize deviation from both goals", "ex2.c", options->useDualPenalty, &options->useDualPenalty, NULL));
     48:   PetscOptionsEnd();
     49:   PetscFunctionReturn(PETSC_SUCCESS);
     50: }

     52: static PetscErrorCode CreateMesh(MPI_Comm comm, AppCtx *user, DM *dm)
     53: {
     54:   PetscFunctionBeginUser;
     55:   PetscCall(DMCreate(comm, dm));
     56:   PetscCall(DMSetType(*dm, DMPLEX));
     57:   PetscCall(DMSetFromOptions(*dm));
     58:   PetscCall(DMViewFromOptions(*dm, NULL, "-dm_view"));
     59:   PetscFunctionReturn(PETSC_SUCCESS);
     60: }

     62: void f0_u(PetscInt dim, PetscInt Nf, PetscInt NfAux, const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[], const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[], PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
     63: {
     64:   f0[0] = (u[0] - (x[0] * x[0] + x[1] * x[1]));
     65: }
     66: void f0_u_full(PetscInt dim, PetscInt Nf, PetscInt NfAux, const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[], const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[], PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
     67: {
     68:   f0[0] = (u[0] - (x[0] * x[0] + x[1] * x[1])) * PetscSqr(u[0] - (sin(2.0 * PETSC_PI * x[0]) * sin(2.0 * PETSC_PI * x[1]))) + PetscSqr(u[0] - (x[0] * x[0] + x[1] * x[1])) * (u[0] - (sin(2.0 * PETSC_PI * x[0]) * sin(2.0 * PETSC_PI * x[1])));
     69: }
     70: void f1_u(PetscInt dim, PetscInt Nf, PetscInt NfAux, const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[], const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[], PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f1[])
     71: {
     72:   PetscInt d;
     73:   for (d = 0; d < dim; ++d) f1[d] = u_x[dim * 2 + d];
     74: }
     75: void g0_uu(PetscInt dim, PetscInt Nf, PetscInt NfAux, const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[], const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[], PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g0[])
     76: {
     77:   g0[0] = 1.0;
     78: }
     79: void g0_uu_full(PetscInt dim, PetscInt Nf, PetscInt NfAux, const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[], const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[], PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g0[])
     80: {
     81:   g0[0] = PetscSqr(u[0] - sin(2.0 * PETSC_PI * x[0]) * sin(2.0 * PETSC_PI * x[1])) + PetscSqr(u[0] - (x[0] * x[0] + x[1] * x[1])) - 2.0 * ((x[0] * x[0] + x[1] * x[1]) + (sin(2.0 * PETSC_PI * x[0]) * sin(2.0 * PETSC_PI * x[1]))) * u[0] + 4.0 * (x[0] * x[0] + x[1] * x[1]) * (sin(2.0 * PETSC_PI * x[0]) * sin(2.0 * PETSC_PI * x[1]));
     82: }
     83: void g3_ul(PetscInt dim, PetscInt Nf, PetscInt NfAux, const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[], const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[], PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g3[])
     84: {
     85:   PetscInt d;
     86:   for (d = 0; d < dim; ++d) g3[d * dim + d] = 1.0;
     87: }

     89: void f0_a(PetscInt dim, PetscInt Nf, PetscInt NfAux, const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[], const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[], PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
     90: {
     91:   f0[0] = u[1] - 4.0 /* 0.0 */ + u[2];
     92: }
     93: void g0_aa(PetscInt dim, PetscInt Nf, PetscInt NfAux, const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[], const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[], PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g0[])
     94: {
     95:   g0[0] = 1.0;
     96: }
     97: void g0_al(PetscInt dim, PetscInt Nf, PetscInt NfAux, const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[], const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[], PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g0[])
     98: {
     99:   g0[0] = 1.0;
    100: }

    102: void f0_l(PetscInt dim, PetscInt Nf, PetscInt NfAux, const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[], const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[], PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
    103: {
    104:   f0[0] = u[1];
    105: }
    106: void f1_l(PetscInt dim, PetscInt Nf, PetscInt NfAux, const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[], const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[], PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f1[])
    107: {
    108:   PetscInt d;
    109:   for (d = 0; d < dim; ++d) f1[d] = u_x[d];
    110: }
    111: void g0_la(PetscInt dim, PetscInt Nf, PetscInt NfAux, const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[], const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[], PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g0[])
    112: {
    113:   g0[0] = 1.0;
    114: }
    115: void g3_lu(PetscInt dim, PetscInt Nf, PetscInt NfAux, const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[], const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[], PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g3[])
    116: {
    117:   PetscInt d;
    118:   for (d = 0; d < dim; ++d) g3[d * dim + d] = 1.0;
    119: }

    121: /*
    122:   In 2D for Dirichlet conditions with a variable coefficient, we use exact solution:

    124:     u   = x^2 + y^2
    125:     a   = 4
    126:     d_A = 4
    127:     d_B = sin(2*pi*x[0]) * sin(2*pi*x[1])

    129:   so that

    131:     -\Delta u + a = -4 + 4 = 0
    132: */
    133: PetscErrorCode quadratic_u_2d(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar *u, void *ctx)
    134: {
    135:   *u = x[0] * x[0] + x[1] * x[1];
    136:   return PETSC_SUCCESS;
    137: }
    138: PetscErrorCode constant_a_2d(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar *a, void *ctx)
    139: {
    140:   *a = 4;
    141:   return PETSC_SUCCESS;
    142: }
    143: PetscErrorCode zero(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar *l, void *ctx)
    144: {
    145:   *l = 0.0;
    146:   return PETSC_SUCCESS;
    147: }

    149: PetscErrorCode SetupProblem(DM dm, AppCtx *user)
    150: {
    151:   PetscDS        ds;
    152:   DMLabel        label;
    153:   const PetscInt id = 1;

    155:   PetscFunctionBeginUser;
    156:   PetscCall(DMGetDS(dm, &ds));
    157:   PetscCall(PetscDSSetResidual(ds, 0, user->useDualPenalty == PETSC_TRUE ? f0_u_full : f0_u, f1_u));
    158:   PetscCall(PetscDSSetResidual(ds, 1, f0_a, NULL));
    159:   PetscCall(PetscDSSetResidual(ds, 2, f0_l, f1_l));
    160:   PetscCall(PetscDSSetJacobian(ds, 0, 0, user->useDualPenalty == PETSC_TRUE ? g0_uu_full : g0_uu, NULL, NULL, NULL));
    161:   PetscCall(PetscDSSetJacobian(ds, 0, 2, NULL, NULL, NULL, g3_ul));
    162:   PetscCall(PetscDSSetJacobian(ds, 1, 1, g0_aa, NULL, NULL, NULL));
    163:   PetscCall(PetscDSSetJacobian(ds, 1, 2, g0_al, NULL, NULL, NULL));
    164:   PetscCall(PetscDSSetJacobian(ds, 2, 1, g0_la, NULL, NULL, NULL));
    165:   PetscCall(PetscDSSetJacobian(ds, 2, 0, NULL, NULL, NULL, g3_lu));

    167:   PetscCall(PetscDSSetExactSolution(ds, 0, quadratic_u_2d, NULL));
    168:   PetscCall(PetscDSSetExactSolution(ds, 1, constant_a_2d, NULL));
    169:   PetscCall(PetscDSSetExactSolution(ds, 2, zero, NULL));
    170:   PetscCall(DMGetLabel(dm, "marker", &label));
    171:   PetscCall(DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 0, 0, NULL, (void (*)())quadratic_u_2d, NULL, user, NULL));
    172:   PetscCall(DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 1, 0, NULL, (void (*)())constant_a_2d, NULL, user, NULL));
    173:   PetscCall(DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 2, 0, NULL, (void (*)())zero, NULL, user, NULL));
    174:   PetscFunctionReturn(PETSC_SUCCESS);
    175: }

    177: PetscErrorCode SetupDiscretization(DM dm, AppCtx *user)
    178: {
    179:   DM             cdm = dm;
    180:   const PetscInt dim = 2;
    181:   PetscFE        fe[3];
    182:   PetscInt       f;
    183:   MPI_Comm       comm;

    185:   PetscFunctionBeginUser;
    186:   /* Create finite element */
    187:   PetscCall(PetscObjectGetComm((PetscObject)dm, &comm));
    188:   PetscCall(PetscFECreateDefault(comm, dim, 1, PETSC_TRUE, "potential_", -1, &fe[0]));
    189:   PetscCall(PetscObjectSetName((PetscObject)fe[0], "potential"));
    190:   PetscCall(PetscFECreateDefault(comm, dim, 1, PETSC_TRUE, "charge_", -1, &fe[1]));
    191:   PetscCall(PetscObjectSetName((PetscObject)fe[1], "charge"));
    192:   PetscCall(PetscFECopyQuadrature(fe[0], fe[1]));
    193:   PetscCall(PetscFECreateDefault(comm, dim, 1, PETSC_TRUE, "multiplier_", -1, &fe[2]));
    194:   PetscCall(PetscObjectSetName((PetscObject)fe[2], "multiplier"));
    195:   PetscCall(PetscFECopyQuadrature(fe[0], fe[2]));
    196:   /* Set discretization and boundary conditions for each mesh */
    197:   for (f = 0; f < 3; ++f) PetscCall(DMSetField(dm, f, NULL, (PetscObject)fe[f]));
    198:   PetscCall(DMCreateDS(cdm));
    199:   PetscCall(SetupProblem(dm, user));
    200:   while (cdm) {
    201:     PetscCall(DMCopyDisc(dm, cdm));
    202:     PetscCall(DMGetCoarseDM(cdm, &cdm));
    203:   }
    204:   for (f = 0; f < 3; ++f) PetscCall(PetscFEDestroy(&fe[f]));
    205:   PetscFunctionReturn(PETSC_SUCCESS);
    206: }

    208: int main(int argc, char **argv)
    209: {
    210:   DM     dm;
    211:   SNES   snes;
    212:   Vec    u, r;
    213:   AppCtx user;

    215:   PetscFunctionBeginUser;
    216:   PetscCall(PetscInitialize(&argc, &argv, NULL, help));
    217:   PetscCall(ProcessOptions(PETSC_COMM_WORLD, &user));
    218:   PetscCall(SNESCreate(PETSC_COMM_WORLD, &snes));
    219:   PetscCall(CreateMesh(PETSC_COMM_WORLD, &user, &dm));
    220:   PetscCall(SNESSetDM(snes, dm));
    221:   PetscCall(SetupDiscretization(dm, &user));

    223:   PetscCall(DMCreateGlobalVector(dm, &u));
    224:   PetscCall(PetscObjectSetName((PetscObject)u, "solution"));
    225:   PetscCall(VecDuplicate(u, &r));
    226:   PetscCall(DMPlexSetSNESLocalFEM(dm, PETSC_FALSE, &user));
    227:   PetscCall(SNESSetFromOptions(snes));

    229:   PetscCall(DMSNESCheckFromOptions(snes, u));
    230:   if (user.runType == RUN_FULL) {
    231:     PetscDS ds;
    232:     PetscErrorCode (*exactFuncs[3])(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar *u, void *ctx);
    233:     PetscErrorCode (*initialGuess[3])(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar u[], void *ctx);
    234:     PetscReal error;

    236:     PetscCall(DMGetDS(dm, &ds));
    237:     PetscCall(PetscDSGetExactSolution(ds, 0, &exactFuncs[0], NULL));
    238:     PetscCall(PetscDSGetExactSolution(ds, 1, &exactFuncs[1], NULL));
    239:     PetscCall(PetscDSGetExactSolution(ds, 2, &exactFuncs[2], NULL));
    240:     initialGuess[0] = zero;
    241:     initialGuess[1] = zero;
    242:     initialGuess[2] = zero;
    243:     PetscCall(DMProjectFunction(dm, 0.0, initialGuess, NULL, INSERT_VALUES, u));
    244:     PetscCall(VecViewFromOptions(u, NULL, "-initial_vec_view"));
    245:     PetscCall(DMComputeL2Diff(dm, 0.0, exactFuncs, NULL, u, &error));
    246:     if (error < 1.0e-11) PetscCall(PetscPrintf(PETSC_COMM_WORLD, "Initial L_2 Error: < 1.0e-11\n"));
    247:     else PetscCall(PetscPrintf(PETSC_COMM_WORLD, "Initial L_2 Error: %g\n", (double)error));
    248:     PetscCall(SNESSolve(snes, NULL, u));
    249:     PetscCall(DMComputeL2Diff(dm, 0.0, exactFuncs, NULL, u, &error));
    250:     if (error < 1.0e-11) PetscCall(PetscPrintf(PETSC_COMM_WORLD, "Final L_2 Error: < 1.0e-11\n"));
    251:     else PetscCall(PetscPrintf(PETSC_COMM_WORLD, "Final L_2 Error: %g\n", (double)error));
    252:   }
    253:   PetscCall(VecViewFromOptions(u, NULL, "-sol_vec_view"));

    255:   PetscCall(VecDestroy(&u));
    256:   PetscCall(VecDestroy(&r));
    257:   PetscCall(SNESDestroy(&snes));
    258:   PetscCall(DMDestroy(&dm));
    259:   PetscCall(PetscFinalize());
    260:   return 0;
    261: }

    263: /*TEST

    265:   build:
    266:     requires: !complex triangle

    268:   test:
    269:     suffix: 0
    270:     args: -run_type test -dmsnes_check -potential_petscspace_degree 2 -charge_petscspace_degree 1 -multiplier_petscspace_degree 1

    272:   test:
    273:     suffix: 1
    274:     args: -potential_petscspace_degree 2 -charge_petscspace_degree 1 -multiplier_petscspace_degree 1 -snes_monitor -snes_converged_reason -pc_type fieldsplit -pc_fieldsplit_0_fields 0,1 -pc_fieldsplit_1_fields 2 -pc_fieldsplit_type schur -pc_fieldsplit_schur_factorization_type full -pc_fieldsplit_schur_precondition selfp -fieldsplit_0_pc_type lu -sol_vec_view

    276:   test:
    277:     suffix: 2
    278:     args: -potential_petscspace_degree 2 -charge_petscspace_degree 1 -multiplier_petscspace_degree 1 -snes_monitor -snes_converged_reason -snes_fd -pc_type fieldsplit -pc_fieldsplit_0_fields 0,1 -pc_fieldsplit_1_fields 2 -pc_fieldsplit_type schur -pc_fieldsplit_schur_factorization_type full -pc_fieldsplit_schur_precondition selfp -fieldsplit_0_pc_type lu -sol_vec_view

    280: TEST*/