Actual source code: ex1.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 $a$ and $u$, given by
      7: \begin{align}
      8:   L(u, a, \lambda) = \frac{1}{2} || Qu - d ||^2 + \frac{1}{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 exact control for the reference $a_r$.

     15: The PDE will be the Laplace equation with homogeneous boundary conditions
     16: \begin{align}
     17:   -nabla \cdot a \nabla u = f
     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: } AppCtx;

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

     39:   PetscFunctionBeginUser;
     40:   options->runType = RUN_FULL;
     41:   PetscOptionsBegin(comm, "", "Inverse Problem Options", "DMPLEX");
     42:   run = options->runType;
     43:   PetscCall(PetscOptionsEList("-run_type", "The run type", "ex1.c", runTypes, 2, runTypes[options->runType], &run, NULL));
     44:   options->runType = (RunType)run;
     45:   PetscOptionsEnd();
     46:   PetscFunctionReturn(PETSC_SUCCESS);
     47: }

     49: static PetscErrorCode CreateMesh(MPI_Comm comm, AppCtx *user, DM *dm)
     50: {
     51:   PetscFunctionBeginUser;
     52:   PetscCall(DMCreate(comm, dm));
     53:   PetscCall(DMSetType(*dm, DMPLEX));
     54:   PetscCall(DMSetFromOptions(*dm));
     55:   PetscCall(DMViewFromOptions(*dm, NULL, "-dm_view"));
     56:   PetscFunctionReturn(PETSC_SUCCESS);
     57: }

     59: /* u - (x^2 + y^2) */
     60: 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[])
     61: {
     62:   f0[0] = u[0] - (x[0] * x[0] + x[1] * x[1]);
     63: }
     64: /* a \nabla\lambda */
     65: 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[])
     66: {
     67:   PetscInt d;
     68:   for (d = 0; d < dim; ++d) f1[d] = u[1] * u_x[dim * 2 + d];
     69: }
     70: /* I */
     71: 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[])
     72: {
     73:   g0[0] = 1.0;
     74: }
     75: /* \nabla */
     76: void g2_ua(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 g2[])
     77: {
     78:   PetscInt d;
     79:   for (d = 0; d < dim; ++d) g2[d] = u_x[dim * 2 + d];
     80: }
     81: /* a */
     82: 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[])
     83: {
     84:   PetscInt d;
     85:   for (d = 0; d < dim; ++d) g3[d * dim + d] = u[1];
     86: }
     87: /* a - (x + y) */
     88: 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[])
     89: {
     90:   f0[0] = u[1] - (x[0] + x[1]);
     91: }
     92: /* \lambda \nabla u */
     93: void f1_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 f1[])
     94: {
     95:   PetscInt d;
     96:   for (d = 0; d < dim; ++d) f1[d] = u[2] * u_x[d];
     97: }
     98: /* I */
     99: 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[])
    100: {
    101:   g0[0] = 1.0;
    102: }
    103: /* 6 (x + y) */
    104: 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[])
    105: {
    106:   f0[0] = 6.0 * (x[0] + x[1]);
    107: }
    108: /* a \nabla u */
    109: 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[])
    110: {
    111:   PetscInt d;
    112:   for (d = 0; d < dim; ++d) f1[d] = u[1] * u_x[d];
    113: }
    114: /* \nabla u */
    115: void g2_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 g2[])
    116: {
    117:   PetscInt d;
    118:   for (d = 0; d < dim; ++d) g2[d] = u_x[d];
    119: }
    120: /* a */
    121: 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[])
    122: {
    123:   PetscInt d;
    124:   for (d = 0; d < dim; ++d) g3[d * dim + d] = u[1];
    125: }

    127: /*
    128:   In 2D for Dirichlet conditions with a variable coefficient, we use exact solution:

    130:     u  = x^2 + y^2
    131:     f  = 6 (x + y)
    132:     kappa(a) = a = (x + y)

    134:   so that

    136:     -\div \kappa(a) \grad u + f = -6 (x + y) + 6 (x + y) = 0
    137: */
    138: PetscErrorCode quadratic_u_2d(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar *u, PetscCtx ctx)
    139: {
    140:   *u = x[0] * x[0] + x[1] * x[1];
    141:   return PETSC_SUCCESS;
    142: }
    143: PetscErrorCode linear_a_2d(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar *a, PetscCtx ctx)
    144: {
    145:   *a = x[0] + x[1];
    146:   return PETSC_SUCCESS;
    147: }
    148: PetscErrorCode zero(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar *l, PetscCtx ctx)
    149: {
    150:   *l = 0.0;
    151:   return PETSC_SUCCESS;
    152: }

    154: PetscErrorCode SetupProblem(DM dm, AppCtx *user)
    155: {
    156:   PetscDS        ds;
    157:   DMLabel        label;
    158:   const PetscInt id = 1;

    160:   PetscFunctionBeginUser;
    161:   PetscCall(DMGetDS(dm, &ds));
    162:   PetscCall(PetscDSSetResidual(ds, 0, f0_u, f1_u));
    163:   PetscCall(PetscDSSetResidual(ds, 1, f0_a, f1_a));
    164:   PetscCall(PetscDSSetResidual(ds, 2, f0_l, f1_l));
    165:   PetscCall(PetscDSSetJacobian(ds, 0, 0, g0_uu, NULL, NULL, NULL));
    166:   PetscCall(PetscDSSetJacobian(ds, 0, 1, NULL, NULL, g2_ua, NULL));
    167:   PetscCall(PetscDSSetJacobian(ds, 0, 2, NULL, NULL, NULL, g3_ul));
    168:   PetscCall(PetscDSSetJacobian(ds, 1, 1, g0_aa, NULL, NULL, NULL));
    169:   PetscCall(PetscDSSetJacobian(ds, 2, 1, NULL, NULL, g2_la, NULL));
    170:   PetscCall(PetscDSSetJacobian(ds, 2, 0, NULL, NULL, NULL, g3_lu));

    172:   PetscCall(PetscDSSetExactSolution(ds, 0, quadratic_u_2d, NULL));
    173:   PetscCall(PetscDSSetExactSolution(ds, 1, linear_a_2d, NULL));
    174:   PetscCall(PetscDSSetExactSolution(ds, 2, zero, NULL));
    175:   PetscCall(DMGetLabel(dm, "marker", &label));
    176:   PetscCall(DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 0, 0, NULL, (PetscVoidFn *)quadratic_u_2d, NULL, user, NULL));
    177:   PetscCall(DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 1, 0, NULL, (PetscVoidFn *)linear_a_2d, NULL, user, NULL));
    178:   PetscCall(DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 2, 0, NULL, (PetscVoidFn *)zero, NULL, user, NULL));
    179:   PetscFunctionReturn(PETSC_SUCCESS);
    180: }

    182: PetscErrorCode SetupDiscretization(DM dm, AppCtx *user)
    183: {
    184:   DM             cdm = dm;
    185:   const PetscInt dim = 2;
    186:   PetscFE        fe[3];
    187:   PetscInt       f;
    188:   MPI_Comm       comm;

    190:   PetscFunctionBeginUser;
    191:   /* Create finite element */
    192:   PetscCall(PetscObjectGetComm((PetscObject)dm, &comm));
    193:   PetscCall(PetscFECreateDefault(comm, dim, 1, PETSC_TRUE, "potential_", -1, &fe[0]));
    194:   PetscCall(PetscObjectSetName((PetscObject)fe[0], "potential"));
    195:   PetscCall(PetscFECreateDefault(comm, dim, 1, PETSC_TRUE, "conductivity_", -1, &fe[1]));
    196:   PetscCall(PetscObjectSetName((PetscObject)fe[1], "conductivity"));
    197:   PetscCall(PetscFECopyQuadrature(fe[0], fe[1]));
    198:   PetscCall(PetscFECreateDefault(comm, dim, 1, PETSC_TRUE, "multiplier_", -1, &fe[2]));
    199:   PetscCall(PetscObjectSetName((PetscObject)fe[2], "multiplier"));
    200:   PetscCall(PetscFECopyQuadrature(fe[0], fe[2]));
    201:   /* Set discretization and boundary conditions for each mesh */
    202:   for (f = 0; f < 3; ++f) PetscCall(DMSetField(dm, f, NULL, (PetscObject)fe[f]));
    203:   PetscCall(DMCreateDS(dm));
    204:   PetscCall(SetupProblem(dm, user));
    205:   while (cdm) {
    206:     PetscCall(DMCopyDisc(dm, cdm));
    207:     PetscCall(DMGetCoarseDM(cdm, &cdm));
    208:   }
    209:   for (f = 0; f < 3; ++f) PetscCall(PetscFEDestroy(&fe[f]));
    210:   PetscFunctionReturn(PETSC_SUCCESS);
    211: }

    213: int main(int argc, char **argv)
    214: {
    215:   DM     dm;
    216:   SNES   snes;
    217:   Vec    u, r;
    218:   AppCtx user;

    220:   PetscFunctionBeginUser;
    221:   PetscCall(PetscInitialize(&argc, &argv, NULL, help));
    222:   PetscCall(ProcessOptions(PETSC_COMM_WORLD, &user));
    223:   PetscCall(SNESCreate(PETSC_COMM_WORLD, &snes));
    224:   PetscCall(CreateMesh(PETSC_COMM_WORLD, &user, &dm));
    225:   PetscCall(SNESSetDM(snes, dm));
    226:   PetscCall(SetupDiscretization(dm, &user));

    228:   PetscCall(DMCreateGlobalVector(dm, &u));
    229:   PetscCall(PetscObjectSetName((PetscObject)u, "solution"));
    230:   PetscCall(VecDuplicate(u, &r));
    231:   PetscCall(DMPlexSetSNESLocalFEM(dm, PETSC_FALSE, &user));
    232:   PetscCall(SNESSetFromOptions(snes));

    234:   PetscCall(DMSNESCheckFromOptions(snes, u));
    235:   if (user.runType == RUN_FULL) {
    236:     PetscDS ds;
    237:     PetscErrorCode (*exactFuncs[3])(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar *u, PetscCtx ctx);
    238:     PetscErrorCode (*initialGuess[3])(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar u[], PetscCtx ctx);
    239:     PetscReal error;

    241:     PetscCall(DMGetDS(dm, &ds));
    242:     PetscCall(PetscDSGetExactSolution(ds, 0, &exactFuncs[0], NULL));
    243:     PetscCall(PetscDSGetExactSolution(ds, 1, &exactFuncs[1], NULL));
    244:     PetscCall(PetscDSGetExactSolution(ds, 2, &exactFuncs[2], NULL));
    245:     initialGuess[0] = zero;
    246:     initialGuess[1] = zero;
    247:     initialGuess[2] = zero;
    248:     PetscCall(DMProjectFunction(dm, 0.0, initialGuess, NULL, INSERT_VALUES, u));
    249:     PetscCall(VecViewFromOptions(u, NULL, "-initial_vec_view"));
    250:     PetscCall(DMComputeL2Diff(dm, 0.0, exactFuncs, NULL, u, &error));
    251:     if (error < 1.0e-11) PetscCall(PetscPrintf(PETSC_COMM_WORLD, "Initial L_2 Error: < 1.0e-11\n"));
    252:     else PetscCall(PetscPrintf(PETSC_COMM_WORLD, "Initial L_2 Error: %g\n", (double)error));
    253:     PetscCall(SNESSolve(snes, NULL, u));
    254:     PetscCall(DMComputeL2Diff(dm, 0.0, exactFuncs, NULL, u, &error));
    255:     if (error < 1.0e-11) PetscCall(PetscPrintf(PETSC_COMM_WORLD, "Final L_2 Error: < 1.0e-11\n"));
    256:     else PetscCall(PetscPrintf(PETSC_COMM_WORLD, "Final L_2 Error: %g\n", (double)error));
    257:   }
    258:   PetscCall(VecViewFromOptions(u, NULL, "-sol_vec_view"));

    260:   PetscCall(VecDestroy(&u));
    261:   PetscCall(VecDestroy(&r));
    262:   PetscCall(SNESDestroy(&snes));
    263:   PetscCall(DMDestroy(&dm));
    264:   PetscCall(PetscFinalize());
    265:   return 0;
    266: }

    268: /*TEST

    270:   build:
    271:     requires: !complex

    273:   test:
    274:     suffix: 0
    275:     requires: triangle
    276:     args: -run_type test -dmsnes_check -potential_petscspace_degree 2 -conductivity_petscspace_degree 1 -multiplier_petscspace_degree 2

    278:   test:
    279:     suffix: 1
    280:     requires: triangle
    281:     args: -potential_petscspace_degree 2 -conductivity_petscspace_degree 1 -multiplier_petscspace_degree 2 -snes_monitor -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 -fieldsplit_multiplier_ksp_rtol 1.0e-10 -fieldsplit_multiplier_pc_type lu -sol_vec_view

    283: TEST*/