Date: 2022-08-02T03:15:17+00:00

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";

We are solving the parameter estimation problem for the Laplacian. We will ask to minimize a Lagrangian
function over $a$ and $u$, given by
\begin{align}
L(u, a, \lambda) = \frac{1}{2} || Qu - d ||^2 + \frac{1}{2} || L (a - a_r) ||^2 + \lambda F(u; a)
\end{align}
where $Q$ is a sampling operator, $L$ is a regularization operator, $F$ defines the PDE.
Currently, we have perfect information, meaning $Q = I$, and then we need no regularization, $L = I$. We
also give the exact control for the reference $a_r$.
The PDE will be the Laplace equation with homogeneous boundary conditions
\begin{align}
-nabla \cdot a \nabla u = f
\end{align}

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

     25: typedef enum {RUN_FULL, RUN_TEST} RunType;

     27: typedef struct {
     28:   RunType runType;  /* Whether to run tests, or solve the full problem */
     29: } AppCtx;

     31: static PetscErrorCode ProcessOptions(MPI_Comm comm, AppCtx *options)
     32: {
     33:   const char    *runTypes[2] = {"full", "test"};
     34:   PetscInt       run;

     38:   options->runType = RUN_FULL;

     40:   PetscOptionsBegin(comm, "", "Inverse Problem Options", "DMPLEX");
     41:   run  = options->runType;
     42:   PetscOptionsEList("-run_type", "The run type", "ex1.c", runTypes, 2, runTypes[options->runType], &run, NULL);
     43:   options->runType = (RunType) run;
     44:   PetscOptionsEnd();
     45:   return 0;
     46: }

     48: static PetscErrorCode CreateMesh(MPI_Comm comm, AppCtx *user, DM *dm)
     49: {
     51:   DMCreate(comm, dm);
     52:   DMSetType(*dm, DMPLEX);
     53:   DMSetFromOptions(*dm);
     54:   DMViewFromOptions(*dm, NULL, "-dm_view");
     55:   return 0;
     56: }

     58: /* u - (x^2 + y^2) */
     59: void f0_u(PetscInt dim, PetscInt Nf, PetscInt NfAux,
     60:           const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
     61:           const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
     62:           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: /* a \nabla\lambda */
     67: void f1_u(PetscInt dim, PetscInt Nf, PetscInt NfAux,
     68:           const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
     69:           const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
     70:           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[1]*u_x[dim*2+d];
     74: }
     75: /* I */
     76: void g0_uu(PetscInt dim, PetscInt Nf, PetscInt NfAux,
     77:            const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
     78:            const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
     79:            PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g0[])
     80: {
     81:   g0[0] = 1.0;
     82: }
     83: /* \nabla */
     84: void g2_ua(PetscInt dim, PetscInt Nf, PetscInt NfAux,
     85:            const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
     86:            const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
     87:            PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g2[])
     88: {
     89:   PetscInt d;
     90:   for (d = 0; d < dim; ++d) g2[d] = u_x[dim*2+d];
     91: }
     92: /* a */
     93: void g3_ul(PetscInt dim, PetscInt Nf, PetscInt NfAux,
     94:            const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
     95:            const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
     96:            PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g3[])
     97: {
     98:   PetscInt d;
     99:   for (d = 0; d < dim; ++d) g3[d*dim+d] = u[1];
    100: }
    101: /* a - (x + y) */
    102: void f0_a(PetscInt dim, PetscInt Nf, PetscInt NfAux,
    103:           const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
    104:           const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
    105:           PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
    106: {
    107:   f0[0] = u[1] - (x[0] + x[1]);
    108: }
    109: /* \lambda \nabla u */
    110: void f1_a(PetscInt dim, PetscInt Nf, PetscInt NfAux,
    111:           const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
    112:           const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
    113:           PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f1[])
    114: {
    115:   PetscInt d;
    116:   for (d = 0; d < dim; ++d) f1[d] = u[2]*u_x[d];
    117: }
    118: /* I */
    119: void g0_aa(PetscInt dim, PetscInt Nf, PetscInt NfAux,
    120:            const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
    121:            const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
    122:            PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g0[])
    123: {
    124:   g0[0] = 1.0;
    125: }
    126: /* 6 (x + y) */
    127: void f0_l(PetscInt dim, PetscInt Nf, PetscInt NfAux,
    128:           const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
    129:           const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
    130:           PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
    131: {
    132:   f0[0] = 6.0*(x[0] + x[1]);
    133: }
    134: /* a \nabla u */
    135: void f1_l(PetscInt dim, PetscInt Nf, PetscInt NfAux,
    136:           const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
    137:           const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
    138:           PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f1[])
    139: {
    140:   PetscInt d;
    141:   for (d = 0; d < dim; ++d) f1[d] = u[1]*u_x[d];
    142: }
    143: /* \nabla u */
    144: void g2_la(PetscInt dim, PetscInt Nf, PetscInt NfAux,
    145:            const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
    146:            const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
    147:            PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g2[])
    148: {
    149:   PetscInt d;
    150:   for (d = 0; d < dim; ++d) g2[d] = u_x[d];
    151: }
    152: /* a */
    153: void g3_lu(PetscInt dim, PetscInt Nf, PetscInt NfAux,
    154:            const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
    155:            const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
    156:            PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g3[])
    157: {
    158:   PetscInt d;
    159:   for (d = 0; d < dim; ++d) g3[d*dim+d] = u[1];
    160: }

    162: /*
    163:   In 2D for Dirichlet conditions with a variable coefficient, we use exact solution:

    165:     u  = x^2 + y^2
    166:     f  = 6 (x + y)
    167:     kappa(a) = a = (x + y)

    169:   so that

    171:     -\div \kappa(a) \grad u + f = -6 (x + y) + 6 (x + y) = 0
    172: */
    173: PetscErrorCode quadratic_u_2d(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar *u, void *ctx)
    174: {
    175:   *u = x[0]*x[0] + x[1]*x[1];
    176:   return 0;
    177: }
    178: PetscErrorCode linear_a_2d(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar *a, void *ctx)
    179: {
    180:   *a = x[0] + x[1];
    181:   return 0;
    182: }
    183: PetscErrorCode zero(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar *l, void *ctx)
    184: {
    185:   *l = 0.0;
    186:   return 0;
    187: }

    189: PetscErrorCode SetupProblem(DM dm, AppCtx *user)
    190: {
    191:   PetscDS        ds;
    192:   DMLabel        label;
    193:   const PetscInt id = 1;

    196:   DMGetDS(dm, &ds);
    197:   PetscDSSetResidual(ds, 0, f0_u, f1_u);
    198:   PetscDSSetResidual(ds, 1, f0_a, f1_a);
    199:   PetscDSSetResidual(ds, 2, f0_l, f1_l);
    200:   PetscDSSetJacobian(ds, 0, 0, g0_uu, NULL, NULL, NULL);
    201:   PetscDSSetJacobian(ds, 0, 1, NULL, NULL, g2_ua, NULL);
    202:   PetscDSSetJacobian(ds, 0, 2, NULL, NULL, NULL, g3_ul);
    203:   PetscDSSetJacobian(ds, 1, 1, g0_aa, NULL, NULL, NULL);
    204:   PetscDSSetJacobian(ds, 2, 1, NULL, NULL, g2_la, NULL);
    205:   PetscDSSetJacobian(ds, 2, 0, NULL, NULL, NULL, g3_lu);

    207:   PetscDSSetExactSolution(ds, 0, quadratic_u_2d, NULL);
    208:   PetscDSSetExactSolution(ds, 1, linear_a_2d, NULL);
    209:   PetscDSSetExactSolution(ds, 2, zero, NULL);
    210:   DMGetLabel(dm, "marker", &label);
    211:   DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 0, 0, NULL, (void (*)(void)) quadratic_u_2d, NULL, user, NULL);
    212:   DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 1, 0, NULL, (void (*)(void)) linear_a_2d, NULL, user, NULL);
    213:   DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 2, 0, NULL, (void (*)(void)) zero, NULL, user, NULL);
    214:   return 0;
    215: }

    217: PetscErrorCode SetupDiscretization(DM dm, AppCtx *user)
    218: {
    219:   DM              cdm = dm;
    220:   const PetscInt  dim = 2;
    221:   PetscFE         fe[3];
    222:   PetscInt        f;
    223:   MPI_Comm        comm;

    226:   /* Create finite element */
    227:   PetscObjectGetComm((PetscObject) dm, &comm);
    228:   PetscFECreateDefault(comm, dim, 1, PETSC_TRUE, "potential_", -1, &fe[0]);
    229:   PetscObjectSetName((PetscObject) fe[0], "potential");
    230:   PetscFECreateDefault(comm, dim, 1, PETSC_TRUE, "conductivity_", -1, &fe[1]);
    231:   PetscObjectSetName((PetscObject) fe[1], "conductivity");
    232:   PetscFECopyQuadrature(fe[0], fe[1]);
    233:   PetscFECreateDefault(comm, dim, 1, PETSC_TRUE, "multiplier_", -1, &fe[2]);
    234:   PetscObjectSetName((PetscObject) fe[2], "multiplier");
    235:   PetscFECopyQuadrature(fe[0], fe[2]);
    236:   /* Set discretization and boundary conditions for each mesh */
    237:   for (f = 0; f < 3; ++f) DMSetField(dm, f, NULL, (PetscObject) fe[f]);
    238:   DMCreateDS(dm);
    239:   SetupProblem(dm, user);
    240:   while (cdm) {
    241:     DMCopyDisc(dm, cdm);
    242:     DMGetCoarseDM(cdm, &cdm);
    243:   }
    244:   for (f = 0; f < 3; ++f) PetscFEDestroy(&fe[f]);
    245:   return 0;
    246: }

    248: int main(int argc, char **argv)
    249: {
    250:   DM             dm;
    251:   SNES           snes;
    252:   Vec            u, r;
    253:   AppCtx         user;

    255:   PetscInitialize(&argc, &argv, NULL,help);
    256:   ProcessOptions(PETSC_COMM_WORLD, &user);
    257:   SNESCreate(PETSC_COMM_WORLD, &snes);
    258:   CreateMesh(PETSC_COMM_WORLD, &user, &dm);
    259:   SNESSetDM(snes, dm);
    260:   SetupDiscretization(dm, &user);

    262:   DMCreateGlobalVector(dm, &u);
    263:   PetscObjectSetName((PetscObject) u, "solution");
    264:   VecDuplicate(u, &r);
    265:   DMPlexSetSNESLocalFEM(dm,&user,&user,&user);
    266:   SNESSetFromOptions(snes);

    268:   DMSNESCheckFromOptions(snes, u);
    269:   if (user.runType == RUN_FULL) {
    270:     PetscDS          ds;
    271:     PetscErrorCode (*exactFuncs[3])(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar *u, void *ctx);
    272:     PetscErrorCode (*initialGuess[3])(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar u[], void *ctx);
    273:     PetscReal        error;

    275:     DMGetDS(dm, &ds);
    276:     PetscDSGetExactSolution(ds, 0, &exactFuncs[0], NULL);
    277:     PetscDSGetExactSolution(ds, 1, &exactFuncs[1], NULL);
    278:     PetscDSGetExactSolution(ds, 2, &exactFuncs[2], NULL);
    279:     initialGuess[0] = zero;
    280:     initialGuess[1] = zero;
    281:     initialGuess[2] = zero;
    282:     DMProjectFunction(dm, 0.0, initialGuess, NULL, INSERT_VALUES, u);
    283:     VecViewFromOptions(u, NULL, "-initial_vec_view");
    284:     DMComputeL2Diff(dm, 0.0, exactFuncs, NULL, u, &error);
    285:     if (error < 1.0e-11) PetscPrintf(PETSC_COMM_WORLD, "Initial L_2 Error: < 1.0e-11\n");
    286:     else                 PetscPrintf(PETSC_COMM_WORLD, "Initial L_2 Error: %g\n", error);
    287:     SNESSolve(snes, NULL, u);
    288:     DMComputeL2Diff(dm, 0.0, exactFuncs, NULL, u, &error);
    289:     if (error < 1.0e-11) PetscPrintf(PETSC_COMM_WORLD, "Final L_2 Error: < 1.0e-11\n");
    290:     else                 PetscPrintf(PETSC_COMM_WORLD, "Final L_2 Error: %g\n", error);
    291:   }
    292:   VecViewFromOptions(u, NULL, "-sol_vec_view");

    294:   VecDestroy(&u);
    295:   VecDestroy(&r);
    296:   SNESDestroy(&snes);
    297:   DMDestroy(&dm);
    298:   PetscFinalize();
    299:   return 0;
    300: }

    302: /*TEST

    304:   build:
    305:     requires: !complex

    307:   test:
    308:     suffix: 0
    309:     requires: triangle
    310:     args: -run_type test -dmsnes_check -potential_petscspace_degree 2 -conductivity_petscspace_degree 1 -multiplier_petscspace_degree 2

    312:   test:
    313:     suffix: 1
    314:     requires: triangle
    315:     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

    317: TEST*/