Date: 2022-03-04T17:39:58+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: {

     53:   DMCreate(comm, dm);
     54:   DMSetType(*dm, DMPLEX);
     55:   DMSetFromOptions(*dm);
     56:   DMViewFromOptions(*dm, NULL, "-dm_view");
     57:   return(0);
     58: }

     60: /* u - (x^2 + y^2) */
     61: void f0_u(PetscInt dim, PetscInt Nf, PetscInt NfAux,
     62:           const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
     63:           const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
     64:           PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
     65: {
     66:   f0[0] = u[0] - (x[0]*x[0] + x[1]*x[1]);
     67: }
     68: /* a \nabla\lambda */
     69: void f1_u(PetscInt dim, PetscInt Nf, PetscInt NfAux,
     70:           const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
     71:           const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
     72:           PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f1[])
     73: {
     74:   PetscInt d;
     75:   for (d = 0; d < dim; ++d) f1[d] = u[1]*u_x[dim*2+d];
     76: }
     77: /* I */
     78: void g0_uu(PetscInt dim, PetscInt Nf, PetscInt NfAux,
     79:            const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
     80:            const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
     81:            PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g0[])
     82: {
     83:   g0[0] = 1.0;
     84: }
     85: /* \nabla */
     86: void g2_ua(PetscInt dim, PetscInt Nf, PetscInt NfAux,
     87:            const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
     88:            const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
     89:            PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g2[])
     90: {
     91:   PetscInt d;
     92:   for (d = 0; d < dim; ++d) g2[d] = u_x[dim*2+d];
     93: }
     94: /* a */
     95: void g3_ul(PetscInt dim, PetscInt Nf, PetscInt NfAux,
     96:            const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
     97:            const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
     98:            PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g3[])
     99: {
    100:   PetscInt d;
    101:   for (d = 0; d < dim; ++d) g3[d*dim+d] = u[1];
    102: }
    103: /* a - (x + y) */
    104: void f0_a(PetscInt dim, PetscInt Nf, PetscInt NfAux,
    105:           const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
    106:           const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
    107:           PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
    108: {
    109:   f0[0] = u[1] - (x[0] + x[1]);
    110: }
    111: /* \lambda \nabla u */
    112: void f1_a(PetscInt dim, PetscInt Nf, PetscInt NfAux,
    113:           const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
    114:           const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
    115:           PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f1[])
    116: {
    117:   PetscInt d;
    118:   for (d = 0; d < dim; ++d) f1[d] = u[2]*u_x[d];
    119: }
    120: /* I */
    121: void g0_aa(PetscInt dim, PetscInt Nf, PetscInt NfAux,
    122:            const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
    123:            const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
    124:            PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g0[])
    125: {
    126:   g0[0] = 1.0;
    127: }
    128: /* 6 (x + y) */
    129: void f0_l(PetscInt dim, PetscInt Nf, PetscInt NfAux,
    130:           const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
    131:           const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
    132:           PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
    133: {
    134:   f0[0] = 6.0*(x[0] + x[1]);
    135: }
    136: /* a \nabla u */
    137: void f1_l(PetscInt dim, PetscInt Nf, PetscInt NfAux,
    138:           const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
    139:           const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
    140:           PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f1[])
    141: {
    142:   PetscInt d;
    143:   for (d = 0; d < dim; ++d) f1[d] = u[1]*u_x[d];
    144: }
    145: /* \nabla u */
    146: void g2_la(PetscInt dim, PetscInt Nf, PetscInt NfAux,
    147:            const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
    148:            const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
    149:            PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g2[])
    150: {
    151:   PetscInt d;
    152:   for (d = 0; d < dim; ++d) g2[d] = u_x[d];
    153: }
    154: /* a */
    155: void g3_lu(PetscInt dim, PetscInt Nf, PetscInt NfAux,
    156:            const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
    157:            const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
    158:            PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g3[])
    159: {
    160:   PetscInt d;
    161:   for (d = 0; d < dim; ++d) g3[d*dim+d] = u[1];
    162: }

    164: /*
    165:   In 2D for Dirichlet conditions with a variable coefficient, we use exact solution:

    167:     u  = x^2 + y^2
    168:     f  = 6 (x + y)
    169:     kappa(a) = a = (x + y)

    171:   so that

    173:     -\div \kappa(a) \grad u + f = -6 (x + y) + 6 (x + y) = 0
    174: */
    175: PetscErrorCode quadratic_u_2d(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar *u, void *ctx)
    176: {
    177:   *u = x[0]*x[0] + x[1]*x[1];
    178:   return 0;
    179: }
    180: PetscErrorCode linear_a_2d(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar *a, void *ctx)
    181: {
    182:   *a = x[0] + x[1];
    183:   return 0;
    184: }
    185: PetscErrorCode zero(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar *l, void *ctx)
    186: {
    187:   *l = 0.0;
    188:   return 0;
    189: }

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

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

    210:   PetscDSSetExactSolution(ds, 0, quadratic_u_2d, NULL);
    211:   PetscDSSetExactSolution(ds, 1, linear_a_2d, NULL);
    212:   PetscDSSetExactSolution(ds, 2, zero, NULL);
    213:   DMGetLabel(dm, "marker", &label);
    214:   DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 0, 0, NULL, (void (*)(void)) quadratic_u_2d, NULL, user, NULL);
    215:   DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 1, 0, NULL, (void (*)(void)) linear_a_2d, NULL, user, NULL);
    216:   DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 2, 0, NULL, (void (*)(void)) zero, NULL, user, NULL);
    217:   return(0);
    218: }

    220: PetscErrorCode SetupDiscretization(DM dm, AppCtx *user)
    221: {
    222:   DM              cdm = dm;
    223:   const PetscInt  dim = 2;
    224:   PetscFE         fe[3];
    225:   PetscInt        f;
    226:   MPI_Comm        comm;
    227:   PetscErrorCode  ierr;

    230:   /* Create finite element */
    231:   PetscObjectGetComm((PetscObject) dm, &comm);
    232:   PetscFECreateDefault(comm, dim, 1, PETSC_TRUE, "potential_", -1, &fe[0]);
    233:   PetscObjectSetName((PetscObject) fe[0], "potential");
    234:   PetscFECreateDefault(comm, dim, 1, PETSC_TRUE, "conductivity_", -1, &fe[1]);
    235:   PetscObjectSetName((PetscObject) fe[1], "conductivity");
    236:   PetscFECopyQuadrature(fe[0], fe[1]);
    237:   PetscFECreateDefault(comm, dim, 1, PETSC_TRUE, "multiplier_", -1, &fe[2]);
    238:   PetscObjectSetName((PetscObject) fe[2], "multiplier");
    239:   PetscFECopyQuadrature(fe[0], fe[2]);
    240:   /* Set discretization and boundary conditions for each mesh */
    241:   for (f = 0; f < 3; ++f) {DMSetField(dm, f, NULL, (PetscObject) fe[f]);}
    242:   DMCreateDS(dm);
    243:   SetupProblem(dm, user);
    244:   while (cdm) {
    245:     DMCopyDisc(dm, cdm);
    246:     DMGetCoarseDM(cdm, &cdm);
    247:   }
    248:   for (f = 0; f < 3; ++f) {PetscFEDestroy(&fe[f]);}
    249:   return(0);
    250: }

    252: int main(int argc, char **argv)
    253: {
    254:   DM             dm;
    255:   SNES           snes;
    256:   Vec            u, r;
    257:   AppCtx         user;

    260:   PetscInitialize(&argc, &argv, NULL,help);if (ierr) return ierr;
    261:   ProcessOptions(PETSC_COMM_WORLD, &user);
    262:   SNESCreate(PETSC_COMM_WORLD, &snes);
    263:   CreateMesh(PETSC_COMM_WORLD, &user, &dm);
    264:   SNESSetDM(snes, dm);
    265:   SetupDiscretization(dm, &user);

    267:   DMCreateGlobalVector(dm, &u);
    268:   PetscObjectSetName((PetscObject) u, "solution");
    269:   VecDuplicate(u, &r);
    270:   DMPlexSetSNESLocalFEM(dm,&user,&user,&user);
    271:   SNESSetFromOptions(snes);

    273:   DMSNESCheckFromOptions(snes, u);
    274:   if (user.runType == RUN_FULL) {
    275:     PetscDS          ds;
    276:     PetscErrorCode (*exactFuncs[3])(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar *u, void *ctx);
    277:     PetscErrorCode (*initialGuess[3])(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar u[], void *ctx);
    278:     PetscReal        error;

    280:     DMGetDS(dm, &ds);
    281:     PetscDSGetExactSolution(ds, 0, &exactFuncs[0], NULL);
    282:     PetscDSGetExactSolution(ds, 1, &exactFuncs[1], NULL);
    283:     PetscDSGetExactSolution(ds, 2, &exactFuncs[2], NULL);
    284:     initialGuess[0] = zero;
    285:     initialGuess[1] = zero;
    286:     initialGuess[2] = zero;
    287:     DMProjectFunction(dm, 0.0, initialGuess, NULL, INSERT_VALUES, u);
    288:     VecViewFromOptions(u, NULL, "-initial_vec_view");
    289:     DMComputeL2Diff(dm, 0.0, exactFuncs, NULL, u, &error);
    290:     if (error < 1.0e-11) {PetscPrintf(PETSC_COMM_WORLD, "Initial L_2 Error: < 1.0e-11\n");}
    291:     else                 {PetscPrintf(PETSC_COMM_WORLD, "Initial L_2 Error: %g\n", error);}
    292:     SNESSolve(snes, NULL, u);
    293:     DMComputeL2Diff(dm, 0.0, exactFuncs, NULL, u, &error);
    294:     if (error < 1.0e-11) {PetscPrintf(PETSC_COMM_WORLD, "Final L_2 Error: < 1.0e-11\n");}
    295:     else                 {PetscPrintf(PETSC_COMM_WORLD, "Final L_2 Error: %g\n", error);}
    296:   }
    297:   VecViewFromOptions(u, NULL, "-sol_vec_view");

    299:   VecDestroy(&u);
    300:   VecDestroy(&r);
    301:   SNESDestroy(&snes);
    302:   DMDestroy(&dm);
    303:   PetscFinalize();
    304:   return ierr;
    305: }

    307: /*TEST

    309:   build:
    310:     requires: !complex

    312:   test:
    313:     suffix: 0
    314:     requires: triangle
    315:     args: -run_type test -dmsnes_check -potential_petscspace_degree 2 -conductivity_petscspace_degree 1 -multiplier_petscspace_degree 2

    317:   test:
    318:     suffix: 1
    319:     requires: triangle
    320:     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

    322: TEST*/