Date: 2022-03-31T04:49:08+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";

We are solving the parameter estimation problem for the Laplacian. We will ask to minimize a Lagrangian
function over $y$ and $u$, given by
\begin{align}
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)
\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 null vector for the reference control $a_r$. Right now $\beta = 1$.
The PDE will be the Laplace equation with homogeneous boundary conditions
\begin{align}
-Delta u = a
\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:   PetscBool useDualPenalty; /* Penalize deviation from both goals */
     30: } AppCtx;

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

     39:   options->runType        = RUN_FULL;
     40:   options->useDualPenalty = PETSC_FALSE;

     42:   PetscOptionsBegin(comm, "", "Inverse Problem Options", "DMPLEX");
     43:   run  = options->runType;
     44:   PetscOptionsEList("-run_type", "The run type", "ex2.c", runTypes, 2, runTypes[options->runType], &run, NULL);
     45:   options->runType = (RunType) run;
     46:   PetscOptionsBool("-use_dual_penalty", "Penalize deviation from both goals", "ex2.c", options->useDualPenalty, &options->useDualPenalty, NULL);
     47:   PetscOptionsEnd();
     48:   return 0;
     49: }

     51: static PetscErrorCode CreateMesh(MPI_Comm comm, AppCtx *user, DM *dm)
     52: {
     54:   DMCreate(comm, dm);
     55:   DMSetType(*dm, DMPLEX);
     56:   DMSetFromOptions(*dm);
     57:   DMViewFromOptions(*dm, NULL, "-dm_view");
     58:   return 0;
     59: }

     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: void f0_u_full(PetscInt dim, PetscInt Nf, PetscInt NfAux,
     69:           const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
     70:           const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
     71:           PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
     72: {
     73:   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]))) +
     74:     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])));
     75: }
     76: void f1_u(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, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f1[])
     80: {
     81:   PetscInt d;
     82:   for (d = 0; d < dim; ++d) f1[d] = u_x[dim*2+d];
     83: }
     84: void g0_uu(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 g0[])
     88: {
     89:   g0[0] = 1.0;
     90: }
     91: void g0_uu_full(PetscInt dim, PetscInt Nf, PetscInt NfAux,
     92:                 const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
     93:                 const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
     94:                 PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g0[])
     95: {
     96:   g0[0] = PetscSqr(u[0] - sin(2.0*PETSC_PI*x[0]) * sin(2.0*PETSC_PI*x[1]))
     97:     + PetscSqr(u[0] - (x[0]*x[0] + x[1]*x[1]))
     98:     - 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]
     99:     + 4.0*(x[0]*x[0] + x[1]*x[1])*(sin(2.0*PETSC_PI*x[0]) * sin(2.0*PETSC_PI*x[1]));
    100: }
    101: void g3_ul(PetscInt dim, PetscInt Nf, PetscInt NfAux,
    102:            const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
    103:            const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
    104:            PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g3[])
    105: {
    106:   PetscInt d;
    107:   for (d = 0; d < dim; ++d) g3[d*dim+d] = 1.0;
    108: }

    110: void f0_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 f0[])
    114: {
    115:   f0[0] = u[1] - 4.0 /* 0.0 */ + u[2];
    116: }
    117: void g0_aa(PetscInt dim, PetscInt Nf, PetscInt NfAux,
    118:            const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
    119:            const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
    120:            PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g0[])
    121: {
    122:   g0[0] = 1.0;
    123: }
    124: void g0_al(PetscInt dim, PetscInt Nf, PetscInt NfAux,
    125:            const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
    126:            const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
    127:            PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g0[])
    128: {
    129:   g0[0] = 1.0;
    130: }

    132: void f0_l(PetscInt dim, PetscInt Nf, PetscInt NfAux,
    133:           const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
    134:           const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
    135:           PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
    136: {
    137:   f0[0] = u[1];
    138: }
    139: void f1_l(PetscInt dim, PetscInt Nf, PetscInt NfAux,
    140:           const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
    141:           const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
    142:           PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f1[])
    143: {
    144:   PetscInt d;
    145:   for (d = 0; d < dim; ++d) f1[d] = u_x[d];
    146: }
    147: void g0_la(PetscInt dim, PetscInt Nf, PetscInt NfAux,
    148:            const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
    149:            const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
    150:            PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g0[])
    151: {
    152:   g0[0] = 1.0;
    153: }
    154: void g3_lu(PetscInt dim, PetscInt Nf, PetscInt NfAux,
    155:            const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
    156:            const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
    157:            PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g3[])
    158: {
    159:   PetscInt d;
    160:   for (d = 0; d < dim; ++d) g3[d*dim+d] = 1.0;
    161: }

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

    166:     u   = x^2 + y^2
    167:     a   = 4
    168:     d_A = 4
    169:     d_B = sin(2*pi*x[0]) * sin(2*pi*x[1])

    171:   so that

    173:     -\Delta u + a = -4 + 4 = 0
    174: */
    175: PetscErrorCode quadratic_u_2d(PetscInt dim, PetscReal time, 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 constant_a_2d(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar *a, void *ctx)
    181: {
    182:   *a = 4;
    183:   return 0;
    184: }
    185: PetscErrorCode zero(PetscInt dim, PetscReal time, 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;

    198:   DMGetDS(dm, &ds);
    199:   PetscDSSetResidual(ds, 0, user->useDualPenalty == PETSC_TRUE ? f0_u_full : f0_u, f1_u);
    200:   PetscDSSetResidual(ds, 1, f0_a, NULL);
    201:   PetscDSSetResidual(ds, 2, f0_l, f1_l);
    202:   PetscDSSetJacobian(ds, 0, 0, user->useDualPenalty == PETSC_TRUE ? g0_uu_full : g0_uu, NULL, NULL, NULL);
    203:   PetscDSSetJacobian(ds, 0, 2, NULL, NULL, NULL, g3_ul);
    204:   PetscDSSetJacobian(ds, 1, 1, g0_aa, NULL, NULL, NULL);
    205:   PetscDSSetJacobian(ds, 1, 2, g0_al, NULL, NULL, NULL);
    206:   PetscDSSetJacobian(ds, 2, 1, g0_la, NULL, NULL, NULL);
    207:   PetscDSSetJacobian(ds, 2, 0, NULL, NULL, NULL, g3_lu);

    209:   PetscDSSetExactSolution(ds, 0, quadratic_u_2d, NULL);
    210:   PetscDSSetExactSolution(ds, 1, constant_a_2d, NULL);
    211:   PetscDSSetExactSolution(ds, 2, zero, NULL);
    212:   DMGetLabel(dm, "marker", &label);
    213:   DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 0, 0, NULL, (void (*)()) quadratic_u_2d, NULL, user, NULL);
    214:   DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 1, 0, NULL, (void (*)()) constant_a_2d, NULL, user, NULL);
    215:   DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 2, 0, NULL, (void (*)()) zero, NULL, user, NULL);
    216:   return 0;
    217: }

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

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

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

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

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

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

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

    296:   VecDestroy(&u);
    297:   VecDestroy(&r);
    298:   SNESDestroy(&snes);
    299:   DMDestroy(&dm);
    300:   PetscFinalize();
    301:   return 0;
    302: }

    304: /*TEST

    306:   build:
    307:     requires: !complex triangle

    309:   test:
    310:     suffix: 0
    311:     args: -run_type test -dmsnes_check -potential_petscspace_degree 2 -charge_petscspace_degree 1 -multiplier_petscspace_degree 1

    313:   test:
    314:     suffix: 1
    315:     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

    317:   test:
    318:     suffix: 2
    319:     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

    321: TEST*/