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: {
     53:   DM             distributedMesh = NULL;

     57:   DMPlexCreateBoxMesh(comm, 2, PETSC_TRUE, NULL, NULL, NULL, NULL, PETSC_TRUE, dm);
     58:   PetscObjectSetName((PetscObject) *dm, "Mesh");
     59:   DMPlexDistribute(*dm, 0, NULL, &distributedMesh);
     60:   if (distributedMesh) {
     61:     DMDestroy(dm);
     62:     *dm  = distributedMesh;
     63:   }
     64:   DMSetFromOptions(*dm);
     65:   DMViewFromOptions(*dm, NULL, "-dm_view");
     66:   return(0);
     67: }

     69: void f0_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 f0[])
     73: {
     74:   f0[0] = (u[0] - (x[0]*x[0] + x[1]*x[1]));
     75: }
     76: void f0_u_full(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 f0[])
     80: {
     81:   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]))) +
     82:     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])));
     83: }
     84: void f1_u(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, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f1[])
     88: {
     89:   PetscInt d;
     90:   for (d = 0; d < dim; ++d) f1[d] = u_x[dim*2+d];
     91: }
     92: void g0_uu(PetscInt dim, PetscInt Nf, PetscInt NfAux,
     93:            const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
     94:            const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
     95:            PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g0[])
     96: {
     97:   g0[0] = 1.0;
     98: }
     99: void g0_uu_full(PetscInt dim, PetscInt Nf, PetscInt NfAux,
    100:                 const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
    101:                 const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
    102:                 PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g0[])
    103: {
    104:   g0[0] = PetscSqr(u[0] - sin(2.0*PETSC_PI*x[0]) * sin(2.0*PETSC_PI*x[1]))
    105:     + PetscSqr(u[0] - (x[0]*x[0] + x[1]*x[1]))
    106:     - 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]
    107:     + 4.0*(x[0]*x[0] + x[1]*x[1])*(sin(2.0*PETSC_PI*x[0]) * sin(2.0*PETSC_PI*x[1]));
    108: }
    109: void g3_ul(PetscInt dim, PetscInt Nf, PetscInt NfAux,
    110:            const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
    111:            const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
    112:            PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g3[])
    113: {
    114:   PetscInt d;
    115:   for (d = 0; d < dim; ++d) g3[d*dim+d] = 1.0;
    116: }

    118: void f0_a(PetscInt dim, PetscInt Nf, PetscInt NfAux,
    119:           const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
    120:           const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
    121:           PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
    122: {
    123:   f0[0] = u[1] - 4.0 /* 0.0 */ + u[2];
    124: }
    125: void g0_aa(PetscInt dim, PetscInt Nf, PetscInt NfAux,
    126:            const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
    127:            const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
    128:            PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g0[])
    129: {
    130:   g0[0] = 1.0;
    131: }
    132: void g0_al(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, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g0[])
    136: {
    137:   g0[0] = 1.0;
    138: }

    140: void f0_l(PetscInt dim, PetscInt Nf, PetscInt NfAux,
    141:           const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
    142:           const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
    143:           PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
    144: {
    145:   f0[0] = u[1];
    146: }
    147: void f1_l(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, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f1[])
    151: {
    152:   PetscInt d;
    153:   for (d = 0; d < dim; ++d) f1[d] = u_x[d];
    154: }
    155: void g0_la(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 g0[])
    159: {
    160:   g0[0] = 1.0;
    161: }
    162: void g3_lu(PetscInt dim, PetscInt Nf, PetscInt NfAux,
    163:            const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
    164:            const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
    165:            PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g3[])
    166: {
    167:   PetscInt d;
    168:   for (d = 0; d < dim; ++d) g3[d*dim+d] = 1.0;
    169: }

    171: /*
    172:   In 2D for Dirichlet conditions with a variable coefficient, we use exact solution:

    174:     u   = x^2 + y^2
    175:     a   = 4
    176:     d_A = 4
    177:     d_B = sin(2*pi*x[0]) * sin(2*pi*x[1])

    179:   so that

    181:     -\Delta u + a = -4 + 4 = 0
    182: */
    183: PetscErrorCode quadratic_u_2d(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar *u, void *ctx)
    184: {
    185:   *u = x[0]*x[0] + x[1]*x[1];
    186:   return 0;
    187: }
    188: PetscErrorCode constant_a_2d(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar *a, void *ctx)
    189: {
    190:   *a = 4;
    191:   return 0;
    192: }
    193: PetscErrorCode zero(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar *l, void *ctx)
    194: {
    195:   *l = 0.0;
    196:   return 0;
    197: }

    199: PetscErrorCode SetupProblem(DM dm, AppCtx *user)
    200: {
    201:   PetscDS        prob;
    202:   const PetscInt id = 1;

    206:   DMGetDS(dm, &prob);
    207:   PetscDSSetResidual(prob, 0, user->useDualPenalty == PETSC_TRUE ? f0_u_full : f0_u, f1_u);
    208:   PetscDSSetResidual(prob, 1, f0_a, NULL);
    209:   PetscDSSetResidual(prob, 2, f0_l, f1_l);
    210:   PetscDSSetJacobian(prob, 0, 0, user->useDualPenalty == PETSC_TRUE ? g0_uu_full : g0_uu, NULL, NULL, NULL);
    211:   PetscDSSetJacobian(prob, 0, 2, NULL, NULL, NULL, g3_ul);
    212:   PetscDSSetJacobian(prob, 1, 1, g0_aa, NULL, NULL, NULL);
    213:   PetscDSSetJacobian(prob, 1, 2, g0_al, NULL, NULL, NULL);
    214:   PetscDSSetJacobian(prob, 2, 1, g0_la, NULL, NULL, NULL);
    215:   PetscDSSetJacobian(prob, 2, 0, NULL, NULL, NULL, g3_lu);

    217:   PetscDSSetExactSolution(prob, 0, quadratic_u_2d, NULL);
    218:   PetscDSSetExactSolution(prob, 1, constant_a_2d, NULL);
    219:   PetscDSSetExactSolution(prob, 2, zero, NULL);
    220:   DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", "marker", 0, 0, NULL, (void (*)()) quadratic_u_2d, NULL, 1, &id, user);
    221:   DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", "marker", 1, 0, NULL, (void (*)()) constant_a_2d, NULL, 1, &id, user);
    222:   DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", "marker", 2, 0, NULL, (void (*)()) zero, NULL, 1, &id, user);
    223:   return(0);
    224: }

    226: PetscErrorCode SetupDiscretization(DM dm, AppCtx *user)
    227: {
    228:   DM              cdm = dm;
    229:   const PetscInt  dim = 2;
    230:   PetscFE         fe[3];
    231:   PetscInt        f;
    232:   MPI_Comm        comm;
    233:   PetscErrorCode  ierr;

    236:   /* Create finite element */
    237:   PetscObjectGetComm((PetscObject) dm, &comm);
    238:   PetscFECreateDefault(comm, dim, 1, PETSC_TRUE, "potential_", -1, &fe[0]);
    239:   PetscObjectSetName((PetscObject) fe[0], "potential");
    240:   PetscFECreateDefault(comm, dim, 1, PETSC_TRUE, "charge_", -1, &fe[1]);
    241:   PetscObjectSetName((PetscObject) fe[1], "charge");
    242:   PetscFECopyQuadrature(fe[0], fe[1]);
    243:   PetscFECreateDefault(comm, dim, 1, PETSC_TRUE, "multiplier_", -1, &fe[2]);
    244:   PetscObjectSetName((PetscObject) fe[2], "multiplier");
    245:   PetscFECopyQuadrature(fe[0], fe[2]);
    246:   /* Set discretization and boundary conditions for each mesh */
    247:   for (f = 0; f < 3; ++f) {DMSetField(dm, f, NULL, (PetscObject) fe[f]);}
    248:   DMCreateDS(cdm);
    249:   SetupProblem(dm, user);
    250:   while (cdm) {
    251:     DMCopyDisc(dm, cdm);
    252:     DMGetCoarseDM(cdm, &cdm);
    253:   }
    254:   for (f = 0; f < 3; ++f) {PetscFEDestroy(&fe[f]);}
    255:   return(0);
    256: }

    258: int main(int argc, char **argv)
    259: {
    260:   DM             dm;
    261:   SNES           snes;
    262:   Vec            u, r;
    263:   AppCtx         user;

    266:   PetscInitialize(&argc, &argv, NULL,help);if (ierr) return ierr;
    267:   ProcessOptions(PETSC_COMM_WORLD, &user);
    268:   SNESCreate(PETSC_COMM_WORLD, &snes);
    269:   CreateMesh(PETSC_COMM_WORLD, &user, &dm);
    270:   SNESSetDM(snes, dm);
    271:   SetupDiscretization(dm, &user);

    273:   DMCreateGlobalVector(dm, &u);
    274:   PetscObjectSetName((PetscObject) u, "solution");
    275:   VecDuplicate(u, &r);
    276:   DMPlexSetSNESLocalFEM(dm,&user,&user,&user);
    277:   SNESSetFromOptions(snes);

    279:   DMSNESCheckFromOptions(snes, u);
    280:   if (user.runType == RUN_FULL) {
    281:     PetscDS          ds;
    282:     PetscErrorCode (*exactFuncs[3])(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar *u, void *ctx);
    283:     PetscErrorCode (*initialGuess[3])(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar u[], void *ctx);
    284:     PetscReal        error;

    286:     DMGetDS(dm, &ds);
    287:     PetscDSGetExactSolution(ds, 0, &exactFuncs[0], NULL);
    288:     PetscDSGetExactSolution(ds, 1, &exactFuncs[1], NULL);
    289:     PetscDSGetExactSolution(ds, 2, &exactFuncs[2], NULL);
    290:     initialGuess[0] = zero;
    291:     initialGuess[1] = zero;
    292:     initialGuess[2] = zero;
    293:     DMProjectFunction(dm, 0.0, initialGuess, NULL, INSERT_VALUES, u);
    294:     VecViewFromOptions(u, NULL, "-initial_vec_view");
    295:     DMComputeL2Diff(dm, 0.0, exactFuncs, NULL, u, &error);
    296:     if (error < 1.0e-11) {PetscPrintf(PETSC_COMM_WORLD, "Initial L_2 Error: < 1.0e-11\n");}
    297:     else                 {PetscPrintf(PETSC_COMM_WORLD, "Initial L_2 Error: %g\n", error);}
    298:     SNESSolve(snes, NULL, u);
    299:     DMComputeL2Diff(dm, 0.0, exactFuncs, NULL, u, &error);
    300:     if (error < 1.0e-11) {PetscPrintf(PETSC_COMM_WORLD, "Final L_2 Error: < 1.0e-11\n");}
    301:     else                 {PetscPrintf(PETSC_COMM_WORLD, "Final L_2 Error: %g\n", error);}
    302:   }
    303:   VecViewFromOptions(u, NULL, "-sol_vec_view");

    305:   VecDestroy(&u);
    306:   VecDestroy(&r);
    307:   SNESDestroy(&snes);
    308:   DMDestroy(&dm);
    309:   PetscFinalize();
    310:   return ierr;
    311: }

    313: /*TEST

    315:   build:
    316:     requires: !complex triangle

    318:   test:
    319:     suffix: 0
    320:     args: -run_type test -dmsnes_check -potential_petscspace_degree 2 -charge_petscspace_degree 1 -multiplier_petscspace_degree 1

    322:   test:
    323:     suffix: 1
    324:     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

    326:   test:
    327:     suffix: 2
    328:     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

    330: TEST*/