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;

 40:   options->runType = RUN_FULL;
 41:   PetscOptionsBegin(comm, "", "Inverse Problem Options", "DMPLEX");
 42:   run = options->runType;
 43:   PetscOptionsEList("-run_type", "The run type", "ex1.c", runTypes, 2, runTypes[options->runType], &run, NULL);
 44:   options->runType = (RunType)run;
 45:   PetscOptionsEnd();
 46:   return 0;
 47: }

 49: static PetscErrorCode CreateMesh(MPI_Comm comm, AppCtx *user, DM *dm)
 50: {
 52:   DMCreate(comm, dm);
 53:   DMSetType(*dm, DMPLEX);
 54:   DMSetFromOptions(*dm);
 55:   DMViewFromOptions(*dm, NULL, "-dm_view");
 56:   return 0;
 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, void *ctx)
139: {
140:   *u = x[0] * x[0] + x[1] * x[1];
141:   return 0;
142: }
143: PetscErrorCode linear_a_2d(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar *a, void *ctx)
144: {
145:   *a = x[0] + x[1];
146:   return 0;
147: }
148: PetscErrorCode zero(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar *l, void *ctx)
149: {
150:   *l = 0.0;
151:   return 0;
152: }

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

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

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

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

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

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

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

234:   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, void *ctx);
238:     PetscErrorCode (*initialGuess[3])(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar u[], void *ctx);
239:     PetscReal error;

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

260:   VecDestroy(&u);
261:   VecDestroy(&r);
262:   SNESDestroy(&snes);
263:   DMDestroy(&dm);
264:   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*/