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";

4: /*F
5:   We are solving the parameter estimation problem for the Laplacian. We will ask to minimize a Lagrangian
6: function over $y$ and $u$, given by
7: \begin{align}
8:   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)
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 null vector for the reference control $a_r$. Right now $\beta = 1$.

15: The PDE will be the Laplace equation with homogeneous boundary conditions
16: \begin{align}
17:   -Delta u = a
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:   PetscBool useDualPenalty; /* Penalize deviation from both goals */
33: } AppCtx;

35: static PetscErrorCode ProcessOptions(MPI_Comm comm, AppCtx *options)
36: {
37:   const char *runTypes[2] = {"full", "test"};
38:   PetscInt    run;

40:   PetscFunctionBeginUser;
41:   options->runType        = RUN_FULL;
42:   options->useDualPenalty = PETSC_FALSE;
43:   PetscOptionsBegin(comm, "", "Inverse Problem Options", "DMPLEX");
44:   run = options->runType;
45:   PetscCall(PetscOptionsEList("-run_type", "The run type", "ex2.c", runTypes, 2, runTypes[options->runType], &run, NULL));
46:   options->runType = (RunType)run;
47:   PetscCall(PetscOptionsBool("-use_dual_penalty", "Penalize deviation from both goals", "ex2.c", options->useDualPenalty, &options->useDualPenalty, NULL));
48:   PetscOptionsEnd();
49:   PetscFunctionReturn(PETSC_SUCCESS);
50: }

52: static PetscErrorCode CreateMesh(MPI_Comm comm, AppCtx *user, DM *dm)
53: {
54:   PetscFunctionBeginUser;
55:   PetscCall(DMCreate(comm, dm));
56:   PetscCall(DMSetType(*dm, DMPLEX));
57:   PetscCall(DMSetFromOptions(*dm));
58:   PetscCall(DMViewFromOptions(*dm, NULL, "-dm_view"));
59:   PetscFunctionReturn(PETSC_SUCCESS);
60: }

62: 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[])
63: {
64:   f0[0] = (u[0] - (x[0] * x[0] + x[1] * x[1]));
65: }
66: void f0_u_full(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[])
67: {
68:   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]))) + 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])));
69: }
70: 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[])
71: {
72:   PetscInt d;
73:   for (d = 0; d < dim; ++d) f1[d] = u_x[dim * 2 + d];
74: }
75: 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[])
76: {
77:   g0[0] = 1.0;
78: }
79: void g0_uu_full(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[])
80: {
81:   g0[0] = PetscSqr(u[0] - sin(2.0 * PETSC_PI * x[0]) * sin(2.0 * PETSC_PI * x[1])) + PetscSqr(u[0] - (x[0] * x[0] + x[1] * x[1])) - 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] + 4.0 * (x[0] * x[0] + x[1] * x[1]) * (sin(2.0 * PETSC_PI * x[0]) * sin(2.0 * PETSC_PI * x[1]));
82: }
83: 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[])
84: {
85:   PetscInt d;
86:   for (d = 0; d < dim; ++d) g3[d * dim + d] = 1.0;
87: }

89: 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[])
90: {
91:   f0[0] = u[1] - 4.0 /* 0.0 */ + u[2];
92: }
93: 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[])
94: {
95:   g0[0] = 1.0;
96: }
97: void g0_al(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[])
98: {
99:   g0[0] = 1.0;
100: }

102: 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[])
103: {
104:   f0[0] = u[1];
105: }
106: 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[])
107: {
108:   PetscInt d;
109:   for (d = 0; d < dim; ++d) f1[d] = u_x[d];
110: }
111: void g0_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 g0[])
112: {
113:   g0[0] = 1.0;
114: }
115: 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[])
116: {
117:   PetscInt d;
118:   for (d = 0; d < dim; ++d) g3[d * dim + d] = 1.0;
119: }

121: /*
122:   In 2D for Dirichlet conditions with a variable coefficient, we use exact solution:

124:     u   = x^2 + y^2
125:     a   = 4
126:     d_A = 4
127:     d_B = sin(2*pi*x[0]) * sin(2*pi*x[1])

129:   so that

131:     -\Delta u + a = -4 + 4 = 0
132: */
133: PetscErrorCode quadratic_u_2d(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar *u, void *ctx)
134: {
135:   *u = x[0] * x[0] + x[1] * x[1];
136:   return PETSC_SUCCESS;
137: }
138: PetscErrorCode constant_a_2d(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar *a, void *ctx)
139: {
140:   *a = 4;
141:   return PETSC_SUCCESS;
142: }
143: PetscErrorCode zero(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar *l, void *ctx)
144: {
145:   *l = 0.0;
146:   return PETSC_SUCCESS;
147: }

149: PetscErrorCode SetupProblem(DM dm, AppCtx *user)
150: {
151:   PetscDS        ds;
152:   DMLabel        label;
153:   const PetscInt id = 1;

155:   PetscFunctionBeginUser;
156:   PetscCall(DMGetDS(dm, &ds));
157:   PetscCall(PetscDSSetResidual(ds, 0, user->useDualPenalty == PETSC_TRUE ? f0_u_full : f0_u, f1_u));
158:   PetscCall(PetscDSSetResidual(ds, 1, f0_a, NULL));
159:   PetscCall(PetscDSSetResidual(ds, 2, f0_l, f1_l));
160:   PetscCall(PetscDSSetJacobian(ds, 0, 0, user->useDualPenalty == PETSC_TRUE ? g0_uu_full : g0_uu, NULL, NULL, NULL));
161:   PetscCall(PetscDSSetJacobian(ds, 0, 2, NULL, NULL, NULL, g3_ul));
162:   PetscCall(PetscDSSetJacobian(ds, 1, 1, g0_aa, NULL, NULL, NULL));
163:   PetscCall(PetscDSSetJacobian(ds, 1, 2, g0_al, NULL, NULL, NULL));
164:   PetscCall(PetscDSSetJacobian(ds, 2, 1, g0_la, NULL, NULL, NULL));
165:   PetscCall(PetscDSSetJacobian(ds, 2, 0, NULL, NULL, NULL, g3_lu));

168:   PetscCall(PetscDSSetExactSolution(ds, 1, constant_a_2d, NULL));
169:   PetscCall(PetscDSSetExactSolution(ds, 2, zero, NULL));
170:   PetscCall(DMGetLabel(dm, "marker", &label));
171:   PetscCall(DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 0, 0, NULL, (void (*)())quadratic_u_2d, NULL, user, NULL));
172:   PetscCall(DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 1, 0, NULL, (void (*)())constant_a_2d, NULL, user, NULL));
173:   PetscCall(DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 2, 0, NULL, (void (*)())zero, NULL, user, NULL));
174:   PetscFunctionReturn(PETSC_SUCCESS);
175: }

177: PetscErrorCode SetupDiscretization(DM dm, AppCtx *user)
178: {
179:   DM             cdm = dm;
180:   const PetscInt dim = 2;
181:   PetscFE        fe[3];
182:   PetscInt       f;
183:   MPI_Comm       comm;

185:   PetscFunctionBeginUser;
186:   /* Create finite element */
187:   PetscCall(PetscObjectGetComm((PetscObject)dm, &comm));
188:   PetscCall(PetscFECreateDefault(comm, dim, 1, PETSC_TRUE, "potential_", -1, &fe[0]));
189:   PetscCall(PetscObjectSetName((PetscObject)fe[0], "potential"));
190:   PetscCall(PetscFECreateDefault(comm, dim, 1, PETSC_TRUE, "charge_", -1, &fe[1]));
191:   PetscCall(PetscObjectSetName((PetscObject)fe[1], "charge"));
193:   PetscCall(PetscFECreateDefault(comm, dim, 1, PETSC_TRUE, "multiplier_", -1, &fe[2]));
194:   PetscCall(PetscObjectSetName((PetscObject)fe[2], "multiplier"));
196:   /* Set discretization and boundary conditions for each mesh */
197:   for (f = 0; f < 3; ++f) PetscCall(DMSetField(dm, f, NULL, (PetscObject)fe[f]));
198:   PetscCall(DMCreateDS(cdm));
199:   PetscCall(SetupProblem(dm, user));
200:   while (cdm) {
201:     PetscCall(DMCopyDisc(dm, cdm));
202:     PetscCall(DMGetCoarseDM(cdm, &cdm));
203:   }
204:   for (f = 0; f < 3; ++f) PetscCall(PetscFEDestroy(&fe[f]));
205:   PetscFunctionReturn(PETSC_SUCCESS);
206: }

208: int main(int argc, char **argv)
209: {
210:   DM     dm;
211:   SNES   snes;
212:   Vec    u, r;
213:   AppCtx user;

215:   PetscFunctionBeginUser;
216:   PetscCall(PetscInitialize(&argc, &argv, NULL, help));
217:   PetscCall(ProcessOptions(PETSC_COMM_WORLD, &user));
218:   PetscCall(SNESCreate(PETSC_COMM_WORLD, &snes));
219:   PetscCall(CreateMesh(PETSC_COMM_WORLD, &user, &dm));
220:   PetscCall(SNESSetDM(snes, dm));
221:   PetscCall(SetupDiscretization(dm, &user));

223:   PetscCall(DMCreateGlobalVector(dm, &u));
224:   PetscCall(PetscObjectSetName((PetscObject)u, "solution"));
225:   PetscCall(VecDuplicate(u, &r));
226:   PetscCall(DMPlexSetSNESLocalFEM(dm, PETSC_FALSE, &user));
227:   PetscCall(SNESSetFromOptions(snes));

229:   PetscCall(DMSNESCheckFromOptions(snes, u));
230:   if (user.runType == RUN_FULL) {
231:     PetscDS ds;
232:     PetscErrorCode (*exactFuncs[3])(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar *u, void *ctx);
233:     PetscErrorCode (*initialGuess[3])(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar u[], void *ctx);
234:     PetscReal error;

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

255:   PetscCall(VecDestroy(&u));
256:   PetscCall(VecDestroy(&r));
257:   PetscCall(SNESDestroy(&snes));
258:   PetscCall(DMDestroy(&dm));
259:   PetscCall(PetscFinalize());
260:   return 0;
261: }

263: /*TEST

265:   build:
266:     requires: !complex triangle

268:   test:
269:     suffix: 0
270:     args: -run_type test -dmsnes_check -potential_petscspace_degree 2 -charge_petscspace_degree 1 -multiplier_petscspace_degree 1

272:   test:
273:     suffix: 1
274:     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

276:   test:
277:     suffix: 2
278:     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

280: TEST*/