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;

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

49: static PetscErrorCode CreateMesh(MPI_Comm comm, AppCtx *user, DM *dm)
50: {
51:   PetscFunctionBeginUser;
52:   PetscCall(DMCreate(comm, dm));
53:   PetscCall(DMSetType(*dm, DMPLEX));
54:   PetscCall(DMSetFromOptions(*dm));
55:   PetscCall(DMViewFromOptions(*dm, NULL, "-dm_view"));
56:   PetscFunctionReturn(PETSC_SUCCESS);
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 PETSC_SUCCESS;
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 PETSC_SUCCESS;
147: }
148: PetscErrorCode zero(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar *l, void *ctx)
149: {
150:   *l = 0.0;
151:   return PETSC_SUCCESS;
152: }

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

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

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

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

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

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

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

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

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