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";

We are solving the parameter estimation problem for the Laplacian. We will ask to minimize a Lagrangian
function over $a$ and $u$, given by
\begin{align}
L(u, a, \lambda) = \frac{1}{2} || Qu - d ||^2 + \frac{1}{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 exact control for the reference $a_r$.

The PDE will be the Laplace equation with homogeneous boundary conditions
\begin{align}
-nabla \cdot a \nabla u = f
\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: } AppCtx;

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

 38:   options->runType = RUN_FULL;

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

 48: static PetscErrorCode CreateMesh(MPI_Comm comm, AppCtx *user, DM *dm)
 49: {
 51:   DMCreate(comm, dm);
 52:   DMSetType(*dm, DMPLEX);
 53:   DMSetFromOptions(*dm);
 54:   DMViewFromOptions(*dm, NULL, "-dm_view");
 55:   return 0;
 56: }

 58: /* u - (x^2 + y^2) */
 59: void f0_u(PetscInt dim, PetscInt Nf, PetscInt NfAux,
 60:           const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
 61:           const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
 62:           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: /* a \nabla\lambda */
 67: void f1_u(PetscInt dim, PetscInt Nf, PetscInt NfAux,
 68:           const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
 69:           const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
 70:           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[1]*u_x[dim*2+d];
 74: }
 75: /* I */
 76: void g0_uu(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, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g0[])
 80: {
 81:   g0[0] = 1.0;
 82: }
 83: /* \nabla */
 84: void g2_ua(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 g2[])
 88: {
 89:   PetscInt d;
 90:   for (d = 0; d < dim; ++d) g2[d] = u_x[dim*2+d];
 91: }
 92: /* a */
 93: void g3_ul(PetscInt dim, PetscInt Nf, PetscInt NfAux,
 94:            const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
 95:            const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
 96:            PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g3[])
 97: {
 98:   PetscInt d;
 99:   for (d = 0; d < dim; ++d) g3[d*dim+d] = u[1];
100: }
101: /* a - (x + y) */
102: void f0_a(PetscInt dim, PetscInt Nf, PetscInt NfAux,
103:           const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
104:           const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
105:           PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
106: {
107:   f0[0] = u[1] - (x[0] + x[1]);
108: }
109: /* \lambda \nabla u */
110: void f1_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 f1[])
114: {
115:   PetscInt d;
116:   for (d = 0; d < dim; ++d) f1[d] = u[2]*u_x[d];
117: }
118: /* I */
119: void g0_aa(PetscInt dim, PetscInt Nf, PetscInt NfAux,
120:            const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
121:            const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
122:            PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g0[])
123: {
124:   g0[0] = 1.0;
125: }
126: /* 6 (x + y) */
127: void f0_l(PetscInt dim, PetscInt Nf, PetscInt NfAux,
128:           const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
129:           const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
130:           PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
131: {
132:   f0[0] = 6.0*(x[0] + x[1]);
133: }
134: /* a \nabla u */
135: void f1_l(PetscInt dim, PetscInt Nf, PetscInt NfAux,
136:           const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
137:           const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
138:           PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f1[])
139: {
140:   PetscInt d;
141:   for (d = 0; d < dim; ++d) f1[d] = u[1]*u_x[d];
142: }
143: /* \nabla u */
144: void g2_la(PetscInt dim, PetscInt Nf, PetscInt NfAux,
145:            const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
146:            const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
147:            PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g2[])
148: {
149:   PetscInt d;
150:   for (d = 0; d < dim; ++d) g2[d] = u_x[d];
151: }
152: /* a */
153: void g3_lu(PetscInt dim, PetscInt Nf, PetscInt NfAux,
154:            const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
155:            const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
156:            PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g3[])
157: {
158:   PetscInt d;
159:   for (d = 0; d < dim; ++d) g3[d*dim+d] = u[1];
160: }

162: /*
163:   In 2D for Dirichlet conditions with a variable coefficient, we use exact solution:

165:     u  = x^2 + y^2
166:     f  = 6 (x + y)
167:     kappa(a) = a = (x + y)

169:   so that

171:     -\div \kappa(a) \grad u + f = -6 (x + y) + 6 (x + y) = 0
172: */
173: PetscErrorCode quadratic_u_2d(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar *u, void *ctx)
174: {
175:   *u = x[0]*x[0] + x[1]*x[1];
176:   return 0;
177: }
178: PetscErrorCode linear_a_2d(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar *a, void *ctx)
179: {
180:   *a = x[0] + x[1];
181:   return 0;
182: }
183: PetscErrorCode zero(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar *l, void *ctx)
184: {
185:   *l = 0.0;
186:   return 0;
187: }

189: PetscErrorCode SetupProblem(DM dm, AppCtx *user)
190: {
191:   PetscDS        ds;
192:   DMLabel        label;
193:   const PetscInt id = 1;

196:   DMGetDS(dm, &ds);
197:   PetscDSSetResidual(ds, 0, f0_u, f1_u);
198:   PetscDSSetResidual(ds, 1, f0_a, f1_a);
199:   PetscDSSetResidual(ds, 2, f0_l, f1_l);
200:   PetscDSSetJacobian(ds, 0, 0, g0_uu, NULL, NULL, NULL);
201:   PetscDSSetJacobian(ds, 0, 1, NULL, NULL, g2_ua, NULL);
202:   PetscDSSetJacobian(ds, 0, 2, NULL, NULL, NULL, g3_ul);
203:   PetscDSSetJacobian(ds, 1, 1, g0_aa, NULL, NULL, NULL);
204:   PetscDSSetJacobian(ds, 2, 1, NULL, NULL, g2_la, NULL);
205:   PetscDSSetJacobian(ds, 2, 0, NULL, NULL, NULL, g3_lu);

207:   PetscDSSetExactSolution(ds, 0, quadratic_u_2d, NULL);
208:   PetscDSSetExactSolution(ds, 1, linear_a_2d, NULL);
209:   PetscDSSetExactSolution(ds, 2, zero, NULL);
210:   DMGetLabel(dm, "marker", &label);
211:   DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 0, 0, NULL, (void (*)(void)) quadratic_u_2d, NULL, user, NULL);
212:   DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 1, 0, NULL, (void (*)(void)) linear_a_2d, NULL, user, NULL);
213:   DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 2, 0, NULL, (void (*)(void)) zero, NULL, user, NULL);
214:   return 0;
215: }

217: PetscErrorCode SetupDiscretization(DM dm, AppCtx *user)
218: {
219:   DM              cdm = dm;
220:   const PetscInt  dim = 2;
221:   PetscFE         fe[3];
222:   PetscInt        f;
223:   MPI_Comm        comm;

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

248: int main(int argc, char **argv)
249: {
250:   DM             dm;
251:   SNES           snes;
252:   Vec            u, r;
253:   AppCtx         user;

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

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

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

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

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

302: /*TEST

304:   build:
305:     requires: !complex

307:   test:
308:     suffix: 0
309:     requires: triangle
310:     args: -run_type test -dmsnes_check -potential_petscspace_degree 2 -conductivity_petscspace_degree 1 -multiplier_petscspace_degree 2

312:   test:
313:     suffix: 1
314:     requires: triangle
315:     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

317: TEST*/