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: {
50:   DM             distributedMesh = NULL;

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

66: /* u - (x^2 + y^2) */
67: void f0_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 f0[])
71: {
72:   f0[0] = u[0] - (x[0]*x[0] + x[1]*x[1]);
73: }
74: /* a \nabla\lambda */
75: void f1_u(PetscInt dim, PetscInt Nf, PetscInt NfAux,
76:           const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
77:           const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
78:           PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f1[])
79: {
80:   PetscInt d;
81:   for (d = 0; d < dim; ++d) f1[d] = u[1]*u_x[dim*2+d];
82: }
83: /* I */
84: void g0_uu(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 g0[])
88: {
89:   g0[0] = 1.0;
90: }
91: /* \nabla */
92: void g2_ua(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 g2[])
96: {
97:   PetscInt d;
98:   for (d = 0; d < dim; ++d) g2[d] = u_x[dim*2+d];
99: }
100: /* a */
101: void g3_ul(PetscInt dim, PetscInt Nf, PetscInt NfAux,
102:            const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
103:            const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
104:            PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g3[])
105: {
106:   PetscInt d;
107:   for (d = 0; d < dim; ++d) g3[d*dim+d] = u[1];
108: }
109: /* a - (x + y) */
110: void f0_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 f0[])
114: {
115:   f0[0] = u[1] - (x[0] + x[1]);
116: }
117: /* \lambda \nabla u */
118: void f1_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 f1[])
122: {
123:   PetscInt d;
124:   for (d = 0; d < dim; ++d) f1[d] = u[2]*u_x[d];
125: }
126: /* I */
127: void g0_aa(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, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g0[])
131: {
132:   g0[0] = 1.0;
133: }
134: /* 6 (x + y) */
135: void f0_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 f0[])
139: {
140:   f0[0] = 6.0*(x[0] + x[1]);
141: }
142: /* a \nabla u */
143: void f1_l(PetscInt dim, PetscInt Nf, PetscInt NfAux,
144:           const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
145:           const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
146:           PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f1[])
147: {
148:   PetscInt d;
149:   for (d = 0; d < dim; ++d) f1[d] = u[1]*u_x[d];
150: }
151: /* \nabla u */
152: void g2_la(PetscInt dim, PetscInt Nf, PetscInt NfAux,
153:            const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
154:            const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
155:            PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g2[])
156: {
157:   PetscInt d;
158:   for (d = 0; d < dim; ++d) g2[d] = u_x[d];
159: }
160: /* a */
161: void g3_lu(PetscInt dim, PetscInt Nf, PetscInt NfAux,
162:            const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
163:            const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
164:            PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g3[])
165: {
166:   PetscInt d;
167:   for (d = 0; d < dim; ++d) g3[d*dim+d] = u[1];
168: }

170: /*
171:   In 2D for Dirichlet conditions with a variable coefficient, we use exact solution:

173:     u  = x^2 + y^2
174:     f  = 6 (x + y)
175:     kappa(a) = a = (x + y)

177:   so that

179:     -\div \kappa(a) \grad u + f = -6 (x + y) + 6 (x + y) = 0
180: */
181: PetscErrorCode quadratic_u_2d(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar *u, void *ctx)
182: {
183:   *u = x[0]*x[0] + x[1]*x[1];
184:   return 0;
185: }
186: PetscErrorCode linear_a_2d(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar *a, void *ctx)
187: {
188:   *a = x[0] + x[1];
189:   return 0;
190: }
191: PetscErrorCode zero(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar *l, void *ctx)
192: {
193:   *l = 0.0;
194:   return 0;
195: }

197: PetscErrorCode SetupProblem(DM dm, AppCtx *user)
198: {
199:   PetscDS        prob;
200:   const PetscInt id = 1;

204:   DMGetDS(dm, &prob);
205:   PetscDSSetResidual(prob, 0, f0_u, f1_u);
206:   PetscDSSetResidual(prob, 1, f0_a, f1_a);
207:   PetscDSSetResidual(prob, 2, f0_l, f1_l);
208:   PetscDSSetJacobian(prob, 0, 0, g0_uu, NULL, NULL, NULL);
209:   PetscDSSetJacobian(prob, 0, 1, NULL, NULL, g2_ua, NULL);
210:   PetscDSSetJacobian(prob, 0, 2, NULL, NULL, NULL, g3_ul);
211:   PetscDSSetJacobian(prob, 1, 1, g0_aa, NULL, NULL, NULL);
212:   PetscDSSetJacobian(prob, 2, 1, NULL, NULL, g2_la, NULL);
213:   PetscDSSetJacobian(prob, 2, 0, NULL, NULL, NULL, g3_lu);

216:   PetscDSSetExactSolution(prob, 1, linear_a_2d, NULL);
217:   PetscDSSetExactSolution(prob, 2, zero, NULL);
218:   DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", "marker", 0, 0, NULL, (void (*)(void)) quadratic_u_2d, NULL, 1, &id, user);
219:   DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", "marker", 1, 0, NULL, (void (*)(void)) linear_a_2d, NULL, 1, &id, user);
220:   DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", "marker", 2, 0, NULL, (void (*)(void)) zero, NULL, 1, &id, user);
221:   return(0);
222: }

224: PetscErrorCode SetupDiscretization(DM dm, AppCtx *user)
225: {
226:   DM              cdm = dm;
227:   const PetscInt  dim = 2;
228:   PetscFE         fe[3];
229:   PetscInt        f;
230:   MPI_Comm        comm;
231:   PetscErrorCode  ierr;

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

256: int main(int argc, char **argv)
257: {
258:   DM             dm;
259:   SNES           snes;
260:   Vec            u, r;
261:   AppCtx         user;

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

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

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

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

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

311: /*TEST

313:   build:
314:     requires: !complex

316:   test:
317:     suffix: 0
318:     requires: triangle
319:     args: -run_type test -dmsnes_check -potential_petscspace_degree 2 -conductivity_petscspace_degree 1 -multiplier_petscspace_degree 2

321:   test:
322:     suffix: 1
323:     requires: triangle
324:     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

326: TEST*/