Actual source code: ex2.c

petsc-3.13.6 2020-09-29

  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 $y$ and $u$, given by
\begin{align}
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)
\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 null vector for the reference control $a_r$. Right now $\beta = 1$.

The PDE will be the Laplace equation with homogeneous boundary conditions
\begin{align}
-Delta u = a
\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:   PetscBool         useDualPenalty; /* Penalize deviation from both goals */
 30:   PetscErrorCode (**exactFuncs)(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar *u, void *ctx);
 31: } AppCtx;

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

 40:   options->runType        = RUN_FULL;
 41:   options->useDualPenalty = PETSC_FALSE;

 43:   PetscOptionsBegin(comm, "", "Inverse Problem Options", "DMPLEX");
 44:   run  = options->runType;
 45:   PetscOptionsEList("-run_type", "The run type", "ex2.c", runTypes, 2, runTypes[options->runType], &run, NULL);
 46:   options->runType = (RunType) run;
 47:   PetscOptionsBool("-use_dual_penalty", "Penalize deviation from both goals", "ex2.c", options->useDualPenalty, &options->useDualPenalty, NULL);
 48:   PetscOptionsEnd();
 49:   return(0);
 50: }

 52: static PetscErrorCode CreateMesh(MPI_Comm comm, AppCtx *user, DM *dm)
 53: {
 54:   DM             distributedMesh = NULL;

 58:   DMPlexCreateBoxMesh(comm, 2, PETSC_TRUE, NULL, NULL, NULL, NULL, PETSC_TRUE, dm);
 59:   PetscObjectSetName((PetscObject) *dm, "Mesh");
 60:   DMPlexDistribute(*dm, 0, NULL, &distributedMesh);
 61:   if (distributedMesh) {
 62:     DMDestroy(dm);
 63:     *dm  = distributedMesh;
 64:   }
 65:   DMSetFromOptions(*dm);
 66:   DMViewFromOptions(*dm, NULL, "-dm_view");
 67:   return(0);
 68: }

 70: void f0_u(PetscInt dim, PetscInt Nf, PetscInt NfAux,
 71:           const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
 72:           const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
 73:           PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
 74: {
 75:   f0[0] = (u[0] - (x[0]*x[0] + x[1]*x[1]));
 76: }
 77: void f0_u_full(PetscInt dim, PetscInt Nf, PetscInt NfAux,
 78:           const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
 79:           const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
 80:           PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
 81: {
 82:   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]))) +
 83:     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])));
 84: }
 85: void f1_u(PetscInt dim, PetscInt Nf, PetscInt NfAux,
 86:           const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
 87:           const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
 88:           PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f1[])
 89: {
 90:   PetscInt d;
 91:   for (d = 0; d < dim; ++d) f1[d] = u_x[dim*2+d];
 92: }
 93: void g0_uu(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 g0[])
 97: {
 98:   g0[0] = 1.0;
 99: }
100: void g0_uu_full(PetscInt dim, PetscInt Nf, PetscInt NfAux,
101:                 const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
102:                 const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
103:                 PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g0[])
104: {
105:   g0[0] = PetscSqr(u[0] - sin(2.0*PETSC_PI*x[0]) * sin(2.0*PETSC_PI*x[1]))
106:     + PetscSqr(u[0] - (x[0]*x[0] + x[1]*x[1]))
107:     - 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]
108:     + 4.0*(x[0]*x[0] + x[1]*x[1])*(sin(2.0*PETSC_PI*x[0]) * sin(2.0*PETSC_PI*x[1]));
109: }
110: void g3_ul(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, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g3[])
114: {
115:   PetscInt d;
116:   for (d = 0; d < dim; ++d) g3[d*dim+d] = 1.0;
117: }

119: void f0_a(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, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
123: {
124:   f0[0] = u[1] - 4.0 /* 0.0 */ + u[2];
125: }
126: void g0_aa(PetscInt dim, PetscInt Nf, PetscInt NfAux,
127:            const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
128:            const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
129:            PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g0[])
130: {
131:   g0[0] = 1.0;
132: }
133: void g0_al(PetscInt dim, PetscInt Nf, PetscInt NfAux,
134:            const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
135:            const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
136:            PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g0[])
137: {
138:   g0[0] = 1.0;
139: }

141: void f0_l(PetscInt dim, PetscInt Nf, PetscInt NfAux,
142:           const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
143:           const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
144:           PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
145: {
146:   f0[0] = u[1];
147: }
148: void f1_l(PetscInt dim, PetscInt Nf, PetscInt NfAux,
149:           const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
150:           const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
151:           PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f1[])
152: {
153:   PetscInt d;
154:   for (d = 0; d < dim; ++d) f1[d] = u_x[d];
155: }
156: void g0_la(PetscInt dim, PetscInt Nf, PetscInt NfAux,
157:            const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
158:            const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
159:            PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g0[])
160: {
161:   g0[0] = 1.0;
162: }
163: void g3_lu(PetscInt dim, PetscInt Nf, PetscInt NfAux,
164:            const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
165:            const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
166:            PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g3[])
167: {
168:   PetscInt d;
169:   for (d = 0; d < dim; ++d) g3[d*dim+d] = 1.0;
170: }

172: /*
173:   In 2D for Dirichlet conditions with a variable coefficient, we use exact solution:

175:     u   = x^2 + y^2
176:     a   = 4
177:     d_A = 4
178:     d_B = sin(2*pi*x[0]) * sin(2*pi*x[1])

180:   so that

182:     -\Delta u + a = -4 + 4 = 0
183: */
184: PetscErrorCode quadratic_u_2d(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar *u, void *ctx)
185: {
186:   *u = x[0]*x[0] + x[1]*x[1];
187:   return 0;
188: }
189: PetscErrorCode constant_a_2d(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar *a, void *ctx)
190: {
191:   *a = 4;
192:   return 0;
193: }
194: PetscErrorCode zero(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar *l, void *ctx)
195: {
196:   *l = 0.0;
197:   return 0;
198: }

200: PetscErrorCode SetupProblem(DM dm, AppCtx *user)
201: {
202:   PetscDS        prob;
203:   const PetscInt id = 1;

207:   DMGetDS(dm, &prob);
208:   PetscDSSetResidual(prob, 0, user->useDualPenalty == PETSC_TRUE ? f0_u_full : f0_u, f1_u);
209:   PetscDSSetResidual(prob, 1, f0_a, NULL);
210:   PetscDSSetResidual(prob, 2, f0_l, f1_l);
211:   PetscDSSetJacobian(prob, 0, 0, user->useDualPenalty == PETSC_TRUE ? g0_uu_full : g0_uu, NULL, NULL, NULL);
212:   PetscDSSetJacobian(prob, 0, 2, NULL, NULL, NULL, g3_ul);
213:   PetscDSSetJacobian(prob, 1, 1, g0_aa, NULL, NULL, NULL);
214:   PetscDSSetJacobian(prob, 1, 2, g0_al, NULL, NULL, NULL);
215:   PetscDSSetJacobian(prob, 2, 1, g0_la, NULL, NULL, NULL);
216:   PetscDSSetJacobian(prob, 2, 0, NULL, NULL, NULL, g3_lu);

218:   user->exactFuncs[0] = quadratic_u_2d;
219:   user->exactFuncs[1] = constant_a_2d;
220:   user->exactFuncs[2] = zero;
221:   PetscDSAddBoundary(prob, DM_BC_ESSENTIAL, "wall", "marker", 0, 0, NULL, (void (*)()) user->exactFuncs[0], 1, &id, user);
222:   PetscDSAddBoundary(prob, DM_BC_ESSENTIAL, "wall", "marker", 1, 0, NULL, (void (*)()) user->exactFuncs[1], 1, &id, user);
223:   PetscDSAddBoundary(prob, DM_BC_ESSENTIAL, "wall", "marker", 2, 0, NULL, (void (*)()) user->exactFuncs[2], 1, &id, user);
224:   return(0);
225: }

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

237:   /* Create finite element */
238:   PetscObjectGetComm((PetscObject) dm, &comm);
239:   PetscFECreateDefault(comm, dim, 1, PETSC_TRUE, "potential_", -1, &fe[0]);
240:   PetscObjectSetName((PetscObject) fe[0], "potential");
241:   PetscFECreateDefault(comm, dim, 1, PETSC_TRUE, "charge_", -1, &fe[1]);
242:   PetscObjectSetName((PetscObject) fe[1], "charge");
243:   PetscFECopyQuadrature(fe[0], fe[1]);
244:   PetscFECreateDefault(comm, dim, 1, PETSC_TRUE, "multiplier_", -1, &fe[2]);
245:   PetscObjectSetName((PetscObject) fe[2], "multiplier");
246:   PetscFECopyQuadrature(fe[0], fe[2]);
247:   /* Set discretization and boundary conditions for each mesh */
248:   for (f = 0; f < 3; ++f) {DMSetField(dm, f, NULL, (PetscObject) fe[f]);}
249:   DMCreateDS(cdm);
250:   SetupProblem(dm, user);
251:   while (cdm) {
252:     DMCopyDisc(dm, cdm);
253:     DMGetCoarseDM(cdm, &cdm);
254:   }
255:   for (f = 0; f < 3; ++f) {PetscFEDestroy(&fe[f]);}
256:   return(0);
257: }

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

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

273:   PetscMalloc(3 * sizeof(void (*)()), &user.exactFuncs);
274:   SetupDiscretization(dm, &user);

276:   DMCreateGlobalVector(dm, &u);
277:   PetscObjectSetName((PetscObject) u, "solution");
278:   VecDuplicate(u, &r);
279:   DMPlexSetSNESLocalFEM(dm,&user,&user,&user);
280:   SNESSetFromOptions(snes);

282:   DMProjectFunction(dm, 0.0, user.exactFuncs, NULL, INSERT_ALL_VALUES, u);
283:   DMSNESCheckFromOptions(snes, u, user.exactFuncs, NULL);
284:   if (user.runType == RUN_FULL) {
285:     PetscErrorCode (*initialGuess[3])(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar u[], void *ctx);
286:     PetscReal        error;

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, user.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, user.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:   PetscFree(user.exactFuncs);
308:   PetscFinalize();
309:   return ierr;
310: }

312: /*TEST

314:   build:
315:     requires: !complex triangle

317:   test:
318:     suffix: 0
319:     args: -run_type test -dmsnes_check -potential_petscspace_degree 2 -charge_petscspace_degree 1 -multiplier_petscspace_degree 1

321:   test:
322:     suffix: 1
323:     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

325:   test:
326:     suffix: 2
327:     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

329: TEST*/