Actual source code: ex1.c

petsc-3.13.6 2020-09-29
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  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:   PetscErrorCode (**exactFuncs)(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar *u, void *ctx);
 30: } AppCtx;

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

 39:   options->runType = RUN_FULL;

 41:   PetscOptionsBegin(comm, "", "Inverse Problem Options", "DMPLEX");
 42:   run  = options->runType;
 43:   PetscOptionsEList("-run_type", "The run type", "ex1.c", runTypes, 2, runTypes[options->runType], &run, NULL);
 44:   options->runType = (RunType) run;
 45:   PetscOptionsEnd();
 46:   return(0);
 47: }

 49: static PetscErrorCode CreateMesh(MPI_Comm comm, AppCtx *user, DM *dm)
 50: {
 51:   DM             distributedMesh = NULL;

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

 67: /* u - (x^2 + y^2) */
 68: void f0_u(PetscInt dim, PetscInt Nf, PetscInt NfAux,
 69:           const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
 70:           const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
 71:           PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
 72: {
 73:   f0[0] = u[0] - (x[0]*x[0] + x[1]*x[1]);
 74: }
 75: /* a \nabla\lambda */
 76: void f1_u(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, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f1[])
 80: {
 81:   PetscInt d;
 82:   for (d = 0; d < dim; ++d) f1[d] = u[1]*u_x[dim*2+d];
 83: }
 84: /* I */
 85: void g0_uu(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, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g0[])
 89: {
 90:   g0[0] = 1.0;
 91: }
 92: /* \nabla */
 93: void g2_ua(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 g2[])
 97: {
 98:   PetscInt d;
 99:   for (d = 0; d < dim; ++d) g2[d] = u_x[dim*2+d];
100: }
101: /* a */
102: void g3_ul(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, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g3[])
106: {
107:   PetscInt d;
108:   for (d = 0; d < dim; ++d) g3[d*dim+d] = u[1];
109: }
110: /* a - (x + y) */
111: void f0_a(PetscInt dim, PetscInt Nf, PetscInt NfAux,
112:           const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
113:           const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
114:           PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
115: {
116:   f0[0] = u[1] - (x[0] + x[1]);
117: }
118: /* \lambda \nabla u */
119: void f1_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 f1[])
123: {
124:   PetscInt d;
125:   for (d = 0; d < dim; ++d) f1[d] = u[2]*u_x[d];
126: }
127: /* I */
128: void g0_aa(PetscInt dim, PetscInt Nf, PetscInt NfAux,
129:            const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
130:            const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
131:            PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g0[])
132: {
133:   g0[0] = 1.0;
134: }
135: /* 6 (x + y) */
136: void f0_l(PetscInt dim, PetscInt Nf, PetscInt NfAux,
137:           const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
138:           const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
139:           PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
140: {
141:   f0[0] = 6.0*(x[0] + x[1]);
142: }
143: /* a \nabla u */
144: void f1_l(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, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f1[])
148: {
149:   PetscInt d;
150:   for (d = 0; d < dim; ++d) f1[d] = u[1]*u_x[d];
151: }
152: /* \nabla u */
153: void g2_la(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 g2[])
157: {
158:   PetscInt d;
159:   for (d = 0; d < dim; ++d) g2[d] = u_x[d];
160: }
161: /* a */
162: void g3_lu(PetscInt dim, PetscInt Nf, PetscInt NfAux,
163:            const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
164:            const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
165:            PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g3[])
166: {
167:   PetscInt d;
168:   for (d = 0; d < dim; ++d) g3[d*dim+d] = u[1];
169: }

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

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

178:   so that

180:     -\div \kappa(a) \grad u + f = -6 (x + y) + 6 (x + y) = 0
181: */
182: PetscErrorCode quadratic_u_2d(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar *u, void *ctx)
183: {
184:   *u = x[0]*x[0] + x[1]*x[1];
185:   return 0;
186: }
187: PetscErrorCode linear_a_2d(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar *a, void *ctx)
188: {
189:   *a = x[0] + x[1];
190:   return 0;
191: }
192: PetscErrorCode zero(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar *l, void *ctx)
193: {
194:   *l = 0.0;
195:   return 0;
196: }

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

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

216:   user->exactFuncs[0] = quadratic_u_2d;
217:   user->exactFuncs[1] = linear_a_2d;
218:   user->exactFuncs[2] = zero;
219:   PetscDSAddBoundary(prob, DM_BC_ESSENTIAL, "wall", "marker", 0, 0, NULL, (void (*)(void)) user->exactFuncs[0], 1, &id, user);
220:   PetscDSAddBoundary(prob, DM_BC_ESSENTIAL, "wall", "marker", 1, 0, NULL, (void (*)(void)) user->exactFuncs[1], 1, &id, user);
221:   PetscDSAddBoundary(prob, DM_BC_ESSENTIAL, "wall", "marker", 2, 0, NULL, (void (*)(void)) user->exactFuncs[2], 1, &id, user);
222:   return(0);
223: }

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

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

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

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

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

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

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

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

301:   VecDestroy(&u);
302:   VecDestroy(&r);
303:   SNESDestroy(&snes);
304:   DMDestroy(&dm);
305:   PetscFree(user.exactFuncs);
306:   PetscFinalize();
307:   return ierr;
308: }

310: /*TEST

312:   build:
313:     requires: !complex

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

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

325: TEST*/