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*/