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: {
53: DMCreate(comm, dm);
54: DMSetType(*dm, DMPLEX);
55: DMSetFromOptions(*dm);
56: DMViewFromOptions(*dm, NULL, "-dm_view");
57: return(0);
58: }
60: /* u - (x^2 + y^2) */
61: void f0_u(PetscInt dim, PetscInt Nf, PetscInt NfAux,
62: const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
63: const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
64: PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
65: {
66: f0[0] = u[0] - (x[0]*x[0] + x[1]*x[1]);
67: }
68: /* a \nabla\lambda */
69: void f1_u(PetscInt dim, PetscInt Nf, PetscInt NfAux,
70: const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
71: const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
72: PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f1[])
73: {
74: PetscInt d;
75: for (d = 0; d < dim; ++d) f1[d] = u[1]*u_x[dim*2+d];
76: }
77: /* I */
78: void g0_uu(PetscInt dim, PetscInt Nf, PetscInt NfAux,
79: const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
80: const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
81: PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g0[])
82: {
83: g0[0] = 1.0;
84: }
85: /* \nabla */
86: void g2_ua(PetscInt dim, PetscInt Nf, PetscInt NfAux,
87: const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
88: const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
89: PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g2[])
90: {
91: PetscInt d;
92: for (d = 0; d < dim; ++d) g2[d] = u_x[dim*2+d];
93: }
94: /* a */
95: void g3_ul(PetscInt dim, PetscInt Nf, PetscInt NfAux,
96: const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
97: const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
98: PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g3[])
99: {
100: PetscInt d;
101: for (d = 0; d < dim; ++d) g3[d*dim+d] = u[1];
102: }
103: /* a - (x + y) */
104: void f0_a(PetscInt dim, PetscInt Nf, PetscInt NfAux,
105: const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
106: const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
107: PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
108: {
109: f0[0] = u[1] - (x[0] + x[1]);
110: }
111: /* \lambda \nabla u */
112: void f1_a(PetscInt dim, PetscInt Nf, PetscInt NfAux,
113: const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
114: const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
115: PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f1[])
116: {
117: PetscInt d;
118: for (d = 0; d < dim; ++d) f1[d] = u[2]*u_x[d];
119: }
120: /* I */
121: void g0_aa(PetscInt dim, PetscInt Nf, PetscInt NfAux,
122: const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
123: const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
124: PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g0[])
125: {
126: g0[0] = 1.0;
127: }
128: /* 6 (x + y) */
129: void f0_l(PetscInt dim, PetscInt Nf, PetscInt NfAux,
130: const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
131: const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
132: PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
133: {
134: f0[0] = 6.0*(x[0] + x[1]);
135: }
136: /* a \nabla u */
137: void f1_l(PetscInt dim, PetscInt Nf, PetscInt NfAux,
138: const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
139: const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
140: PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f1[])
141: {
142: PetscInt d;
143: for (d = 0; d < dim; ++d) f1[d] = u[1]*u_x[d];
144: }
145: /* \nabla u */
146: void g2_la(PetscInt dim, PetscInt Nf, PetscInt NfAux,
147: const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
148: const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
149: PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g2[])
150: {
151: PetscInt d;
152: for (d = 0; d < dim; ++d) g2[d] = u_x[d];
153: }
154: /* a */
155: void g3_lu(PetscInt dim, PetscInt Nf, PetscInt NfAux,
156: const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
157: const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
158: PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g3[])
159: {
160: PetscInt d;
161: for (d = 0; d < dim; ++d) g3[d*dim+d] = u[1];
162: }
164: /*
165: In 2D for Dirichlet conditions with a variable coefficient, we use exact solution:
167: u = x^2 + y^2
168: f = 6 (x + y)
169: kappa(a) = a = (x + y)
171: so that
173: -\div \kappa(a) \grad u + f = -6 (x + y) + 6 (x + y) = 0
174: */
175: PetscErrorCode quadratic_u_2d(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar *u, void *ctx)
176: {
177: *u = x[0]*x[0] + x[1]*x[1];
178: return 0;
179: }
180: PetscErrorCode linear_a_2d(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar *a, void *ctx)
181: {
182: *a = x[0] + x[1];
183: return 0;
184: }
185: PetscErrorCode zero(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar *l, void *ctx)
186: {
187: *l = 0.0;
188: return 0;
189: }
191: PetscErrorCode SetupProblem(DM dm, AppCtx *user)
192: {
193: PetscDS ds;
194: DMLabel label;
195: const PetscInt id = 1;
199: DMGetDS(dm, &ds);
200: PetscDSSetResidual(ds, 0, f0_u, f1_u);
201: PetscDSSetResidual(ds, 1, f0_a, f1_a);
202: PetscDSSetResidual(ds, 2, f0_l, f1_l);
203: PetscDSSetJacobian(ds, 0, 0, g0_uu, NULL, NULL, NULL);
204: PetscDSSetJacobian(ds, 0, 1, NULL, NULL, g2_ua, NULL);
205: PetscDSSetJacobian(ds, 0, 2, NULL, NULL, NULL, g3_ul);
206: PetscDSSetJacobian(ds, 1, 1, g0_aa, NULL, NULL, NULL);
207: PetscDSSetJacobian(ds, 2, 1, NULL, NULL, g2_la, NULL);
208: PetscDSSetJacobian(ds, 2, 0, NULL, NULL, NULL, g3_lu);
210: PetscDSSetExactSolution(ds, 0, quadratic_u_2d, NULL);
211: PetscDSSetExactSolution(ds, 1, linear_a_2d, NULL);
212: PetscDSSetExactSolution(ds, 2, zero, NULL);
213: DMGetLabel(dm, "marker", &label);
214: DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 0, 0, NULL, (void (*)(void)) quadratic_u_2d, NULL, user, NULL);
215: DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 1, 0, NULL, (void (*)(void)) linear_a_2d, NULL, user, NULL);
216: DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 2, 0, NULL, (void (*)(void)) zero, NULL, user, NULL);
217: return(0);
218: }
220: PetscErrorCode SetupDiscretization(DM dm, AppCtx *user)
221: {
222: DM cdm = dm;
223: const PetscInt dim = 2;
224: PetscFE fe[3];
225: PetscInt f;
226: MPI_Comm comm;
227: PetscErrorCode ierr;
230: /* Create finite element */
231: PetscObjectGetComm((PetscObject) dm, &comm);
232: PetscFECreateDefault(comm, dim, 1, PETSC_TRUE, "potential_", -1, &fe[0]);
233: PetscObjectSetName((PetscObject) fe[0], "potential");
234: PetscFECreateDefault(comm, dim, 1, PETSC_TRUE, "conductivity_", -1, &fe[1]);
235: PetscObjectSetName((PetscObject) fe[1], "conductivity");
236: PetscFECopyQuadrature(fe[0], fe[1]);
237: PetscFECreateDefault(comm, dim, 1, PETSC_TRUE, "multiplier_", -1, &fe[2]);
238: PetscObjectSetName((PetscObject) fe[2], "multiplier");
239: PetscFECopyQuadrature(fe[0], fe[2]);
240: /* Set discretization and boundary conditions for each mesh */
241: for (f = 0; f < 3; ++f) {DMSetField(dm, f, NULL, (PetscObject) fe[f]);}
242: DMCreateDS(dm);
243: SetupProblem(dm, user);
244: while (cdm) {
245: DMCopyDisc(dm, cdm);
246: DMGetCoarseDM(cdm, &cdm);
247: }
248: for (f = 0; f < 3; ++f) {PetscFEDestroy(&fe[f]);}
249: return(0);
250: }
252: int main(int argc, char **argv)
253: {
254: DM dm;
255: SNES snes;
256: Vec u, r;
257: AppCtx user;
260: PetscInitialize(&argc, &argv, NULL,help);if (ierr) return ierr;
261: ProcessOptions(PETSC_COMM_WORLD, &user);
262: SNESCreate(PETSC_COMM_WORLD, &snes);
263: CreateMesh(PETSC_COMM_WORLD, &user, &dm);
264: SNESSetDM(snes, dm);
265: SetupDiscretization(dm, &user);
267: DMCreateGlobalVector(dm, &u);
268: PetscObjectSetName((PetscObject) u, "solution");
269: VecDuplicate(u, &r);
270: DMPlexSetSNESLocalFEM(dm,&user,&user,&user);
271: SNESSetFromOptions(snes);
273: DMSNESCheckFromOptions(snes, u);
274: if (user.runType == RUN_FULL) {
275: PetscDS ds;
276: PetscErrorCode (*exactFuncs[3])(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar *u, void *ctx);
277: PetscErrorCode (*initialGuess[3])(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar u[], void *ctx);
278: PetscReal error;
280: DMGetDS(dm, &ds);
281: PetscDSGetExactSolution(ds, 0, &exactFuncs[0], NULL);
282: PetscDSGetExactSolution(ds, 1, &exactFuncs[1], NULL);
283: PetscDSGetExactSolution(ds, 2, &exactFuncs[2], NULL);
284: initialGuess[0] = zero;
285: initialGuess[1] = zero;
286: initialGuess[2] = zero;
287: DMProjectFunction(dm, 0.0, initialGuess, NULL, INSERT_VALUES, u);
288: VecViewFromOptions(u, NULL, "-initial_vec_view");
289: DMComputeL2Diff(dm, 0.0, exactFuncs, NULL, u, &error);
290: if (error < 1.0e-11) {PetscPrintf(PETSC_COMM_WORLD, "Initial L_2 Error: < 1.0e-11\n");}
291: else {PetscPrintf(PETSC_COMM_WORLD, "Initial L_2 Error: %g\n", error);}
292: SNESSolve(snes, NULL, u);
293: DMComputeL2Diff(dm, 0.0, exactFuncs, NULL, u, &error);
294: if (error < 1.0e-11) {PetscPrintf(PETSC_COMM_WORLD, "Final L_2 Error: < 1.0e-11\n");}
295: else {PetscPrintf(PETSC_COMM_WORLD, "Final L_2 Error: %g\n", error);}
296: }
297: VecViewFromOptions(u, NULL, "-sol_vec_view");
299: VecDestroy(&u);
300: VecDestroy(&r);
301: SNESDestroy(&snes);
302: DMDestroy(&dm);
303: PetscFinalize();
304: return ierr;
305: }
307: /*TEST
309: build:
310: requires: !complex
312: test:
313: suffix: 0
314: requires: triangle
315: args: -run_type test -dmsnes_check -potential_petscspace_degree 2 -conductivity_petscspace_degree 1 -multiplier_petscspace_degree 2
317: test:
318: suffix: 1
319: requires: triangle
320: 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
322: TEST*/