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"; 3 4 /*F 5 We are solving the parameter estimation problem for the Laplacian. We will ask to minimize a Lagrangian 6 function over $a$ and $u$, given by 7 \begin{align} 8 L(u, a, \lambda) = \frac{1}{2} || Qu - d ||^2 + \frac{1}{2} || L (a - a_r) ||^2 + \lambda F(u; a) 9 \end{align} 10 where $Q$ is a sampling operator, $L$ is a regularization operator, $F$ defines the PDE. 11 12 Currently, we have perfect information, meaning $Q = I$, and then we need no regularization, $L = I$. We 13 also give the exact control for the reference $a_r$. 14 15 The PDE will be the Laplace equation with homogeneous boundary conditions 16 \begin{align} 17 -nabla \cdot a \nabla u = f 18 \end{align} 19 20 F*/ 21 22 #include <petsc.h> 23 #include <petscfe.h> 24 25 typedef enum {RUN_FULL, RUN_TEST} RunType; 26 27 typedef struct { 28 RunType runType; /* Whether to run tests, or solve the full problem */ 29 } AppCtx; 30 31 static PetscErrorCode ProcessOptions(MPI_Comm comm, AppCtx *options) 32 { 33 const char *runTypes[2] = {"full", "test"}; 34 PetscInt run; 35 PetscErrorCode ierr; 36 37 PetscFunctionBeginUser; 38 options->runType = RUN_FULL; 39 40 ierr = PetscOptionsBegin(comm, "", "Inverse Problem Options", "DMPLEX");CHKERRQ(ierr); 41 run = options->runType; 42 ierr = PetscOptionsEList("-run_type", "The run type", "ex1.c", runTypes, 2, runTypes[options->runType], &run, NULL);CHKERRQ(ierr); 43 options->runType = (RunType) run; 44 ierr = PetscOptionsEnd();CHKERRQ(ierr); 45 PetscFunctionReturn(0); 46 } 47 48 static PetscErrorCode CreateMesh(MPI_Comm comm, AppCtx *user, DM *dm) 49 { 50 DM distributedMesh = NULL; 51 PetscErrorCode ierr; 52 53 PetscFunctionBeginUser; 54 ierr = DMPlexCreateBoxMesh(comm, 2, PETSC_TRUE, NULL, NULL, NULL, NULL, PETSC_TRUE, dm);CHKERRQ(ierr); 55 ierr = PetscObjectSetName((PetscObject) *dm, "Mesh");CHKERRQ(ierr); 56 ierr = DMPlexDistribute(*dm, 0, NULL, &distributedMesh);CHKERRQ(ierr); 57 if (distributedMesh) { 58 ierr = DMDestroy(dm);CHKERRQ(ierr); 59 *dm = distributedMesh; 60 } 61 ierr = DMSetFromOptions(*dm);CHKERRQ(ierr); 62 ierr = DMViewFromOptions(*dm, NULL, "-dm_view");CHKERRQ(ierr); 63 PetscFunctionReturn(0); 64 } 65 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 } 169 170 /* 171 In 2D for Dirichlet conditions with a variable coefficient, we use exact solution: 172 173 u = x^2 + y^2 174 f = 6 (x + y) 175 kappa(a) = a = (x + y) 176 177 so that 178 179 -\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 } 196 197 PetscErrorCode SetupProblem(DM dm, AppCtx *user) 198 { 199 PetscDS ds; 200 DMLabel label; 201 const PetscInt id = 1; 202 PetscErrorCode ierr; 203 204 PetscFunctionBeginUser; 205 ierr = DMGetDS(dm, &ds);CHKERRQ(ierr); 206 ierr = PetscDSSetResidual(ds, 0, f0_u, f1_u);CHKERRQ(ierr); 207 ierr = PetscDSSetResidual(ds, 1, f0_a, f1_a);CHKERRQ(ierr); 208 ierr = PetscDSSetResidual(ds, 2, f0_l, f1_l);CHKERRQ(ierr); 209 ierr = PetscDSSetJacobian(ds, 0, 0, g0_uu, NULL, NULL, NULL);CHKERRQ(ierr); 210 ierr = PetscDSSetJacobian(ds, 0, 1, NULL, NULL, g2_ua, NULL);CHKERRQ(ierr); 211 ierr = PetscDSSetJacobian(ds, 0, 2, NULL, NULL, NULL, g3_ul);CHKERRQ(ierr); 212 ierr = PetscDSSetJacobian(ds, 1, 1, g0_aa, NULL, NULL, NULL);CHKERRQ(ierr); 213 ierr = PetscDSSetJacobian(ds, 2, 1, NULL, NULL, g2_la, NULL);CHKERRQ(ierr); 214 ierr = PetscDSSetJacobian(ds, 2, 0, NULL, NULL, NULL, g3_lu);CHKERRQ(ierr); 215 216 ierr = PetscDSSetExactSolution(ds, 0, quadratic_u_2d, NULL);CHKERRQ(ierr); 217 ierr = PetscDSSetExactSolution(ds, 1, linear_a_2d, NULL);CHKERRQ(ierr); 218 ierr = PetscDSSetExactSolution(ds, 2, zero, NULL);CHKERRQ(ierr); 219 ierr = DMGetLabel(dm, "marker", &label);CHKERRQ(ierr); 220 ierr = DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 0, 0, NULL, (void (*)(void)) quadratic_u_2d, NULL, user, NULL);CHKERRQ(ierr); 221 ierr = DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 1, 0, NULL, (void (*)(void)) linear_a_2d, NULL, user, NULL);CHKERRQ(ierr); 222 ierr = DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 2, 0, NULL, (void (*)(void)) zero, NULL, user, NULL);CHKERRQ(ierr); 223 PetscFunctionReturn(0); 224 } 225 226 PetscErrorCode SetupDiscretization(DM dm, AppCtx *user) 227 { 228 DM cdm = dm; 229 const PetscInt dim = 2; 230 PetscFE fe[3]; 231 PetscInt f; 232 MPI_Comm comm; 233 PetscErrorCode ierr; 234 235 PetscFunctionBeginUser; 236 /* Create finite element */ 237 ierr = PetscObjectGetComm((PetscObject) dm, &comm);CHKERRQ(ierr); 238 ierr = PetscFECreateDefault(comm, dim, 1, PETSC_TRUE, "potential_", -1, &fe[0]);CHKERRQ(ierr); 239 ierr = PetscObjectSetName((PetscObject) fe[0], "potential");CHKERRQ(ierr); 240 ierr = PetscFECreateDefault(comm, dim, 1, PETSC_TRUE, "conductivity_", -1, &fe[1]);CHKERRQ(ierr); 241 ierr = PetscObjectSetName((PetscObject) fe[1], "conductivity");CHKERRQ(ierr); 242 ierr = PetscFECopyQuadrature(fe[0], fe[1]);CHKERRQ(ierr); 243 ierr = PetscFECreateDefault(comm, dim, 1, PETSC_TRUE, "multiplier_", -1, &fe[2]);CHKERRQ(ierr); 244 ierr = PetscObjectSetName((PetscObject) fe[2], "multiplier");CHKERRQ(ierr); 245 ierr = PetscFECopyQuadrature(fe[0], fe[2]);CHKERRQ(ierr); 246 /* Set discretization and boundary conditions for each mesh */ 247 for (f = 0; f < 3; ++f) {ierr = DMSetField(dm, f, NULL, (PetscObject) fe[f]);CHKERRQ(ierr);} 248 ierr = DMCreateDS(dm);CHKERRQ(ierr); 249 ierr = SetupProblem(dm, user);CHKERRQ(ierr); 250 while (cdm) { 251 ierr = DMCopyDisc(dm, cdm);CHKERRQ(ierr); 252 ierr = DMGetCoarseDM(cdm, &cdm);CHKERRQ(ierr); 253 } 254 for (f = 0; f < 3; ++f) {ierr = PetscFEDestroy(&fe[f]);CHKERRQ(ierr);} 255 PetscFunctionReturn(0); 256 } 257 258 int main(int argc, char **argv) 259 { 260 DM dm; 261 SNES snes; 262 Vec u, r; 263 AppCtx user; 264 PetscErrorCode ierr; 265 266 ierr = PetscInitialize(&argc, &argv, NULL,help);if (ierr) return ierr; 267 ierr = ProcessOptions(PETSC_COMM_WORLD, &user);CHKERRQ(ierr); 268 ierr = SNESCreate(PETSC_COMM_WORLD, &snes);CHKERRQ(ierr); 269 ierr = CreateMesh(PETSC_COMM_WORLD, &user, &dm);CHKERRQ(ierr); 270 ierr = SNESSetDM(snes, dm);CHKERRQ(ierr); 271 ierr = SetupDiscretization(dm, &user);CHKERRQ(ierr); 272 273 ierr = DMCreateGlobalVector(dm, &u);CHKERRQ(ierr); 274 ierr = PetscObjectSetName((PetscObject) u, "solution");CHKERRQ(ierr); 275 ierr = VecDuplicate(u, &r);CHKERRQ(ierr); 276 ierr = DMPlexSetSNESLocalFEM(dm,&user,&user,&user);CHKERRQ(ierr); 277 ierr = SNESSetFromOptions(snes);CHKERRQ(ierr); 278 279 ierr = DMSNESCheckFromOptions(snes, u);CHKERRQ(ierr); 280 if (user.runType == RUN_FULL) { 281 PetscDS ds; 282 PetscErrorCode (*exactFuncs[3])(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar *u, void *ctx); 283 PetscErrorCode (*initialGuess[3])(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar u[], void *ctx); 284 PetscReal error; 285 286 ierr = DMGetDS(dm, &ds);CHKERRQ(ierr); 287 ierr = PetscDSGetExactSolution(ds, 0, &exactFuncs[0], NULL);CHKERRQ(ierr); 288 ierr = PetscDSGetExactSolution(ds, 1, &exactFuncs[1], NULL);CHKERRQ(ierr); 289 ierr = PetscDSGetExactSolution(ds, 2, &exactFuncs[2], NULL);CHKERRQ(ierr); 290 initialGuess[0] = zero; 291 initialGuess[1] = zero; 292 initialGuess[2] = zero; 293 ierr = DMProjectFunction(dm, 0.0, initialGuess, NULL, INSERT_VALUES, u);CHKERRQ(ierr); 294 ierr = VecViewFromOptions(u, NULL, "-initial_vec_view");CHKERRQ(ierr); 295 ierr = DMComputeL2Diff(dm, 0.0, exactFuncs, NULL, u, &error);CHKERRQ(ierr); 296 if (error < 1.0e-11) {ierr = PetscPrintf(PETSC_COMM_WORLD, "Initial L_2 Error: < 1.0e-11\n");CHKERRQ(ierr);} 297 else {ierr = PetscPrintf(PETSC_COMM_WORLD, "Initial L_2 Error: %g\n", error);CHKERRQ(ierr);} 298 ierr = SNESSolve(snes, NULL, u);CHKERRQ(ierr); 299 ierr = DMComputeL2Diff(dm, 0.0, exactFuncs, NULL, u, &error);CHKERRQ(ierr); 300 if (error < 1.0e-11) {ierr = PetscPrintf(PETSC_COMM_WORLD, "Final L_2 Error: < 1.0e-11\n");CHKERRQ(ierr);} 301 else {ierr = PetscPrintf(PETSC_COMM_WORLD, "Final L_2 Error: %g\n", error);CHKERRQ(ierr);} 302 } 303 ierr = VecViewFromOptions(u, NULL, "-sol_vec_view");CHKERRQ(ierr); 304 305 ierr = VecDestroy(&u);CHKERRQ(ierr); 306 ierr = VecDestroy(&r);CHKERRQ(ierr); 307 ierr = SNESDestroy(&snes);CHKERRQ(ierr); 308 ierr = DMDestroy(&dm);CHKERRQ(ierr); 309 ierr = PetscFinalize(); 310 return ierr; 311 } 312 313 /*TEST 314 315 build: 316 requires: !complex 317 318 test: 319 suffix: 0 320 requires: triangle 321 args: -run_type test -dmsnes_check -potential_petscspace_degree 2 -conductivity_petscspace_degree 1 -multiplier_petscspace_degree 2 322 323 test: 324 suffix: 1 325 requires: triangle 326 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 327 328 TEST*/ 329