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 PetscErrorCode ierr; 51 52 PetscFunctionBeginUser; 53 ierr = DMCreate(comm, dm);CHKERRQ(ierr); 54 ierr = DMSetType(*dm, DMPLEX);CHKERRQ(ierr); 55 ierr = DMSetFromOptions(*dm);CHKERRQ(ierr); 56 ierr = DMViewFromOptions(*dm, NULL, "-dm_view");CHKERRQ(ierr); 57 PetscFunctionReturn(0); 58 } 59 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 } 163 164 /* 165 In 2D for Dirichlet conditions with a variable coefficient, we use exact solution: 166 167 u = x^2 + y^2 168 f = 6 (x + y) 169 kappa(a) = a = (x + y) 170 171 so that 172 173 -\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 } 190 191 PetscErrorCode SetupProblem(DM dm, AppCtx *user) 192 { 193 PetscDS ds; 194 DMLabel label; 195 const PetscInt id = 1; 196 PetscErrorCode ierr; 197 198 PetscFunctionBeginUser; 199 ierr = DMGetDS(dm, &ds);CHKERRQ(ierr); 200 ierr = PetscDSSetResidual(ds, 0, f0_u, f1_u);CHKERRQ(ierr); 201 ierr = PetscDSSetResidual(ds, 1, f0_a, f1_a);CHKERRQ(ierr); 202 ierr = PetscDSSetResidual(ds, 2, f0_l, f1_l);CHKERRQ(ierr); 203 ierr = PetscDSSetJacobian(ds, 0, 0, g0_uu, NULL, NULL, NULL);CHKERRQ(ierr); 204 ierr = PetscDSSetJacobian(ds, 0, 1, NULL, NULL, g2_ua, NULL);CHKERRQ(ierr); 205 ierr = PetscDSSetJacobian(ds, 0, 2, NULL, NULL, NULL, g3_ul);CHKERRQ(ierr); 206 ierr = PetscDSSetJacobian(ds, 1, 1, g0_aa, NULL, NULL, NULL);CHKERRQ(ierr); 207 ierr = PetscDSSetJacobian(ds, 2, 1, NULL, NULL, g2_la, NULL);CHKERRQ(ierr); 208 ierr = PetscDSSetJacobian(ds, 2, 0, NULL, NULL, NULL, g3_lu);CHKERRQ(ierr); 209 210 ierr = PetscDSSetExactSolution(ds, 0, quadratic_u_2d, NULL);CHKERRQ(ierr); 211 ierr = PetscDSSetExactSolution(ds, 1, linear_a_2d, NULL);CHKERRQ(ierr); 212 ierr = PetscDSSetExactSolution(ds, 2, zero, NULL);CHKERRQ(ierr); 213 ierr = DMGetLabel(dm, "marker", &label);CHKERRQ(ierr); 214 ierr = DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 0, 0, NULL, (void (*)(void)) quadratic_u_2d, NULL, user, NULL);CHKERRQ(ierr); 215 ierr = DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 1, 0, NULL, (void (*)(void)) linear_a_2d, NULL, user, NULL);CHKERRQ(ierr); 216 ierr = DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 2, 0, NULL, (void (*)(void)) zero, NULL, user, NULL);CHKERRQ(ierr); 217 PetscFunctionReturn(0); 218 } 219 220 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; 228 229 PetscFunctionBeginUser; 230 /* Create finite element */ 231 ierr = PetscObjectGetComm((PetscObject) dm, &comm);CHKERRQ(ierr); 232 ierr = PetscFECreateDefault(comm, dim, 1, PETSC_TRUE, "potential_", -1, &fe[0]);CHKERRQ(ierr); 233 ierr = PetscObjectSetName((PetscObject) fe[0], "potential");CHKERRQ(ierr); 234 ierr = PetscFECreateDefault(comm, dim, 1, PETSC_TRUE, "conductivity_", -1, &fe[1]);CHKERRQ(ierr); 235 ierr = PetscObjectSetName((PetscObject) fe[1], "conductivity");CHKERRQ(ierr); 236 ierr = PetscFECopyQuadrature(fe[0], fe[1]);CHKERRQ(ierr); 237 ierr = PetscFECreateDefault(comm, dim, 1, PETSC_TRUE, "multiplier_", -1, &fe[2]);CHKERRQ(ierr); 238 ierr = PetscObjectSetName((PetscObject) fe[2], "multiplier");CHKERRQ(ierr); 239 ierr = PetscFECopyQuadrature(fe[0], fe[2]);CHKERRQ(ierr); 240 /* Set discretization and boundary conditions for each mesh */ 241 for (f = 0; f < 3; ++f) {ierr = DMSetField(dm, f, NULL, (PetscObject) fe[f]);CHKERRQ(ierr);} 242 ierr = DMCreateDS(dm);CHKERRQ(ierr); 243 ierr = SetupProblem(dm, user);CHKERRQ(ierr); 244 while (cdm) { 245 ierr = DMCopyDisc(dm, cdm);CHKERRQ(ierr); 246 ierr = DMGetCoarseDM(cdm, &cdm);CHKERRQ(ierr); 247 } 248 for (f = 0; f < 3; ++f) {ierr = PetscFEDestroy(&fe[f]);CHKERRQ(ierr);} 249 PetscFunctionReturn(0); 250 } 251 252 int main(int argc, char **argv) 253 { 254 DM dm; 255 SNES snes; 256 Vec u, r; 257 AppCtx user; 258 PetscErrorCode ierr; 259 260 ierr = PetscInitialize(&argc, &argv, NULL,help);if (ierr) return ierr; 261 ierr = ProcessOptions(PETSC_COMM_WORLD, &user);CHKERRQ(ierr); 262 ierr = SNESCreate(PETSC_COMM_WORLD, &snes);CHKERRQ(ierr); 263 ierr = CreateMesh(PETSC_COMM_WORLD, &user, &dm);CHKERRQ(ierr); 264 ierr = SNESSetDM(snes, dm);CHKERRQ(ierr); 265 ierr = SetupDiscretization(dm, &user);CHKERRQ(ierr); 266 267 ierr = DMCreateGlobalVector(dm, &u);CHKERRQ(ierr); 268 ierr = PetscObjectSetName((PetscObject) u, "solution");CHKERRQ(ierr); 269 ierr = VecDuplicate(u, &r);CHKERRQ(ierr); 270 ierr = DMPlexSetSNESLocalFEM(dm,&user,&user,&user);CHKERRQ(ierr); 271 ierr = SNESSetFromOptions(snes);CHKERRQ(ierr); 272 273 ierr = DMSNESCheckFromOptions(snes, u);CHKERRQ(ierr); 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; 279 280 ierr = DMGetDS(dm, &ds);CHKERRQ(ierr); 281 ierr = PetscDSGetExactSolution(ds, 0, &exactFuncs[0], NULL);CHKERRQ(ierr); 282 ierr = PetscDSGetExactSolution(ds, 1, &exactFuncs[1], NULL);CHKERRQ(ierr); 283 ierr = PetscDSGetExactSolution(ds, 2, &exactFuncs[2], NULL);CHKERRQ(ierr); 284 initialGuess[0] = zero; 285 initialGuess[1] = zero; 286 initialGuess[2] = zero; 287 ierr = DMProjectFunction(dm, 0.0, initialGuess, NULL, INSERT_VALUES, u);CHKERRQ(ierr); 288 ierr = VecViewFromOptions(u, NULL, "-initial_vec_view");CHKERRQ(ierr); 289 ierr = DMComputeL2Diff(dm, 0.0, exactFuncs, NULL, u, &error);CHKERRQ(ierr); 290 if (error < 1.0e-11) {ierr = PetscPrintf(PETSC_COMM_WORLD, "Initial L_2 Error: < 1.0e-11\n");CHKERRQ(ierr);} 291 else {ierr = PetscPrintf(PETSC_COMM_WORLD, "Initial L_2 Error: %g\n", error);CHKERRQ(ierr);} 292 ierr = SNESSolve(snes, NULL, u);CHKERRQ(ierr); 293 ierr = DMComputeL2Diff(dm, 0.0, exactFuncs, NULL, u, &error);CHKERRQ(ierr); 294 if (error < 1.0e-11) {ierr = PetscPrintf(PETSC_COMM_WORLD, "Final L_2 Error: < 1.0e-11\n");CHKERRQ(ierr);} 295 else {ierr = PetscPrintf(PETSC_COMM_WORLD, "Final L_2 Error: %g\n", error);CHKERRQ(ierr);} 296 } 297 ierr = VecViewFromOptions(u, NULL, "-sol_vec_view");CHKERRQ(ierr); 298 299 ierr = VecDestroy(&u);CHKERRQ(ierr); 300 ierr = VecDestroy(&r);CHKERRQ(ierr); 301 ierr = SNESDestroy(&snes);CHKERRQ(ierr); 302 ierr = DMDestroy(&dm);CHKERRQ(ierr); 303 ierr = PetscFinalize(); 304 return ierr; 305 } 306 307 /*TEST 308 309 build: 310 requires: !complex 311 312 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 2 316 317 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 321 322 TEST*/ 323