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 { 26 RUN_FULL, 27 RUN_TEST 28 } RunType; 29 30 typedef struct { 31 RunType runType; /* Whether to run tests, or solve the full problem */ 32 } AppCtx; 33 34 static PetscErrorCode ProcessOptions(MPI_Comm comm, AppCtx *options) 35 { 36 const char *runTypes[2] = {"full", "test"}; 37 PetscInt run; 38 39 PetscFunctionBeginUser; 40 options->runType = RUN_FULL; 41 PetscOptionsBegin(comm, "", "Inverse Problem Options", "DMPLEX"); 42 run = options->runType; 43 PetscCall(PetscOptionsEList("-run_type", "The run type", "ex1.c", runTypes, 2, runTypes[options->runType], &run, NULL)); 44 options->runType = (RunType)run; 45 PetscOptionsEnd(); 46 PetscFunctionReturn(PETSC_SUCCESS); 47 } 48 49 static PetscErrorCode CreateMesh(MPI_Comm comm, AppCtx *user, DM *dm) 50 { 51 PetscFunctionBeginUser; 52 PetscCall(DMCreate(comm, dm)); 53 PetscCall(DMSetType(*dm, DMPLEX)); 54 PetscCall(DMSetFromOptions(*dm)); 55 PetscCall(DMViewFromOptions(*dm, NULL, "-dm_view")); 56 PetscFunctionReturn(PETSC_SUCCESS); 57 } 58 59 /* u - (x^2 + y^2) */ 60 void f0_u(PetscInt dim, PetscInt Nf, PetscInt NfAux, const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[], const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[], PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[]) 61 { 62 f0[0] = u[0] - (x[0] * x[0] + x[1] * x[1]); 63 } 64 /* a \nabla\lambda */ 65 void f1_u(PetscInt dim, PetscInt Nf, PetscInt NfAux, const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[], const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[], PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f1[]) 66 { 67 PetscInt d; 68 for (d = 0; d < dim; ++d) f1[d] = u[1] * u_x[dim * 2 + d]; 69 } 70 /* I */ 71 void g0_uu(PetscInt dim, PetscInt Nf, PetscInt NfAux, const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[], const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[], PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g0[]) 72 { 73 g0[0] = 1.0; 74 } 75 /* \nabla */ 76 void g2_ua(PetscInt dim, PetscInt Nf, PetscInt NfAux, const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[], const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[], PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g2[]) 77 { 78 PetscInt d; 79 for (d = 0; d < dim; ++d) g2[d] = u_x[dim * 2 + d]; 80 } 81 /* a */ 82 void g3_ul(PetscInt dim, PetscInt Nf, PetscInt NfAux, const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[], const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[], PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g3[]) 83 { 84 PetscInt d; 85 for (d = 0; d < dim; ++d) g3[d * dim + d] = u[1]; 86 } 87 /* a - (x + y) */ 88 void f0_a(PetscInt dim, PetscInt Nf, PetscInt NfAux, const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[], const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[], PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[]) 89 { 90 f0[0] = u[1] - (x[0] + x[1]); 91 } 92 /* \lambda \nabla u */ 93 void f1_a(PetscInt dim, PetscInt Nf, PetscInt NfAux, const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[], const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[], PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f1[]) 94 { 95 PetscInt d; 96 for (d = 0; d < dim; ++d) f1[d] = u[2] * u_x[d]; 97 } 98 /* I */ 99 void g0_aa(PetscInt dim, PetscInt Nf, PetscInt NfAux, const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[], const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[], PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g0[]) 100 { 101 g0[0] = 1.0; 102 } 103 /* 6 (x + y) */ 104 void f0_l(PetscInt dim, PetscInt Nf, PetscInt NfAux, const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[], const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[], PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[]) 105 { 106 f0[0] = 6.0 * (x[0] + x[1]); 107 } 108 /* a \nabla u */ 109 void f1_l(PetscInt dim, PetscInt Nf, PetscInt NfAux, const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[], const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[], PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f1[]) 110 { 111 PetscInt d; 112 for (d = 0; d < dim; ++d) f1[d] = u[1] * u_x[d]; 113 } 114 /* \nabla u */ 115 void g2_la(PetscInt dim, PetscInt Nf, PetscInt NfAux, const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[], const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[], PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g2[]) 116 { 117 PetscInt d; 118 for (d = 0; d < dim; ++d) g2[d] = u_x[d]; 119 } 120 /* a */ 121 void g3_lu(PetscInt dim, PetscInt Nf, PetscInt NfAux, const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[], const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[], PetscReal t, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g3[]) 122 { 123 PetscInt d; 124 for (d = 0; d < dim; ++d) g3[d * dim + d] = u[1]; 125 } 126 127 /* 128 In 2D for Dirichlet conditions with a variable coefficient, we use exact solution: 129 130 u = x^2 + y^2 131 f = 6 (x + y) 132 kappa(a) = a = (x + y) 133 134 so that 135 136 -\div \kappa(a) \grad u + f = -6 (x + y) + 6 (x + y) = 0 137 */ 138 PetscErrorCode quadratic_u_2d(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar *u, void *ctx) 139 { 140 *u = x[0] * x[0] + x[1] * x[1]; 141 return PETSC_SUCCESS; 142 } 143 PetscErrorCode linear_a_2d(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar *a, void *ctx) 144 { 145 *a = x[0] + x[1]; 146 return PETSC_SUCCESS; 147 } 148 PetscErrorCode zero(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar *l, void *ctx) 149 { 150 *l = 0.0; 151 return PETSC_SUCCESS; 152 } 153 154 PetscErrorCode SetupProblem(DM dm, AppCtx *user) 155 { 156 PetscDS ds; 157 DMLabel label; 158 const PetscInt id = 1; 159 160 PetscFunctionBeginUser; 161 PetscCall(DMGetDS(dm, &ds)); 162 PetscCall(PetscDSSetResidual(ds, 0, f0_u, f1_u)); 163 PetscCall(PetscDSSetResidual(ds, 1, f0_a, f1_a)); 164 PetscCall(PetscDSSetResidual(ds, 2, f0_l, f1_l)); 165 PetscCall(PetscDSSetJacobian(ds, 0, 0, g0_uu, NULL, NULL, NULL)); 166 PetscCall(PetscDSSetJacobian(ds, 0, 1, NULL, NULL, g2_ua, NULL)); 167 PetscCall(PetscDSSetJacobian(ds, 0, 2, NULL, NULL, NULL, g3_ul)); 168 PetscCall(PetscDSSetJacobian(ds, 1, 1, g0_aa, NULL, NULL, NULL)); 169 PetscCall(PetscDSSetJacobian(ds, 2, 1, NULL, NULL, g2_la, NULL)); 170 PetscCall(PetscDSSetJacobian(ds, 2, 0, NULL, NULL, NULL, g3_lu)); 171 172 PetscCall(PetscDSSetExactSolution(ds, 0, quadratic_u_2d, NULL)); 173 PetscCall(PetscDSSetExactSolution(ds, 1, linear_a_2d, NULL)); 174 PetscCall(PetscDSSetExactSolution(ds, 2, zero, NULL)); 175 PetscCall(DMGetLabel(dm, "marker", &label)); 176 PetscCall(DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 0, 0, NULL, (PetscVoidFn *)quadratic_u_2d, NULL, user, NULL)); 177 PetscCall(DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 1, 0, NULL, (PetscVoidFn *)linear_a_2d, NULL, user, NULL)); 178 PetscCall(DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 2, 0, NULL, (PetscVoidFn *)zero, NULL, user, NULL)); 179 PetscFunctionReturn(PETSC_SUCCESS); 180 } 181 182 PetscErrorCode SetupDiscretization(DM dm, AppCtx *user) 183 { 184 DM cdm = dm; 185 const PetscInt dim = 2; 186 PetscFE fe[3]; 187 PetscInt f; 188 MPI_Comm comm; 189 190 PetscFunctionBeginUser; 191 /* Create finite element */ 192 PetscCall(PetscObjectGetComm((PetscObject)dm, &comm)); 193 PetscCall(PetscFECreateDefault(comm, dim, 1, PETSC_TRUE, "potential_", -1, &fe[0])); 194 PetscCall(PetscObjectSetName((PetscObject)fe[0], "potential")); 195 PetscCall(PetscFECreateDefault(comm, dim, 1, PETSC_TRUE, "conductivity_", -1, &fe[1])); 196 PetscCall(PetscObjectSetName((PetscObject)fe[1], "conductivity")); 197 PetscCall(PetscFECopyQuadrature(fe[0], fe[1])); 198 PetscCall(PetscFECreateDefault(comm, dim, 1, PETSC_TRUE, "multiplier_", -1, &fe[2])); 199 PetscCall(PetscObjectSetName((PetscObject)fe[2], "multiplier")); 200 PetscCall(PetscFECopyQuadrature(fe[0], fe[2])); 201 /* Set discretization and boundary conditions for each mesh */ 202 for (f = 0; f < 3; ++f) PetscCall(DMSetField(dm, f, NULL, (PetscObject)fe[f])); 203 PetscCall(DMCreateDS(dm)); 204 PetscCall(SetupProblem(dm, user)); 205 while (cdm) { 206 PetscCall(DMCopyDisc(dm, cdm)); 207 PetscCall(DMGetCoarseDM(cdm, &cdm)); 208 } 209 for (f = 0; f < 3; ++f) PetscCall(PetscFEDestroy(&fe[f])); 210 PetscFunctionReturn(PETSC_SUCCESS); 211 } 212 213 int main(int argc, char **argv) 214 { 215 DM dm; 216 SNES snes; 217 Vec u, r; 218 AppCtx user; 219 220 PetscFunctionBeginUser; 221 PetscCall(PetscInitialize(&argc, &argv, NULL, help)); 222 PetscCall(ProcessOptions(PETSC_COMM_WORLD, &user)); 223 PetscCall(SNESCreate(PETSC_COMM_WORLD, &snes)); 224 PetscCall(CreateMesh(PETSC_COMM_WORLD, &user, &dm)); 225 PetscCall(SNESSetDM(snes, dm)); 226 PetscCall(SetupDiscretization(dm, &user)); 227 228 PetscCall(DMCreateGlobalVector(dm, &u)); 229 PetscCall(PetscObjectSetName((PetscObject)u, "solution")); 230 PetscCall(VecDuplicate(u, &r)); 231 PetscCall(DMPlexSetSNESLocalFEM(dm, PETSC_FALSE, &user)); 232 PetscCall(SNESSetFromOptions(snes)); 233 234 PetscCall(DMSNESCheckFromOptions(snes, u)); 235 if (user.runType == RUN_FULL) { 236 PetscDS ds; 237 PetscErrorCode (*exactFuncs[3])(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar *u, void *ctx); 238 PetscErrorCode (*initialGuess[3])(PetscInt dim, PetscReal t, const PetscReal x[], PetscInt Nf, PetscScalar u[], void *ctx); 239 PetscReal error; 240 241 PetscCall(DMGetDS(dm, &ds)); 242 PetscCall(PetscDSGetExactSolution(ds, 0, &exactFuncs[0], NULL)); 243 PetscCall(PetscDSGetExactSolution(ds, 1, &exactFuncs[1], NULL)); 244 PetscCall(PetscDSGetExactSolution(ds, 2, &exactFuncs[2], NULL)); 245 initialGuess[0] = zero; 246 initialGuess[1] = zero; 247 initialGuess[2] = zero; 248 PetscCall(DMProjectFunction(dm, 0.0, initialGuess, NULL, INSERT_VALUES, u)); 249 PetscCall(VecViewFromOptions(u, NULL, "-initial_vec_view")); 250 PetscCall(DMComputeL2Diff(dm, 0.0, exactFuncs, NULL, u, &error)); 251 if (error < 1.0e-11) PetscCall(PetscPrintf(PETSC_COMM_WORLD, "Initial L_2 Error: < 1.0e-11\n")); 252 else PetscCall(PetscPrintf(PETSC_COMM_WORLD, "Initial L_2 Error: %g\n", (double)error)); 253 PetscCall(SNESSolve(snes, NULL, u)); 254 PetscCall(DMComputeL2Diff(dm, 0.0, exactFuncs, NULL, u, &error)); 255 if (error < 1.0e-11) PetscCall(PetscPrintf(PETSC_COMM_WORLD, "Final L_2 Error: < 1.0e-11\n")); 256 else PetscCall(PetscPrintf(PETSC_COMM_WORLD, "Final L_2 Error: %g\n", (double)error)); 257 } 258 PetscCall(VecViewFromOptions(u, NULL, "-sol_vec_view")); 259 260 PetscCall(VecDestroy(&u)); 261 PetscCall(VecDestroy(&r)); 262 PetscCall(SNESDestroy(&snes)); 263 PetscCall(DMDestroy(&dm)); 264 PetscCall(PetscFinalize()); 265 return 0; 266 } 267 268 /*TEST 269 270 build: 271 requires: !complex 272 273 test: 274 suffix: 0 275 requires: triangle 276 args: -run_type test -dmsnes_check -potential_petscspace_degree 2 -conductivity_petscspace_degree 1 -multiplier_petscspace_degree 2 277 278 test: 279 suffix: 1 280 requires: triangle 281 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 282 283 TEST*/ 284