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