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 $y$ and $u$, given by 7 \begin{align} 8 L(u, a, \lambda) = \frac{1}{2} || Qu - d_A ||^2 || Qu - d_B ||^2 + \frac{\beta}{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 null vector for the reference control $a_r$. Right now $\beta = 1$. 14 15 The PDE will be the Laplace equation with homogeneous boundary conditions 16 \begin{align} 17 -Delta u = a 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 PetscBool useDualPenalty; /* Penalize deviation from both goals */ 33 } AppCtx; 34 35 static PetscErrorCode ProcessOptions(MPI_Comm comm, AppCtx *options) 36 { 37 const char *runTypes[2] = {"full", "test"}; 38 PetscInt run; 39 40 PetscFunctionBeginUser; 41 options->runType = RUN_FULL; 42 options->useDualPenalty = PETSC_FALSE; 43 PetscOptionsBegin(comm, "", "Inverse Problem Options", "DMPLEX"); 44 run = options->runType; 45 PetscCall(PetscOptionsEList("-run_type", "The run type", "ex2.c", runTypes, 2, runTypes[options->runType], &run, NULL)); 46 options->runType = (RunType)run; 47 PetscCall(PetscOptionsBool("-use_dual_penalty", "Penalize deviation from both goals", "ex2.c", options->useDualPenalty, &options->useDualPenalty, NULL)); 48 PetscOptionsEnd(); 49 PetscFunctionReturn(PETSC_SUCCESS); 50 } 51 52 static PetscErrorCode CreateMesh(MPI_Comm comm, AppCtx *user, DM *dm) 53 { 54 PetscFunctionBeginUser; 55 PetscCall(DMCreate(comm, dm)); 56 PetscCall(DMSetType(*dm, DMPLEX)); 57 PetscCall(DMSetFromOptions(*dm)); 58 PetscCall(DMViewFromOptions(*dm, NULL, "-dm_view")); 59 PetscFunctionReturn(PETSC_SUCCESS); 60 } 61 62 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[]) 63 { 64 f0[0] = (u[0] - (x[0] * x[0] + x[1] * x[1])); 65 } 66 void f0_u_full(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[]) 67 { 68 f0[0] = (u[0] - (x[0] * x[0] + x[1] * x[1])) * PetscSqr(u[0] - (sin(2.0 * PETSC_PI * x[0]) * sin(2.0 * PETSC_PI * x[1]))) + PetscSqr(u[0] - (x[0] * x[0] + x[1] * x[1])) * (u[0] - (sin(2.0 * PETSC_PI * x[0]) * sin(2.0 * PETSC_PI * x[1]))); 69 } 70 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[]) 71 { 72 PetscInt d; 73 for (d = 0; d < dim; ++d) f1[d] = u_x[dim * 2 + d]; 74 } 75 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[]) 76 { 77 g0[0] = 1.0; 78 } 79 void g0_uu_full(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[]) 80 { 81 g0[0] = PetscSqr(u[0] - sin(2.0 * PETSC_PI * x[0]) * sin(2.0 * PETSC_PI * x[1])) + PetscSqr(u[0] - (x[0] * x[0] + x[1] * x[1])) - 2.0 * ((x[0] * x[0] + x[1] * x[1]) + (sin(2.0 * PETSC_PI * x[0]) * sin(2.0 * PETSC_PI * x[1]))) * u[0] + 4.0 * (x[0] * x[0] + x[1] * x[1]) * (sin(2.0 * PETSC_PI * x[0]) * sin(2.0 * PETSC_PI * x[1])); 82 } 83 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[]) 84 { 85 PetscInt d; 86 for (d = 0; d < dim; ++d) g3[d * dim + d] = 1.0; 87 } 88 89 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[]) 90 { 91 f0[0] = u[1] - 4.0 /* 0.0 */ + u[2]; 92 } 93 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[]) 94 { 95 g0[0] = 1.0; 96 } 97 void g0_al(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[]) 98 { 99 g0[0] = 1.0; 100 } 101 102 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[]) 103 { 104 f0[0] = u[1]; 105 } 106 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[]) 107 { 108 PetscInt d; 109 for (d = 0; d < dim; ++d) f1[d] = u_x[d]; 110 } 111 void g0_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 g0[]) 112 { 113 g0[0] = 1.0; 114 } 115 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[]) 116 { 117 PetscInt d; 118 for (d = 0; d < dim; ++d) g3[d * dim + d] = 1.0; 119 } 120 121 /* 122 In 2D for Dirichlet conditions with a variable coefficient, we use exact solution: 123 124 u = x^2 + y^2 125 a = 4 126 d_A = 4 127 d_B = sin(2*pi*x[0]) * sin(2*pi*x[1]) 128 129 so that 130 131 -\Delta u + a = -4 + 4 = 0 132 */ 133 PetscErrorCode quadratic_u_2d(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar *u, void *ctx) 134 { 135 *u = x[0] * x[0] + x[1] * x[1]; 136 return PETSC_SUCCESS; 137 } 138 PetscErrorCode constant_a_2d(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar *a, void *ctx) 139 { 140 *a = 4; 141 return PETSC_SUCCESS; 142 } 143 PetscErrorCode zero(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar *l, void *ctx) 144 { 145 *l = 0.0; 146 return PETSC_SUCCESS; 147 } 148 149 PetscErrorCode SetupProblem(DM dm, AppCtx *user) 150 { 151 PetscDS ds; 152 DMLabel label; 153 const PetscInt id = 1; 154 155 PetscFunctionBeginUser; 156 PetscCall(DMGetDS(dm, &ds)); 157 PetscCall(PetscDSSetResidual(ds, 0, user->useDualPenalty == PETSC_TRUE ? f0_u_full : f0_u, f1_u)); 158 PetscCall(PetscDSSetResidual(ds, 1, f0_a, NULL)); 159 PetscCall(PetscDSSetResidual(ds, 2, f0_l, f1_l)); 160 PetscCall(PetscDSSetJacobian(ds, 0, 0, user->useDualPenalty == PETSC_TRUE ? g0_uu_full : g0_uu, NULL, NULL, NULL)); 161 PetscCall(PetscDSSetJacobian(ds, 0, 2, NULL, NULL, NULL, g3_ul)); 162 PetscCall(PetscDSSetJacobian(ds, 1, 1, g0_aa, NULL, NULL, NULL)); 163 PetscCall(PetscDSSetJacobian(ds, 1, 2, g0_al, NULL, NULL, NULL)); 164 PetscCall(PetscDSSetJacobian(ds, 2, 1, g0_la, NULL, NULL, NULL)); 165 PetscCall(PetscDSSetJacobian(ds, 2, 0, NULL, NULL, NULL, g3_lu)); 166 167 PetscCall(PetscDSSetExactSolution(ds, 0, quadratic_u_2d, NULL)); 168 PetscCall(PetscDSSetExactSolution(ds, 1, constant_a_2d, NULL)); 169 PetscCall(PetscDSSetExactSolution(ds, 2, zero, NULL)); 170 PetscCall(DMGetLabel(dm, "marker", &label)); 171 PetscCall(DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 0, 0, NULL, (void (*)())quadratic_u_2d, NULL, user, NULL)); 172 PetscCall(DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 1, 0, NULL, (void (*)())constant_a_2d, NULL, user, NULL)); 173 PetscCall(DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 2, 0, NULL, (void (*)())zero, NULL, user, NULL)); 174 PetscFunctionReturn(PETSC_SUCCESS); 175 } 176 177 PetscErrorCode SetupDiscretization(DM dm, AppCtx *user) 178 { 179 DM cdm = dm; 180 const PetscInt dim = 2; 181 PetscFE fe[3]; 182 PetscInt f; 183 MPI_Comm comm; 184 185 PetscFunctionBeginUser; 186 /* Create finite element */ 187 PetscCall(PetscObjectGetComm((PetscObject)dm, &comm)); 188 PetscCall(PetscFECreateDefault(comm, dim, 1, PETSC_TRUE, "potential_", -1, &fe[0])); 189 PetscCall(PetscObjectSetName((PetscObject)fe[0], "potential")); 190 PetscCall(PetscFECreateDefault(comm, dim, 1, PETSC_TRUE, "charge_", -1, &fe[1])); 191 PetscCall(PetscObjectSetName((PetscObject)fe[1], "charge")); 192 PetscCall(PetscFECopyQuadrature(fe[0], fe[1])); 193 PetscCall(PetscFECreateDefault(comm, dim, 1, PETSC_TRUE, "multiplier_", -1, &fe[2])); 194 PetscCall(PetscObjectSetName((PetscObject)fe[2], "multiplier")); 195 PetscCall(PetscFECopyQuadrature(fe[0], fe[2])); 196 /* Set discretization and boundary conditions for each mesh */ 197 for (f = 0; f < 3; ++f) PetscCall(DMSetField(dm, f, NULL, (PetscObject)fe[f])); 198 PetscCall(DMCreateDS(cdm)); 199 PetscCall(SetupProblem(dm, user)); 200 while (cdm) { 201 PetscCall(DMCopyDisc(dm, cdm)); 202 PetscCall(DMGetCoarseDM(cdm, &cdm)); 203 } 204 for (f = 0; f < 3; ++f) PetscCall(PetscFEDestroy(&fe[f])); 205 PetscFunctionReturn(PETSC_SUCCESS); 206 } 207 208 int main(int argc, char **argv) 209 { 210 DM dm; 211 SNES snes; 212 Vec u, r; 213 AppCtx user; 214 215 PetscFunctionBeginUser; 216 PetscCall(PetscInitialize(&argc, &argv, NULL, help)); 217 PetscCall(ProcessOptions(PETSC_COMM_WORLD, &user)); 218 PetscCall(SNESCreate(PETSC_COMM_WORLD, &snes)); 219 PetscCall(CreateMesh(PETSC_COMM_WORLD, &user, &dm)); 220 PetscCall(SNESSetDM(snes, dm)); 221 PetscCall(SetupDiscretization(dm, &user)); 222 223 PetscCall(DMCreateGlobalVector(dm, &u)); 224 PetscCall(PetscObjectSetName((PetscObject)u, "solution")); 225 PetscCall(VecDuplicate(u, &r)); 226 PetscCall(DMPlexSetSNESLocalFEM(dm, PETSC_FALSE, &user)); 227 PetscCall(SNESSetFromOptions(snes)); 228 229 PetscCall(DMSNESCheckFromOptions(snes, u)); 230 if (user.runType == RUN_FULL) { 231 PetscDS ds; 232 PetscErrorCode (*exactFuncs[3])(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar *u, void *ctx); 233 PetscErrorCode (*initialGuess[3])(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar u[], void *ctx); 234 PetscReal error; 235 236 PetscCall(DMGetDS(dm, &ds)); 237 PetscCall(PetscDSGetExactSolution(ds, 0, &exactFuncs[0], NULL)); 238 PetscCall(PetscDSGetExactSolution(ds, 1, &exactFuncs[1], NULL)); 239 PetscCall(PetscDSGetExactSolution(ds, 2, &exactFuncs[2], NULL)); 240 initialGuess[0] = zero; 241 initialGuess[1] = zero; 242 initialGuess[2] = zero; 243 PetscCall(DMProjectFunction(dm, 0.0, initialGuess, NULL, INSERT_VALUES, u)); 244 PetscCall(VecViewFromOptions(u, NULL, "-initial_vec_view")); 245 PetscCall(DMComputeL2Diff(dm, 0.0, exactFuncs, NULL, u, &error)); 246 if (error < 1.0e-11) PetscCall(PetscPrintf(PETSC_COMM_WORLD, "Initial L_2 Error: < 1.0e-11\n")); 247 else PetscCall(PetscPrintf(PETSC_COMM_WORLD, "Initial L_2 Error: %g\n", (double)error)); 248 PetscCall(SNESSolve(snes, NULL, u)); 249 PetscCall(DMComputeL2Diff(dm, 0.0, exactFuncs, NULL, u, &error)); 250 if (error < 1.0e-11) PetscCall(PetscPrintf(PETSC_COMM_WORLD, "Final L_2 Error: < 1.0e-11\n")); 251 else PetscCall(PetscPrintf(PETSC_COMM_WORLD, "Final L_2 Error: %g\n", (double)error)); 252 } 253 PetscCall(VecViewFromOptions(u, NULL, "-sol_vec_view")); 254 255 PetscCall(VecDestroy(&u)); 256 PetscCall(VecDestroy(&r)); 257 PetscCall(SNESDestroy(&snes)); 258 PetscCall(DMDestroy(&dm)); 259 PetscCall(PetscFinalize()); 260 return 0; 261 } 262 263 /*TEST 264 265 build: 266 requires: !complex triangle 267 268 test: 269 suffix: 0 270 args: -run_type test -dmsnes_check -potential_petscspace_degree 2 -charge_petscspace_degree 1 -multiplier_petscspace_degree 1 271 272 test: 273 suffix: 1 274 args: -potential_petscspace_degree 2 -charge_petscspace_degree 1 -multiplier_petscspace_degree 1 -snes_monitor -snes_converged_reason -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 -sol_vec_view 275 276 test: 277 suffix: 2 278 args: -potential_petscspace_degree 2 -charge_petscspace_degree 1 -multiplier_petscspace_degree 1 -snes_monitor -snes_converged_reason -snes_fd -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 -sol_vec_view 279 280 TEST*/ 281