static char help[] = "One-Shot Multigrid for Parameter Estimation Problem for the Poisson Equation.\n\ Using the Interior Point Method.\n\n\n"; /*F We are solving the parameter estimation problem for the Laplacian. We will ask to minimize a Lagrangian function over $y$ and $u$, given by \begin{align} 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) \end{align} where $Q$ is a sampling operator, $L$ is a regularization operator, $F$ defines the PDE. Currently, we have perfect information, meaning $Q = I$, and then we need no regularization, $L = I$. We also give the null vector for the reference control $a_r$. Right now $\beta = 1$. The PDE will be the Laplace equation with homogeneous boundary conditions \begin{align} -Delta u = a \end{align} F*/ #include #include typedef enum {RUN_FULL, RUN_TEST} RunType; typedef struct { RunType runType; /* Whether to run tests, or solve the full problem */ PetscBool useDualPenalty; /* Penalize deviation from both goals */ } AppCtx; static PetscErrorCode ProcessOptions(MPI_Comm comm, AppCtx *options) { const char *runTypes[2] = {"full", "test"}; PetscInt run; PetscErrorCode ierr; PetscFunctionBeginUser; options->runType = RUN_FULL; options->useDualPenalty = PETSC_FALSE; ierr = PetscOptionsBegin(comm, "", "Inverse Problem Options", "DMPLEX");CHKERRQ(ierr); run = options->runType; ierr = PetscOptionsEList("-run_type", "The run type", "ex2.c", runTypes, 2, runTypes[options->runType], &run, NULL);CHKERRQ(ierr); options->runType = (RunType) run; ierr = PetscOptionsBool("-use_dual_penalty", "Penalize deviation from both goals", "ex2.c", options->useDualPenalty, &options->useDualPenalty, NULL);CHKERRQ(ierr); ierr = PetscOptionsEnd();CHKERRQ(ierr); PetscFunctionReturn(0); } static PetscErrorCode CreateMesh(MPI_Comm comm, AppCtx *user, DM *dm) { DM distributedMesh = NULL; PetscErrorCode ierr; PetscFunctionBeginUser; ierr = DMPlexCreateBoxMesh(comm, 2, PETSC_TRUE, NULL, NULL, NULL, NULL, PETSC_TRUE, dm);CHKERRQ(ierr); ierr = PetscObjectSetName((PetscObject) *dm, "Mesh");CHKERRQ(ierr); ierr = DMPlexDistribute(*dm, 0, NULL, &distributedMesh);CHKERRQ(ierr); if (distributedMesh) { ierr = DMDestroy(dm);CHKERRQ(ierr); *dm = distributedMesh; } ierr = DMSetFromOptions(*dm);CHKERRQ(ierr); ierr = DMViewFromOptions(*dm, NULL, "-dm_view");CHKERRQ(ierr); PetscFunctionReturn(0); } 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[]) { f0[0] = (u[0] - (x[0]*x[0] + x[1]*x[1])); } 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[]) { 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]))); } 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[]) { PetscInt d; for (d = 0; d < dim; ++d) f1[d] = u_x[dim*2+d]; } 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[]) { g0[0] = 1.0; } 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[]) { 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])); } 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[]) { PetscInt d; for (d = 0; d < dim; ++d) g3[d*dim+d] = 1.0; } 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[]) { f0[0] = u[1] - 4.0 /* 0.0 */ + u[2]; } 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[]) { g0[0] = 1.0; } 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[]) { g0[0] = 1.0; } 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[]) { f0[0] = u[1]; } 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[]) { PetscInt d; for (d = 0; d < dim; ++d) f1[d] = u_x[d]; } 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[]) { g0[0] = 1.0; } 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[]) { PetscInt d; for (d = 0; d < dim; ++d) g3[d*dim+d] = 1.0; } /* In 2D for Dirichlet conditions with a variable coefficient, we use exact solution: u = x^2 + y^2 a = 4 d_A = 4 d_B = sin(2*pi*x[0]) * sin(2*pi*x[1]) so that -\Delta u + a = -4 + 4 = 0 */ PetscErrorCode quadratic_u_2d(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar *u, void *ctx) { *u = x[0]*x[0] + x[1]*x[1]; return 0; } PetscErrorCode constant_a_2d(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar *a, void *ctx) { *a = 4; return 0; } PetscErrorCode zero(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar *l, void *ctx) { *l = 0.0; return 0; } PetscErrorCode SetupProblem(DM dm, AppCtx *user) { PetscDS ds; DMLabel label; const PetscInt id = 1; PetscErrorCode ierr; PetscFunctionBeginUser; ierr = DMGetDS(dm, &ds);CHKERRQ(ierr); ierr = PetscDSSetResidual(ds, 0, user->useDualPenalty == PETSC_TRUE ? f0_u_full : f0_u, f1_u);CHKERRQ(ierr); ierr = PetscDSSetResidual(ds, 1, f0_a, NULL);CHKERRQ(ierr); ierr = PetscDSSetResidual(ds, 2, f0_l, f1_l);CHKERRQ(ierr); ierr = PetscDSSetJacobian(ds, 0, 0, user->useDualPenalty == PETSC_TRUE ? g0_uu_full : g0_uu, NULL, NULL, NULL);CHKERRQ(ierr); ierr = PetscDSSetJacobian(ds, 0, 2, NULL, NULL, NULL, g3_ul);CHKERRQ(ierr); ierr = PetscDSSetJacobian(ds, 1, 1, g0_aa, NULL, NULL, NULL);CHKERRQ(ierr); ierr = PetscDSSetJacobian(ds, 1, 2, g0_al, NULL, NULL, NULL);CHKERRQ(ierr); ierr = PetscDSSetJacobian(ds, 2, 1, g0_la, NULL, NULL, NULL);CHKERRQ(ierr); ierr = PetscDSSetJacobian(ds, 2, 0, NULL, NULL, NULL, g3_lu);CHKERRQ(ierr); ierr = PetscDSSetExactSolution(ds, 0, quadratic_u_2d, NULL);CHKERRQ(ierr); ierr = PetscDSSetExactSolution(ds, 1, constant_a_2d, NULL);CHKERRQ(ierr); ierr = PetscDSSetExactSolution(ds, 2, zero, NULL);CHKERRQ(ierr); ierr = DMGetLabel(dm, "marker", &label);CHKERRQ(ierr); ierr = DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 0, 0, NULL, (void (*)()) quadratic_u_2d, NULL, user, NULL);CHKERRQ(ierr); ierr = DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 1, 0, NULL, (void (*)()) constant_a_2d, NULL, user, NULL);CHKERRQ(ierr); ierr = DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 2, 0, NULL, (void (*)()) zero, NULL, user, NULL);CHKERRQ(ierr); PetscFunctionReturn(0); } PetscErrorCode SetupDiscretization(DM dm, AppCtx *user) { DM cdm = dm; const PetscInt dim = 2; PetscFE fe[3]; PetscInt f; MPI_Comm comm; PetscErrorCode ierr; PetscFunctionBeginUser; /* Create finite element */ ierr = PetscObjectGetComm((PetscObject) dm, &comm);CHKERRQ(ierr); ierr = PetscFECreateDefault(comm, dim, 1, PETSC_TRUE, "potential_", -1, &fe[0]);CHKERRQ(ierr); ierr = PetscObjectSetName((PetscObject) fe[0], "potential");CHKERRQ(ierr); ierr = PetscFECreateDefault(comm, dim, 1, PETSC_TRUE, "charge_", -1, &fe[1]);CHKERRQ(ierr); ierr = PetscObjectSetName((PetscObject) fe[1], "charge");CHKERRQ(ierr); ierr = PetscFECopyQuadrature(fe[0], fe[1]);CHKERRQ(ierr); ierr = PetscFECreateDefault(comm, dim, 1, PETSC_TRUE, "multiplier_", -1, &fe[2]);CHKERRQ(ierr); ierr = PetscObjectSetName((PetscObject) fe[2], "multiplier");CHKERRQ(ierr); ierr = PetscFECopyQuadrature(fe[0], fe[2]);CHKERRQ(ierr); /* Set discretization and boundary conditions for each mesh */ for (f = 0; f < 3; ++f) {ierr = DMSetField(dm, f, NULL, (PetscObject) fe[f]);CHKERRQ(ierr);} ierr = DMCreateDS(cdm);CHKERRQ(ierr); ierr = SetupProblem(dm, user);CHKERRQ(ierr); while (cdm) { ierr = DMCopyDisc(dm, cdm);CHKERRQ(ierr); ierr = DMGetCoarseDM(cdm, &cdm);CHKERRQ(ierr); } for (f = 0; f < 3; ++f) {ierr = PetscFEDestroy(&fe[f]);CHKERRQ(ierr);} PetscFunctionReturn(0); } int main(int argc, char **argv) { DM dm; SNES snes; Vec u, r; AppCtx user; PetscErrorCode ierr; ierr = PetscInitialize(&argc, &argv, NULL,help);if (ierr) return ierr; ierr = ProcessOptions(PETSC_COMM_WORLD, &user);CHKERRQ(ierr); ierr = SNESCreate(PETSC_COMM_WORLD, &snes);CHKERRQ(ierr); ierr = CreateMesh(PETSC_COMM_WORLD, &user, &dm);CHKERRQ(ierr); ierr = SNESSetDM(snes, dm);CHKERRQ(ierr); ierr = SetupDiscretization(dm, &user);CHKERRQ(ierr); ierr = DMCreateGlobalVector(dm, &u);CHKERRQ(ierr); ierr = PetscObjectSetName((PetscObject) u, "solution");CHKERRQ(ierr); ierr = VecDuplicate(u, &r);CHKERRQ(ierr); ierr = DMPlexSetSNESLocalFEM(dm,&user,&user,&user);CHKERRQ(ierr); ierr = SNESSetFromOptions(snes);CHKERRQ(ierr); ierr = DMSNESCheckFromOptions(snes, u);CHKERRQ(ierr); if (user.runType == RUN_FULL) { PetscDS ds; PetscErrorCode (*exactFuncs[3])(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar *u, void *ctx); PetscErrorCode (*initialGuess[3])(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar u[], void *ctx); PetscReal error; ierr = DMGetDS(dm, &ds);CHKERRQ(ierr); ierr = PetscDSGetExactSolution(ds, 0, &exactFuncs[0], NULL);CHKERRQ(ierr); ierr = PetscDSGetExactSolution(ds, 1, &exactFuncs[1], NULL);CHKERRQ(ierr); ierr = PetscDSGetExactSolution(ds, 2, &exactFuncs[2], NULL);CHKERRQ(ierr); initialGuess[0] = zero; initialGuess[1] = zero; initialGuess[2] = zero; ierr = DMProjectFunction(dm, 0.0, initialGuess, NULL, INSERT_VALUES, u);CHKERRQ(ierr); ierr = VecViewFromOptions(u, NULL, "-initial_vec_view");CHKERRQ(ierr); ierr = DMComputeL2Diff(dm, 0.0, exactFuncs, NULL, u, &error);CHKERRQ(ierr); if (error < 1.0e-11) {ierr = PetscPrintf(PETSC_COMM_WORLD, "Initial L_2 Error: < 1.0e-11\n");CHKERRQ(ierr);} else {ierr = PetscPrintf(PETSC_COMM_WORLD, "Initial L_2 Error: %g\n", error);CHKERRQ(ierr);} ierr = SNESSolve(snes, NULL, u);CHKERRQ(ierr); ierr = DMComputeL2Diff(dm, 0.0, exactFuncs, NULL, u, &error);CHKERRQ(ierr); if (error < 1.0e-11) {ierr = PetscPrintf(PETSC_COMM_WORLD, "Final L_2 Error: < 1.0e-11\n");CHKERRQ(ierr);} else {ierr = PetscPrintf(PETSC_COMM_WORLD, "Final L_2 Error: %g\n", error);CHKERRQ(ierr);} } ierr = VecViewFromOptions(u, NULL, "-sol_vec_view");CHKERRQ(ierr); ierr = VecDestroy(&u);CHKERRQ(ierr); ierr = VecDestroy(&r);CHKERRQ(ierr); ierr = SNESDestroy(&snes);CHKERRQ(ierr); ierr = DMDestroy(&dm);CHKERRQ(ierr); ierr = PetscFinalize(); return ierr; } /*TEST build: requires: !complex triangle test: suffix: 0 args: -run_type test -dmsnes_check -potential_petscspace_degree 2 -charge_petscspace_degree 1 -multiplier_petscspace_degree 1 test: suffix: 1 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 test: suffix: 2 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 TEST*/