Actual source code: ex22.c


  2: static const char help[] = "Solves PDE optimization problem using full-space method, interlaces state and adjoint variables.\n\n";

  4: #include <petscdm.h>
  5: #include <petscdmda.h>
  6: #include <petscdmredundant.h>
  7: #include <petscdmcomposite.h>
  8: #include <petscpf.h>
  9: #include <petscsnes.h>
 10: #include <petsc/private/dmimpl.h>

 12: /*

 14:        w - design variables (what we change to get an optimal solution)
 15:        u - state variables (i.e. the PDE solution)
 16:        lambda - the Lagrange multipliers

 18:             U = (w [u_0 lambda_0 u_1 lambda_1 .....])

 20:        fu, fw, flambda contain the gradient of L(w,u,lambda)

 22:             FU = (fw [fu_0 flambda_0 .....])

 24:        In this example the PDE is
 25:                              Uxx = 2,
 26:                             u(0) = w(0), thus this is the free parameter
 27:                             u(1) = 0
 28:        the function we wish to minimize is
 29:                             \integral u^{2}

 31:        The exact solution for u is given by u(x) = x*x - 1.25*x + .25

 33:        Use the usual centered finite differences.

 35:        Note we treat the problem as non-linear though it happens to be linear

 37:        See ex21.c for the same code, but that does NOT interlaces the u and the lambda

 39:        The vectors u_lambda and fu_lambda contain the u and the lambda interlaced
 40: */

 42: typedef struct {
 43:   PetscViewer u_lambda_viewer;
 44:   PetscViewer fu_lambda_viewer;
 45: } UserCtx;

 47: extern PetscErrorCode ComputeFunction(SNES, Vec, Vec, void *);
 48: extern PetscErrorCode ComputeJacobian_MF(SNES, Vec, Mat, Mat, void *);
 49: extern PetscErrorCode Monitor(SNES, PetscInt, PetscReal, void *);

 51: /*
 52:     Uses full multigrid preconditioner with GMRES (with no preconditioner inside the GMRES) as the
 53:   smoother on all levels. This is because (1) in the matrix free case no matrix entries are
 54:   available for doing Jacobi or SOR preconditioning and (2) the explicit matrix case the diagonal
 55:   entry for the control variable is zero which means default SOR will not work.

 57: */
 58: char common_options[] = "-ksp_type fgmres\
 59:                          -snes_grid_sequence 2 \
 60:                          -pc_type mg\
 61:                          -mg_levels_pc_type none \
 62:                          -mg_coarse_pc_type none \
 63:                          -pc_mg_type full \
 64:                          -mg_coarse_ksp_type gmres \
 65:                          -mg_levels_ksp_type gmres \
 66:                          -mg_coarse_ksp_max_it 6 \
 67:                          -mg_levels_ksp_max_it 3";

 69: char matrix_free_options[] = "-mat_mffd_compute_normu no \
 70:                               -mat_mffd_type wp";

 72: extern PetscErrorCode DMCreateMatrix_MF(DM, Mat *);

 74: int main(int argc, char **argv)
 75: {
 76:   UserCtx   user;
 77:   DM        red, da;
 78:   SNES      snes;
 79:   DM        packer;
 80:   PetscBool use_monitor = PETSC_FALSE;

 82:   PetscFunctionBeginUser;
 83:   PetscCall(PetscInitialize(&argc, &argv, NULL, help));

 85:   /* Hardwire several options; can be changed at command line */
 86:   PetscCall(PetscOptionsInsertString(NULL, common_options));
 87:   PetscCall(PetscOptionsInsertString(NULL, matrix_free_options));
 88:   PetscCall(PetscOptionsInsert(NULL, &argc, &argv, NULL));
 89:   PetscCall(PetscOptionsGetBool(NULL, NULL, "-use_monitor", &use_monitor, PETSC_IGNORE));

 91:   /* Create a global vector that includes a single redundant array and two da arrays */
 92:   PetscCall(DMCompositeCreate(PETSC_COMM_WORLD, &packer));
 93:   PetscCall(DMRedundantCreate(PETSC_COMM_WORLD, 0, 1, &red));
 94:   PetscCall(DMSetOptionsPrefix(red, "red_"));
 95:   PetscCall(DMCompositeAddDM(packer, red));
 96:   PetscCall(DMDACreate1d(PETSC_COMM_WORLD, DM_BOUNDARY_NONE, 5, 2, 1, NULL, &da));
 97:   PetscCall(DMSetOptionsPrefix(red, "da_"));
 98:   PetscCall(DMSetFromOptions(da));
 99:   PetscCall(DMSetUp(da));
100:   PetscCall(DMDASetFieldName(da, 0, "u"));
101:   PetscCall(DMDASetFieldName(da, 1, "lambda"));
102:   PetscCall(DMCompositeAddDM(packer, (DM)da));
103:   PetscCall(DMSetApplicationContext(packer, &user));

105:   packer->ops->creatematrix = DMCreateMatrix_MF;

107:   /* create nonlinear multi-level solver */
108:   PetscCall(SNESCreate(PETSC_COMM_WORLD, &snes));
109:   PetscCall(SNESSetDM(snes, packer));
110:   PetscCall(SNESSetFunction(snes, NULL, ComputeFunction, NULL));
111:   PetscCall(SNESSetJacobian(snes, NULL, NULL, ComputeJacobian_MF, NULL));

113:   PetscCall(SNESSetFromOptions(snes));

115:   if (use_monitor) {
116:     /* create graphics windows */
117:     PetscCall(PetscViewerDrawOpen(PETSC_COMM_WORLD, 0, "u_lambda - state variables and Lagrange multipliers", -1, -1, -1, -1, &user.u_lambda_viewer));
118:     PetscCall(PetscViewerDrawOpen(PETSC_COMM_WORLD, 0, "fu_lambda - derivative w.r.t. state variables and Lagrange multipliers", -1, -1, -1, -1, &user.fu_lambda_viewer));
119:     PetscCall(SNESMonitorSet(snes, Monitor, 0, 0));
120:   }

122:   PetscCall(SNESSolve(snes, NULL, NULL));
123:   PetscCall(SNESDestroy(&snes));

125:   PetscCall(DMDestroy(&red));
126:   PetscCall(DMDestroy(&da));
127:   PetscCall(DMDestroy(&packer));
128:   if (use_monitor) {
129:     PetscCall(PetscViewerDestroy(&user.u_lambda_viewer));
130:     PetscCall(PetscViewerDestroy(&user.fu_lambda_viewer));
131:   }
132:   PetscCall(PetscFinalize());
133:   return 0;
134: }

136: typedef struct {
137:   PetscScalar u;
138:   PetscScalar lambda;
139: } ULambda;

141: /*
142:       Evaluates FU = Gradient(L(w,u,lambda))

144:      This local function acts on the ghosted version of U (accessed via DMCompositeGetLocalVectors() and
145:    DMCompositeScatter()) BUT the global, nonghosted version of FU (via DMCompositeGetAccess()).

147: */
148: PetscErrorCode ComputeFunction(SNES snes, Vec U, Vec FU, void *ctx)
149: {
150:   PetscInt    xs, xm, i, N;
151:   ULambda    *u_lambda, *fu_lambda;
152:   PetscScalar d, h, *w, *fw;
153:   Vec         vw, vfw, vu_lambda, vfu_lambda;
154:   DM          packer, red, da;

156:   PetscFunctionBeginUser;
157:   PetscCall(VecGetDM(U, &packer));
158:   PetscCall(DMCompositeGetEntries(packer, &red, &da));
159:   PetscCall(DMCompositeGetLocalVectors(packer, &vw, &vu_lambda));
160:   PetscCall(DMCompositeScatter(packer, U, vw, vu_lambda));
161:   PetscCall(DMCompositeGetAccess(packer, FU, &vfw, &vfu_lambda));

163:   PetscCall(DMDAGetCorners(da, &xs, NULL, NULL, &xm, NULL, NULL));
164:   PetscCall(DMDAGetInfo(da, 0, &N, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0));
165:   PetscCall(VecGetArray(vw, &w));
166:   PetscCall(VecGetArray(vfw, &fw));
167:   PetscCall(DMDAVecGetArray(da, vu_lambda, &u_lambda));
168:   PetscCall(DMDAVecGetArray(da, vfu_lambda, &fu_lambda));
169:   d = N - 1.0;
170:   h = 1.0 / d;

172:   /* derivative of L() w.r.t. w */
173:   if (xs == 0) { /* only first processor computes this */
174:     fw[0] = -2.0 * d * u_lambda[0].lambda;
175:   }

177:   /* derivative of L() w.r.t. u */
178:   for (i = xs; i < xs + xm; i++) {
179:     if (i == 0) fu_lambda[0].lambda = h * u_lambda[0].u + 2. * d * u_lambda[0].lambda - d * u_lambda[1].lambda;
180:     else if (i == 1) fu_lambda[1].lambda = 2. * h * u_lambda[1].u + 2. * d * u_lambda[1].lambda - d * u_lambda[2].lambda;
181:     else if (i == N - 1) fu_lambda[N - 1].lambda = h * u_lambda[N - 1].u + 2. * d * u_lambda[N - 1].lambda - d * u_lambda[N - 2].lambda;
182:     else if (i == N - 2) fu_lambda[N - 2].lambda = 2. * h * u_lambda[N - 2].u + 2. * d * u_lambda[N - 2].lambda - d * u_lambda[N - 3].lambda;
183:     else fu_lambda[i].lambda = 2. * h * u_lambda[i].u - d * (u_lambda[i + 1].lambda - 2.0 * u_lambda[i].lambda + u_lambda[i - 1].lambda);
184:   }

186:   /* derivative of L() w.r.t. lambda */
187:   for (i = xs; i < xs + xm; i++) {
188:     if (i == 0) fu_lambda[0].u = 2.0 * d * (u_lambda[0].u - w[0]);
189:     else if (i == N - 1) fu_lambda[N - 1].u = 2.0 * d * u_lambda[N - 1].u;
190:     else fu_lambda[i].u = -(d * (u_lambda[i + 1].u - 2.0 * u_lambda[i].u + u_lambda[i - 1].u) - 2.0 * h);
191:   }

193:   PetscCall(VecRestoreArray(vw, &w));
194:   PetscCall(VecRestoreArray(vfw, &fw));
195:   PetscCall(DMDAVecRestoreArray(da, vu_lambda, &u_lambda));
196:   PetscCall(DMDAVecRestoreArray(da, vfu_lambda, &fu_lambda));
197:   PetscCall(DMCompositeRestoreLocalVectors(packer, &vw, &vu_lambda));
198:   PetscCall(DMCompositeRestoreAccess(packer, FU, &vfw, &vfu_lambda));
199:   PetscCall(PetscLogFlops(13.0 * N));
200:   PetscFunctionReturn(PETSC_SUCCESS);
201: }

203: /*
204:     Computes the exact solution
205: */
206: PetscErrorCode u_solution(void *dummy, PetscInt n, const PetscScalar *x, PetscScalar *u)
207: {
208:   PetscInt i;

210:   PetscFunctionBeginUser;
211:   for (i = 0; i < n; i++) u[2 * i] = x[i] * x[i] - 1.25 * x[i] + .25;
212:   PetscFunctionReturn(PETSC_SUCCESS);
213: }

215: PetscErrorCode ExactSolution(DM packer, Vec U)
216: {
217:   PF           pf;
218:   Vec          x, u_global;
219:   PetscScalar *w;
220:   DM           da;
221:   PetscInt     m;

223:   PetscFunctionBeginUser;
224:   PetscCall(DMCompositeGetEntries(packer, &m, &da));

226:   PetscCall(PFCreate(PETSC_COMM_WORLD, 1, 2, &pf));
227:   /* The cast through PETSC_UINTPTR_T is so that compilers will warn about casting to void * from void(*)(void) */
228:   PetscCall(PFSetType(pf, PFQUICK, (void *)(PETSC_UINTPTR_T)u_solution));
229:   PetscCall(DMGetCoordinates(da, &x));
230:   if (!x) {
231:     PetscCall(DMDASetUniformCoordinates(da, 0.0, 1.0, 0.0, 1.0, 0.0, 1.0));
232:     PetscCall(DMGetCoordinates(da, &x));
233:   }
234:   PetscCall(DMCompositeGetAccess(packer, U, &w, &u_global, 0));
235:   if (w) w[0] = .25;
236:   PetscCall(PFApplyVec(pf, x, u_global));
237:   PetscCall(PFDestroy(&pf));
238:   PetscCall(DMCompositeRestoreAccess(packer, U, &w, &u_global, 0));
239:   PetscFunctionReturn(PETSC_SUCCESS);
240: }

242: PetscErrorCode Monitor(SNES snes, PetscInt its, PetscReal rnorm, void *dummy)
243: {
244:   UserCtx     *user;
245:   PetscInt     m, N;
246:   PetscScalar *w, *dw;
247:   Vec          u_lambda, U, F, Uexact;
248:   DM           packer;
249:   PetscReal    norm;
250:   DM           da;

252:   PetscFunctionBeginUser;
253:   PetscCall(SNESGetDM(snes, &packer));
254:   PetscCall(DMGetApplicationContext(packer, &user));
255:   PetscCall(SNESGetSolution(snes, &U));
256:   PetscCall(DMCompositeGetAccess(packer, U, &w, &u_lambda));
257:   PetscCall(VecView(u_lambda, user->u_lambda_viewer));
258:   PetscCall(DMCompositeRestoreAccess(packer, U, &w, &u_lambda));

260:   PetscCall(SNESGetFunction(snes, &F, 0, 0));
261:   PetscCall(DMCompositeGetAccess(packer, F, &w, &u_lambda));
262:   /* ierr = VecView(u_lambda,user->fu_lambda_viewer); */
263:   PetscCall(DMCompositeRestoreAccess(packer, U, &w, &u_lambda));

265:   PetscCall(DMCompositeGetEntries(packer, &m, &da));
266:   PetscCall(DMDAGetInfo(da, 0, &N, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0));
267:   PetscCall(VecDuplicate(U, &Uexact));
268:   PetscCall(ExactSolution(packer, Uexact));
269:   PetscCall(VecAXPY(Uexact, -1.0, U));
270:   PetscCall(DMCompositeGetAccess(packer, Uexact, &dw, &u_lambda));
271:   PetscCall(VecStrideNorm(u_lambda, 0, NORM_2, &norm));
272:   norm = norm / PetscSqrtReal((PetscReal)N - 1.);
273:   if (dw) PetscCall(PetscPrintf(PETSC_COMM_WORLD, "Norm of error %g Error at x = 0 %g\n", (double)norm, (double)PetscRealPart(dw[0])));
274:   PetscCall(VecView(u_lambda, user->fu_lambda_viewer));
275:   PetscCall(DMCompositeRestoreAccess(packer, Uexact, &dw, &u_lambda));
276:   PetscCall(VecDestroy(&Uexact));
277:   PetscFunctionReturn(PETSC_SUCCESS);
278: }

280: PetscErrorCode DMCreateMatrix_MF(DM packer, Mat *A)
281: {
282:   Vec      t;
283:   PetscInt m;

285:   PetscFunctionBeginUser;
286:   PetscCall(DMGetGlobalVector(packer, &t));
287:   PetscCall(VecGetLocalSize(t, &m));
288:   PetscCall(DMRestoreGlobalVector(packer, &t));
289:   PetscCall(MatCreateMFFD(PETSC_COMM_WORLD, m, m, PETSC_DETERMINE, PETSC_DETERMINE, A));
290:   PetscCall(MatSetUp(*A));
291:   PetscCall(MatSetDM(*A, packer));
292:   PetscFunctionReturn(PETSC_SUCCESS);
293: }

295: PetscErrorCode ComputeJacobian_MF(SNES snes, Vec x, Mat A, Mat B, void *ctx)
296: {
297:   PetscFunctionBeginUser;
298:   PetscCall(MatMFFDSetFunction(A, (PetscErrorCode(*)(void *, Vec, Vec))SNESComputeFunction, snes));
299:   PetscCall(MatMFFDSetBase(A, x, NULL));
300:   PetscFunctionReturn(PETSC_SUCCESS);
301: }

303: /*TEST

305:    test:
306:       nsize: 2
307:       args: -da_grid_x 10 -snes_converged_reason -ksp_converged_reason -snes_view
308:       requires: !single

310: TEST*/