Date: 2022-03-31T04:48:34+00:00

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:   PetscInitialize(&argc,&argv,NULL,help);

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

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

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

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

    112:   SNESSetFromOptions(snes);

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

    121:   SNESSolve(snes,NULL,NULL);
    122:   SNESDestroy(&snes);

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

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

    140: /*
    141:       Evaluates FU = Gradiant(L(w,u,lambda))

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

    146: */
    147: PetscErrorCode ComputeFunction(SNES snes,Vec U,Vec FU,void *ctx)
    148: {
    149:   PetscInt       xs,xm,i,N;
    150:   ULambda        *u_lambda,*fu_lambda;
    151:   PetscScalar    d,h,*w,*fw;
    152:   Vec            vw,vfw,vu_lambda,vfu_lambda;
    153:   DM             packer,red,da;

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

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

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

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

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

    192:   VecRestoreArray(vw,&w);
    193:   VecRestoreArray(vfw,&fw);
    194:   DMDAVecRestoreArray(da,vu_lambda,&u_lambda);
    195:   DMDAVecRestoreArray(da,vfu_lambda,&fu_lambda);
    196:   DMCompositeRestoreLocalVectors(packer,&vw,&vu_lambda);
    197:   DMCompositeRestoreAccess(packer,FU,&vfw,&vfu_lambda);
    198:   PetscLogFlops(13.0*N);
    199:   return 0;
    200: }

    202: /*
    203:     Computes the exact solution
    204: */
    205: PetscErrorCode u_solution(void *dummy,PetscInt n,const PetscScalar *x,PetscScalar *u)
    206: {
    207:   PetscInt i;

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

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

    223:   DMCompositeGetEntries(packer,&m,&da);

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

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

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

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

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

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

    285:   DMGetGlobalVector(packer,&t);
    286:   VecGetLocalSize(t,&m);
    287:   DMRestoreGlobalVector(packer,&t);
    288:   MatCreateMFFD(PETSC_COMM_WORLD,m,m,PETSC_DETERMINE,PETSC_DETERMINE,A);
    289:   MatSetUp(*A);
    290:   MatSetDM(*A,packer);
    291:   return 0;
    292: }

    294: PetscErrorCode ComputeJacobian_MF(SNES snes,Vec x,Mat A,Mat B,void *ctx)
    295: {
    297:   MatMFFDSetFunction(A,(PetscErrorCode (*)(void*,Vec,Vec))SNESComputeFunction,snes);
    298:   MatMFFDSetBase(A,x,NULL);
    299:   return 0;
    300: }

    302: /*TEST

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

    309: TEST*/