Actual source code: ex14.c


  2: static char help[] = "Solves a nonlinear system in parallel with a user-defined Newton method.\n\
  3: Uses KSP to solve the linearized Newton systems.  This solver\n\
  4: is a very simplistic inexact Newton method.  The intent of this code is to\n\
  5: demonstrate the repeated solution of linear systems with the same nonzero pattern.\n\
  6: \n\
  7: This is NOT the recommended approach for solving nonlinear problems with PETSc!\n\
  8: We urge users to employ the SNES component for solving nonlinear problems whenever\n\
  9: possible, as it offers many advantages over coding nonlinear solvers independently.\n\
 10: \n\
 11: We solve the  Bratu (SFI - solid fuel ignition) problem in a 2D rectangular\n\
 12: domain, using distributed arrays (DMDAs) to partition the parallel grid.\n\
 13: The command line options include:\n\
 14:   -par <parameter>, where <parameter> indicates the problem's nonlinearity\n\
 15:      problem SFI:  <parameter> = Bratu parameter (0 <= par <= 6.81)\n\
 16:   -mx <xg>, where <xg> = number of grid points in the x-direction\n\
 17:   -my <yg>, where <yg> = number of grid points in the y-direction\n\
 18:   -Nx <npx>, where <npx> = number of processors in the x-direction\n\
 19:   -Ny <npy>, where <npy> = number of processors in the y-direction\n\n";

 21: /*T
 22:    Concepts: KSP^writing a user-defined nonlinear solver (parallel Bratu example);
 23:    Concepts: DMDA^using distributed arrays;
 24:    Processors: n
 25: T*/

 27: /* ------------------------------------------------------------------------

 29:     Solid Fuel Ignition (SFI) problem.  This problem is modeled by
 30:     the partial differential equation

 32:             -Laplacian u - lambda*exp(u) = 0,  0 < x,y < 1,

 34:     with boundary conditions

 36:              u = 0  for  x = 0, x = 1, y = 0, y = 1.

 38:     A finite difference approximation with the usual 5-point stencil
 39:     is used to discretize the boundary value problem to obtain a nonlinear
 40:     system of equations.

 42:     The SNES version of this problem is:  snes/tutorials/ex5.c
 43:     We urge users to employ the SNES component for solving nonlinear
 44:     problems whenever possible, as it offers many advantages over coding
 45:     nonlinear solvers independently.

 47:   ------------------------------------------------------------------------- */

 49: /*
 50:    Include "petscdmda.h" so that we can use distributed arrays (DMDAs).
 51:    Include "petscksp.h" so that we can use KSP solvers.  Note that this
 52:    file automatically includes:
 53:      petscsys.h       - base PETSc routines   petscvec.h - vectors
 54:      petscmat.h - matrices
 55:      petscis.h     - index sets            petscksp.h - Krylov subspace methods
 56:      petscviewer.h - viewers               petscpc.h  - preconditioners
 57: */
 58: #include <petscdm.h>
 59: #include <petscdmda.h>
 60: #include <petscksp.h>

 62: /*
 63:    User-defined application context - contains data needed by the
 64:    application-provided call-back routines, ComputeJacobian() and
 65:    ComputeFunction().
 66: */
 67: typedef struct {
 68:   PetscReal param;             /* test problem parameter */
 69:   PetscInt  mx,my;             /* discretization in x,y directions */
 70:   Vec       localX;           /* ghosted local vector */
 71:   DM        da;                /* distributed array data structure */
 72: } AppCtx;

 74: /*
 75:    User-defined routines
 76: */
 77: extern PetscErrorCode ComputeFunction(AppCtx*,Vec,Vec),FormInitialGuess(AppCtx*,Vec);
 78: extern PetscErrorCode ComputeJacobian(AppCtx*,Vec,Mat);

 80: int main(int argc,char **argv)
 81: {
 82:   /* -------------- Data to define application problem ---------------- */
 83:   MPI_Comm       comm;                /* communicator */
 84:   KSP            ksp;                /* linear solver */
 85:   Vec            X,Y,F;             /* solution, update, residual vectors */
 86:   Mat            J;                   /* Jacobian matrix */
 87:   AppCtx         user;                /* user-defined work context */
 88:   PetscInt       Nx,Ny;              /* number of preocessors in x- and y- directions */
 89:   PetscMPIInt    size;                /* number of processors */
 90:   PetscReal      bratu_lambda_max = 6.81,bratu_lambda_min = 0.;
 91:   PetscInt       m,N;

 93:   /* --------------- Data to define nonlinear solver -------------- */
 94:   PetscReal    rtol = 1.e-8;          /* relative convergence tolerance */
 95:   PetscReal    xtol = 1.e-8;          /* step convergence tolerance */
 96:   PetscReal    ttol;                  /* convergence tolerance */
 97:   PetscReal    fnorm,ynorm,xnorm;     /* various vector norms */
 98:   PetscInt     max_nonlin_its = 3;   /* maximum number of iterations for nonlinear solver */
 99:   PetscInt     max_functions  = 50;   /* maximum number of function evaluations */
100:   PetscInt     lin_its;               /* number of linear solver iterations for each step */
101:   PetscInt     i;                     /* nonlinear solve iteration number */
102:   PetscBool    no_output = PETSC_FALSE;             /* flag indicating whether to surpress output */

104:   PetscInitialize(&argc,&argv,(char*)0,help);
105:   comm = PETSC_COMM_WORLD;
106:   PetscOptionsGetBool(NULL,NULL,"-no_output",&no_output,NULL);

108:   /*
109:      Initialize problem parameters
110:   */
111:   user.mx = 4; user.my = 4; user.param = 6.0;

113:   PetscOptionsGetInt(NULL,NULL,"-mx",&user.mx,NULL);
114:   PetscOptionsGetInt(NULL,NULL,"-my",&user.my,NULL);
115:   PetscOptionsGetReal(NULL,NULL,"-par",&user.param,NULL);
117:   N = user.mx*user.my;

119:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
120:      Create linear solver context
121:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

123:   KSPCreate(comm,&ksp);

125:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
126:      Create vector data structures
127:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

129:   /*
130:      Create distributed array (DMDA) to manage parallel grid and vectors
131:   */
132:   MPI_Comm_size(comm,&size);
133:   Nx   = PETSC_DECIDE; Ny = PETSC_DECIDE;
134:   PetscOptionsGetInt(NULL,NULL,"-Nx",&Nx,NULL);
135:   PetscOptionsGetInt(NULL,NULL,"-Ny",&Ny,NULL);
137:   DMDACreate2d(PETSC_COMM_WORLD,DM_BOUNDARY_NONE,DM_BOUNDARY_NONE,DMDA_STENCIL_STAR,user.mx,user.my,Nx,Ny,1,1,NULL,NULL,&user.da);
138:   DMSetFromOptions(user.da);
139:   DMSetUp(user.da);

141:   /*
142:      Extract global and local vectors from DMDA; then duplicate for remaining
143:      vectors that are the same types
144:   */
145:   DMCreateGlobalVector(user.da,&X);
146:   DMCreateLocalVector(user.da,&user.localX);
147:   VecDuplicate(X,&F);
148:   VecDuplicate(X,&Y);

150:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
151:      Create matrix data structure for Jacobian
152:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
153:   /*
154:      Note:  For the parallel case, vectors and matrices MUST be partitioned
155:      accordingly.  When using distributed arrays (DMDAs) to create vectors,
156:      the DMDAs determine the problem partitioning.  We must explicitly
157:      specify the local matrix dimensions upon its creation for compatibility
158:      with the vector distribution.  Thus, the generic MatCreate() routine
159:      is NOT sufficient when working with distributed arrays.

161:      Note: Here we only approximately preallocate storage space for the
162:      Jacobian.  See the users manual for a discussion of better techniques
163:      for preallocating matrix memory.
164:   */
165:   if (size == 1) {
166:     MatCreateSeqAIJ(comm,N,N,5,NULL,&J);
167:   } else {
168:     VecGetLocalSize(X,&m);
169:     MatCreateAIJ(comm,m,m,N,N,5,NULL,3,NULL,&J);
170:   }

172:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
173:      Customize linear solver; set runtime options
174:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

176:   /*
177:      Set runtime options (e.g.,-ksp_monitor -ksp_rtol <rtol> -ksp_type <type>)
178:   */
179:   KSPSetFromOptions(ksp);

181:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
182:      Evaluate initial guess
183:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

185:   FormInitialGuess(&user,X);
186:   ComputeFunction(&user,X,F);   /* Compute F(X)    */
187:   VecNorm(F,NORM_2,&fnorm);     /* fnorm = || F || */
188:   ttol = fnorm*rtol;
189:   if (!no_output) PetscPrintf(comm,"Initial function norm = %g\n",(double)fnorm);

191:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
192:      Solve nonlinear system with a user-defined method
193:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

195:   /*
196:       This solver is a very simplistic inexact Newton method, with no
197:       no damping strategies or bells and whistles. The intent of this code
198:       is  merely to demonstrate the repeated solution with KSP of linear
199:       systems with the same nonzero structure.

201:       This is NOT the recommended approach for solving nonlinear problems
202:       with PETSc!  We urge users to employ the SNES component for solving
203:       nonlinear problems whenever possible with application codes, as it
204:       offers many advantages over coding nonlinear solvers independently.
205:    */

207:   for (i=0; i<max_nonlin_its; i++) {

209:     /*
210:         Compute the Jacobian matrix.
211:      */
212:     ComputeJacobian(&user,X,J);

214:     /*
215:         Solve J Y = F, where J is the Jacobian matrix.
216:           - First, set the KSP linear operators.  Here the matrix that
217:             defines the linear system also serves as the preconditioning
218:             matrix.
219:           - Then solve the Newton system.
220:      */
221:     KSPSetOperators(ksp,J,J);
222:     KSPSolve(ksp,F,Y);
223:     KSPGetIterationNumber(ksp,&lin_its);

225:     /*
226:        Compute updated iterate
227:      */
228:     VecNorm(Y,NORM_2,&ynorm);       /* ynorm = || Y || */
229:     VecAYPX(Y,-1.0,X);              /* Y <- X - Y      */
230:     VecCopy(Y,X);                   /* X <- Y          */
231:     VecNorm(X,NORM_2,&xnorm);       /* xnorm = || X || */
232:     if (!no_output) {
233:       PetscPrintf(comm,"   linear solve iterations = %D, xnorm=%g, ynorm=%g\n",lin_its,(double)xnorm,(double)ynorm);
234:     }

236:     /*
237:        Evaluate new nonlinear function
238:      */
239:     ComputeFunction(&user,X,F);     /* Compute F(X)    */
240:     VecNorm(F,NORM_2,&fnorm);       /* fnorm = || F || */
241:     if (!no_output) {
242:       PetscPrintf(comm,"Iteration %D, function norm = %g\n",i+1,(double)fnorm);
243:     }

245:     /*
246:        Test for convergence
247:      */
248:     if (fnorm <= ttol) {
249:       if (!no_output) {
250:         PetscPrintf(comm,"Converged due to function norm %g < %g (relative tolerance)\n",(double)fnorm,(double)ttol);
251:       }
252:       break;
253:     }
254:     if (ynorm < xtol*(xnorm)) {
255:       if (!no_output) {
256:         PetscPrintf(comm,"Converged due to small update length: %g < %g * %g\n",(double)ynorm,(double)xtol,(double)xnorm);
257:       }
258:       break;
259:     }
260:     if (i > max_functions) {
261:       if (!no_output) {
262:         PetscPrintf(comm,"Exceeded maximum number of function evaluations: %D > %D\n",i,max_functions);
263:       }
264:       break;
265:     }
266:   }
267:   PetscPrintf(comm,"Number of nonlinear iterations = %D\n",i);

269:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
270:      Free work space.  All PETSc objects should be destroyed when they
271:      are no longer needed.
272:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

274:   MatDestroy(&J));           PetscCall(VecDestroy(&Y);
275:   VecDestroy(&user.localX)); PetscCall(VecDestroy(&X);
276:   VecDestroy(&F);
277:   KSPDestroy(&ksp));  PetscCall(DMDestroy(&user.da);
278:   PetscFinalize();
279:   return 0;
280: }
281: /* ------------------------------------------------------------------- */
282: /*
283:    FormInitialGuess - Forms initial approximation.

285:    Input Parameters:
286:    user - user-defined application context
287:    X - vector

289:    Output Parameter:
290:    X - vector
291:  */
292: PetscErrorCode FormInitialGuess(AppCtx *user,Vec X)
293: {
294:   PetscInt    i,j,row,mx,my,xs,ys,xm,ym,gxm,gym,gxs,gys;
295:   PetscReal   one = 1.0,lambda,temp1,temp,hx,hy;
296:   PetscScalar *x;

298:   mx    = user->mx;            my = user->my;            lambda = user->param;
299:   hx    = one/(PetscReal)(mx-1);  hy = one/(PetscReal)(my-1);
300:   temp1 = lambda/(lambda + one);

302:   /*
303:      Get a pointer to vector data.
304:        - For default PETSc vectors, VecGetArray() returns a pointer to
305:          the data array.  Otherwise, the routine is implementation dependent.
306:        - You MUST call VecRestoreArray() when you no longer need access to
307:          the array.
308:   */
309:   VecGetArray(X,&x);

311:   /*
312:      Get local grid boundaries (for 2-dimensional DMDA):
313:        xs, ys   - starting grid indices (no ghost points)
314:        xm, ym   - widths of local grid (no ghost points)
315:        gxs, gys - starting grid indices (including ghost points)
316:        gxm, gym - widths of local grid (including ghost points)
317:   */
318:   DMDAGetCorners(user->da,&xs,&ys,NULL,&xm,&ym,NULL);
319:   DMDAGetGhostCorners(user->da,&gxs,&gys,NULL,&gxm,&gym,NULL);

321:   /*
322:      Compute initial guess over the locally owned part of the grid
323:   */
324:   for (j=ys; j<ys+ym; j++) {
325:     temp = (PetscReal)(PetscMin(j,my-j-1))*hy;
326:     for (i=xs; i<xs+xm; i++) {
327:       row = i - gxs + (j - gys)*gxm;
328:       if (i == 0 || j == 0 || i == mx-1 || j == my-1) {
329:         x[row] = 0.0;
330:         continue;
331:       }
332:       x[row] = temp1*PetscSqrtReal(PetscMin((PetscReal)(PetscMin(i,mx-i-1))*hx,temp));
333:     }
334:   }

336:   /*
337:      Restore vector
338:   */
339:   VecRestoreArray(X,&x);
340:   return 0;
341: }
342: /* ------------------------------------------------------------------- */
343: /*
344:    ComputeFunction - Evaluates nonlinear function, F(x).

346:    Input Parameters:
347: .  X - input vector
348: .  user - user-defined application context

350:    Output Parameter:
351: .  F - function vector
352:  */
353: PetscErrorCode ComputeFunction(AppCtx *user,Vec X,Vec F)
354: {
355:   PetscInt       i,j,row,mx,my,xs,ys,xm,ym,gxs,gys,gxm,gym;
356:   PetscReal      two = 2.0,one = 1.0,lambda,hx,hy,hxdhy,hydhx,sc;
357:   PetscScalar    u,uxx,uyy,*x,*f;
358:   Vec            localX = user->localX;

360:   mx = user->mx;            my = user->my;            lambda = user->param;
361:   hx = one/(PetscReal)(mx-1);  hy = one/(PetscReal)(my-1);
362:   sc = hx*hy*lambda;        hxdhy = hx/hy;            hydhx = hy/hx;

364:   /*
365:      Scatter ghost points to local vector, using the 2-step process
366:         DMGlobalToLocalBegin(), DMGlobalToLocalEnd().
367:      By placing code between these two statements, computations can be
368:      done while messages are in transition.
369:   */
370:   DMGlobalToLocalBegin(user->da,X,INSERT_VALUES,localX);
371:   DMGlobalToLocalEnd(user->da,X,INSERT_VALUES,localX);

373:   /*
374:      Get pointers to vector data
375:   */
376:   VecGetArray(localX,&x);
377:   VecGetArray(F,&f);

379:   /*
380:      Get local grid boundaries
381:   */
382:   DMDAGetCorners(user->da,&xs,&ys,NULL,&xm,&ym,NULL);
383:   DMDAGetGhostCorners(user->da,&gxs,&gys,NULL,&gxm,&gym,NULL);

385:   /*
386:      Compute function over the locally owned part of the grid
387:   */
388:   for (j=ys; j<ys+ym; j++) {
389:     row = (j - gys)*gxm + xs - gxs - 1;
390:     for (i=xs; i<xs+xm; i++) {
391:       row++;
392:       if (i == 0 || j == 0 || i == mx-1 || j == my-1) {
393:         f[row] = x[row];
394:         continue;
395:       }
396:       u      = x[row];
397:       uxx    = (two*u - x[row-1] - x[row+1])*hydhx;
398:       uyy    = (two*u - x[row-gxm] - x[row+gxm])*hxdhy;
399:       f[row] = uxx + uyy - sc*PetscExpScalar(u);
400:     }
401:   }

403:   /*
404:      Restore vectors
405:   */
406:   VecRestoreArray(localX,&x);
407:   VecRestoreArray(F,&f);
408:   PetscLogFlops(11.0*ym*xm);
409:   return 0;
410: }
411: /* ------------------------------------------------------------------- */
412: /*
413:    ComputeJacobian - Evaluates Jacobian matrix.

415:    Input Parameters:
416: .  x - input vector
417: .  user - user-defined application context

419:    Output Parameters:
420: .  jac - Jacobian matrix
421: .  flag - flag indicating matrix structure

423:    Notes:
424:    Due to grid point reordering with DMDAs, we must always work
425:    with the local grid points, and then transform them to the new
426:    global numbering with the "ltog" mapping
427:    We cannot work directly with the global numbers for the original
428:    uniprocessor grid!
429: */
430: PetscErrorCode ComputeJacobian(AppCtx *user,Vec X,Mat jac)
431: {
432:   Vec                    localX = user->localX;   /* local vector */
433:   const PetscInt         *ltog;                   /* local-to-global mapping */
434:   PetscInt               i,j,row,mx,my,col[5];
435:   PetscInt               xs,ys,xm,ym,gxs,gys,gxm,gym,grow;
436:   PetscScalar            two = 2.0,one = 1.0,lambda,v[5],hx,hy,hxdhy,hydhx,sc,*x;
437:   ISLocalToGlobalMapping ltogm;

439:   mx = user->mx;            my = user->my;            lambda = user->param;
440:   hx = one/(PetscReal)(mx-1);  hy = one/(PetscReal)(my-1);
441:   sc = hx*hy;               hxdhy = hx/hy;            hydhx = hy/hx;

443:   /*
444:      Scatter ghost points to local vector, using the 2-step process
445:         DMGlobalToLocalBegin(), DMGlobalToLocalEnd().
446:      By placing code between these two statements, computations can be
447:      done while messages are in transition.
448:   */
449:   DMGlobalToLocalBegin(user->da,X,INSERT_VALUES,localX);
450:   DMGlobalToLocalEnd(user->da,X,INSERT_VALUES,localX);

452:   /*
453:      Get pointer to vector data
454:   */
455:   VecGetArray(localX,&x);

457:   /*
458:      Get local grid boundaries
459:   */
460:   DMDAGetCorners(user->da,&xs,&ys,NULL,&xm,&ym,NULL);
461:   DMDAGetGhostCorners(user->da,&gxs,&gys,NULL,&gxm,&gym,NULL);

463:   /*
464:      Get the global node numbers for all local nodes, including ghost points
465:   */
466:   DMGetLocalToGlobalMapping(user->da,&ltogm);
467:   ISLocalToGlobalMappingGetIndices(ltogm,&ltog);

469:   /*
470:      Compute entries for the locally owned part of the Jacobian.
471:       - Currently, all PETSc parallel matrix formats are partitioned by
472:         contiguous chunks of rows across the processors. The "grow"
473:         parameter computed below specifies the global row number
474:         corresponding to each local grid point.
475:       - Each processor needs to insert only elements that it owns
476:         locally (but any non-local elements will be sent to the
477:         appropriate processor during matrix assembly).
478:       - Always specify global row and columns of matrix entries.
479:       - Here, we set all entries for a particular row at once.
480:   */
481:   for (j=ys; j<ys+ym; j++) {
482:     row = (j - gys)*gxm + xs - gxs - 1;
483:     for (i=xs; i<xs+xm; i++) {
484:       row++;
485:       grow = ltog[row];
486:       /* boundary points */
487:       if (i == 0 || j == 0 || i == mx-1 || j == my-1) {
488:         MatSetValues(jac,1,&grow,1,&grow,&one,INSERT_VALUES);
489:         continue;
490:       }
491:       /* interior grid points */
492:       v[0] = -hxdhy; col[0] = ltog[row - gxm];
493:       v[1] = -hydhx; col[1] = ltog[row - 1];
494:       v[2] = two*(hydhx + hxdhy) - sc*lambda*PetscExpScalar(x[row]); col[2] = grow;
495:       v[3] = -hydhx; col[3] = ltog[row + 1];
496:       v[4] = -hxdhy; col[4] = ltog[row + gxm];
497:       MatSetValues(jac,1,&grow,5,col,v,INSERT_VALUES);
498:     }
499:   }
500:   ISLocalToGlobalMappingRestoreIndices(ltogm,&ltog);

502:   /*
503:      Assemble matrix, using the 2-step process:
504:        MatAssemblyBegin(), MatAssemblyEnd().
505:      By placing code between these two statements, computations can be
506:      done while messages are in transition.
507:   */
508:   MatAssemblyBegin(jac,MAT_FINAL_ASSEMBLY);
509:   VecRestoreArray(localX,&x);
510:   MatAssemblyEnd(jac,MAT_FINAL_ASSEMBLY);

512:   return 0;
513: }

515: /*TEST

517:     test:

519: TEST*/