Actual source code: ex1.c


  2: static char help[] = "Solves the nonlinear system, the Bratu (SFI - solid fuel ignition) problem in a 2D rectangular domain.\n\
  3: This example also illustrates the use of matrix coloring.  Runtime options include:\n\
  4:   -par <parameter>, where <parameter> indicates the problem's nonlinearity\n\
  5:      problem SFI:  <parameter> = Bratu parameter (0 <= par <= 6.81)\n\
  6:   -mx <xg>, where <xg> = number of grid points in the x-direction\n\
  7:   -my <yg>, where <yg> = number of grid points in the y-direction\n\n";

  9: /*T
 10:    Concepts: SNES^sequential Bratu example
 11:    Processors: 1
 12: T*/

 14: /* ------------------------------------------------------------------------

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

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

 21:     with boundary conditions

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

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

 29:     The parallel version of this code is snes/tutorials/ex5.c

 31:   ------------------------------------------------------------------------- */

 33: /*
 34:    Include "petscsnes.h" so that we can use SNES solvers.  Note that
 35:    this file automatically includes:
 36:      petscsys.h       - base PETSc routines   petscvec.h - vectors
 37:      petscmat.h - matrices
 38:      petscis.h     - index sets            petscksp.h - Krylov subspace methods
 39:      petscviewer.h - viewers               petscpc.h  - preconditioners
 40:      petscksp.h   - linear solvers
 41: */

 43: #include <petscsnes.h>

 45: /*
 46:    User-defined application context - contains data needed by the
 47:    application-provided call-back routines, FormJacobian() and
 48:    FormFunction().
 49: */
 50: typedef struct {
 51:   PetscReal param;              /* test problem parameter */
 52:   PetscInt  mx;                 /* Discretization in x-direction */
 53:   PetscInt  my;                 /* Discretization in y-direction */
 54: } AppCtx;

 56: /*
 57:    User-defined routines
 58: */
 59: extern PetscErrorCode FormJacobian(SNES,Vec,Mat,Mat,void*);
 60: extern PetscErrorCode FormFunction(SNES,Vec,Vec,void*);
 61: extern PetscErrorCode FormInitialGuess(AppCtx*,Vec);
 62: extern PetscErrorCode ConvergenceTest(KSP,PetscInt,PetscReal,KSPConvergedReason*,void*);
 63: extern PetscErrorCode ConvergenceDestroy(void*);
 64: extern PetscErrorCode postcheck(SNES,Vec,Vec,Vec,PetscBool*,PetscBool*,void*);

 66: int main(int argc,char **argv)
 67: {
 68:   SNES           snes;                 /* nonlinear solver context */
 69:   Vec            x,r;                 /* solution, residual vectors */
 70:   Mat            J;                    /* Jacobian matrix */
 71:   AppCtx         user;                 /* user-defined application context */
 72:   PetscInt       i,its,N,hist_its[50];
 73:   PetscMPIInt    size;
 74:   PetscReal      bratu_lambda_max = 6.81,bratu_lambda_min = 0.,history[50];
 75:   MatFDColoring  fdcoloring;
 76:   PetscBool      matrix_free = PETSC_FALSE,flg,fd_coloring = PETSC_FALSE, use_convergence_test = PETSC_FALSE,pc = PETSC_FALSE;
 77:   KSP            ksp;
 78:   PetscInt       *testarray;

 80:   PetscInitialize(&argc,&argv,(char*)0,help);
 81:   MPI_Comm_size(PETSC_COMM_WORLD,&size);

 84:   /*
 85:      Initialize problem parameters
 86:   */
 87:   user.mx = 4; user.my = 4; user.param = 6.0;
 88:   PetscOptionsGetInt(NULL,NULL,"-mx",&user.mx,NULL);
 89:   PetscOptionsGetInt(NULL,NULL,"-my",&user.my,NULL);
 90:   PetscOptionsGetReal(NULL,NULL,"-par",&user.param,NULL);
 91:   PetscOptionsGetBool(NULL,NULL,"-pc",&pc,NULL);
 93:   N = user.mx*user.my;
 94:   PetscOptionsGetBool(NULL,NULL,"-use_convergence_test",&use_convergence_test,NULL);

 96:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 97:      Create nonlinear solver context
 98:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

100:   SNESCreate(PETSC_COMM_WORLD,&snes);

102:   if (pc) {
103:     SNESSetType(snes,SNESNEWTONTR);
104:     SNESNewtonTRSetPostCheck(snes, postcheck,NULL);
105:   }

107:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
108:      Create vector data structures; set function evaluation routine
109:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

111:   VecCreate(PETSC_COMM_WORLD,&x);
112:   VecSetSizes(x,PETSC_DECIDE,N);
113:   VecSetFromOptions(x);
114:   VecDuplicate(x,&r);

116:   /*
117:      Set function evaluation routine and vector.  Whenever the nonlinear
118:      solver needs to evaluate the nonlinear function, it will call this
119:      routine.
120:       - Note that the final routine argument is the user-defined
121:         context that provides application-specific data for the
122:         function evaluation routine.
123:   */
124:   SNESSetFunction(snes,r,FormFunction,(void*)&user);

126:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
127:      Create matrix data structure; set Jacobian evaluation routine
128:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

130:   /*
131:      Create matrix. Here we only approximately preallocate storage space
132:      for the Jacobian.  See the users manual for a discussion of better
133:      techniques for preallocating matrix memory.
134:   */
135:   PetscOptionsGetBool(NULL,NULL,"-snes_mf",&matrix_free,NULL);
136:   if (!matrix_free) {
137:     PetscBool matrix_free_operator = PETSC_FALSE;
138:     PetscOptionsGetBool(NULL,NULL,"-snes_mf_operator",&matrix_free_operator,NULL);
139:     if (matrix_free_operator) matrix_free = PETSC_FALSE;
140:   }
141:   if (!matrix_free) {
142:     MatCreateSeqAIJ(PETSC_COMM_WORLD,N,N,5,NULL,&J);
143:   }

145:   /*
146:      This option will cause the Jacobian to be computed via finite differences
147:     efficiently using a coloring of the columns of the matrix.
148:   */
149:   PetscOptionsGetBool(NULL,NULL,"-snes_fd_coloring",&fd_coloring,NULL);

152:   if (fd_coloring) {
153:     ISColoring   iscoloring;
154:     MatColoring  mc;

156:     /*
157:       This initializes the nonzero structure of the Jacobian. This is artificial
158:       because clearly if we had a routine to compute the Jacobian we won't need
159:       to use finite differences.
160:     */
161:     FormJacobian(snes,x,J,J,&user);

163:     /*
164:        Color the matrix, i.e. determine groups of columns that share no common
165:       rows. These columns in the Jacobian can all be computed simultaneously.
166:     */
167:     MatColoringCreate(J,&mc);
168:     MatColoringSetType(mc,MATCOLORINGSL);
169:     MatColoringSetFromOptions(mc);
170:     MatColoringApply(mc,&iscoloring);
171:     MatColoringDestroy(&mc);
172:     /*
173:        Create the data structure that SNESComputeJacobianDefaultColor() uses
174:        to compute the actual Jacobians via finite differences.
175:     */
176:     MatFDColoringCreate(J,iscoloring,&fdcoloring);
177:     MatFDColoringSetFunction(fdcoloring,(PetscErrorCode (*)(void))FormFunction,&user);
178:     MatFDColoringSetFromOptions(fdcoloring);
179:     MatFDColoringSetUp(J,iscoloring,fdcoloring);
180:     /*
181:         Tell SNES to use the routine SNESComputeJacobianDefaultColor()
182:       to compute Jacobians.
183:     */
184:     SNESSetJacobian(snes,J,J,SNESComputeJacobianDefaultColor,fdcoloring);
185:     ISColoringDestroy(&iscoloring);
186:   }
187:   /*
188:      Set Jacobian matrix data structure and default Jacobian evaluation
189:      routine.  Whenever the nonlinear solver needs to compute the
190:      Jacobian matrix, it will call this routine.
191:       - Note that the final routine argument is the user-defined
192:         context that provides application-specific data for the
193:         Jacobian evaluation routine.
194:       - The user can override with:
195:          -snes_fd : default finite differencing approximation of Jacobian
196:          -snes_mf : matrix-free Newton-Krylov method with no preconditioning
197:                     (unless user explicitly sets preconditioner)
198:          -snes_mf_operator : form preconditioning matrix as set by the user,
199:                              but use matrix-free approx for Jacobian-vector
200:                              products within Newton-Krylov method
201:   */
202:   else if (!matrix_free) {
203:     SNESSetJacobian(snes,J,J,FormJacobian,(void*)&user);
204:   }

206:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
207:      Customize nonlinear solver; set runtime options
208:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

210:   /*
211:      Set runtime options (e.g., -snes_monitor -snes_rtol <rtol> -ksp_type <type>)
212:   */
213:   SNESSetFromOptions(snes);

215:   /*
216:      Set array that saves the function norms.  This array is intended
217:      when the user wants to save the convergence history for later use
218:      rather than just to view the function norms via -snes_monitor.
219:   */
220:   SNESSetConvergenceHistory(snes,history,hist_its,50,PETSC_TRUE);

222:   /*
223:       Add a user provided convergence test; this is to test that SNESNEWTONTR properly calls the
224:       user provided test before the specialized test. The convergence context is just an array to
225:       test that it gets properly freed at the end
226:   */
227:   if (use_convergence_test) {
228:     SNESGetKSP(snes,&ksp);
229:     PetscMalloc1(5,&testarray);
230:     KSPSetConvergenceTest(ksp,ConvergenceTest,testarray,ConvergenceDestroy);
231:   }

233:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
234:      Evaluate initial guess; then solve nonlinear system
235:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
236:   /*
237:      Note: The user should initialize the vector, x, with the initial guess
238:      for the nonlinear solver prior to calling SNESSolve().  In particular,
239:      to employ an initial guess of zero, the user should explicitly set
240:      this vector to zero by calling VecSet().
241:   */
242:   FormInitialGuess(&user,x);
243:   SNESSolve(snes,NULL,x);
244:   SNESGetIterationNumber(snes,&its);
245:   PetscPrintf(PETSC_COMM_WORLD,"Number of SNES iterations = %D\n",its);

247:   /*
248:      Print the convergence history.  This is intended just to demonstrate
249:      use of the data attained via SNESSetConvergenceHistory().
250:   */
251:   PetscOptionsHasName(NULL,NULL,"-print_history",&flg);
252:   if (flg) {
253:     for (i=0; i<its+1; i++) {
254:       PetscPrintf(PETSC_COMM_WORLD,"iteration %D: Linear iterations %D Function norm = %g\n",i,hist_its[i],(double)history[i]);
255:     }
256:   }

258:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
259:      Free work space.  All PETSc objects should be destroyed when they
260:      are no longer needed.
261:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

263:   if (!matrix_free) {
264:     MatDestroy(&J);
265:   }
266:   if (fd_coloring) {
267:     MatFDColoringDestroy(&fdcoloring);
268:   }
269:   VecDestroy(&x);
270:   VecDestroy(&r);
271:   SNESDestroy(&snes);
272:   PetscFinalize();
273:   return 0;
274: }
275: /* ------------------------------------------------------------------- */
276: /*
277:    FormInitialGuess - Forms initial approximation.

279:    Input Parameters:
280:    user - user-defined application context
281:    X - vector

283:    Output Parameter:
284:    X - vector
285:  */
286: PetscErrorCode FormInitialGuess(AppCtx *user,Vec X)
287: {
288:   PetscInt       i,j,row,mx,my;
289:   PetscReal      lambda,temp1,temp,hx,hy;
290:   PetscScalar    *x;

292:   mx     = user->mx;
293:   my     = user->my;
294:   lambda = user->param;

296:   hx = 1.0 / (PetscReal)(mx-1);
297:   hy = 1.0 / (PetscReal)(my-1);

299:   /*
300:      Get a pointer to vector data.
301:        - For default PETSc vectors, VecGetArray() returns a pointer to
302:          the data array.  Otherwise, the routine is implementation dependent.
303:        - You MUST call VecRestoreArray() when you no longer need access to
304:          the array.
305:   */
306:   VecGetArray(X,&x);
307:   temp1 = lambda/(lambda + 1.0);
308:   for (j=0; j<my; j++) {
309:     temp = (PetscReal)(PetscMin(j,my-j-1))*hy;
310:     for (i=0; i<mx; i++) {
311:       row = i + j*mx;
312:       if (i == 0 || j == 0 || i == mx-1 || j == my-1) {
313:         x[row] = 0.0;
314:         continue;
315:       }
316:       x[row] = temp1*PetscSqrtReal(PetscMin((PetscReal)(PetscMin(i,mx-i-1))*hx,temp));
317:     }
318:   }

320:   /*
321:      Restore vector
322:   */
323:   VecRestoreArray(X,&x);
324:   return 0;
325: }
326: /* ------------------------------------------------------------------- */
327: /*
328:    FormFunction - Evaluates nonlinear function, F(x).

330:    Input Parameters:
331: .  snes - the SNES context
332: .  X - input vector
333: .  ptr - optional user-defined context, as set by SNESSetFunction()

335:    Output Parameter:
336: .  F - function vector
337:  */
338: PetscErrorCode FormFunction(SNES snes,Vec X,Vec F,void *ptr)
339: {
340:   AppCtx            *user = (AppCtx*)ptr;
341:   PetscInt          i,j,row,mx,my;
342:   PetscReal         two = 2.0,one = 1.0,lambda,hx,hy,hxdhy,hydhx;
343:   PetscScalar       ut,ub,ul,ur,u,uxx,uyy,sc,*f;
344:   const PetscScalar *x;

346:   mx     = user->mx;
347:   my     = user->my;
348:   lambda = user->param;
349:   hx     = one / (PetscReal)(mx-1);
350:   hy     = one / (PetscReal)(my-1);
351:   sc     = hx*hy;
352:   hxdhy  = hx/hy;
353:   hydhx  = hy/hx;

355:   /*
356:      Get pointers to vector data
357:   */
358:   VecGetArrayRead(X,&x);
359:   VecGetArray(F,&f);

361:   /*
362:      Compute function
363:   */
364:   for (j=0; j<my; j++) {
365:     for (i=0; i<mx; i++) {
366:       row = i + j*mx;
367:       if (i == 0 || j == 0 || i == mx-1 || j == my-1) {
368:         f[row] = x[row];
369:         continue;
370:       }
371:       u      = x[row];
372:       ub     = x[row - mx];
373:       ul     = x[row - 1];
374:       ut     = x[row + mx];
375:       ur     = x[row + 1];
376:       uxx    = (-ur + two*u - ul)*hydhx;
377:       uyy    = (-ut + two*u - ub)*hxdhy;
378:       f[row] = uxx + uyy - sc*lambda*PetscExpScalar(u);
379:     }
380:   }

382:   /*
383:      Restore vectors
384:   */
385:   VecRestoreArrayRead(X,&x);
386:   VecRestoreArray(F,&f);
387:   return 0;
388: }
389: /* ------------------------------------------------------------------- */
390: /*
391:    FormJacobian - Evaluates Jacobian matrix.

393:    Input Parameters:
394: .  snes - the SNES context
395: .  x - input vector
396: .  ptr - optional user-defined context, as set by SNESSetJacobian()

398:    Output Parameters:
399: .  A - Jacobian matrix
400: .  B - optionally different preconditioning matrix
401: .  flag - flag indicating matrix structure
402: */
403: PetscErrorCode FormJacobian(SNES snes,Vec X,Mat J,Mat jac,void *ptr)
404: {
405:   AppCtx            *user = (AppCtx*)ptr;   /* user-defined applicatin context */
406:   PetscInt          i,j,row,mx,my,col[5];
407:   PetscScalar       two = 2.0,one = 1.0,lambda,v[5],sc;
408:   const PetscScalar *x;
409:   PetscReal         hx,hy,hxdhy,hydhx;

411:   mx     = user->mx;
412:   my     = user->my;
413:   lambda = user->param;
414:   hx     = 1.0 / (PetscReal)(mx-1);
415:   hy     = 1.0 / (PetscReal)(my-1);
416:   sc     = hx*hy;
417:   hxdhy  = hx/hy;
418:   hydhx  = hy/hx;

420:   /*
421:      Get pointer to vector data
422:   */
423:   VecGetArrayRead(X,&x);

425:   /*
426:      Compute entries of the Jacobian
427:   */
428:   for (j=0; j<my; j++) {
429:     for (i=0; i<mx; i++) {
430:       row = i + j*mx;
431:       if (i == 0 || j == 0 || i == mx-1 || j == my-1) {
432:         MatSetValues(jac,1,&row,1,&row,&one,INSERT_VALUES);
433:         continue;
434:       }
435:       v[0] = -hxdhy; col[0] = row - mx;
436:       v[1] = -hydhx; col[1] = row - 1;
437:       v[2] = two*(hydhx + hxdhy) - sc*lambda*PetscExpScalar(x[row]); col[2] = row;
438:       v[3] = -hydhx; col[3] = row + 1;
439:       v[4] = -hxdhy; col[4] = row + mx;
440:       MatSetValues(jac,1,&row,5,col,v,INSERT_VALUES);
441:     }
442:   }

444:   /*
445:      Restore vector
446:   */
447:   VecRestoreArrayRead(X,&x);

449:   /*
450:      Assemble matrix
451:   */
452:   MatAssemblyBegin(jac,MAT_FINAL_ASSEMBLY);
453:   MatAssemblyEnd(jac,MAT_FINAL_ASSEMBLY);

455:   if (jac != J) {
456:     MatAssemblyBegin(J,MAT_FINAL_ASSEMBLY);
457:     MatAssemblyEnd(J,MAT_FINAL_ASSEMBLY);
458:   }

460:   return 0;
461: }

463: PetscErrorCode ConvergenceTest(KSP ksp,PetscInt it,PetscReal nrm,KSPConvergedReason *reason,void *ctx)
464: {
465:   *reason = KSP_CONVERGED_ITERATING;
466:   if (it > 1) {
467:     *reason = KSP_CONVERGED_ITS;
468:     PetscInfo(NULL,"User provided convergence test returning after 2 iterations\n");
469:   }
470:   return 0;
471: }

473: PetscErrorCode ConvergenceDestroy(void* ctx)
474: {
475:   PetscInfo(NULL,"User provided convergence destroy called\n");
476:   PetscFree(ctx);
477:   return 0;
478: }

480: PetscErrorCode postcheck(SNES snes,Vec x,Vec y,Vec w,PetscBool *changed_y,PetscBool *changed_w,void *ctx)
481: {
482:   PetscReal      norm;
483:   Vec            tmp;

485:   VecDuplicate(x,&tmp);
486:   VecWAXPY(tmp,-1.0,x,w);
487:   VecNorm(tmp,NORM_2,&norm);
488:   VecDestroy(&tmp);
489:   PetscPrintf(PETSC_COMM_WORLD,"Norm of search step %g\n",(double)norm);
490:   return 0;
491: }

493: /*TEST

495:    build:
496:       requires: !single

498:    test:
499:       args: -ksp_gmres_cgs_refinement_type refine_always

501:    test:
502:       suffix: 2
503:       args: -snes_monitor_short -snes_type newtontr -ksp_gmres_cgs_refinement_type refine_always

505:    test:
506:       suffix: 2a
507:       filter: grep -i KSPConvergedDefault > /dev/null && echo "Found KSPConvergedDefault"
508:       args: -snes_monitor_short -snes_type newtontr -ksp_gmres_cgs_refinement_type refine_always -info
509:       requires: defined(PETSC_USE_INFO)

511:    test:
512:       suffix: 2b
513:       filter: grep -i  "User provided convergence test" > /dev/null  && echo "Found User provided convergence test"
514:       args: -snes_monitor_short -snes_type newtontr -ksp_gmres_cgs_refinement_type refine_always -use_convergence_test -info
515:       requires: defined(PETSC_USE_INFO)

517:    test:
518:       suffix: 3
519:       args: -snes_monitor_short -mat_coloring_type sl -snes_fd_coloring -mx 8 -my 11 -ksp_gmres_cgs_refinement_type refine_always

521:    test:
522:       suffix: 4
523:       args: -pc -par 6.807 -snes_monitor -snes_converged_reason

525: TEST*/