Actual source code: ex5.c

  1: /*$Id: ex5.c,v 1.29 2001/08/07 21:31:12 bsmith Exp $*/

  3: static char help[] = "Solves a nonlinear system in parallel with SNES.\n\
  4: We solve the modified Bratu problem in a 2D rectangular domain,\n\
  5: using distributed arrays (DAs) to partition the parallel grid.\n\
  6: The command line options include:\n\
  7:   -lambda <parameter>, where <parameter> indicates the problem's nonlinearity\n\
  8:   -kappa  <parameter>, where <parameter> indicates the problem's nonlinearity\n\
  9:   -mx <xg>, where <xg> = number of grid points in the x-direction\n\
 10:   -my <yg>, where <yg> = number of grid points in the y-direction\n\
 11:   -Nx <npx>, where <npx> = number of processors in the x-direction\n\
 12:   -Ny <npy>, where <npy> = number of processors in the y-direction\n\n";

 14: /*T
 15:    Concepts: SNES^solving a system of nonlinear equations (parallel Bratu example);
 16:    Concepts: DA^using distributed arrays;
 17:    Processors: n
 18: T*/

 20: /* ------------------------------------------------------------------------

 22:     Modified Solid Fuel Ignition problem.  This problem is modeled by
 23:     the partial differential equation

 25:         -Laplacian u - kappa*\PartDer{u}{x} - lambda*exp(u) = 0,

 27:     where

 29:          0 < x,y < 1,
 30:   
 31:     with boundary conditions
 32:    
 33:              u = 0  for  x = 0, x = 1, y = 0, y = 1.
 34:   
 35:     A finite difference approximation with the usual 5-point stencil
 36:     is used to discretize the boundary value problem to obtain a nonlinear 
 37:     system of equations.

 39:   ------------------------------------------------------------------------- */

 41: /* 
 42:    Include "petscda.h" so that we can use distributed arrays (DAs).
 43:    Include "petscsnes.h" so that we can use SNES solvers.  Note that this
 44:    file automatically includes:
 45:      petsc.h       - base PETSc routines   petscvec.h - vectors
 46:      petscsys.h    - system routines       petscmat.h - matrices
 47:      petscis.h     - index sets            petscksp.h - Krylov subspace methods
 48:      petscviewer.h - viewers               petscpc.h  - preconditioners
 49:      petscsles.h   - linear solvers
 50: */
 51:  #include petscda.h
 52:  #include petscsnes.h

 54: /* 
 55:    User-defined application context - contains data needed by the 
 56:    application-provided call-back routines, FormJacobian() and
 57:    FormFunction().
 58: */
 59: typedef struct {
 60:    PetscReal   param;          /* test problem parameter */
 61:    PetscReal   param2;         /* test problem parameter */
 62:    int         mx,my;          /* discretization in x, y directions */
 63:    Vec         localX,localF; /* ghosted local vector */
 64:    DA          da;             /* distributed array data structure */
 65:    int         rank;           /* processor rank */
 66: } AppCtx;

 68: /* 
 69:    User-defined routines
 70: */
 71: extern int FormFunction(SNES,Vec,Vec,void*),FormInitialGuess(AppCtx*,Vec);
 72: extern int FormJacobian(SNES,Vec,Mat*,Mat*,MatStructure*,void*);

 76: int main(int argc,char **argv)
 77: {
 78:   SNES       snes;                /* nonlinear solver */
 79:   Vec        x,r;                /* solution, residual vectors */
 80:   Mat        J;                   /* Jacobian matrix */
 81:   AppCtx     user;                /* user-defined work context */
 82:   int        its;                 /* iterations for convergence */
 83:   int        Nx,Ny;              /* number of preocessors in x- and y- directions */
 84:   PetscTruth matrix_free;         /* flag - 1 indicates matrix-free version */
 85:   int        size;                /* number of processors */
 86:   int        m,N,ierr;
 87:   PetscReal  bratu_lambda_max = 6.81,bratu_lambda_min = 0.;
 88:   PetscReal  bratu_kappa_max = 10000,bratu_kappa_min = 0.;

 90:   PetscInitialize(&argc,&argv,(char *)0,help);
 91:   MPI_Comm_rank(PETSC_COMM_WORLD,&user.rank);

 93:   /*
 94:      Initialize problem parameters
 95:   */
 96:   user.mx = 4; user.my = 4; user.param = 6.0; user.param2 = 0.0;
 97:   PetscOptionsGetInt(PETSC_NULL,"-mx",&user.mx,PETSC_NULL);
 98:   PetscOptionsGetInt(PETSC_NULL,"-my",&user.my,PETSC_NULL);
 99:   PetscOptionsGetReal(PETSC_NULL,"-lambda",&user.param,PETSC_NULL);
100:   if (user.param >= bratu_lambda_max || user.param <= bratu_lambda_min) {
101:     SETERRQ(1,"Lambda is out of range");
102:   }
103:   PetscOptionsGetReal(PETSC_NULL,"-kappa",&user.param2,PETSC_NULL);
104:   if (user.param2 >= bratu_kappa_max || user.param2 < bratu_kappa_min) {
105:     SETERRQ(1,"Kappa is out of range");
106:   }
107:   PetscPrintf(PETSC_COMM_WORLD,"Solving the Bratu problem with lambda=%g, kappa=%g\n",user.param,user.param2);

109:   N = user.mx*user.my;

111:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
112:      Create nonlinear solver context
113:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

115:   SNESCreate(PETSC_COMM_WORLD,&snes);

117:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
118:      Create vector data structures; set function evaluation routine
119:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

121:   /*
122:      Create distributed array (DA) to manage parallel grid and vectors
123:   */
124:   MPI_Comm_size(PETSC_COMM_WORLD,&size);
125:   Nx = PETSC_DECIDE; Ny = PETSC_DECIDE;
126:   PetscOptionsGetInt(PETSC_NULL,"-Nx",&Nx,PETSC_NULL);
127:   PetscOptionsGetInt(PETSC_NULL,"-Ny",&Ny,PETSC_NULL);
128:   if (Nx*Ny != size && (Nx != PETSC_DECIDE || Ny != PETSC_DECIDE))
129:     SETERRQ(1,"Incompatible number of processors:  Nx * Ny != size");
130:   DACreate2d(PETSC_COMM_WORLD,DA_NONPERIODIC,DA_STENCIL_STAR,user.mx,user.my,Nx,Ny,1,1,PETSC_NULL,PETSC_NULL,&user.da);

132:   /*
133:      Visualize the distribution of the array across the processors
134:   */
135:   /*  DAView(user.da,PETSC_VIEWER_DRAW_WORLD); */


138:   /*
139:      Extract global and local vectors from DA; then duplicate for remaining
140:      vectors that are the same types
141:   */
142:   DACreateGlobalVector(user.da,&x);
143:   DACreateLocalVector(user.da,&user.localX);
144:   VecDuplicate(x,&r);
145:   VecDuplicate(user.localX,&user.localF);

147:   /* 
148:      Set function evaluation routine and vector
149:   */
150:   SNESSetFunction(snes,r,FormFunction,(void*)&user);

152:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
153:      Create matrix data structure; set Jacobian evaluation routine
154:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

156:   /* 
157:      Set Jacobian matrix data structure and default Jacobian evaluation
158:      routine. User can override with:
159:      -snes_fd : default finite differencing approximation of Jacobian
160:      -snes_mf : matrix-free Newton-Krylov method with no preconditioning
161:                 (unless user explicitly sets preconditioner) 
162:      -snes_mf_operator : form preconditioning matrix as set by the user,
163:                          but use matrix-free approx for Jacobian-vector
164:                          products within Newton-Krylov method

166:      Note:  For the parallel case, vectors and matrices MUST be partitioned
167:      accordingly.  When using distributed arrays (DAs) to create vectors,
168:      the DAs determine the problem partitioning.  We must explicitly
169:      specify the local matrix dimensions upon its creation for compatibility
170:      with the vector distribution.  Thus, the generic MatCreate() routine
171:      is NOT sufficient when working with distributed arrays.

173:      Note: Here we only approximately preallocate storage space for the
174:      Jacobian.  See the users manual for a discussion of better techniques
175:      for preallocating matrix memory.
176:   */
177:   PetscOptionsHasName(PETSC_NULL,"-snes_mf",&matrix_free);
178:   if (!matrix_free) {
179:     if (size == 1) {
180:       MatCreateSeqAIJ(PETSC_COMM_WORLD,N,N,5,PETSC_NULL,&J);
181:     } else {
182:       VecGetLocalSize(x,&m);
183:       MatCreateMPIAIJ(PETSC_COMM_WORLD,m,m,N,N,5,PETSC_NULL,3,PETSC_NULL,&J);
184:     }
185:     SNESSetJacobian(snes,J,J,FormJacobian,&user);
186:   }

188:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
189:      Customize nonlinear solver; set runtime options
190:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

192:   /*
193:      Set runtime options (e.g., -snes_monitor -snes_rtol <rtol> -ksp_type <type>)
194:   */
195:   SNESSetFromOptions(snes);

197:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
198:      Evaluate initial guess; then solve nonlinear system
199:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
200:   /*
201:      Note: The user should initialize the vector, x, with the initial guess
202:      for the nonlinear solver prior to calling SNESSolve().  In particular,
203:      to employ an initial guess of zero, the user should explicitly set
204:      this vector to zero by calling VecSet().
205:   */
206:   FormInitialGuess(&user,x);
207:   SNESSolve(snes,x,&its);
208:   PetscPrintf(PETSC_COMM_WORLD,"Number of Newton iterations = %d\n",its);

210:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
211:      Free work space.  All PETSc objects should be destroyed when they
212:      are no longer needed.
213:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

215:   if (!matrix_free) {
216:     MatDestroy(J);
217:   }
218:   VecDestroy(user.localX); VecDestroy(x);
219:   VecDestroy(user.localF); VecDestroy(r);
220:   SNESDestroy(snes);  DADestroy(user.da);
221:   PetscFinalize();

223:   return 0;
224: }
225: /* ------------------------------------------------------------------- */
228: /* 
229:    FormInitialGuess - Forms initial approximation.

231:    Input Parameters:
232:    user - user-defined application context
233:    X - vector

235:    Output Parameter:
236:    X - vector
237:  */
238: int FormInitialGuess(AppCtx *user,Vec X)
239: {
240:   int          i,j,row,mx,my,ierr,xs,ys,xm,ym,gxm,gym,gxs,gys;
241:   PetscReal    one = 1.0,lambda,temp1,temp,hx,hy,hxdhy,hydhx,sc;
242:   PetscScalar  *x;
243:   Vec          localX = user->localX;

245:   mx = user->mx;            my = user->my;            lambda = user->param;
246:   hx = one/(PetscReal)(mx-1);  hy = one/(PetscReal)(my-1);
247:   sc = hx*hy*lambda;        hxdhy = hx/hy;            hydhx = hy/hx;
248:   temp1 = lambda/(lambda + one);

250:   /*
251:      Get a pointer to vector data.
252:        - For default PETSc vectors,VecGetArray() returns a pointer to
253:          the data array.  Otherwise, the routine is implementation dependent.
254:        - You MUST call VecRestoreArray() when you no longer need access to
255:          the array.
256:   */
257:   VecGetArray(localX,&x);

259:   /*
260:      Get local grid boundaries (for 2-dimensional DA):
261:        xs, ys   - starting grid indices (no ghost points)
262:        xm, ym   - widths of local grid (no ghost points)
263:        gxs, gys - starting grid indices (including ghost points)
264:        gxm, gym - widths of local grid (including ghost points)
265:   */
266:   DAGetCorners(user->da,&xs,&ys,PETSC_NULL,&xm,&ym,PETSC_NULL);
267:   DAGetGhostCorners(user->da,&gxs,&gys,PETSC_NULL,&gxm,&gym,PETSC_NULL);

269:   /*
270:      Compute initial guess over the locally owned part of the grid
271:   */
272:   for (j=ys; j<ys+ym; j++) {
273:     temp = (PetscReal)(PetscMin(j,my-j-1))*hy;
274:     for (i=xs; i<xs+xm; i++) {
275:       row = i - gxs + (j - gys)*gxm;
276:       if (i == 0 || j == 0 || i == mx-1 || j == my-1) {
277:         x[row] = 0.0;
278:         continue;
279:       }
280:       x[row] = temp1*sqrt(PetscMin((PetscReal)(PetscMin(i,mx-i-1))*hx,temp));
281:     }
282:   }

284:   /*
285:      Restore vector
286:   */
287:   VecRestoreArray(localX,&x);

289:   /*
290:      Insert values into global vector
291:   */
292:   DALocalToGlobal(user->da,localX,INSERT_VALUES,X);
293:   return 0;
294: }
295: /* ------------------------------------------------------------------- */
298: /* 
299:    FormFunction - Evaluates nonlinear function, F(x).

301:    Input Parameters:
302: .  snes - the SNES context
303: .  X - input vector
304: .  ptr - optional user-defined context, as set by SNESSetFunction()

306:    Output Parameter:
307: .  F - function vector
308:  */
309: int FormFunction(SNES snes,Vec X,Vec F,void *ptr)
310: {
311:   AppCtx      *user = (AppCtx*)ptr;
312:   int         ierr,i,j,row,mx,my,xs,ys,xm,ym,gxs,gys,gxm,gym;
313:   PetscReal   two = 2.0,one = 1.0,half = 0.5;
314:   PetscReal   lambda,hx,hy,hxdhy,hydhx,sc;
315:   PetscScalar u,ux,uxx,uyy,*x,*f,kappa;
316:   Vec         localX = user->localX,localF = user->localF;

318:   mx = user->mx;            my = user->my;            lambda = user->param;
319:   hx = one/(PetscReal)(mx-1);  hy = one/(PetscReal)(my-1);
320:   sc = hx*hy*lambda;        hxdhy = hx/hy;            hydhx = hy/hx;
321:   kappa = user->param2;

323:   /*
324:      Scatter ghost points to local vector, using the 2-step process
325:         DAGlobalToLocalBegin(), DAGlobalToLocalEnd().
326:      By placing code between these two statements, computations can be
327:      done while messages are in transition.
328:   */
329:   DAGlobalToLocalBegin(user->da,X,INSERT_VALUES,localX);
330:   DAGlobalToLocalEnd(user->da,X,INSERT_VALUES,localX);

332:   /*
333:      Get pointers to vector data
334:   */
335:   VecGetArray(localX,&x);
336:   VecGetArray(localF,&f);

338:   /*
339:      Get local grid boundaries
340:   */
341:   DAGetCorners(user->da,&xs,&ys,PETSC_NULL,&xm,&ym,PETSC_NULL);
342:   DAGetGhostCorners(user->da,&gxs,&gys,PETSC_NULL,&gxm,&gym,PETSC_NULL);

344:   /*
345:      Compute function over the locally owned part of the grid
346:   */
347:   for (j=ys; j<ys+ym; j++) {
348:     row = (j - gys)*gxm + xs - gxs - 1;
349:     for (i=xs; i<xs+xm; i++) {
350:       row++;
351:       if (i == 0 || j == 0 || i == mx-1 || j == my-1) {
352:         f[row] = x[row];
353:         continue;
354:       }
355:       u = x[row];
356:       ux  = (x[row+1] - x[row-1])*half*hy;
357:       uxx = (two*u - x[row-1] - x[row+1])*hydhx;
358:       uyy = (two*u - x[row-gxm] - x[row+gxm])*hxdhy;
359:       f[row] = uxx + uyy - kappa*ux - sc*exp(u);
360:     }
361:   }

363:   /*
364:      Restore vectors
365:   */
366:   VecRestoreArray(localX,&x);
367:   VecRestoreArray(localF,&f);

369:   /*
370:      Insert values into global vector
371:   */
372:   DALocalToGlobal(user->da,localF,INSERT_VALUES,F);
373:   PetscLogFlops(11*ym*xm);
374:   return 0;
375: }
376: /* ------------------------------------------------------------------- */
379: /*
380:    FormJacobian - Evaluates Jacobian matrix.

382:    Input Parameters:
383: .  snes - the SNES context
384: .  x - input vector
385: .  ptr - optional user-defined context, as set by SNESSetJacobian()

387:    Output Parameters:
388: .  A - Jacobian matrix
389: .  B - optionally different preconditioning matrix
390: .  flag - flag indicating matrix structure

392:    Notes:
393:    Due to grid point reordering with DAs, we must always work
394:    with the local grid points, and then transform them to the new
395:    global numbering with the "ltog" mapping (via DAGetGlobalIndices()).
396:    We cannot work directly with the global numbers for the original
397:    uniprocessor grid!
398: */
399: int FormJacobian(SNES snes,Vec X,Mat *J,Mat *B,MatStructure *flag,void *ptr)
400: {
401:   AppCtx  *user = (AppCtx*)ptr;  /* user-defined application context */
402:   Mat     jac = *B;                /* Jacobian matrix */
403:   Vec     localX = user->localX;   /* local vector */
404:   int     *ltog;                   /* local-to-global mapping */
405:   int     ierr,i,j,row,mx,my,col[5];
406:   int     nloc,xs,ys,xm,ym,gxs,gys,gxm,gym,grow;
407:   PetscScalar  two = 2.0,one = 1.0,lambda,v[5],hx,hy,hxdhy,hydhx,sc,*x;

409:   mx = user->mx;            my = user->my;            lambda = user->param;
410:   hx = one/(PetscReal)(mx-1);  hy = one/(PetscReal)(my-1);
411:   sc = hx*hy;               hxdhy = hx/hy;            hydhx = hy/hx;

413:   /*
414:      Scatter ghost points to local vector,using the 2-step process
415:         DAGlobalToLocalBegin(),DAGlobalToLocalEnd().
416:      By placing code between these two statements, computations can be
417:      done while messages are in transition.
418:   */
419:   DAGlobalToLocalBegin(user->da,X,INSERT_VALUES,localX);
420:   DAGlobalToLocalEnd(user->da,X,INSERT_VALUES,localX);

422:   /*
423:      Get pointer to vector data
424:   */
425:   VecGetArray(localX,&x);

427:   /*
428:      Get local grid boundaries
429:   */
430:   DAGetCorners(user->da,&xs,&ys,PETSC_NULL,&xm,&ym,PETSC_NULL);
431:   DAGetGhostCorners(user->da,&gxs,&gys,PETSC_NULL,&gxm,&gym,PETSC_NULL);

433:   /*
434:      Get the global node numbers for all local nodes, including ghost points
435:   */
436:   DAGetGlobalIndices(user->da,&nloc,&ltog);

438:   /* 
439:      Compute entries for the locally owned part of the Jacobian.
440:       - Currently, all PETSc parallel matrix formats are partitioned by
441:         contiguous chunks of rows across the processors. The "grow"
442:         parameter computed below specifies the global row number 
443:         corresponding to each local grid point.
444:       - Each processor needs to insert only elements that it owns
445:         locally (but any non-local elements will be sent to the
446:         appropriate processor during matrix assembly). 
447:       - Always specify global row and columns of matrix entries.
448:       - Here, we set all entries for a particular row at once.
449:   */
450:   for (j=ys; j<ys+ym; j++) {
451:     row = (j - gys)*gxm + xs - gxs - 1;
452:     for (i=xs; i<xs+xm; i++) {
453:       row++;
454:       grow = ltog[row];
455:       /* boundary points */
456:       if (i == 0 || j == 0 || i == mx-1 || j == my-1) {
457:         MatSetValues(jac,1,&grow,1,&grow,&one,INSERT_VALUES);
458:         continue;
459:       }
460:       /* interior grid points */
461:       v[0] = -hxdhy; col[0] = ltog[row - gxm];
462:       v[1] = -hydhx; col[1] = ltog[row - 1];
463:       v[2] = two*(hydhx + hxdhy) - sc*lambda*exp(x[row]); col[2] = grow;
464:       v[3] = -hydhx; col[3] = ltog[row + 1];
465:       v[4] = -hxdhy; col[4] = ltog[row + gxm];
466:       MatSetValues(jac,1,&grow,5,col,v,INSERT_VALUES);
467:     }
468:   }

470:   /* 
471:      Assemble matrix, using the 2-step process:
472:        MatAssemblyBegin(), MatAssemblyEnd().
473:      By placing code between these two statements, computations can be
474:      done while messages are in transition.
475:   */
476:   MatAssemblyBegin(jac,MAT_FINAL_ASSEMBLY);
477:   VecRestoreArray(localX,&x);
478:   MatAssemblyEnd(jac,MAT_FINAL_ASSEMBLY);

480:   /*
481:      Set flag to indicate that the Jacobian matrix retains an identical
482:      nonzero structure throughout all nonlinear iterations (although the
483:      values of the entries change). Thus, we can save some work in setting
484:      up the preconditioner (e.g., no need to redo symbolic factorization for
485:      ILU/ICC preconditioners).
486:       - If the nonzero structure of the matrix is different during
487:         successive linear solves, then the flag DIFFERENT_NONZERO_PATTERN
488:         must be used instead.  If you are unsure whether the matrix
489:         structure has changed or not, use the flag DIFFERENT_NONZERO_PATTERN.
490:       - Caution:  If you specify SAME_NONZERO_PATTERN, PETSc
491:         believes your assertion and does not check the structure
492:         of the matrix.  If you erroneously claim that the structure
493:         is the same when it actually is not, the new preconditioner
494:         will not function correctly.  Thus, use this optimization
495:         feature with caution!
496:   */
497:   *flag = SAME_NONZERO_PATTERN;
498:   /*
499:       Tell the matrix we will never add a new nonzero location to the
500:     matrix. If we do it will generate an error.
501:   */
502:   /* MatSetOption(jac,MAT_NEW_NONZERO_LOCATION_ERR); */
503:   return 0;
504: }