Actual source code: ex1.c

  1: /*$Id: ex1.c,v 1.87 2001/08/27 17:31:28 bsmith Exp $*/

  3: /* Program usage:  ex4 [-help] [all PETSc options] */

  5: static char help[] = "Solves a nonlinear system on 1 processor with SNES. We\n\
  6: solve the Bratu (SFI - solid fuel ignition) problem in a 2D rectangular domain.\n\
  7: This example also illustrates the use of matrix coloring.  Runtime options include:\n\
  8:   -par <parameter>, where <parameter> indicates the problem's nonlinearity\n\
  9:      problem SFI:  <parameter> = Bratu parameter (0 <= par <= 6.81)\n\
 10:   -mx <xg>, where <xg> = number of grid points in the x-direction\n\
 11:   -my <yg>, where <yg> = number of grid points in the y-direction\n\n";

 13: /*T
 14:    Concepts: SNES^sequential Bratu example
 15:    Processors: 1
 16: T*/

 18: /* ------------------------------------------------------------------------

 20:     Solid Fuel Ignition (SFI) problem.  This problem is modeled by
 21:     the partial differential equation
 22:   
 23:             -Laplacian u - lambda*exp(u) = 0,  0 < x,y < 1,
 24:   
 25:     with boundary conditions
 26:    
 27:              u = 0  for  x = 0, x = 1, y = 0, y = 1.
 28:   
 29:     A finite difference approximation with the usual 5-point stencil
 30:     is used to discretize the boundary value problem to obtain a nonlinear 
 31:     system of equations.

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

 35:   ------------------------------------------------------------------------- */

 37: /* 
 38:    Include "petscsnes.h" so that we can use SNES solvers.  Note that
 39:    this file automatically includes:
 40:      petsc.h       - base PETSc routines   petscvec.h - vectors
 41:      petscsys.h    - system routines       petscmat.h - matrices
 42:      petscis.h     - index sets            petscksp.h - Krylov subspace methods
 43:      petscviewer.h - viewers               petscpc.h  - preconditioners
 44:      petscsles.h   - linear solvers
 45: */

 47:  #include petscsnes.h

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

 60: /* 
 61:    User-defined routines
 62: */
 63: extern int FormJacobian(SNES,Vec,Mat*,Mat*,MatStructure*,void*);
 64: extern int FormFunction(SNES,Vec,Vec,void*);
 65: extern int FormInitialGuess(AppCtx*,Vec);

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

 80:   PetscInitialize(&argc,&argv,(char *)0,help);
 81:   MPI_Comm_size(PETSC_COMM_WORLD,&size);
 82:   if (size != 1) SETERRQ(1,"This is a uniprocessor example only!");

 84:   /*
 85:      Initialize problem parameters
 86:   */
 87:   user.mx = 4; user.my = 4; user.param = 6.0;
 88:   PetscOptionsGetInt(PETSC_NULL,"-mx",&user.mx,PETSC_NULL);
 89:   PetscOptionsGetInt(PETSC_NULL,"-my",&user.my,PETSC_NULL);
 90:   PetscOptionsGetReal(PETSC_NULL,"-par",&user.param,PETSC_NULL);
 91:   if (user.param >= bratu_lambda_max || user.param <= bratu_lambda_min) {
 92:     SETERRQ(1,"Lambda is out of range");
 93:   }
 94:   N = user.mx*user.my;
 95: 
 96:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 97:      Create nonlinear solver context
 98:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

100:   SNESCreate(PETSC_COMM_WORLD,&snes);

102:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
103:      Create vector data structures; set function evaluation routine
104:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

106:   VecCreate(PETSC_COMM_WORLD,&x);
107:   VecSetSizes(x,PETSC_DECIDE,N);
108:   VecSetFromOptions(x);
109:   VecDuplicate(x,&r);

111:   /* 
112:      Set function evaluation routine and vector.  Whenever the nonlinear
113:      solver needs to evaluate the nonlinear function, it will call this
114:      routine.
115:       - Note that the final routine argument is the user-defined
116:         context that provides application-specific data for the
117:         function evaluation routine.
118:   */
119:   SNESSetFunction(snes,r,FormFunction,(void *)&user);

121:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
122:      Create matrix data structure; set Jacobian evaluation routine
123:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

125:   /*
126:      Create matrix. Here we only approximately preallocate storage space
127:      for the Jacobian.  See the users manual for a discussion of better 
128:      techniques for preallocating matrix memory.
129:   */
130:   PetscOptionsHasName(PETSC_NULL,"-snes_mf",&matrix_free);
131:   if (!matrix_free) {
132:     MatCreateSeqAIJ(PETSC_COMM_WORLD,N,N,5,PETSC_NULL,&J);
133:   }

135:   /*
136:      This option will cause the Jacobian to be computed via finite differences
137:     efficiently using a coloring of the columns of the matrix.
138:   */
139:   PetscOptionsHasName(PETSC_NULL,"-snes_fd_coloring",&fd_coloring);

141:   if (matrix_free && fd_coloring)  SETERRQ(1,"Use only one of -snes_mf, -snes_fd_coloring options!\n\
142:                                                 You can do -snes_mf_operator -snes_fd_coloring");

144:   if (fd_coloring) {
145:     ISColoring   iscoloring;
146:     MatStructure str;

148:     /* 
149:       This initializes the nonzero structure of the Jacobian. This is artificial
150:       because clearly if we had a routine to compute the Jacobian we won't need
151:       to use finite differences.
152:     */
153:     FormJacobian(snes,x,&J,&J,&str,&user);

155:     /*
156:        Color the matrix, i.e. determine groups of columns that share no common 
157:       rows. These columns in the Jacobian can all be computed simulataneously.
158:     */
159:     MatGetColoring(J,MATCOLORING_NATURAL,&iscoloring);
160:     /*
161:        Create the data structure that SNESDefaultComputeJacobianColor() uses
162:        to compute the actual Jacobians via finite differences.
163:     */
164:     MatFDColoringCreate(J,iscoloring,&fdcoloring);
165:     MatFDColoringSetFunction(fdcoloring,(int (*)(void))FormFunction,&user);
166:     MatFDColoringSetFromOptions(fdcoloring);
167:     /*
168:         Tell SNES to use the routine SNESDefaultComputeJacobianColor()
169:       to compute Jacobians.
170:     */
171:     SNESSetJacobian(snes,J,J,SNESDefaultComputeJacobianColor,fdcoloring);
172:     ISColoringDestroy(iscoloring);
173:   }
174:   /* 
175:      Set Jacobian matrix data structure and default Jacobian evaluation
176:      routine.  Whenever the nonlinear solver needs to compute the
177:      Jacobian matrix, it will call this routine.
178:       - Note that the final routine argument is the user-defined
179:         context that provides application-specific data for the
180:         Jacobian evaluation routine.
181:       - The user can override with:
182:          -snes_fd : default finite differencing approximation of Jacobian
183:          -snes_mf : matrix-free Newton-Krylov method with no preconditioning
184:                     (unless user explicitly sets preconditioner) 
185:          -snes_mf_operator : form preconditioning matrix as set by the user,
186:                              but use matrix-free approx for Jacobian-vector
187:                              products within Newton-Krylov method
188:   */
189:   else if (!matrix_free) {
190:     SNESSetJacobian(snes,J,J,FormJacobian,(void*)&user);
191:   }

193:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
194:      Customize nonlinear solver; set runtime options
195:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

197:   /*
198:      Set runtime options (e.g., -snes_monitor -snes_rtol <rtol> -ksp_type <type>)
199:   */
200:   SNESSetFromOptions(snes);

202:   /*
203:      Set array that saves the function norms.  This array is intended
204:      when the user wants to save the convergence history for later use
205:      rather than just to view the function norms via -snes_monitor.
206:   */
207:   SNESSetConvergenceHistory(snes,history,hist_its,50,PETSC_TRUE);

209:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
210:      Evaluate initial guess; then solve nonlinear system
211:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
212:   /*
213:      Note: The user should initialize the vector, x, with the initial guess
214:      for the nonlinear solver prior to calling SNESSolve().  In particular,
215:      to employ an initial guess of zero, the user should explicitly set
216:      this vector to zero by calling VecSet().
217:   */
218:   FormInitialGuess(&user,x);
219:   SNESSolve(snes,x,&its);
220:   PetscPrintf(PETSC_COMM_WORLD,"Number of Newton iterations = %d\n",its);


223:   /* 
224:      Print the convergence history.  This is intended just to demonstrate
225:      use of the data attained via SNESSetConvergenceHistory().  
226:   */
227:   PetscOptionsHasName(PETSC_NULL,"-print_history",&flg);
228:   if (flg) {
229:     for (i=0; i<its+1; i++) {
230:       PetscPrintf(PETSC_COMM_WORLD,"iteration %d: Linear iterations %d Function norm = %g\n",i,hist_its[i],history[i]);
231:     }
232:   }

234:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
235:      Free work space.  All PETSc objects should be destroyed when they
236:      are no longer needed.
237:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

239:   if (!matrix_free) {
240:     MatDestroy(J);
241:   }
242:   if (fd_coloring) {
243:     MatFDColoringDestroy(fdcoloring);
244:   }
245:   VecDestroy(x);
246:   VecDestroy(r);
247:   SNESDestroy(snes);
248:   PetscFinalize();

250:   return 0;
251: }
252: /* ------------------------------------------------------------------- */
255: /* 
256:    FormInitialGuess - Forms initial approximation.

258:    Input Parameters:
259:    user - user-defined application context
260:    X - vector

262:    Output Parameter:
263:    X - vector
264:  */
265: int FormInitialGuess(AppCtx *user,Vec X)
266: {
267:   int         i,j,row,mx,my,ierr;
268:   PetscReal   lambda,temp1,temp,hx,hy;
269:   PetscScalar *x;

271:   mx         = user->mx;
272:   my         = user->my;
273:   lambda = user->param;

275:   hx    = 1.0 / (PetscReal)(mx-1);
276:   hy    = 1.0 / (PetscReal)(my-1);

278:   /*
279:      Get a pointer to vector data.
280:        - For default PETSc vectors, VecGetArray() returns a pointer to
281:          the data array.  Otherwise, the routine is implementation dependent.
282:        - You MUST call VecRestoreArray() when you no longer need access to
283:          the array.
284:   */
285:   VecGetArray(X,&x);
286:   temp1 = lambda/(lambda + 1.0);
287:   for (j=0; j<my; j++) {
288:     temp = (PetscReal)(PetscMin(j,my-j-1))*hy;
289:     for (i=0; i<mx; i++) {
290:       row = i + j*mx;
291:       if (i == 0 || j == 0 || i == mx-1 || j == my-1) {
292:         x[row] = 0.0;
293:         continue;
294:       }
295:       x[row] = temp1*sqrt(PetscMin((PetscReal)(PetscMin(i,mx-i-1))*hx,temp));
296:     }
297:   }

299:   /*
300:      Restore vector
301:   */
302:   VecRestoreArray(X,&x);
303:   return 0;
304: }
305: /* ------------------------------------------------------------------- */
308: /* 
309:    FormFunction - Evaluates nonlinear function, F(x).

311:    Input Parameters:
312: .  snes - the SNES context
313: .  X - input vector
314: .  ptr - optional user-defined context, as set by SNESSetFunction()

316:    Output Parameter:
317: .  F - function vector
318:  */
319: int FormFunction(SNES snes,Vec X,Vec F,void *ptr)
320: {
321:   AppCtx       *user = (AppCtx*)ptr;
322:   int          ierr,i,j,row,mx,my;
323:   PetscReal    two = 2.0,one = 1.0,lambda,hx,hy,hxdhy,hydhx;
324:   PetscScalar  ut,ub,ul,ur,u,uxx,uyy,sc,*x,*f;

326:   mx         = user->mx;
327:   my         = user->my;
328:   lambda = user->param;
329:   hx     = one / (PetscReal)(mx-1);
330:   hy     = one / (PetscReal)(my-1);
331:   sc     = hx*hy;
332:   hxdhy  = hx/hy;
333:   hydhx  = hy/hx;

335:   /*
336:      Get pointers to vector data
337:   */
338:   VecGetArray(X,&x);
339:   VecGetArray(F,&f);

341:   /*
342:      Compute function 
343:   */
344:   for (j=0; j<my; j++) {
345:     for (i=0; i<mx; i++) {
346:       row = i + j*mx;
347:       if (i == 0 || j == 0 || i == mx-1 || j == my-1) {
348:         f[row] = x[row];
349:         continue;
350:       }
351:       u = x[row];
352:       ub = x[row - mx];
353:       ul = x[row - 1];
354:       ut = x[row + mx];
355:       ur = x[row + 1];
356:       uxx = (-ur + two*u - ul)*hydhx;
357:       uyy = (-ut + two*u - ub)*hxdhy;
358:       f[row] = uxx + uyy - sc*lambda*PetscExpScalar(u);
359:     }
360:   }

362:   /*
363:      Restore vectors
364:   */
365:   VecRestoreArray(X,&x);
366:   VecRestoreArray(F,&f);
367:   return 0;
368: }
369: /* ------------------------------------------------------------------- */
372: /*
373:    FormJacobian - Evaluates Jacobian matrix.

375:    Input Parameters:
376: .  snes - the SNES context
377: .  x - input vector
378: .  ptr - optional user-defined context, as set by SNESSetJacobian()

380:    Output Parameters:
381: .  A - Jacobian matrix
382: .  B - optionally different preconditioning matrix
383: .  flag - flag indicating matrix structure
384: */
385: int FormJacobian(SNES snes,Vec X,Mat *J,Mat *B,MatStructure *flag,void *ptr)
386: {
387:   AppCtx      *user = (AppCtx*)ptr;   /* user-defined applicatin context */
388:   Mat         jac = *J;                /* Jacobian matrix */
389:   int         i,j,row,mx,my,col[5],ierr;
390:   PetscScalar two = 2.0,one = 1.0,lambda,v[5],sc,*x;
391:   PetscReal   hx,hy,hxdhy,hydhx;

393:   mx         = user->mx;
394:   my         = user->my;
395:   lambda = user->param;
396:   hx     = 1.0 / (PetscReal)(mx-1);
397:   hy     = 1.0 / (PetscReal)(my-1);
398:   sc     = hx*hy;
399:   hxdhy  = hx/hy;
400:   hydhx  = hy/hx;

402:   /*
403:      Get pointer to vector data
404:   */
405:   VecGetArray(X,&x);

407:   /* 
408:      Compute entries of the Jacobian
409:   */
410:   for (j=0; j<my; j++) {
411:     for (i=0; i<mx; i++) {
412:       row = i + j*mx;
413:       if (i == 0 || j == 0 || i == mx-1 || j == my-1) {
414:         MatSetValues(jac,1,&row,1,&row,&one,INSERT_VALUES);
415:         continue;
416:       }
417:       v[0] = -hxdhy; col[0] = row - mx;
418:       v[1] = -hydhx; col[1] = row - 1;
419:       v[2] = two*(hydhx + hxdhy) - sc*lambda*PetscExpScalar(x[row]); col[2] = row;
420:       v[3] = -hydhx; col[3] = row + 1;
421:       v[4] = -hxdhy; col[4] = row + mx;
422:       MatSetValues(jac,1,&row,5,col,v,INSERT_VALUES);
423:     }
424:   }

426:   /*
427:      Restore vector
428:   */
429:   VecRestoreArray(X,&x);

431:   /* 
432:      Assemble matrix
433:   */
434:   MatAssemblyBegin(jac,MAT_FINAL_ASSEMBLY);
435:   MatAssemblyEnd(jac,MAT_FINAL_ASSEMBLY);

437:   /*
438:      Set flag to indicate that the Jacobian matrix retains an identical
439:      nonzero structure throughout all nonlinear iterations (although the
440:      values of the entries change). Thus, we can save some work in setting
441:      up the preconditioner (e.g., no need to redo symbolic factorization for
442:      ILU/ICC preconditioners).
443:       - If the nonzero structure of the matrix is different during
444:         successive linear solves, then the flag DIFFERENT_NONZERO_PATTERN
445:         must be used instead.  If you are unsure whether the matrix
446:         structure has changed or not, use the flag DIFFERENT_NONZERO_PATTERN.
447:       - Caution:  If you specify SAME_NONZERO_PATTERN, PETSc
448:         believes your assertion and does not check the structure
449:         of the matrix.  If you erroneously claim that the structure
450:         is the same when it actually is not, the new preconditioner
451:         will not function correctly.  Thus, use this optimization
452:         feature with caution!
453:   */
454:   *flag = SAME_NONZERO_PATTERN;
455:   return 0;
456: }