Actual source code: ex14.c
2: /* Program usage: mpiexec -n <procs> ex14 [-help] [all PETSc options] */
4: static char help[] = "Solves a nonlinear system in parallel with a user-defined Newton method.\n\
5: Uses KSP to solve the linearized Newton sytems. This solver\n\
6: is a very simplistic inexact Newton method. The intent of this code is to\n\
7: demonstrate the repeated solution of linear sytems with the same nonzero pattern.\n\
8: \n\
9: This is NOT the recommended approach for solving nonlinear problems with PETSc!\n\
10: We urge users to employ the SNES component for solving nonlinear problems whenever\n\
11: possible, as it offers many advantages over coding nonlinear solvers independently.\n\
12: \n\
13: We solve the Bratu (SFI - solid fuel ignition) problem in a 2D rectangular\n\
14: domain, using distributed arrays (DAs) to partition the parallel grid.\n\
15: The command line options include:\n\
16: -par <parameter>, where <parameter> indicates the problem's nonlinearity\n\
17: problem SFI: <parameter> = Bratu parameter (0 <= par <= 6.81)\n\
18: -mx <xg>, where <xg> = number of grid points in the x-direction\n\
19: -my <yg>, where <yg> = number of grid points in the y-direction\n\
20: -Nx <npx>, where <npx> = number of processors in the x-direction\n\
21: -Ny <npy>, where <npy> = number of processors in the y-direction\n\n";
23: /*T
24: Concepts: KSP^writing a user-defined nonlinear solver (parallel Bratu example);
25: Concepts: DA^using distributed arrays;
26: Processors: n
27: T*/
29: /* ------------------------------------------------------------------------
31: Solid Fuel Ignition (SFI) problem. This problem is modeled by
32: the partial differential equation
33:
34: -Laplacian u - lambda*exp(u) = 0, 0 < x,y < 1,
35:
36: with boundary conditions
37:
38: u = 0 for x = 0, x = 1, y = 0, y = 1.
39:
40: A finite difference approximation with the usual 5-point stencil
41: is used to discretize the boundary value problem to obtain a nonlinear
42: system of equations.
44: The SNES version of this problem is: snes/examples/tutorials/ex5.c
45: We urge users to employ the SNES component for solving nonlinear
46: problems whenever possible, as it offers many advantages over coding
47: nonlinear solvers independently.
49: ------------------------------------------------------------------------- */
51: /*
52: Include "petscda.h" so that we can use distributed arrays (DAs).
53: Include "petscksp.h" so that we can use KSP solvers. Note that this
54: file automatically includes:
55: petsc.h - base PETSc routines petscvec.h - vectors
56: petscsys.h - system routines petscmat.h - matrices
57: petscis.h - index sets petscksp.h - Krylov subspace methods
58: petscviewer.h - viewers petscpc.h - preconditioners
59: */
60: #include petscda.h
61: #include petscksp.h
63: /*
64: User-defined application context - contains data needed by the
65: application-provided call-back routines, ComputeJacobian() and
66: ComputeFunction().
67: */
68: typedef struct {
69: PetscReal param; /* test problem parameter */
70: PetscInt mx,my; /* discretization in x,y directions */
71: Vec localX,localF; /* ghosted local vector */
72: DA da; /* distributed array data structure */
73: PetscInt rank; /* processor rank */
74: } AppCtx;
76: /*
77: User-defined routines
78: */
84: int main(int argc,char **argv)
85: {
86: /* -------------- Data to define application problem ---------------- */
87: MPI_Comm comm; /* communicator */
88: KSP ksp; /* linear solver */
89: Vec X,Y,F; /* solution, update, residual vectors */
90: Mat J; /* Jacobian matrix */
91: AppCtx user; /* user-defined work context */
92: PetscInt Nx,Ny; /* number of preocessors in x- and y- directions */
93: PetscMPIInt size; /* number of processors */
94: PetscReal bratu_lambda_max = 6.81,bratu_lambda_min = 0.;
95: PetscInt m,N;
98: /* --------------- Data to define nonlinear solver -------------- */
99: PetscReal rtol = 1.e-8; /* relative convergence tolerance */
100: PetscReal xtol = 1.e-8; /* step convergence tolerance */
101: PetscReal ttol; /* convergence tolerance */
102: PetscReal fnorm,ynorm,xnorm; /* various vector norms */
103: PetscInt max_nonlin_its = 10; /* maximum number of iterations for nonlinear solver */
104: PetscInt max_functions = 50; /* maximum number of function evaluations */
105: PetscInt lin_its; /* number of linear solver iterations for each step */
106: PetscInt i; /* nonlinear solve iteration number */
107: MatStructure mat_flag; /* flag indicating structure of preconditioner matrix */
108: PetscTruth no_output; /* flag indicating whether to surpress output */
110: PetscInitialize(&argc,&argv,(char *)0,help);
111: comm = PETSC_COMM_WORLD;
112: MPI_Comm_rank(comm,&user.rank);
113: PetscOptionsHasName(PETSC_NULL,"-no_output",&no_output);
115: /*
116: Initialize problem parameters
117: */
118: user.mx = 4; user.my = 4; user.param = 6.0;
119: PetscOptionsGetInt(PETSC_NULL,"-mx",&user.mx,PETSC_NULL);
120: PetscOptionsGetInt(PETSC_NULL,"-my",&user.my,PETSC_NULL);
121: PetscOptionsGetReal(PETSC_NULL,"-par",&user.param,PETSC_NULL);
122: if (user.param >= bratu_lambda_max || user.param <= bratu_lambda_min) {
123: SETERRQ(1,"Lambda is out of range");
124: }
125: N = user.mx*user.my;
127: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
128: Create linear solver context
129: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
131: KSPCreate(comm,&ksp);
133: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
134: Create vector data structures
135: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
137: /*
138: Create distributed array (DA) to manage parallel grid and vectors
139: */
140: MPI_Comm_size(comm,&size);
141: Nx = PETSC_DECIDE; Ny = PETSC_DECIDE;
142: PetscOptionsGetInt(PETSC_NULL,"-Nx",&Nx,PETSC_NULL);
143: PetscOptionsGetInt(PETSC_NULL,"-Ny",&Ny,PETSC_NULL);
144: if (Nx*Ny != size && (Nx != PETSC_DECIDE || Ny != PETSC_DECIDE))
145: SETERRQ(1,"Incompatible number of processors: Nx * Ny != size");
146: DACreate2d(comm,DA_NONPERIODIC,DA_STENCIL_STAR,user.mx,
147: user.my,Nx,Ny,1,1,PETSC_NULL,PETSC_NULL,&user.da);
149: /*
150: Extract global and local vectors from DA; then duplicate for remaining
151: vectors that are the same types
152: */
153: DACreateGlobalVector(user.da,&X);
154: DACreateLocalVector(user.da,&user.localX);
155: VecDuplicate(X,&F);
156: VecDuplicate(X,&Y);
157: VecDuplicate(user.localX,&user.localF);
160: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
161: Create matrix data structure for Jacobian
162: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
163: /*
164: Note: For the parallel case, vectors and matrices MUST be partitioned
165: accordingly. When using distributed arrays (DAs) to create vectors,
166: the DAs determine the problem partitioning. We must explicitly
167: specify the local matrix dimensions upon its creation for compatibility
168: with the vector distribution. Thus, the generic MatCreate() routine
169: is NOT sufficient when working with distributed arrays.
171: Note: Here we only approximately preallocate storage space for the
172: Jacobian. See the users manual for a discussion of better techniques
173: for preallocating matrix memory.
174: */
175: if (size == 1) {
176: MatCreateSeqAIJ(comm,N,N,5,PETSC_NULL,&J);
177: } else {
178: VecGetLocalSize(X,&m);
179: MatCreateMPIAIJ(comm,m,m,N,N,5,PETSC_NULL,3,PETSC_NULL,&J);
180: }
182: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
183: Customize linear solver; set runtime options
184: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
186: /*
187: Set runtime options (e.g.,-ksp_monitor -ksp_rtol <rtol> -ksp_type <type>)
188: */
189: KSPSetFromOptions(ksp);
191: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
192: Evaluate initial guess
193: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
195: FormInitialGuess(&user,X);
196: ComputeFunction(&user,X,F); /* Compute F(X) */
197: VecNorm(F,NORM_2,&fnorm); /* fnorm = || F || */
198: ttol = fnorm*rtol;
199: if (!no_output) PetscPrintf(comm,"Initial function norm = %G\n",fnorm);
201: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
202: Solve nonlinear system with a user-defined method
203: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
205: /*
206: This solver is a very simplistic inexact Newton method, with no
207: no damping strategies or bells and whistles. The intent of this code
208: is merely to demonstrate the repeated solution with KSP of linear
209: sytems with the same nonzero structure.
211: This is NOT the recommended approach for solving nonlinear problems
212: with PETSc! We urge users to employ the SNES component for solving
213: nonlinear problems whenever possible with application codes, as it
214: offers many advantages over coding nonlinear solvers independently.
215: */
217: for (i=0; i<max_nonlin_its; i++) {
219: /*
220: Compute the Jacobian matrix. See the comments in this routine for
221: important information about setting the flag mat_flag.
222: */
223: ComputeJacobian(&user,X,J,&mat_flag);
225: /*
226: Solve J Y = F, where J is the Jacobian matrix.
227: - First, set the KSP linear operators. Here the matrix that
228: defines the linear system also serves as the preconditioning
229: matrix.
230: - Then solve the Newton system.
231: */
232: KSPSetOperators(ksp,J,J,mat_flag);
233: KSPSolve(ksp,F,Y);
234: KSPGetIterationNumber(ksp,&lin_its);
236: /*
237: Compute updated iterate
238: */
239: VecNorm(Y,NORM_2,&ynorm); /* ynorm = || Y || */
240: VecAYPX(Y,-1.0,X); /* Y <- X - Y */
241: VecCopy(Y,X); /* X <- Y */
242: VecNorm(X,NORM_2,&xnorm); /* xnorm = || X || */
243: if (!no_output) {
244: PetscPrintf(comm," linear solve iterations = %D, xnorm=%G, ynorm=%G\n",lin_its,xnorm,ynorm);
245: }
247: /*
248: Evaluate new nonlinear function
249: */
250: ComputeFunction(&user,X,F); /* Compute F(X) */
251: VecNorm(F,NORM_2,&fnorm); /* fnorm = || F || */
252: if (!no_output) {
253: PetscPrintf(comm,"Iteration %D, function norm = %G\n",i+1,fnorm);
254: }
256: /*
257: Test for convergence
258: */
259: if (fnorm <= ttol) {
260: if (!no_output) {
261: PetscPrintf(comm,"Converged due to function norm %G < %G (relative tolerance)\n",fnorm,ttol);
262: }
263: break;
264: }
265: if (ynorm < xtol*(xnorm)) {
266: if (!no_output) {
267: PetscPrintf(comm,"Converged due to small update length: %G < %G * %G\n",ynorm,xtol,xnorm);
268: }
269: break;
270: }
271: if (i > max_functions) {
272: if (!no_output) {
273: PetscPrintf(comm,"Exceeded maximum number of function evaluations: %D > %D\n",i,max_functions);
274: }
275: break;
276: }
277: }
278: PetscPrintf(comm,"Number of Newton iterations = %D\n",i+1);
280: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
281: Free work space. All PETSc objects should be destroyed when they
282: are no longer needed.
283: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
285: MatDestroy(J); VecDestroy(Y);
286: VecDestroy(user.localX); VecDestroy(X);
287: VecDestroy(user.localF); VecDestroy(F);
288: KSPDestroy(ksp); DADestroy(user.da);
289: PetscFinalize();
291: return 0;
292: }
293: /* ------------------------------------------------------------------- */
296: /*
297: FormInitialGuess - Forms initial approximation.
299: Input Parameters:
300: user - user-defined application context
301: X - vector
303: Output Parameter:
304: X - vector
305: */
306: PetscErrorCode FormInitialGuess(AppCtx *user,Vec X)
307: {
308: PetscInt i,j,row,mx,my,ierr,xs,ys,xm,ym,gxm,gym,gxs,gys;
309: PetscReal one = 1.0,lambda,temp1,temp,hx,hy;
310: PetscScalar *x;
311: Vec localX = user->localX;
313: mx = user->mx; my = user->my; lambda = user->param;
314: hx = one/(PetscReal)(mx-1); hy = one/(PetscReal)(my-1);
315: temp1 = lambda/(lambda + one);
317: /*
318: Get a pointer to vector data.
319: - For default PETSc vectors, VecGetArray() returns a pointer to
320: the data array. Otherwise, the routine is implementation dependent.
321: - You MUST call VecRestoreArray() when you no longer need access to
322: the array.
323: */
324: VecGetArray(localX,&x);
326: /*
327: Get local grid boundaries (for 2-dimensional DA):
328: xs, ys - starting grid indices (no ghost points)
329: xm, ym - widths of local grid (no ghost points)
330: gxs, gys - starting grid indices (including ghost points)
331: gxm, gym - widths of local grid (including ghost points)
332: */
333: DAGetCorners(user->da,&xs,&ys,PETSC_NULL,&xm,&ym,PETSC_NULL);
334: DAGetGhostCorners(user->da,&gxs,&gys,PETSC_NULL,&gxm,&gym,PETSC_NULL);
336: /*
337: Compute initial guess over the locally owned part of the grid
338: */
339: for (j=ys; j<ys+ym; j++) {
340: temp = (PetscReal)(PetscMin(j,my-j-1))*hy;
341: for (i=xs; i<xs+xm; i++) {
342: row = i - gxs + (j - gys)*gxm;
343: if (i == 0 || j == 0 || i == mx-1 || j == my-1) {
344: x[row] = 0.0;
345: continue;
346: }
347: x[row] = temp1*sqrt(PetscMin((PetscReal)(PetscMin(i,mx-i-1))*hx,temp));
348: }
349: }
351: /*
352: Restore vector
353: */
354: VecRestoreArray(localX,&x);
356: /*
357: Insert values into global vector
358: */
359: DALocalToGlobal(user->da,localX,INSERT_VALUES,X);
360: return 0;
361: }
362: /* ------------------------------------------------------------------- */
365: /*
366: ComputeFunction - Evaluates nonlinear function, F(x).
368: Input Parameters:
369: . X - input vector
370: . user - user-defined application context
372: Output Parameter:
373: . F - function vector
374: */
375: PetscErrorCode ComputeFunction(AppCtx *user,Vec X,Vec F)
376: {
378: PetscInt i,j,row,mx,my,xs,ys,xm,ym,gxs,gys,gxm,gym;
379: PetscReal two = 2.0,one = 1.0,lambda,hx,hy,hxdhy,hydhx,sc;
380: PetscScalar u,uxx,uyy,*x,*f;
381: Vec localX = user->localX,localF = user->localF;
383: mx = user->mx; my = user->my; lambda = user->param;
384: hx = one/(PetscReal)(mx-1); hy = one/(PetscReal)(my-1);
385: sc = hx*hy*lambda; hxdhy = hx/hy; hydhx = hy/hx;
387: /*
388: Scatter ghost points to local vector, using the 2-step process
389: DAGlobalToLocalBegin(), DAGlobalToLocalEnd().
390: By placing code between these two statements, computations can be
391: done while messages are in transition.
392: */
393: DAGlobalToLocalBegin(user->da,X,INSERT_VALUES,localX);
394: DAGlobalToLocalEnd(user->da,X,INSERT_VALUES,localX);
396: /*
397: Get pointers to vector data
398: */
399: VecGetArray(localX,&x);
400: VecGetArray(localF,&f);
402: /*
403: Get local grid boundaries
404: */
405: DAGetCorners(user->da,&xs,&ys,PETSC_NULL,&xm,&ym,PETSC_NULL);
406: DAGetGhostCorners(user->da,&gxs,&gys,PETSC_NULL,&gxm,&gym,PETSC_NULL);
408: /*
409: Compute function over the locally owned part of the grid
410: */
411: for (j=ys; j<ys+ym; j++) {
412: row = (j - gys)*gxm + xs - gxs - 1;
413: for (i=xs; i<xs+xm; i++) {
414: row++;
415: if (i == 0 || j == 0 || i == mx-1 || j == my-1) {
416: f[row] = x[row];
417: continue;
418: }
419: u = x[row];
420: uxx = (two*u - x[row-1] - x[row+1])*hydhx;
421: uyy = (two*u - x[row-gxm] - x[row+gxm])*hxdhy;
422: f[row] = uxx + uyy - sc*PetscExpScalar(u);
423: }
424: }
426: /*
427: Restore vectors
428: */
429: VecRestoreArray(localX,&x);
430: VecRestoreArray(localF,&f);
432: /*
433: Insert values into global vector
434: */
435: DALocalToGlobal(user->da,localF,INSERT_VALUES,F);
436: PetscLogFlops(11*ym*xm);
437: return 0;
438: }
439: /* ------------------------------------------------------------------- */
442: /*
443: ComputeJacobian - Evaluates Jacobian matrix.
445: Input Parameters:
446: . x - input vector
447: . user - user-defined application context
449: Output Parameters:
450: . jac - Jacobian matrix
451: . flag - flag indicating matrix structure
453: Notes:
454: Due to grid point reordering with DAs, we must always work
455: with the local grid points, and then transform them to the new
456: global numbering with the "ltog" mapping (via DAGetGlobalIndices()).
457: We cannot work directly with the global numbers for the original
458: uniprocessor grid!
459: */
460: PetscErrorCode ComputeJacobian(AppCtx *user,Vec X,Mat jac,MatStructure *flag)
461: {
463: Vec localX = user->localX; /* local vector */
464: PetscInt *ltog; /* local-to-global mapping */
465: PetscInt i,j,row,mx,my,col[5];
466: PetscInt nloc,xs,ys,xm,ym,gxs,gys,gxm,gym,grow;
467: PetscScalar two = 2.0,one = 1.0,lambda,v[5],hx,hy,hxdhy,hydhx,sc,*x;
469: mx = user->mx; my = user->my; lambda = user->param;
470: hx = one/(PetscReal)(mx-1); hy = one/(PetscReal)(my-1);
471: sc = hx*hy; hxdhy = hx/hy; hydhx = hy/hx;
473: /*
474: Scatter ghost points to local vector, using the 2-step process
475: DAGlobalToLocalBegin(), DAGlobalToLocalEnd().
476: By placing code between these two statements, computations can be
477: done while messages are in transition.
478: */
479: DAGlobalToLocalBegin(user->da,X,INSERT_VALUES,localX);
480: DAGlobalToLocalEnd(user->da,X,INSERT_VALUES,localX);
482: /*
483: Get pointer to vector data
484: */
485: VecGetArray(localX,&x);
487: /*
488: Get local grid boundaries
489: */
490: DAGetCorners(user->da,&xs,&ys,PETSC_NULL,&xm,&ym,PETSC_NULL);
491: DAGetGhostCorners(user->da,&gxs,&gys,PETSC_NULL,&gxm,&gym,PETSC_NULL);
493: /*
494: Get the global node numbers for all local nodes, including ghost points
495: */
496: DAGetGlobalIndices(user->da,&nloc,<og);
498: /*
499: Compute entries for the locally owned part of the Jacobian.
500: - Currently, all PETSc parallel matrix formats are partitioned by
501: contiguous chunks of rows across the processors. The "grow"
502: parameter computed below specifies the global row number
503: corresponding to each local grid point.
504: - Each processor needs to insert only elements that it owns
505: locally (but any non-local elements will be sent to the
506: appropriate processor during matrix assembly).
507: - Always specify global row and columns of matrix entries.
508: - Here, we set all entries for a particular row at once.
509: */
510: for (j=ys; j<ys+ym; j++) {
511: row = (j - gys)*gxm + xs - gxs - 1;
512: for (i=xs; i<xs+xm; i++) {
513: row++;
514: grow = ltog[row];
515: /* boundary points */
516: if (i == 0 || j == 0 || i == mx-1 || j == my-1) {
517: MatSetValues(jac,1,&grow,1,&grow,&one,INSERT_VALUES);
518: continue;
519: }
520: /* interior grid points */
521: v[0] = -hxdhy; col[0] = ltog[row - gxm];
522: v[1] = -hydhx; col[1] = ltog[row - 1];
523: v[2] = two*(hydhx + hxdhy) - sc*lambda*PetscExpScalar(x[row]); col[2] = grow;
524: v[3] = -hydhx; col[3] = ltog[row + 1];
525: v[4] = -hxdhy; col[4] = ltog[row + gxm];
526: MatSetValues(jac,1,&grow,5,col,v,INSERT_VALUES);
527: }
528: }
530: /*
531: Assemble matrix, using the 2-step process:
532: MatAssemblyBegin(), MatAssemblyEnd().
533: By placing code between these two statements, computations can be
534: done while messages are in transition.
535: */
536: MatAssemblyBegin(jac,MAT_FINAL_ASSEMBLY);
537: VecRestoreArray(localX,&x);
538: MatAssemblyEnd(jac,MAT_FINAL_ASSEMBLY);
540: /*
541: Set flag to indicate that the Jacobian matrix retains an identical
542: nonzero structure throughout all nonlinear iterations (although the
543: values of the entries change). Thus, we can save some work in setting
544: up the preconditioner (e.g., no need to redo symbolic factorization for
545: ILU/ICC preconditioners).
546: - If the nonzero structure of the matrix is different during
547: successive linear solves, then the flag DIFFERENT_NONZERO_PATTERN
548: must be used instead. If you are unsure whether the matrix
549: structure has changed or not, use the flag DIFFERENT_NONZERO_PATTERN.
550: - Caution: If you specify SAME_NONZERO_PATTERN, PETSc
551: believes your assertion and does not check the structure
552: of the matrix. If you erroneously claim that the structure
553: is the same when it actually is not, the new preconditioner
554: will not function correctly. Thus, use this optimization
555: feature with caution!
556: */
557: *flag = SAME_NONZERO_PATTERN;
558: return 0;
559: }