Actual source code: ex14.c
petsc-3.5.4 2015-05-23
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 sytems. 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 sytems 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/examples/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,localF; /* 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);
82: int main(int argc,char **argv)
83: {
84: /* -------------- Data to define application problem ---------------- */
85: MPI_Comm comm; /* communicator */
86: KSP ksp; /* linear solver */
87: Vec X,Y,F; /* solution, update, residual vectors */
88: Mat J; /* Jacobian matrix */
89: AppCtx user; /* user-defined work context */
90: PetscInt Nx,Ny; /* number of preocessors in x- and y- directions */
91: PetscMPIInt size; /* number of processors */
92: PetscReal bratu_lambda_max = 6.81,bratu_lambda_min = 0.;
93: PetscInt m,N;
96: /* --------------- Data to define nonlinear solver -------------- */
97: PetscReal rtol = 1.e-8; /* relative convergence tolerance */
98: PetscReal xtol = 1.e-8; /* step convergence tolerance */
99: PetscReal ttol; /* convergence tolerance */
100: PetscReal fnorm,ynorm,xnorm; /* various vector norms */
101: PetscInt max_nonlin_its = 10; /* maximum number of iterations for nonlinear solver */
102: PetscInt max_functions = 50; /* maximum number of function evaluations */
103: PetscInt lin_its; /* number of linear solver iterations for each step */
104: PetscInt i; /* nonlinear solve iteration number */
105: PetscBool no_output = PETSC_FALSE; /* flag indicating whether to surpress output */
107: PetscInitialize(&argc,&argv,(char*)0,help);
108: comm = PETSC_COMM_WORLD;
109: PetscOptionsGetBool(NULL,"-no_output",&no_output,NULL);
111: /*
112: Initialize problem parameters
113: */
114: user.mx = 4; user.my = 4; user.param = 6.0;
116: PetscOptionsGetInt(NULL,"-mx",&user.mx,NULL);
117: PetscOptionsGetInt(NULL,"-my",&user.my,NULL);
118: PetscOptionsGetReal(NULL,"-par",&user.param,NULL);
119: if (user.param >= bratu_lambda_max || user.param <= bratu_lambda_min) SETERRQ(PETSC_COMM_WORLD,1,"Lambda is out of range");
120: N = user.mx*user.my;
122: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
123: Create linear solver context
124: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
126: KSPCreate(comm,&ksp);
128: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
129: Create vector data structures
130: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
132: /*
133: Create distributed array (DMDA) to manage parallel grid and vectors
134: */
135: MPI_Comm_size(comm,&size);
136: Nx = PETSC_DECIDE; Ny = PETSC_DECIDE;
137: PetscOptionsGetInt(NULL,"-Nx",&Nx,NULL);
138: PetscOptionsGetInt(NULL,"-Ny",&Ny,NULL);
139: if (Nx*Ny != size && (Nx != PETSC_DECIDE || Ny != PETSC_DECIDE)) SETERRQ(PETSC_COMM_WORLD,1,"Incompatible number of processors: Nx * Ny != size");
140: 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);
142: /*
143: Extract global and local vectors from DMDA; then duplicate for remaining
144: vectors that are the same types
145: */
146: DMCreateGlobalVector(user.da,&X);
147: DMCreateLocalVector(user.da,&user.localX);
148: VecDuplicate(X,&F);
149: VecDuplicate(X,&Y);
150: VecDuplicate(user.localX,&user.localF);
153: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
154: Create matrix data structure for Jacobian
155: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
156: /*
157: Note: For the parallel case, vectors and matrices MUST be partitioned
158: accordingly. When using distributed arrays (DMDAs) to create vectors,
159: the DMDAs determine the problem partitioning. We must explicitly
160: specify the local matrix dimensions upon its creation for compatibility
161: with the vector distribution. Thus, the generic MatCreate() routine
162: is NOT sufficient when working with distributed arrays.
164: Note: Here we only approximately preallocate storage space for the
165: Jacobian. See the users manual for a discussion of better techniques
166: for preallocating matrix memory.
167: */
168: if (size == 1) {
169: MatCreateSeqAIJ(comm,N,N,5,NULL,&J);
170: } else {
171: VecGetLocalSize(X,&m);
172: MatCreateAIJ(comm,m,m,N,N,5,NULL,3,NULL,&J);
173: }
175: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
176: Customize linear solver; set runtime options
177: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
179: /*
180: Set runtime options (e.g.,-ksp_monitor -ksp_rtol <rtol> -ksp_type <type>)
181: */
182: KSPSetFromOptions(ksp);
184: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
185: Evaluate initial guess
186: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
188: FormInitialGuess(&user,X);
189: ComputeFunction(&user,X,F); /* Compute F(X) */
190: VecNorm(F,NORM_2,&fnorm); /* fnorm = || F || */
191: ttol = fnorm*rtol;
192: if (!no_output) PetscPrintf(comm,"Initial function norm = %g\n",(double)fnorm);
194: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
195: Solve nonlinear system with a user-defined method
196: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
198: /*
199: This solver is a very simplistic inexact Newton method, with no
200: no damping strategies or bells and whistles. The intent of this code
201: is merely to demonstrate the repeated solution with KSP of linear
202: sytems with the same nonzero structure.
204: This is NOT the recommended approach for solving nonlinear problems
205: with PETSc! We urge users to employ the SNES component for solving
206: nonlinear problems whenever possible with application codes, as it
207: offers many advantages over coding nonlinear solvers independently.
208: */
210: for (i=0; i<max_nonlin_its; i++) {
212: /*
213: Compute the Jacobian matrix.
214: */
215: ComputeJacobian(&user,X,J);
217: /*
218: Solve J Y = F, where J is the Jacobian matrix.
219: - First, set the KSP linear operators. Here the matrix that
220: defines the linear system also serves as the preconditioning
221: matrix.
222: - Then solve the Newton system.
223: */
224: KSPSetOperators(ksp,J,J);
225: KSPSolve(ksp,F,Y);
226: KSPGetIterationNumber(ksp,&lin_its);
228: /*
229: Compute updated iterate
230: */
231: VecNorm(Y,NORM_2,&ynorm); /* ynorm = || Y || */
232: VecAYPX(Y,-1.0,X); /* Y <- X - Y */
233: VecCopy(Y,X); /* X <- Y */
234: VecNorm(X,NORM_2,&xnorm); /* xnorm = || X || */
235: if (!no_output) {
236: PetscPrintf(comm," linear solve iterations = %D, xnorm=%g, ynorm=%g\n",lin_its,(double)xnorm,(double)ynorm);
237: }
239: /*
240: Evaluate new nonlinear function
241: */
242: ComputeFunction(&user,X,F); /* Compute F(X) */
243: VecNorm(F,NORM_2,&fnorm); /* fnorm = || F || */
244: if (!no_output) {
245: PetscPrintf(comm,"Iteration %D, function norm = %g\n",i+1,(double)fnorm);
246: }
248: /*
249: Test for convergence
250: */
251: if (fnorm <= ttol) {
252: if (!no_output) {
253: PetscPrintf(comm,"Converged due to function norm %g < %g (relative tolerance)\n",(double)fnorm,(double)ttol);
254: }
255: break;
256: }
257: if (ynorm < xtol*(xnorm)) {
258: if (!no_output) {
259: PetscPrintf(comm,"Converged due to small update length: %g < %g * %g\n",(double)ynorm,(double)xtol,(double)xnorm);
260: }
261: break;
262: }
263: if (i > max_functions) {
264: if (!no_output) {
265: PetscPrintf(comm,"Exceeded maximum number of function evaluations: %D > %D\n",i,max_functions);
266: }
267: break;
268: }
269: }
270: PetscPrintf(comm,"Number of SNES iterations = %D\n",i+1);
272: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
273: Free work space. All PETSc objects should be destroyed when they
274: are no longer needed.
275: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
277: MatDestroy(&J); VecDestroy(&Y);
278: VecDestroy(&user.localX); VecDestroy(&X);
279: VecDestroy(&user.localF); VecDestroy(&F);
280: KSPDestroy(&ksp); DMDestroy(&user.da);
281: PetscFinalize();
283: return 0;
284: }
285: /* ------------------------------------------------------------------- */
288: /*
289: FormInitialGuess - Forms initial approximation.
291: Input Parameters:
292: user - user-defined application context
293: X - vector
295: Output Parameter:
296: X - vector
297: */
298: PetscErrorCode FormInitialGuess(AppCtx *user,Vec X)
299: {
300: PetscInt i,j,row,mx,my,ierr,xs,ys,xm,ym,gxm,gym,gxs,gys;
301: PetscReal one = 1.0,lambda,temp1,temp,hx,hy;
302: PetscScalar *x;
303: Vec localX = user->localX;
305: mx = user->mx; my = user->my; lambda = user->param;
306: hx = one/(PetscReal)(mx-1); hy = one/(PetscReal)(my-1);
307: temp1 = lambda/(lambda + one);
309: /*
310: Get a pointer to vector data.
311: - For default PETSc vectors, VecGetArray() returns a pointer to
312: the data array. Otherwise, the routine is implementation dependent.
313: - You MUST call VecRestoreArray() when you no longer need access to
314: the array.
315: */
316: VecGetArray(localX,&x);
318: /*
319: Get local grid boundaries (for 2-dimensional DMDA):
320: xs, ys - starting grid indices (no ghost points)
321: xm, ym - widths of local grid (no ghost points)
322: gxs, gys - starting grid indices (including ghost points)
323: gxm, gym - widths of local grid (including ghost points)
324: */
325: DMDAGetCorners(user->da,&xs,&ys,NULL,&xm,&ym,NULL);
326: DMDAGetGhostCorners(user->da,&gxs,&gys,NULL,&gxm,&gym,NULL);
328: /*
329: Compute initial guess over the locally owned part of the grid
330: */
331: for (j=ys; j<ys+ym; j++) {
332: temp = (PetscReal)(PetscMin(j,my-j-1))*hy;
333: for (i=xs; i<xs+xm; i++) {
334: row = i - gxs + (j - gys)*gxm;
335: if (i == 0 || j == 0 || i == mx-1 || j == my-1) {
336: x[row] = 0.0;
337: continue;
338: }
339: x[row] = temp1*PetscSqrtReal(PetscMin((PetscReal)(PetscMin(i,mx-i-1))*hx,temp));
340: }
341: }
343: /*
344: Restore vector
345: */
346: VecRestoreArray(localX,&x);
348: /*
349: Insert values into global vector
350: */
351: DMLocalToGlobalBegin(user->da,localX,INSERT_VALUES,X);
352: DMLocalToGlobalEnd(user->da,localX,INSERT_VALUES,X);
353: return 0;
354: }
355: /* ------------------------------------------------------------------- */
358: /*
359: ComputeFunction - Evaluates nonlinear function, F(x).
361: Input Parameters:
362: . X - input vector
363: . user - user-defined application context
365: Output Parameter:
366: . F - function vector
367: */
368: PetscErrorCode ComputeFunction(AppCtx *user,Vec X,Vec F)
369: {
371: PetscInt i,j,row,mx,my,xs,ys,xm,ym,gxs,gys,gxm,gym;
372: PetscReal two = 2.0,one = 1.0,lambda,hx,hy,hxdhy,hydhx,sc;
373: PetscScalar u,uxx,uyy,*x,*f;
374: Vec localX = user->localX,localF = user->localF;
376: mx = user->mx; my = user->my; lambda = user->param;
377: hx = one/(PetscReal)(mx-1); hy = one/(PetscReal)(my-1);
378: sc = hx*hy*lambda; hxdhy = hx/hy; hydhx = hy/hx;
380: /*
381: Scatter ghost points to local vector, using the 2-step process
382: DMGlobalToLocalBegin(), DMGlobalToLocalEnd().
383: By placing code between these two statements, computations can be
384: done while messages are in transition.
385: */
386: DMGlobalToLocalBegin(user->da,X,INSERT_VALUES,localX);
387: DMGlobalToLocalEnd(user->da,X,INSERT_VALUES,localX);
389: /*
390: Get pointers to vector data
391: */
392: VecGetArray(localX,&x);
393: VecGetArray(localF,&f);
395: /*
396: Get local grid boundaries
397: */
398: DMDAGetCorners(user->da,&xs,&ys,NULL,&xm,&ym,NULL);
399: DMDAGetGhostCorners(user->da,&gxs,&gys,NULL,&gxm,&gym,NULL);
401: /*
402: Compute function over the locally owned part of the grid
403: */
404: for (j=ys; j<ys+ym; j++) {
405: row = (j - gys)*gxm + xs - gxs - 1;
406: for (i=xs; i<xs+xm; i++) {
407: row++;
408: if (i == 0 || j == 0 || i == mx-1 || j == my-1) {
409: f[row] = x[row];
410: continue;
411: }
412: u = x[row];
413: uxx = (two*u - x[row-1] - x[row+1])*hydhx;
414: uyy = (two*u - x[row-gxm] - x[row+gxm])*hxdhy;
415: f[row] = uxx + uyy - sc*PetscExpScalar(u);
416: }
417: }
419: /*
420: Restore vectors
421: */
422: VecRestoreArray(localX,&x);
423: VecRestoreArray(localF,&f);
425: /*
426: Insert values into global vector
427: */
428: DMLocalToGlobalBegin(user->da,localF,INSERT_VALUES,F);
429: DMLocalToGlobalEnd(user->da,localF,INSERT_VALUES,F);
430: PetscLogFlops(11.0*ym*xm);
431: return 0;
432: }
433: /* ------------------------------------------------------------------- */
436: /*
437: ComputeJacobian - Evaluates Jacobian matrix.
439: Input Parameters:
440: . x - input vector
441: . user - user-defined application context
443: Output Parameters:
444: . jac - Jacobian matrix
445: . flag - flag indicating matrix structure
447: Notes:
448: Due to grid point reordering with DMDAs, we must always work
449: with the local grid points, and then transform them to the new
450: global numbering with the "ltog" mapping
451: We cannot work directly with the global numbers for the original
452: uniprocessor grid!
453: */
454: PetscErrorCode ComputeJacobian(AppCtx *user,Vec X,Mat jac)
455: {
456: PetscErrorCode ierr;
457: Vec localX = user->localX; /* local vector */
458: const PetscInt *ltog; /* local-to-global mapping */
459: PetscInt i,j,row,mx,my,col[5];
460: PetscInt xs,ys,xm,ym,gxs,gys,gxm,gym,grow;
461: PetscScalar two = 2.0,one = 1.0,lambda,v[5],hx,hy,hxdhy,hydhx,sc,*x;
462: ISLocalToGlobalMapping ltogm;
464: mx = user->mx; my = user->my; lambda = user->param;
465: hx = one/(PetscReal)(mx-1); hy = one/(PetscReal)(my-1);
466: sc = hx*hy; hxdhy = hx/hy; hydhx = hy/hx;
468: /*
469: Scatter ghost points to local vector, using the 2-step process
470: DMGlobalToLocalBegin(), DMGlobalToLocalEnd().
471: By placing code between these two statements, computations can be
472: done while messages are in transition.
473: */
474: DMGlobalToLocalBegin(user->da,X,INSERT_VALUES,localX);
475: DMGlobalToLocalEnd(user->da,X,INSERT_VALUES,localX);
477: /*
478: Get pointer to vector data
479: */
480: VecGetArray(localX,&x);
482: /*
483: Get local grid boundaries
484: */
485: DMDAGetCorners(user->da,&xs,&ys,NULL,&xm,&ym,NULL);
486: DMDAGetGhostCorners(user->da,&gxs,&gys,NULL,&gxm,&gym,NULL);
488: /*
489: Get the global node numbers for all local nodes, including ghost points
490: */
491: DMGetLocalToGlobalMapping(user->da,<ogm);
492: ISLocalToGlobalMappingGetIndices(ltogm,<og);
494: /*
495: Compute entries for the locally owned part of the Jacobian.
496: - Currently, all PETSc parallel matrix formats are partitioned by
497: contiguous chunks of rows across the processors. The "grow"
498: parameter computed below specifies the global row number
499: corresponding to each local grid point.
500: - Each processor needs to insert only elements that it owns
501: locally (but any non-local elements will be sent to the
502: appropriate processor during matrix assembly).
503: - Always specify global row and columns of matrix entries.
504: - Here, we set all entries for a particular row at once.
505: */
506: for (j=ys; j<ys+ym; j++) {
507: row = (j - gys)*gxm + xs - gxs - 1;
508: for (i=xs; i<xs+xm; i++) {
509: row++;
510: grow = ltog[row];
511: /* boundary points */
512: if (i == 0 || j == 0 || i == mx-1 || j == my-1) {
513: MatSetValues(jac,1,&grow,1,&grow,&one,INSERT_VALUES);
514: continue;
515: }
516: /* interior grid points */
517: v[0] = -hxdhy; col[0] = ltog[row - gxm];
518: v[1] = -hydhx; col[1] = ltog[row - 1];
519: v[2] = two*(hydhx + hxdhy) - sc*lambda*PetscExpScalar(x[row]); col[2] = grow;
520: v[3] = -hydhx; col[3] = ltog[row + 1];
521: v[4] = -hxdhy; col[4] = ltog[row + gxm];
522: MatSetValues(jac,1,&grow,5,col,v,INSERT_VALUES);
523: }
524: }
525: ISLocalToGlobalMappingRestoreIndices(ltogm,<og);
527: /*
528: Assemble matrix, using the 2-step process:
529: MatAssemblyBegin(), MatAssemblyEnd().
530: By placing code between these two statements, computations can be
531: done while messages are in transition.
532: */
533: MatAssemblyBegin(jac,MAT_FINAL_ASSEMBLY);
534: VecRestoreArray(localX,&x);
535: MatAssemblyEnd(jac,MAT_FINAL_ASSEMBLY);
537: return 0;
538: }