Actual source code: ex14.c
petsc-3.7.1 2016-05-15
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; /* 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 = 3; /* 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,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,NULL,"-mx",&user.mx,NULL);
117: PetscOptionsGetInt(NULL,NULL,"-my",&user.my,NULL);
118: PetscOptionsGetReal(NULL,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,NULL,"-Nx",&Nx,NULL);
138: PetscOptionsGetInt(NULL,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);
152: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
153: Create matrix data structure for Jacobian
154: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
155: /*
156: Note: For the parallel case, vectors and matrices MUST be partitioned
157: accordingly. When using distributed arrays (DMDAs) to create vectors,
158: the DMDAs determine the problem partitioning. We must explicitly
159: specify the local matrix dimensions upon its creation for compatibility
160: with the vector distribution. Thus, the generic MatCreate() routine
161: is NOT sufficient when working with distributed arrays.
163: Note: Here we only approximately preallocate storage space for the
164: Jacobian. See the users manual for a discussion of better techniques
165: for preallocating matrix memory.
166: */
167: if (size == 1) {
168: MatCreateSeqAIJ(comm,N,N,5,NULL,&J);
169: } else {
170: VecGetLocalSize(X,&m);
171: MatCreateAIJ(comm,m,m,N,N,5,NULL,3,NULL,&J);
172: }
174: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
175: Customize linear solver; set runtime options
176: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
178: /*
179: Set runtime options (e.g.,-ksp_monitor -ksp_rtol <rtol> -ksp_type <type>)
180: */
181: KSPSetFromOptions(ksp);
183: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
184: Evaluate initial guess
185: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
187: FormInitialGuess(&user,X);
188: ComputeFunction(&user,X,F); /* Compute F(X) */
189: VecNorm(F,NORM_2,&fnorm); /* fnorm = || F || */
190: ttol = fnorm*rtol;
191: if (!no_output) PetscPrintf(comm,"Initial function norm = %g\n",(double)fnorm);
193: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
194: Solve nonlinear system with a user-defined method
195: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
197: /*
198: This solver is a very simplistic inexact Newton method, with no
199: no damping strategies or bells and whistles. The intent of this code
200: is merely to demonstrate the repeated solution with KSP of linear
201: sytems with the same nonzero structure.
203: This is NOT the recommended approach for solving nonlinear problems
204: with PETSc! We urge users to employ the SNES component for solving
205: nonlinear problems whenever possible with application codes, as it
206: offers many advantages over coding nonlinear solvers independently.
207: */
209: for (i=0; i<max_nonlin_its; i++) {
211: /*
212: Compute the Jacobian matrix.
213: */
214: ComputeJacobian(&user,X,J);
216: /*
217: Solve J Y = F, where J is the Jacobian matrix.
218: - First, set the KSP linear operators. Here the matrix that
219: defines the linear system also serves as the preconditioning
220: matrix.
221: - Then solve the Newton system.
222: */
223: KSPSetOperators(ksp,J,J);
224: KSPSolve(ksp,F,Y);
225: KSPGetIterationNumber(ksp,&lin_its);
227: /*
228: Compute updated iterate
229: */
230: VecNorm(Y,NORM_2,&ynorm); /* ynorm = || Y || */
231: VecAYPX(Y,-1.0,X); /* Y <- X - Y */
232: VecCopy(Y,X); /* X <- Y */
233: VecNorm(X,NORM_2,&xnorm); /* xnorm = || X || */
234: if (!no_output) {
235: PetscPrintf(comm," linear solve iterations = %D, xnorm=%g, ynorm=%g\n",lin_its,(double)xnorm,(double)ynorm);
236: }
238: /*
239: Evaluate new nonlinear function
240: */
241: ComputeFunction(&user,X,F); /* Compute F(X) */
242: VecNorm(F,NORM_2,&fnorm); /* fnorm = || F || */
243: if (!no_output) {
244: PetscPrintf(comm,"Iteration %D, function norm = %g\n",i+1,(double)fnorm);
245: }
247: /*
248: Test for convergence
249: */
250: if (fnorm <= ttol) {
251: if (!no_output) {
252: PetscPrintf(comm,"Converged due to function norm %g < %g (relative tolerance)\n",(double)fnorm,(double)ttol);
253: }
254: break;
255: }
256: if (ynorm < xtol*(xnorm)) {
257: if (!no_output) {
258: PetscPrintf(comm,"Converged due to small update length: %g < %g * %g\n",(double)ynorm,(double)xtol,(double)xnorm);
259: }
260: break;
261: }
262: if (i > max_functions) {
263: if (!no_output) {
264: PetscPrintf(comm,"Exceeded maximum number of function evaluations: %D > %D\n",i,max_functions);
265: }
266: break;
267: }
268: }
269: PetscPrintf(comm,"Number of nonlinear iterations = %D\n",i);
271: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
272: Free work space. All PETSc objects should be destroyed when they
273: are no longer needed.
274: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
276: MatDestroy(&J); VecDestroy(&Y);
277: VecDestroy(&user.localX); VecDestroy(&X);
278: VecDestroy(&F);
279: KSPDestroy(&ksp); DMDestroy(&user.da);
280: PetscFinalize();
282: return 0;
283: }
284: /* ------------------------------------------------------------------- */
287: /*
288: FormInitialGuess - Forms initial approximation.
290: Input Parameters:
291: user - user-defined application context
292: X - vector
294: Output Parameter:
295: X - vector
296: */
297: PetscErrorCode FormInitialGuess(AppCtx *user,Vec X)
298: {
299: PetscInt i,j,row,mx,my,ierr,xs,ys,xm,ym,gxm,gym,gxs,gys;
300: PetscReal one = 1.0,lambda,temp1,temp,hx,hy;
301: PetscScalar *x;
303: mx = user->mx; my = user->my; lambda = user->param;
304: hx = one/(PetscReal)(mx-1); hy = one/(PetscReal)(my-1);
305: temp1 = lambda/(lambda + one);
307: /*
308: Get a pointer to vector data.
309: - For default PETSc vectors, VecGetArray() returns a pointer to
310: the data array. Otherwise, the routine is implementation dependent.
311: - You MUST call VecRestoreArray() when you no longer need access to
312: the array.
313: */
314: VecGetArray(X,&x);
316: /*
317: Get local grid boundaries (for 2-dimensional DMDA):
318: xs, ys - starting grid indices (no ghost points)
319: xm, ym - widths of local grid (no ghost points)
320: gxs, gys - starting grid indices (including ghost points)
321: gxm, gym - widths of local grid (including ghost points)
322: */
323: DMDAGetCorners(user->da,&xs,&ys,NULL,&xm,&ym,NULL);
324: DMDAGetGhostCorners(user->da,&gxs,&gys,NULL,&gxm,&gym,NULL);
326: /*
327: Compute initial guess over the locally owned part of the grid
328: */
329: for (j=ys; j<ys+ym; j++) {
330: temp = (PetscReal)(PetscMin(j,my-j-1))*hy;
331: for (i=xs; i<xs+xm; i++) {
332: row = i - gxs + (j - gys)*gxm;
333: if (i == 0 || j == 0 || i == mx-1 || j == my-1) {
334: x[row] = 0.0;
335: continue;
336: }
337: x[row] = temp1*PetscSqrtReal(PetscMin((PetscReal)(PetscMin(i,mx-i-1))*hx,temp));
338: }
339: }
341: /*
342: Restore vector
343: */
344: VecRestoreArray(X,&x);
345: return 0;
346: }
347: /* ------------------------------------------------------------------- */
350: /*
351: ComputeFunction - Evaluates nonlinear function, F(x).
353: Input Parameters:
354: . X - input vector
355: . user - user-defined application context
357: Output Parameter:
358: . F - function vector
359: */
360: PetscErrorCode ComputeFunction(AppCtx *user,Vec X,Vec F)
361: {
363: PetscInt i,j,row,mx,my,xs,ys,xm,ym,gxs,gys,gxm,gym;
364: PetscReal two = 2.0,one = 1.0,lambda,hx,hy,hxdhy,hydhx,sc;
365: PetscScalar u,uxx,uyy,*x,*f;
366: Vec localX = user->localX;
368: mx = user->mx; my = user->my; lambda = user->param;
369: hx = one/(PetscReal)(mx-1); hy = one/(PetscReal)(my-1);
370: sc = hx*hy*lambda; hxdhy = hx/hy; hydhx = hy/hx;
372: /*
373: Scatter ghost points to local vector, using the 2-step process
374: DMGlobalToLocalBegin(), DMGlobalToLocalEnd().
375: By placing code between these two statements, computations can be
376: done while messages are in transition.
377: */
378: DMGlobalToLocalBegin(user->da,X,INSERT_VALUES,localX);
379: DMGlobalToLocalEnd(user->da,X,INSERT_VALUES,localX);
381: /*
382: Get pointers to vector data
383: */
384: VecGetArray(localX,&x);
385: VecGetArray(F,&f);
387: /*
388: Get local grid boundaries
389: */
390: DMDAGetCorners(user->da,&xs,&ys,NULL,&xm,&ym,NULL);
391: DMDAGetGhostCorners(user->da,&gxs,&gys,NULL,&gxm,&gym,NULL);
393: /*
394: Compute function over the locally owned part of the grid
395: */
396: for (j=ys; j<ys+ym; j++) {
397: row = (j - gys)*gxm + xs - gxs - 1;
398: for (i=xs; i<xs+xm; i++) {
399: row++;
400: if (i == 0 || j == 0 || i == mx-1 || j == my-1) {
401: f[row] = x[row];
402: continue;
403: }
404: u = x[row];
405: uxx = (two*u - x[row-1] - x[row+1])*hydhx;
406: uyy = (two*u - x[row-gxm] - x[row+gxm])*hxdhy;
407: f[row] = uxx + uyy - sc*PetscExpScalar(u);
408: }
409: }
411: /*
412: Restore vectors
413: */
414: VecRestoreArray(localX,&x);
415: VecRestoreArray(F,&f);
416: PetscLogFlops(11.0*ym*xm);
417: return 0;
418: }
419: /* ------------------------------------------------------------------- */
422: /*
423: ComputeJacobian - Evaluates Jacobian matrix.
425: Input Parameters:
426: . x - input vector
427: . user - user-defined application context
429: Output Parameters:
430: . jac - Jacobian matrix
431: . flag - flag indicating matrix structure
433: Notes:
434: Due to grid point reordering with DMDAs, we must always work
435: with the local grid points, and then transform them to the new
436: global numbering with the "ltog" mapping
437: We cannot work directly with the global numbers for the original
438: uniprocessor grid!
439: */
440: PetscErrorCode ComputeJacobian(AppCtx *user,Vec X,Mat jac)
441: {
442: PetscErrorCode ierr;
443: Vec localX = user->localX; /* local vector */
444: const PetscInt *ltog; /* local-to-global mapping */
445: PetscInt i,j,row,mx,my,col[5];
446: PetscInt xs,ys,xm,ym,gxs,gys,gxm,gym,grow;
447: PetscScalar two = 2.0,one = 1.0,lambda,v[5],hx,hy,hxdhy,hydhx,sc,*x;
448: ISLocalToGlobalMapping ltogm;
450: mx = user->mx; my = user->my; lambda = user->param;
451: hx = one/(PetscReal)(mx-1); hy = one/(PetscReal)(my-1);
452: sc = hx*hy; hxdhy = hx/hy; hydhx = hy/hx;
454: /*
455: Scatter ghost points to local vector, using the 2-step process
456: DMGlobalToLocalBegin(), DMGlobalToLocalEnd().
457: By placing code between these two statements, computations can be
458: done while messages are in transition.
459: */
460: DMGlobalToLocalBegin(user->da,X,INSERT_VALUES,localX);
461: DMGlobalToLocalEnd(user->da,X,INSERT_VALUES,localX);
463: /*
464: Get pointer to vector data
465: */
466: VecGetArray(localX,&x);
468: /*
469: Get local grid boundaries
470: */
471: DMDAGetCorners(user->da,&xs,&ys,NULL,&xm,&ym,NULL);
472: DMDAGetGhostCorners(user->da,&gxs,&gys,NULL,&gxm,&gym,NULL);
474: /*
475: Get the global node numbers for all local nodes, including ghost points
476: */
477: DMGetLocalToGlobalMapping(user->da,<ogm);
478: ISLocalToGlobalMappingGetIndices(ltogm,<og);
480: /*
481: Compute entries for the locally owned part of the Jacobian.
482: - Currently, all PETSc parallel matrix formats are partitioned by
483: contiguous chunks of rows across the processors. The "grow"
484: parameter computed below specifies the global row number
485: corresponding to each local grid point.
486: - Each processor needs to insert only elements that it owns
487: locally (but any non-local elements will be sent to the
488: appropriate processor during matrix assembly).
489: - Always specify global row and columns of matrix entries.
490: - Here, we set all entries for a particular row at once.
491: */
492: for (j=ys; j<ys+ym; j++) {
493: row = (j - gys)*gxm + xs - gxs - 1;
494: for (i=xs; i<xs+xm; i++) {
495: row++;
496: grow = ltog[row];
497: /* boundary points */
498: if (i == 0 || j == 0 || i == mx-1 || j == my-1) {
499: MatSetValues(jac,1,&grow,1,&grow,&one,INSERT_VALUES);
500: continue;
501: }
502: /* interior grid points */
503: v[0] = -hxdhy; col[0] = ltog[row - gxm];
504: v[1] = -hydhx; col[1] = ltog[row - 1];
505: v[2] = two*(hydhx + hxdhy) - sc*lambda*PetscExpScalar(x[row]); col[2] = grow;
506: v[3] = -hydhx; col[3] = ltog[row + 1];
507: v[4] = -hxdhy; col[4] = ltog[row + gxm];
508: MatSetValues(jac,1,&grow,5,col,v,INSERT_VALUES);
509: }
510: }
511: ISLocalToGlobalMappingRestoreIndices(ltogm,<og);
513: /*
514: Assemble matrix, using the 2-step process:
515: MatAssemblyBegin(), MatAssemblyEnd().
516: By placing code between these two statements, computations can be
517: done while messages are in transition.
518: */
519: MatAssemblyBegin(jac,MAT_FINAL_ASSEMBLY);
520: VecRestoreArray(localX,&x);
521: MatAssemblyEnd(jac,MAT_FINAL_ASSEMBLY);
523: return 0;
524: }