Actual source code: ex1.c
2: /* Program usage: ex4 [-help] [all PETSc options] */
4: static char help[] = "Solves a nonlinear system on 1 processor with SNES. We\n\
5: solve the Bratu (SFI - solid fuel ignition) problem in a 2D rectangular domain.\n\
6: This example also illustrates the use of matrix coloring. Runtime options include:\n\
7: -par <parameter>, where <parameter> indicates the problem's nonlinearity\n\
8: problem SFI: <parameter> = Bratu parameter (0 <= par <= 6.81)\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\n";
12: /*T
13: Concepts: SNES^sequential Bratu example
14: Processors: 1
15: T*/
17: /* ------------------------------------------------------------------------
19: Solid Fuel Ignition (SFI) problem. This problem is modeled by
20: the partial differential equation
21:
22: -Laplacian u - lambda*exp(u) = 0, 0 < x,y < 1,
23:
24: with boundary conditions
25:
26: u = 0 for x = 0, x = 1, y = 0, y = 1.
27:
28: A finite difference approximation with the usual 5-point stencil
29: is used to discretize the boundary value problem to obtain a nonlinear
30: system of equations.
32: The parallel version of this code is snes/examples/tutorials/ex5.c
34: ------------------------------------------------------------------------- */
36: /*
37: Include "petscsnes.h" so that we can use SNES solvers. Note that
38: this file automatically includes:
39: petsc.h - base PETSc routines petscvec.h - vectors
40: petscsys.h - system routines petscmat.h - matrices
41: petscis.h - index sets petscksp.h - Krylov subspace methods
42: petscviewer.h - viewers petscpc.h - preconditioners
43: petscksp.h - linear solvers
44: */
46: #include petscsnes.h
48: /*
49: User-defined application context - contains data needed by the
50: application-provided call-back routines, FormJacobian() and
51: FormFunction().
52: */
53: typedef struct {
54: PetscReal param; /* test problem parameter */
55: PetscInt mx; /* Discretization in x-direction */
56: PetscInt my; /* Discretization in y-direction */
57: } AppCtx;
59: /*
60: User-defined routines
61: */
68: int main(int argc,char **argv)
69: {
70: SNES snes; /* nonlinear solver context */
71: Vec x,r; /* solution, residual vectors */
72: Mat J; /* Jacobian matrix */
73: AppCtx user; /* user-defined application context */
75: PetscInt i,its,N,hist_its[50];
76: PetscMPIInt size;
77: PetscReal bratu_lambda_max = 6.81,bratu_lambda_min = 0.,history[50];
78: MatFDColoring fdcoloring;
79: PetscTruth matrix_free,flg,fd_coloring;
81: PetscInitialize(&argc,&argv,(char *)0,help);
82: MPI_Comm_size(PETSC_COMM_WORLD,&size);
83: if (size != 1) SETERRQ(1,"This is a uniprocessor example only!");
85: /*
86: Initialize problem parameters
87: */
88: user.mx = 4; user.my = 4; user.param = 6.0;
89: PetscOptionsGetInt(PETSC_NULL,"-mx",&user.mx,PETSC_NULL);
90: PetscOptionsGetInt(PETSC_NULL,"-my",&user.my,PETSC_NULL);
91: PetscOptionsGetReal(PETSC_NULL,"-par",&user.param,PETSC_NULL);
92: if (user.param >= bratu_lambda_max || user.param <= bratu_lambda_min) {
93: SETERRQ(1,"Lambda is out of range");
94: }
95: N = user.mx*user.my;
96:
97: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
98: Create nonlinear solver context
99: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
101: SNESCreate(PETSC_COMM_WORLD,&snes);
103: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
104: Create vector data structures; set function evaluation routine
105: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
107: VecCreate(PETSC_COMM_WORLD,&x);
108: VecSetSizes(x,PETSC_DECIDE,N);
109: VecSetFromOptions(x);
110: VecDuplicate(x,&r);
112: /*
113: Set function evaluation routine and vector. Whenever the nonlinear
114: solver needs to evaluate the nonlinear function, it will call this
115: routine.
116: - Note that the final routine argument is the user-defined
117: context that provides application-specific data for the
118: function evaluation routine.
119: */
120: SNESSetFunction(snes,r,FormFunction,(void*)&user);
122: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
123: Create matrix data structure; set Jacobian evaluation routine
124: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
126: /*
127: Create matrix. Here we only approximately preallocate storage space
128: for the Jacobian. See the users manual for a discussion of better
129: techniques for preallocating matrix memory.
130: */
131: PetscOptionsHasName(PETSC_NULL,"-snes_mf",&matrix_free);
132: if (!matrix_free) {
133: MatCreateSeqAIJ(PETSC_COMM_WORLD,N,N,5,PETSC_NULL,&J);
134: }
136: /*
137: This option will cause the Jacobian to be computed via finite differences
138: efficiently using a coloring of the columns of the matrix.
139: */
140: PetscOptionsHasName(PETSC_NULL,"-snes_fd_coloring",&fd_coloring);
142: if (matrix_free && fd_coloring) SETERRQ(1,"Use only one of -snes_mf, -snes_fd_coloring options!\n\
143: You can do -snes_mf_operator -snes_fd_coloring");
145: if (fd_coloring) {
146: ISColoring iscoloring;
147: MatStructure str;
149: /*
150: This initializes the nonzero structure of the Jacobian. This is artificial
151: because clearly if we had a routine to compute the Jacobian we won't need
152: to use finite differences.
153: */
154: FormJacobian(snes,x,&J,&J,&str,&user);
156: /*
157: Color the matrix, i.e. determine groups of columns that share no common
158: rows. These columns in the Jacobian can all be computed simulataneously.
159: */
160: MatGetColoring(J,MATCOLORING_NATURAL,&iscoloring);
161: /*
162: Create the data structure that SNESDefaultComputeJacobianColor() uses
163: to compute the actual Jacobians via finite differences.
164: */
165: MatFDColoringCreate(J,iscoloring,&fdcoloring);
166: MatFDColoringSetFunction(fdcoloring,(PetscErrorCode (*)(void))FormFunction,&user);
167: MatFDColoringSetFromOptions(fdcoloring);
168: /*
169: Tell SNES to use the routine SNESDefaultComputeJacobianColor()
170: to compute Jacobians.
171: */
172: SNESSetJacobian(snes,J,J,SNESDefaultComputeJacobianColor,fdcoloring);
173: ISColoringDestroy(iscoloring);
174: }
175: /*
176: Set Jacobian matrix data structure and default Jacobian evaluation
177: routine. Whenever the nonlinear solver needs to compute the
178: Jacobian matrix, it will call this routine.
179: - Note that the final routine argument is the user-defined
180: context that provides application-specific data for the
181: Jacobian evaluation routine.
182: - The user can override with:
183: -snes_fd : default finite differencing approximation of Jacobian
184: -snes_mf : matrix-free Newton-Krylov method with no preconditioning
185: (unless user explicitly sets preconditioner)
186: -snes_mf_operator : form preconditioning matrix as set by the user,
187: but use matrix-free approx for Jacobian-vector
188: products within Newton-Krylov method
189: */
190: else if (!matrix_free) {
191: SNESSetJacobian(snes,J,J,FormJacobian,(void*)&user);
192: }
194: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
195: Customize nonlinear solver; set runtime options
196: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
198: /*
199: Set runtime options (e.g., -snes_monitor -snes_rtol <rtol> -ksp_type <type>)
200: */
201: SNESSetFromOptions(snes);
203: /*
204: Set array that saves the function norms. This array is intended
205: when the user wants to save the convergence history for later use
206: rather than just to view the function norms via -snes_monitor.
207: */
208: SNESSetConvergenceHistory(snes,history,hist_its,50,PETSC_TRUE);
210: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
211: Evaluate initial guess; then solve nonlinear system
212: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
213: /*
214: Note: The user should initialize the vector, x, with the initial guess
215: for the nonlinear solver prior to calling SNESSolve(). In particular,
216: to employ an initial guess of zero, the user should explicitly set
217: this vector to zero by calling VecSet().
218: */
219: FormInitialGuess(&user,x);
220: SNESSolve(snes,PETSC_NULL,x);
221: SNESGetIterationNumber(snes,&its);
222: PetscPrintf(PETSC_COMM_WORLD,"Number of Newton iterations = %D\n",its);
225: /*
226: Print the convergence history. This is intended just to demonstrate
227: use of the data attained via SNESSetConvergenceHistory().
228: */
229: PetscOptionsHasName(PETSC_NULL,"-print_history",&flg);
230: if (flg) {
231: for (i=0; i<its+1; i++) {
232: PetscPrintf(PETSC_COMM_WORLD,"iteration %D: Linear iterations %D Function norm = %G\n",i,hist_its[i],history[i]);
233: }
234: }
236: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
237: Free work space. All PETSc objects should be destroyed when they
238: are no longer needed.
239: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
241: if (!matrix_free) {
242: MatDestroy(J);
243: }
244: if (fd_coloring) {
245: MatFDColoringDestroy(fdcoloring);
246: }
247: VecDestroy(x);
248: VecDestroy(r);
249: SNESDestroy(snes);
250: PetscFinalize();
252: return 0;
253: }
254: /* ------------------------------------------------------------------- */
257: /*
258: FormInitialGuess - Forms initial approximation.
260: Input Parameters:
261: user - user-defined application context
262: X - vector
264: Output Parameter:
265: X - vector
266: */
267: PetscErrorCode FormInitialGuess(AppCtx *user,Vec X)
268: {
269: PetscInt i,j,row,mx,my;
271: PetscReal lambda,temp1,temp,hx,hy;
272: PetscScalar *x;
274: mx = user->mx;
275: my = user->my;
276: lambda = user->param;
278: hx = 1.0 / (PetscReal)(mx-1);
279: hy = 1.0 / (PetscReal)(my-1);
281: /*
282: Get a pointer to vector data.
283: - For default PETSc vectors, VecGetArray() returns a pointer to
284: the data array. Otherwise, the routine is implementation dependent.
285: - You MUST call VecRestoreArray() when you no longer need access to
286: the array.
287: */
288: VecGetArray(X,&x);
289: temp1 = lambda/(lambda + 1.0);
290: for (j=0; j<my; j++) {
291: temp = (PetscReal)(PetscMin(j,my-j-1))*hy;
292: for (i=0; i<mx; i++) {
293: row = i + j*mx;
294: if (i == 0 || j == 0 || i == mx-1 || j == my-1) {
295: x[row] = 0.0;
296: continue;
297: }
298: x[row] = temp1*sqrt(PetscMin((PetscReal)(PetscMin(i,mx-i-1))*hx,temp));
299: }
300: }
302: /*
303: Restore vector
304: */
305: VecRestoreArray(X,&x);
306: return 0;
307: }
308: /* ------------------------------------------------------------------- */
311: /*
312: FormFunction - Evaluates nonlinear function, F(x).
314: Input Parameters:
315: . snes - the SNES context
316: . X - input vector
317: . ptr - optional user-defined context, as set by SNESSetFunction()
319: Output Parameter:
320: . F - function vector
321: */
322: PetscErrorCode FormFunction(SNES snes,Vec X,Vec F,void *ptr)
323: {
324: AppCtx *user = (AppCtx*)ptr;
325: PetscInt i,j,row,mx,my;
327: PetscReal two = 2.0,one = 1.0,lambda,hx,hy,hxdhy,hydhx;
328: PetscScalar ut,ub,ul,ur,u,uxx,uyy,sc,*x,*f;
330: mx = user->mx;
331: my = user->my;
332: lambda = user->param;
333: hx = one / (PetscReal)(mx-1);
334: hy = one / (PetscReal)(my-1);
335: sc = hx*hy;
336: hxdhy = hx/hy;
337: hydhx = hy/hx;
339: /*
340: Get pointers to vector data
341: */
342: VecGetArray(X,&x);
343: VecGetArray(F,&f);
345: /*
346: Compute function
347: */
348: for (j=0; j<my; j++) {
349: for (i=0; i<mx; i++) {
350: row = i + j*mx;
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: ub = x[row - mx];
357: ul = x[row - 1];
358: ut = x[row + mx];
359: ur = x[row + 1];
360: uxx = (-ur + two*u - ul)*hydhx;
361: uyy = (-ut + two*u - ub)*hxdhy;
362: f[row] = uxx + uyy - sc*lambda*PetscExpScalar(u);
363: }
364: }
366: /*
367: Restore vectors
368: */
369: VecRestoreArray(X,&x);
370: VecRestoreArray(F,&f);
371: return 0;
372: }
373: /* ------------------------------------------------------------------- */
376: /*
377: FormJacobian - Evaluates Jacobian matrix.
379: Input Parameters:
380: . snes - the SNES context
381: . x - input vector
382: . ptr - optional user-defined context, as set by SNESSetJacobian()
384: Output Parameters:
385: . A - Jacobian matrix
386: . B - optionally different preconditioning matrix
387: . flag - flag indicating matrix structure
388: */
389: PetscErrorCode FormJacobian(SNES snes,Vec X,Mat *J,Mat *B,MatStructure *flag,void *ptr)
390: {
391: AppCtx *user = (AppCtx*)ptr; /* user-defined applicatin context */
392: Mat jac = *J; /* Jacobian matrix */
393: PetscInt i,j,row,mx,my,col[5];
395: PetscScalar two = 2.0,one = 1.0,lambda,v[5],sc,*x;
396: PetscReal hx,hy,hxdhy,hydhx;
398: mx = user->mx;
399: my = user->my;
400: lambda = user->param;
401: hx = 1.0 / (PetscReal)(mx-1);
402: hy = 1.0 / (PetscReal)(my-1);
403: sc = hx*hy;
404: hxdhy = hx/hy;
405: hydhx = hy/hx;
407: /*
408: Get pointer to vector data
409: */
410: VecGetArray(X,&x);
412: /*
413: Compute entries of the Jacobian
414: */
415: for (j=0; j<my; j++) {
416: for (i=0; i<mx; i++) {
417: row = i + j*mx;
418: if (i == 0 || j == 0 || i == mx-1 || j == my-1) {
419: MatSetValues(jac,1,&row,1,&row,&one,INSERT_VALUES);
420: continue;
421: }
422: v[0] = -hxdhy; col[0] = row - mx;
423: v[1] = -hydhx; col[1] = row - 1;
424: v[2] = two*(hydhx + hxdhy) - sc*lambda*PetscExpScalar(x[row]); col[2] = row;
425: v[3] = -hydhx; col[3] = row + 1;
426: v[4] = -hxdhy; col[4] = row + mx;
427: MatSetValues(jac,1,&row,5,col,v,INSERT_VALUES);
428: }
429: }
431: /*
432: Restore vector
433: */
434: VecRestoreArray(X,&x);
436: /*
437: Assemble matrix
438: */
439: MatAssemblyBegin(jac,MAT_FINAL_ASSEMBLY);
440: MatAssemblyEnd(jac,MAT_FINAL_ASSEMBLY);
442: /*
443: Set flag to indicate that the Jacobian matrix retains an identical
444: nonzero structure throughout all nonlinear iterations (although the
445: values of the entries change). Thus, we can save some work in setting
446: up the preconditioner (e.g., no need to redo symbolic factorization for
447: ILU/ICC preconditioners).
448: - If the nonzero structure of the matrix is different during
449: successive linear solves, then the flag DIFFERENT_NONZERO_PATTERN
450: must be used instead. If you are unsure whether the matrix
451: structure has changed or not, use the flag DIFFERENT_NONZERO_PATTERN.
452: - Caution: If you specify SAME_NONZERO_PATTERN, PETSc
453: believes your assertion and does not check the structure
454: of the matrix. If you erroneously claim that the structure
455: is the same when it actually is not, the new preconditioner
456: will not function correctly. Thus, use this optimization
457: feature with caution!
458: */
459: *flag = SAME_NONZERO_PATTERN;
460: return 0;
461: }