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: petscsys.h - base PETSc routines petscvec.h - vectors
40: 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 = PETSC_FALSE,flg,fd_coloring = PETSC_FALSE;
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: PetscOptionsGetTruth(PETSC_NULL,"-snes_mf",&matrix_free,PETSC_NULL);
132: if (!matrix_free) {
133: PetscTruth matrix_free_operator = PETSC_FALSE;
134: PetscOptionsGetTruth(PETSC_NULL,"-snes_mf_operator",&matrix_free_operator,PETSC_NULL);
135: if (matrix_free_operator) matrix_free = PETSC_FALSE;
136: }
137: if (!matrix_free) {
138: MatCreateSeqAIJ(PETSC_COMM_WORLD,N,N,5,PETSC_NULL,&J);
139: }
141: /*
142: This option will cause the Jacobian to be computed via finite differences
143: efficiently using a coloring of the columns of the matrix.
144: */
145: PetscOptionsGetTruth(PETSC_NULL,"-snes_fd_coloring",&fd_coloring,PETSC_NULL);
147: if (matrix_free && fd_coloring) SETERRQ(1,"Use only one of -snes_mf, -snes_fd_coloring options!\n\
148: You can do -snes_mf_operator -snes_fd_coloring");
150: if (fd_coloring) {
151: ISColoring iscoloring;
152: MatStructure str;
154: /*
155: This initializes the nonzero structure of the Jacobian. This is artificial
156: because clearly if we had a routine to compute the Jacobian we won't need
157: to use finite differences.
158: */
159: FormJacobian(snes,x,&J,&J,&str,&user);
161: /*
162: Color the matrix, i.e. determine groups of columns that share no common
163: rows. These columns in the Jacobian can all be computed simulataneously.
164: */
165: MatGetColoring(J,MATCOLORING_NATURAL,&iscoloring);
166: /*
167: Create the data structure that SNESDefaultComputeJacobianColor() uses
168: to compute the actual Jacobians via finite differences.
169: */
170: MatFDColoringCreate(J,iscoloring,&fdcoloring);
171: MatFDColoringSetFunction(fdcoloring,(PetscErrorCode (*)(void))FormFunction,&user);
172: MatFDColoringSetFromOptions(fdcoloring);
173: /*
174: Tell SNES to use the routine SNESDefaultComputeJacobianColor()
175: to compute Jacobians.
176: */
177: SNESSetJacobian(snes,J,J,SNESDefaultComputeJacobianColor,fdcoloring);
178: ISColoringDestroy(iscoloring);
179: }
180: /*
181: Set Jacobian matrix data structure and default Jacobian evaluation
182: routine. Whenever the nonlinear solver needs to compute the
183: Jacobian matrix, it will call this routine.
184: - Note that the final routine argument is the user-defined
185: context that provides application-specific data for the
186: Jacobian evaluation routine.
187: - The user can override with:
188: -snes_fd : default finite differencing approximation of Jacobian
189: -snes_mf : matrix-free Newton-Krylov method with no preconditioning
190: (unless user explicitly sets preconditioner)
191: -snes_mf_operator : form preconditioning matrix as set by the user,
192: but use matrix-free approx for Jacobian-vector
193: products within Newton-Krylov method
194: */
195: else if (!matrix_free) {
196: SNESSetJacobian(snes,J,J,FormJacobian,(void*)&user);
197: }
199: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
200: Customize nonlinear solver; set runtime options
201: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
203: /*
204: Set runtime options (e.g., -snes_monitor -snes_rtol <rtol> -ksp_type <type>)
205: */
206: SNESSetFromOptions(snes);
208: /*
209: Set array that saves the function norms. This array is intended
210: when the user wants to save the convergence history for later use
211: rather than just to view the function norms via -snes_monitor.
212: */
213: SNESSetConvergenceHistory(snes,history,hist_its,50,PETSC_TRUE);
215: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
216: Evaluate initial guess; then solve nonlinear system
217: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
218: /*
219: Note: The user should initialize the vector, x, with the initial guess
220: for the nonlinear solver prior to calling SNESSolve(). In particular,
221: to employ an initial guess of zero, the user should explicitly set
222: this vector to zero by calling VecSet().
223: */
224: FormInitialGuess(&user,x);
225: SNESSolve(snes,PETSC_NULL,x);
226: SNESGetIterationNumber(snes,&its);
227: PetscPrintf(PETSC_COMM_WORLD,"Number of Newton iterations = %D\n",its);
230: /*
231: Print the convergence history. This is intended just to demonstrate
232: use of the data attained via SNESSetConvergenceHistory().
233: */
234: PetscOptionsHasName(PETSC_NULL,"-print_history",&flg);
235: if (flg) {
236: for (i=0; i<its+1; i++) {
237: PetscPrintf(PETSC_COMM_WORLD,"iteration %D: Linear iterations %D Function norm = %G\n",i,hist_its[i],history[i]);
238: }
239: }
241: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
242: Free work space. All PETSc objects should be destroyed when they
243: are no longer needed.
244: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
246: if (!matrix_free) {
247: MatDestroy(J);
248: }
249: if (fd_coloring) {
250: MatFDColoringDestroy(fdcoloring);
251: }
252: VecDestroy(x);
253: VecDestroy(r);
254: SNESDestroy(snes);
255: PetscFinalize();
257: return 0;
258: }
259: /* ------------------------------------------------------------------- */
262: /*
263: FormInitialGuess - Forms initial approximation.
265: Input Parameters:
266: user - user-defined application context
267: X - vector
269: Output Parameter:
270: X - vector
271: */
272: PetscErrorCode FormInitialGuess(AppCtx *user,Vec X)
273: {
274: PetscInt i,j,row,mx,my;
276: PetscReal lambda,temp1,temp,hx,hy;
277: PetscScalar *x;
279: mx = user->mx;
280: my = user->my;
281: lambda = user->param;
283: hx = 1.0 / (PetscReal)(mx-1);
284: hy = 1.0 / (PetscReal)(my-1);
286: /*
287: Get a pointer to vector data.
288: - For default PETSc vectors, VecGetArray() returns a pointer to
289: the data array. Otherwise, the routine is implementation dependent.
290: - You MUST call VecRestoreArray() when you no longer need access to
291: the array.
292: */
293: VecGetArray(X,&x);
294: temp1 = lambda/(lambda + 1.0);
295: for (j=0; j<my; j++) {
296: temp = (PetscReal)(PetscMin(j,my-j-1))*hy;
297: for (i=0; i<mx; i++) {
298: row = i + j*mx;
299: if (i == 0 || j == 0 || i == mx-1 || j == my-1) {
300: x[row] = 0.0;
301: continue;
302: }
303: x[row] = temp1*sqrt(PetscMin((PetscReal)(PetscMin(i,mx-i-1))*hx,temp));
304: }
305: }
307: /*
308: Restore vector
309: */
310: VecRestoreArray(X,&x);
311: return 0;
312: }
313: /* ------------------------------------------------------------------- */
316: /*
317: FormFunction - Evaluates nonlinear function, F(x).
319: Input Parameters:
320: . snes - the SNES context
321: . X - input vector
322: . ptr - optional user-defined context, as set by SNESSetFunction()
324: Output Parameter:
325: . F - function vector
326: */
327: PetscErrorCode FormFunction(SNES snes,Vec X,Vec F,void *ptr)
328: {
329: AppCtx *user = (AppCtx*)ptr;
330: PetscInt i,j,row,mx,my;
332: PetscReal two = 2.0,one = 1.0,lambda,hx,hy,hxdhy,hydhx;
333: PetscScalar ut,ub,ul,ur,u,uxx,uyy,sc,*x,*f;
335: mx = user->mx;
336: my = user->my;
337: lambda = user->param;
338: hx = one / (PetscReal)(mx-1);
339: hy = one / (PetscReal)(my-1);
340: sc = hx*hy;
341: hxdhy = hx/hy;
342: hydhx = hy/hx;
344: /*
345: Get pointers to vector data
346: */
347: VecGetArray(X,&x);
348: VecGetArray(F,&f);
350: /*
351: Compute function
352: */
353: for (j=0; j<my; j++) {
354: for (i=0; i<mx; i++) {
355: row = i + j*mx;
356: if (i == 0 || j == 0 || i == mx-1 || j == my-1) {
357: f[row] = x[row];
358: continue;
359: }
360: u = x[row];
361: ub = x[row - mx];
362: ul = x[row - 1];
363: ut = x[row + mx];
364: ur = x[row + 1];
365: uxx = (-ur + two*u - ul)*hydhx;
366: uyy = (-ut + two*u - ub)*hxdhy;
367: f[row] = uxx + uyy - sc*lambda*PetscExpScalar(u);
368: }
369: }
371: /*
372: Restore vectors
373: */
374: VecRestoreArray(X,&x);
375: VecRestoreArray(F,&f);
376: return 0;
377: }
378: /* ------------------------------------------------------------------- */
381: /*
382: FormJacobian - Evaluates Jacobian matrix.
384: Input Parameters:
385: . snes - the SNES context
386: . x - input vector
387: . ptr - optional user-defined context, as set by SNESSetJacobian()
389: Output Parameters:
390: . A - Jacobian matrix
391: . B - optionally different preconditioning matrix
392: . flag - flag indicating matrix structure
393: */
394: PetscErrorCode FormJacobian(SNES snes,Vec X,Mat *J,Mat *B,MatStructure *flag,void *ptr)
395: {
396: AppCtx *user = (AppCtx*)ptr; /* user-defined applicatin context */
397: Mat jac = *B; /* Jacobian matrix */
398: PetscInt i,j,row,mx,my,col[5];
400: PetscScalar two = 2.0,one = 1.0,lambda,v[5],sc,*x;
401: PetscReal hx,hy,hxdhy,hydhx;
403: mx = user->mx;
404: my = user->my;
405: lambda = user->param;
406: hx = 1.0 / (PetscReal)(mx-1);
407: hy = 1.0 / (PetscReal)(my-1);
408: sc = hx*hy;
409: hxdhy = hx/hy;
410: hydhx = hy/hx;
412: /*
413: Get pointer to vector data
414: */
415: VecGetArray(X,&x);
417: /*
418: Compute entries of the Jacobian
419: */
420: for (j=0; j<my; j++) {
421: for (i=0; i<mx; i++) {
422: row = i + j*mx;
423: if (i == 0 || j == 0 || i == mx-1 || j == my-1) {
424: MatSetValues(jac,1,&row,1,&row,&one,INSERT_VALUES);
425: continue;
426: }
427: v[0] = -hxdhy; col[0] = row - mx;
428: v[1] = -hydhx; col[1] = row - 1;
429: v[2] = two*(hydhx + hxdhy) - sc*lambda*PetscExpScalar(x[row]); col[2] = row;
430: v[3] = -hydhx; col[3] = row + 1;
431: v[4] = -hxdhy; col[4] = row + mx;
432: MatSetValues(jac,1,&row,5,col,v,INSERT_VALUES);
433: }
434: }
436: /*
437: Restore vector
438: */
439: VecRestoreArray(X,&x);
441: /*
442: Assemble matrix
443: */
444: MatAssemblyBegin(jac,MAT_FINAL_ASSEMBLY);
445: MatAssemblyEnd(jac,MAT_FINAL_ASSEMBLY);
447: if (jac != *J) {
448: MatAssemblyBegin(*J,MAT_FINAL_ASSEMBLY);
449: MatAssemblyEnd(*J,MAT_FINAL_ASSEMBLY);
450: }
452: /*
453: Set flag to indicate that the Jacobian matrix retains an identical
454: nonzero structure throughout all nonlinear iterations (although the
455: values of the entries change). Thus, we can save some work in setting
456: up the preconditioner (e.g., no need to redo symbolic factorization for
457: ILU/ICC preconditioners).
458: - If the nonzero structure of the matrix is different during
459: successive linear solves, then the flag DIFFERENT_NONZERO_PATTERN
460: must be used instead. If you are unsure whether the matrix
461: structure has changed or not, use the flag DIFFERENT_NONZERO_PATTERN.
462: - Caution: If you specify SAME_NONZERO_PATTERN, PETSc
463: believes your assertion and does not check the structure
464: of the matrix. If you erroneously claim that the structure
465: is the same when it actually is not, the new preconditioner
466: will not function correctly. Thus, use this optimization
467: feature with caution!
468: */
469: *flag = SAME_NONZERO_PATTERN;
470: return 0;
471: }