Actual source code: minsurf1.c
1: #include "tao.h"
3: /*
4: Include "tao.h" so we can use the TAO solvers with PETSc support.
5: */
6: static char help[] =
7: "This example demonstrates use of the TAO package to\n\
8: solve an unconstrained system of equations. This example is based on a\n\
9: problem from the MINPACK-2 test suite. Given a rectangular 2-D domain and\n\
10: boundary values along the edges of the domain, the objective is to find the\n\
11: surface with the minimal area that satisfies the boundary conditions.\n\
12: This application solves this problem using complimentarity -- We are actually\n\
13: solving the system (grad f)_i >= 0, if x_i == l_i \n\
14: (grad f)_i = 0, if l_i < x_i < u_i \n\
15: (grad f)_i <= 0, if x_i == u_i \n\
16: where f is the function to be minimized. \n\
17: \n\
18: The command line options are:\n\
19: -mx <xg>, where <xg> = number of grid points in the 1st coordinate direction\n\
20: -my <yg>, where <yg> = number of grid points in the 2nd coordinate direction\n\
21: -start <st>, where <st> =0 for zero vector, and an average of the boundary conditions otherwise \n\n";
23: /*T
24: Concepts: TAO - Solving a complementarity problem
25: Routines: TaoInitialize(); TaoFinalize();
26: Routines: TaoCreate(); TaoDestroy();
27: Routines: TaoApplicationCreate(); TaoAppDestroy();
28: Routines: TaoSetConstraintRoutine(); TaoSetJacobianRoutine();
29: Routines: TaoGetVariableBounds(); TaoAppSetInitialSolutionVec();
30: Routines: TaoSetOptions();
31: Routines: TaoSolve();
32: Processors: 1
33: T*/
36: /*
37: User-defined application context - contains data needed by the
38: application-provided call-back routines, FormFunctionGradient(),
39: FormHessian().
40: */
41: typedef struct {
42: int mx, my;
43: double *bottom, *top, *left, *right;
44: } AppCtx;
47: /* -------- User-defined Routines --------- */
49: static int MSA_BoundaryConditions(AppCtx *);
50: static int MSA_InitialPoint(AppCtx *, Vec);
51: int FormConstraints(TAO_APPLICATION, Vec, Vec, void *);
52: int FormJacobian(TAO_APPLICATION, Vec, Mat *, Mat*, MatStructure*,void *);
56: int main(int argc, char **argv)
57: {
58: int info; /* used to check for functions returning nonzeros */
59: Vec x; /* solution vector */
60: Vec c; /* Constraints function vector */
61: Vec xl,xu; /* Bounds on the variables */
62: PetscTruth flg; /* A return variable when checking for user options */
63: TAO_SOLVER tao; /* TAO_SOLVER solver context */
64: TAO_APPLICATION my_app; /* The PETSc application */
65: Mat J; /* Jacobian matrix */
66: int N; /* Number of elements in vector */
67: PetscScalar lb = -TAO_INFINITY; /* lower bound constant */
68: PetscScalar ub = TAO_INFINITY; /* upper bound constant */
69: AppCtx user; /* user-defined work context */
71: /* Initialize PETSc, TAO */
72: PetscInitialize(&argc, &argv, (char *)0, help );
73: TaoInitialize(&argc, &argv, (char *)0, help );
75: /* Specify default dimension of the problem */
76: user.mx = 4; user.my = 4;
78: /* Check for any command line arguments that override defaults */
79: info = PetscOptionsGetInt(PETSC_NULL, "-mx", &user.mx, &flg); CHKERRQ(info);
80: info = PetscOptionsGetInt(PETSC_NULL, "-my", &user.my, &flg); CHKERRQ(info);
82: /* Calculate any derived values from parameters */
83: N = user.mx*user.my;
85:
86: PetscPrintf(PETSC_COMM_SELF,"\n---- Minimum Surface Area Problem -----\n");
87: PetscPrintf(PETSC_COMM_SELF,"mx:%d, my:%d\n", user.mx,user.my);
90: /* Create appropriate vectors and matrices */
91: info = VecCreateSeq(MPI_COMM_SELF, N, &x);
92: info = VecDuplicate(x, &c); CHKERRQ(info);
93: info = MatCreateSeqAIJ(MPI_COMM_SELF, N, N, 7, PETSC_NULL, &J); CHKERRQ(info);
95: /* The TAO code begins here */
97: /* Create TAO solver and set desired solution method */
98: info = TaoCreate(PETSC_COMM_SELF, "tao_ssils", &tao); CHKERRQ(info);
99: info = TaoApplicationCreate(PETSC_COMM_SELF, &my_app); CHKERRQ(info);
101: /* Set data structure */
102: info = TaoAppSetInitialSolutionVec(my_app, x); CHKERRQ(info);
104: /* Set routines for constraints function and Jacobian evaluation */
105: info = TaoAppSetConstraintRoutine(my_app, FormConstraints, (void *)&user);
106: CHKERRQ(info);
107: info = TaoAppSetJacobianRoutine(my_app, FormJacobian, (void *)&user); CHKERRQ(info);
108: info = TaoAppSetJacobianMat(my_app, J, J); CHKERRQ(info);
110: /* Set the variable bounds */
111: info = MSA_BoundaryConditions(&user); CHKERRQ(info);
113: /* Set initial solution guess */
114: info = MSA_InitialPoint(&user, x); CHKERRQ(info);
116: /* Now that the PETSc application is all set, attach to tao context */
117: info = TaoSetupApplicationSolver(my_app,tao); CHKERRQ(info);
119: /* Set Bounds on variables */
120: info = VecDuplicate(x, &xl); CHKERRQ(info);
121: info = VecDuplicate(x, &xu); CHKERRQ(info);
122: info = VecSet(xl, lb); CHKERRQ(info);
123: info = VecSet(xu, ub); CHKERRQ(info);
124: info = TaoAppSetVariableBounds(my_app,xl,xu); CHKERRQ(info);
126: /* Check for any tao command line options */
127: info = TaoSetOptions(my_app,tao); CHKERRQ(info);
129: /* Solve the application */
130: info = TaoSolveApplication(my_app,tao); CHKERRQ(info);
133: /* Free Tao data structures */
134: info = TaoDestroy(tao); CHKERRQ(info);
135: info = TaoAppDestroy(my_app); CHKERRQ(info);
137: /* Free PETSc data structures */
138: info = VecDestroy(x); CHKERRQ(info);
139: info = VecDestroy(xl); CHKERRQ(info);
140: info = VecDestroy(xu); CHKERRQ(info);
141: info = VecDestroy(c); CHKERRQ(info);
142: info = MatDestroy(J); CHKERRQ(info);
144: /* Free user-created data structures */
145: info = PetscFree(user.bottom); CHKERRQ(info);
146: info = PetscFree(user.top); CHKERRQ(info);
147: info = PetscFree(user.left); CHKERRQ(info);
148: info = PetscFree(user.right); CHKERRQ(info);
150: /* Finalize TAO and PETSc */
151: PetscFinalize();
152: TaoFinalize();
154: return 0;
155: }
157: /* -------------------------------------------------------------------- */
161: /* FormConstraints - Evaluates gradient of f.
163: Input Parameters:
164: . tao - the TAO_APPLICATION context
165: . X - input vector
166: . ptr - optional user-defined context, as set by TaoAppSetConstraintRoutine()
167:
168: Output Parameters:
169: . G - vector containing the newly evaluated gradient
170: */
171: int FormConstraints(TAO_APPLICATION tao, Vec X, Vec G, void *ptr){
172: AppCtx *user = (AppCtx *) ptr;
173: int info,i,j,row;
174: int mx=user->mx, my=user->my;
175: double hx=1.0/(mx+1),hy=1.0/(my+1), hydhx=hy/hx, hxdhy=hx/hy;
176: double f1,f2,f3,f4,f5,f6,d1,d2,d3,d4,d5,d6,d7,d8,xc,xl,xr,xt,xb,xlt,xrb;
177: double df1dxc,df2dxc,df3dxc,df4dxc,df5dxc,df6dxc;
178: PetscScalar zero=0.0;
179: PetscScalar *g, *x;
181: /* Initialize vector to zero */
182: info = VecSet(G, zero); CHKERRQ(info);
184: /* Get pointers to vector data */
185: info = VecGetArray(X, &x); CHKERRQ(info);
186: info = VecGetArray(G, &g); CHKERRQ(info);
188: /* Compute function over the locally owned part of the mesh */
189: for (j=0; j<my; j++){
190: for (i=0; i< mx; i++){
191: row= j*mx + i;
192:
193: xc = x[row];
194: xlt=xrb=xl=xr=xb=xt=xc;
195:
196: if (i==0){ /* left side */
197: xl= user->left[j+1];
198: xlt = user->left[j+2];
199: } else {
200: xl = x[row-1];
201: }
203: if (j==0){ /* bottom side */
204: xb=user->bottom[i+1];
205: xrb = user->bottom[i+2];
206: } else {
207: xb = x[row-mx];
208: }
209:
210: if (i+1 == mx){ /* right side */
211: xr=user->right[j+1];
212: xrb = user->right[j];
213: } else {
214: xr = x[row+1];
215: }
217: if (j+1==0+my){ /* top side */
218: xt=user->top[i+1];
219: xlt = user->top[i];
220: }else {
221: xt = x[row+mx];
222: }
224: if (i>0 && j+1<my){
225: xlt = x[row-1+mx];
226: }
227: if (j>0 && i+1<mx){
228: xrb = x[row+1-mx];
229: }
231: d1 = (xc-xl);
232: d2 = (xc-xr);
233: d3 = (xc-xt);
234: d4 = (xc-xb);
235: d5 = (xr-xrb);
236: d6 = (xrb-xb);
237: d7 = (xlt-xl);
238: d8 = (xt-xlt);
239:
240: df1dxc = d1*hydhx;
241: df2dxc = ( d1*hydhx + d4*hxdhy );
242: df3dxc = d3*hxdhy;
243: df4dxc = ( d2*hydhx + d3*hxdhy );
244: df5dxc = d2*hydhx;
245: df6dxc = d4*hxdhy;
247: d1 /= hx;
248: d2 /= hx;
249: d3 /= hy;
250: d4 /= hy;
251: d5 /= hy;
252: d6 /= hx;
253: d7 /= hy;
254: d8 /= hx;
256: f1 = sqrt( 1.0 + d1*d1 + d7*d7);
257: f2 = sqrt( 1.0 + d1*d1 + d4*d4);
258: f3 = sqrt( 1.0 + d3*d3 + d8*d8);
259: f4 = sqrt( 1.0 + d3*d3 + d2*d2);
260: f5 = sqrt( 1.0 + d2*d2 + d5*d5);
261: f6 = sqrt( 1.0 + d4*d4 + d6*d6);
262:
263: df1dxc /= f1;
264: df2dxc /= f2;
265: df3dxc /= f3;
266: df4dxc /= f4;
267: df5dxc /= f5;
268: df6dxc /= f6;
270: g[row] = (df1dxc+df2dxc+df3dxc+df4dxc+df5dxc+df6dxc )/2.0;
271:
272: }
273: }
274:
275: /* Restore vectors */
276: info = VecRestoreArray(X, &x); CHKERRQ(info);
277: info = VecRestoreArray(G, &g); CHKERRQ(info);
278: info = PetscLogFlops(67*mx*my); CHKERRQ(info);
279: return 0;
280: }
282: /* ------------------------------------------------------------------- */
285: /*
286: FormJacobian - Evaluates Jacobian matrix.
288: Input Parameters:
289: . tao - the TAO_APPLICATION context
290: . X - input vector
291: . ptr - optional user-defined context, as set by TaoSetJacobian()
293: Output Parameters:
294: . tH - Jacobian matrix
296: */
297: int FormJacobian(TAO_APPLICATION tao, Vec X, Mat *tH, Mat* tHPre, MatStructure* flag, void *ptr)
298: {
299: AppCtx *user = (AppCtx *) ptr;
300: Mat H = *tH;
301: int i,j,k,row,info;
302: int mx=user->mx, my=user->my;
303: int col[7];
304: double hx=1.0/(mx+1), hy=1.0/(my+1), hydhx=hy/hx, hxdhy=hx/hy;
305: double f1,f2,f3,f4,f5,f6,d1,d2,d3,d4,d5,d6,d7,d8,xc,xl,xr,xt,xb,xlt,xrb;
306: double hl,hr,ht,hb,hc,htl,hbr;
307: PetscScalar *x, v[7];
308: PetscTruth assembled;
310: /* Set various matrix options */
311: info = MatSetOption(H,MAT_IGNORE_OFF_PROC_ENTRIES); CHKERRQ(info);
312: info = MatSetOption(H,MAT_COLUMNS_SORTED); CHKERRQ(info);
313: info = MatSetOption(H,MAT_ROWS_SORTED); CHKERRQ(info);
314: info = MatAssembled(H,&assembled); CHKERRQ(info);
315: if (assembled){info = MatZeroEntries(H); CHKERRQ(info);}
316: *flag=SAME_NONZERO_PATTERN;
318: /* Get pointers to vector data */
319: info = VecGetArray(X, &x); CHKERRQ(info);
321: /* Compute Jacobian over the locally owned part of the mesh */
322: for (i=0; i< mx; i++){
323: for (j=0; j<my; j++){
324: row= j*mx + i;
325:
326: xc = x[row];
327: xlt=xrb=xl=xr=xb=xt=xc;
329: /* Left side */
330: if (i==0){
331: xl= user->left[j+1];
332: xlt = user->left[j+2];
333: } else {
334: xl = x[row-1];
335: }
336:
337: if (j==0){
338: xb=user->bottom[i+1];
339: xrb = user->bottom[i+2];
340: } else {
341: xb = x[row-mx];
342: }
343:
344: if (i+1 == mx){
345: xr=user->right[j+1];
346: xrb = user->right[j];
347: } else {
348: xr = x[row+1];
349: }
351: if (j+1==my){
352: xt=user->top[i+1];
353: xlt = user->top[i];
354: }else {
355: xt = x[row+mx];
356: }
358: if (i>0 && j+1<my){
359: xlt = x[row-1+mx];
360: }
361: if (j>0 && i+1<mx){
362: xrb = x[row+1-mx];
363: }
366: d1 = (xc-xl)/hx;
367: d2 = (xc-xr)/hx;
368: d3 = (xc-xt)/hy;
369: d4 = (xc-xb)/hy;
370: d5 = (xrb-xr)/hy;
371: d6 = (xrb-xb)/hx;
372: d7 = (xlt-xl)/hy;
373: d8 = (xlt-xt)/hx;
374:
375: f1 = sqrt( 1.0 + d1*d1 + d7*d7);
376: f2 = sqrt( 1.0 + d1*d1 + d4*d4);
377: f3 = sqrt( 1.0 + d3*d3 + d8*d8);
378: f4 = sqrt( 1.0 + d3*d3 + d2*d2);
379: f5 = sqrt( 1.0 + d2*d2 + d5*d5);
380: f6 = sqrt( 1.0 + d4*d4 + d6*d6);
383: hl = (-hydhx*(1.0+d7*d7)+d1*d7)/(f1*f1*f1)+
384: (-hydhx*(1.0+d4*d4)+d1*d4)/(f2*f2*f2);
385: hr = (-hydhx*(1.0+d5*d5)+d2*d5)/(f5*f5*f5)+
386: (-hydhx*(1.0+d3*d3)+d2*d3)/(f4*f4*f4);
387: ht = (-hxdhy*(1.0+d8*d8)+d3*d8)/(f3*f3*f3)+
388: (-hxdhy*(1.0+d2*d2)+d2*d3)/(f4*f4*f4);
389: hb = (-hxdhy*(1.0+d6*d6)+d4*d6)/(f6*f6*f6)+
390: (-hxdhy*(1.0+d1*d1)+d1*d4)/(f2*f2*f2);
392: hbr = -d2*d5/(f5*f5*f5) - d4*d6/(f6*f6*f6);
393: htl = -d1*d7/(f1*f1*f1) - d3*d8/(f3*f3*f3);
395: hc = hydhx*(1.0+d7*d7)/(f1*f1*f1) + hxdhy*(1.0+d8*d8)/(f3*f3*f3) +
396: hydhx*(1.0+d5*d5)/(f5*f5*f5) + hxdhy*(1.0+d6*d6)/(f6*f6*f6) +
397: (hxdhy*(1.0+d1*d1)+hydhx*(1.0+d4*d4)-2*d1*d4)/(f2*f2*f2) +
398: (hxdhy*(1.0+d2*d2)+hydhx*(1.0+d3*d3)-2*d2*d3)/(f4*f4*f4);
400: hl/=2.0; hr/=2.0; ht/=2.0; hb/=2.0; hbr/=2.0; htl/=2.0; hc/=2.0;
402: k=0;
403: if (j>0){
404: v[k]=hb; col[k]=row - mx; k++;
405: }
406:
407: if (j>0 && i < mx -1){
408: v[k]=hbr; col[k]=row - mx+1; k++;
409: }
410:
411: if (i>0){
412: v[k]= hl; col[k]=row - 1; k++;
413: }
414:
415: v[k]= hc; col[k]=row; k++;
416:
417: if (i < mx-1 ){
418: v[k]= hr; col[k]=row+1; k++;
419: }
420:
421: if (i>0 && j < my-1 ){
422: v[k]= htl; col[k] = row+mx-1; k++;
423: }
424:
425: if (j < my-1 ){
426: v[k]= ht; col[k] = row+mx; k++;
427: }
428:
429: /*
430: Set matrix values using local numbering, which was defined
431: earlier, in the main routine.
432: */
433: info = MatSetValues(H,1,&row,k,col,v,INSERT_VALUES);
434: CHKERRQ(info);
435: }
436: }
437:
438: /* Restore vectors */
439: info = VecRestoreArray(X,&x); CHKERRQ(info);
441: /* Assemble the matrix */
442: info = MatAssemblyBegin(H,MAT_FINAL_ASSEMBLY); CHKERRQ(info);
443: info = MatAssemblyEnd(H,MAT_FINAL_ASSEMBLY); CHKERRQ(info);
444: info = PetscLogFlops(199*mx*my); CHKERRQ(info);
445: return 0;
446: }
448: /* ------------------------------------------------------------------- */
451: /*
452: MSA_BoundaryConditions - Calculates the boundary conditions for
453: the region.
455: Input Parameter:
456: . user - user-defined application context
458: Output Parameter:
459: . user - user-defined application context
460: */
461: static int MSA_BoundaryConditions(AppCtx * user)
462: {
463: int i,j,k,limit=0,info,maxits=5;
464: int mx=user->mx,my=user->my;
465: int bsize=0, lsize=0, tsize=0, rsize=0;
466: double one=1.0, two=2.0, three=3.0, tol=1e-10;
467: double fnorm,det,hx,hy,xt=0,yt=0;
468: double u1,u2,nf1,nf2,njac11,njac12,njac21,njac22;
469: double b=-0.5, t=0.5, l=-0.5, r=0.5;
470: double *boundary;
472: bsize=mx+2; lsize=my+2; rsize=my+2; tsize=mx+2;
474: info = PetscMalloc(bsize*sizeof(double), &user->bottom);CHKERRQ(info);
475: info = PetscMalloc(tsize*sizeof(double), &user->top);CHKERRQ(info);
476: info = PetscMalloc(lsize*sizeof(double), &user->left);CHKERRQ(info);
477: info = PetscMalloc(rsize*sizeof(double), &user->right);CHKERRQ(info);
479: hx= (r-l)/(mx+1); hy=(t-b)/(my+1);
481: for (j=0; j<4; j++){
482: if (j==0){
483: yt=b;
484: xt=l;
485: limit=bsize;
486: boundary=user->bottom;
487: } else if (j==1){
488: yt=t;
489: xt=l;
490: limit=tsize;
491: boundary=user->top;
492: } else if (j==2){
493: yt=b;
494: xt=l;
495: limit=lsize;
496: boundary=user->left;
497: } else { // if (j==3)
498: yt=b;
499: xt=r;
500: limit=rsize;
501: boundary=user->right;
502: }
504: for (i=0; i<limit; i++){
505: u1=xt;
506: u2=-yt;
507: for (k=0; k<maxits; k++){
508: nf1=u1 + u1*u2*u2 - u1*u1*u1/three-xt;
509: nf2=-u2 - u1*u1*u2 + u2*u2*u2/three-yt;
510: fnorm=sqrt(nf1*nf1+nf2*nf2);
511: if (fnorm <= tol) break;
512: njac11=one+u2*u2-u1*u1;
513: njac12=two*u1*u2;
514: njac21=-two*u1*u2;
515: njac22=-one - u1*u1 + u2*u2;
516: det = njac11*njac22-njac21*njac12;
517: u1 = u1-(njac22*nf1-njac12*nf2)/det;
518: u2 = u2-(njac11*nf2-njac21*nf1)/det;
519: }
521: boundary[i]=u1*u1-u2*u2;
522: if (j==0 || j==1) {
523: xt=xt+hx;
524: } else { // if (j==2 || j==3)
525: yt=yt+hy;
526: }
527: }
528: }
530: return 0;
531: }
533: /* ------------------------------------------------------------------- */
536: /*
537: MSA_InitialPoint - Calculates the initial guess in one of three ways.
539: Input Parameters:
540: . user - user-defined application context
541: . X - vector for initial guess
543: Output Parameters:
544: . X - newly computed initial guess
545: */
546: static int MSA_InitialPoint(AppCtx * user, Vec X)
547: {
548: int start=-1,i,j,info;
549: PetscScalar zero=0.0;
550: PetscTruth flg;
552: info = PetscOptionsGetInt(PETSC_NULL,"-start",&start,&flg); CHKERRQ(info);
554: if (flg && start==0){ /* The zero vector is reasonable */
555:
556: info = VecSet(X, zero); CHKERRQ(info);
557: /* PLogInfo(user,"Min. Surface Area Problem: Start with 0 vector \n"); */
560: } else { /* Take an average of the boundary conditions */
562: int row;
563: int mx=user->mx,my=user->my;
564: PetscScalar *x;
565:
566: /* Get pointers to vector data */
567: info = VecGetArray(X,&x); CHKERRQ(info);
569: /* Perform local computations */
570: for (j=0; j<my; j++){
571: for (i=0; i< mx; i++){
572: row=(j)*mx + (i);
573: x[row] = ( ((j+1)*user->bottom[i+1]+(my-j+1)*user->top[i+1])/(my+2)+
574: ((i+1)*user->left[j+1]+(mx-i+1)*user->right[j+1])/(mx+2))/2.0;
575: }
576: }
577:
578: /* Restore vectors */
579: info = VecRestoreArray(X,&x); CHKERRQ(info);
580:
581: }
582: return 0;
583: }