1 #include <petsctao.h>
2
3 static char help[] =
4 "This example demonstrates use of the TAO package to\n\
5 solve an unconstrained system of equations. This example is based on a\n\
6 problem from the MINPACK-2 test suite. Given a rectangular 2-D domain and\n\
7 boundary values along the edges of the domain, the objective is to find the\n\
8 surface with the minimal area that satisfies the boundary conditions.\n\
9 This application solves this problem using complimentarity -- We are actually\n\
10 solving the system (grad f)_i >= 0, if x_i == l_i \n\
11 (grad f)_i = 0, if l_i < x_i < u_i \n\
12 (grad f)_i <= 0, if x_i == u_i \n\
13 where f is the function to be minimized. \n\
14 \n\
15 The command line options are:\n\
16 -mx <xg>, where <xg> = number of grid points in the 1st coordinate direction\n\
17 -my <yg>, where <yg> = number of grid points in the 2nd coordinate direction\n\
18 -start <st>, where <st> =0 for zero vector, and an average of the boundary conditions otherwise \n\n";
19
20 /*T
21 Concepts: TAO^Solving a complementarity problem
22 Routines: TaoCreate(); TaoDestroy();
23
24 Processors: 1
25 T*/
26
27
28
29
30 /*
31 User-defined application context - contains data needed by the
32 application-provided call-back routines, FormFunctionGradient(),
33 FormHessian().
34 */
35 typedef struct {
36 PetscInt mx, my;
37 PetscReal *bottom, *top, *left, *right;
38 } AppCtx;
39
40
41 /* -------- User-defined Routines --------- */
42
43 static PetscErrorCode MSA_BoundaryConditions(AppCtx *);
44 static PetscErrorCode MSA_InitialPoint(AppCtx *, Vec);
45 PetscErrorCode FormConstraints(Tao, Vec, Vec, void *);
46 PetscErrorCode FormJacobian(Tao, Vec, Mat, Mat, void *);
47
main(int argc,char ** argv)48 int main(int argc, char **argv)
49 {
50 PetscErrorCode ierr; /* used to check for functions returning nonzeros */
51 Vec x; /* solution vector */
52 Vec c; /* Constraints function vector */
53 Vec xl,xu; /* Bounds on the variables */
54 PetscBool flg; /* A return variable when checking for user options */
55 Tao tao; /* TAO solver context */
56 Mat J; /* Jacobian matrix */
57 PetscInt N; /* Number of elements in vector */
58 PetscScalar lb = PETSC_NINFINITY; /* lower bound constant */
59 PetscScalar ub = PETSC_INFINITY; /* upper bound constant */
60 AppCtx user; /* user-defined work context */
61
62 /* Initialize PETSc, TAO */
63 ierr = PetscInitialize(&argc, &argv, (char *)0, help);if (ierr) return ierr;
64
65 /* Specify default dimension of the problem */
66 user.mx = 4; user.my = 4;
67
68 /* Check for any command line arguments that override defaults */
69 ierr = PetscOptionsGetInt(NULL,NULL, "-mx", &user.mx, &flg);CHKERRQ(ierr);
70 ierr = PetscOptionsGetInt(NULL,NULL, "-my", &user.my, &flg);CHKERRQ(ierr);
71
72 /* Calculate any derived values from parameters */
73 N = user.mx*user.my;
74
75 ierr = PetscPrintf(PETSC_COMM_SELF,"\n---- Minimum Surface Area Problem -----\n");CHKERRQ(ierr);
76 ierr = PetscPrintf(PETSC_COMM_SELF,"mx:%D, my:%D\n", user.mx,user.my);CHKERRQ(ierr);
77
78 /* Create appropriate vectors and matrices */
79 ierr = VecCreateSeq(MPI_COMM_SELF, N, &x);CHKERRQ(ierr);
80 ierr = VecDuplicate(x, &c);CHKERRQ(ierr);
81 ierr = MatCreateSeqAIJ(MPI_COMM_SELF, N, N, 7, NULL, &J);CHKERRQ(ierr);
82
83 /* The TAO code begins here */
84
85 /* Create TAO solver and set desired solution method */
86 ierr = TaoCreate(PETSC_COMM_SELF,&tao);CHKERRQ(ierr);
87 ierr = TaoSetType(tao,TAOSSILS);CHKERRQ(ierr);
88
89 /* Set data structure */
90 ierr = TaoSetInitialVector(tao, x);CHKERRQ(ierr);
91
92 /* Set routines for constraints function and Jacobian evaluation */
93 ierr = TaoSetConstraintsRoutine(tao, c, FormConstraints, (void *)&user);CHKERRQ(ierr);
94 ierr = TaoSetJacobianRoutine(tao, J, J, FormJacobian, (void *)&user);CHKERRQ(ierr);
95
96 /* Set the variable bounds */
97 ierr = MSA_BoundaryConditions(&user);CHKERRQ(ierr);
98
99 /* Set initial solution guess */
100 ierr = MSA_InitialPoint(&user, x);CHKERRQ(ierr);
101
102 /* Set Bounds on variables */
103 ierr = VecDuplicate(x, &xl);CHKERRQ(ierr);
104 ierr = VecDuplicate(x, &xu);CHKERRQ(ierr);
105 ierr = VecSet(xl, lb);CHKERRQ(ierr);
106 ierr = VecSet(xu, ub);CHKERRQ(ierr);
107 ierr = TaoSetVariableBounds(tao,xl,xu);CHKERRQ(ierr);
108
109 /* Check for any tao command line options */
110 ierr = TaoSetFromOptions(tao);CHKERRQ(ierr);
111
112 /* Solve the application */
113 ierr = TaoSolve(tao);CHKERRQ(ierr);
114
115 /* Free Tao data structures */
116 ierr = TaoDestroy(&tao);CHKERRQ(ierr);
117
118 /* Free PETSc data structures */
119 ierr = VecDestroy(&x);CHKERRQ(ierr);
120 ierr = VecDestroy(&xl);CHKERRQ(ierr);
121 ierr = VecDestroy(&xu);CHKERRQ(ierr);
122 ierr = VecDestroy(&c);CHKERRQ(ierr);
123 ierr = MatDestroy(&J);CHKERRQ(ierr);
124
125 /* Free user-created data structures */
126 ierr = PetscFree(user.bottom);CHKERRQ(ierr);
127 ierr = PetscFree(user.top);CHKERRQ(ierr);
128 ierr = PetscFree(user.left);CHKERRQ(ierr);
129 ierr = PetscFree(user.right);CHKERRQ(ierr);
130
131 ierr = PetscFinalize();
132 return ierr;
133 }
134
135 /* -------------------------------------------------------------------- */
136
137 /* FormConstraints - Evaluates gradient of f.
138
139 Input Parameters:
140 . tao - the TAO_APPLICATION context
141 . X - input vector
142 . ptr - optional user-defined context, as set by TaoSetConstraintsRoutine()
143
144 Output Parameters:
145 . G - vector containing the newly evaluated gradient
146 */
FormConstraints(Tao tao,Vec X,Vec G,void * ptr)147 PetscErrorCode FormConstraints(Tao tao, Vec X, Vec G, void *ptr)
148 {
149 AppCtx *user = (AppCtx *) ptr;
150 PetscErrorCode ierr;
151 PetscInt i,j,row;
152 PetscInt mx=user->mx, my=user->my;
153 PetscReal hx=1.0/(mx+1),hy=1.0/(my+1), hydhx=hy/hx, hxdhy=hx/hy;
154 PetscReal f1,f2,f3,f4,f5,f6,d1,d2,d3,d4,d5,d6,d7,d8,xc,xl,xr,xt,xb,xlt,xrb;
155 PetscReal df1dxc,df2dxc,df3dxc,df4dxc,df5dxc,df6dxc;
156 PetscScalar zero=0.0;
157 PetscScalar *g, *x;
158
159 PetscFunctionBegin;
160 /* Initialize vector to zero */
161 ierr = VecSet(G, zero);CHKERRQ(ierr);
162
163 /* Get pointers to vector data */
164 ierr = VecGetArray(X, &x);CHKERRQ(ierr);
165 ierr = VecGetArray(G, &g);CHKERRQ(ierr);
166
167 /* Compute function over the locally owned part of the mesh */
168 for (j=0; j<my; j++){
169 for (i=0; i< mx; i++){
170 row= j*mx + i;
171
172 xc = x[row];
173 xlt=xrb=xl=xr=xb=xt=xc;
174
175 if (i==0){ /* left side */
176 xl= user->left[j+1];
177 xlt = user->left[j+2];
178 } else {
179 xl = x[row-1];
180 }
181
182 if (j==0){ /* bottom side */
183 xb=user->bottom[i+1];
184 xrb = user->bottom[i+2];
185 } else {
186 xb = x[row-mx];
187 }
188
189 if (i+1 == mx){ /* right side */
190 xr=user->right[j+1];
191 xrb = user->right[j];
192 } else {
193 xr = x[row+1];
194 }
195
196 if (j+1==0+my){ /* top side */
197 xt=user->top[i+1];
198 xlt = user->top[i];
199 }else {
200 xt = x[row+mx];
201 }
202
203 if (i>0 && j+1<my){
204 xlt = x[row-1+mx];
205 }
206 if (j>0 && i+1<mx){
207 xrb = x[row+1-mx];
208 }
209
210 d1 = (xc-xl);
211 d2 = (xc-xr);
212 d3 = (xc-xt);
213 d4 = (xc-xb);
214 d5 = (xr-xrb);
215 d6 = (xrb-xb);
216 d7 = (xlt-xl);
217 d8 = (xt-xlt);
218
219 df1dxc = d1*hydhx;
220 df2dxc = (d1*hydhx + d4*hxdhy);
221 df3dxc = d3*hxdhy;
222 df4dxc = (d2*hydhx + d3*hxdhy);
223 df5dxc = d2*hydhx;
224 df6dxc = d4*hxdhy;
225
226 d1 /= hx;
227 d2 /= hx;
228 d3 /= hy;
229 d4 /= hy;
230 d5 /= hy;
231 d6 /= hx;
232 d7 /= hy;
233 d8 /= hx;
234
235 f1 = PetscSqrtScalar(1.0 + d1*d1 + d7*d7);
236 f2 = PetscSqrtScalar(1.0 + d1*d1 + d4*d4);
237 f3 = PetscSqrtScalar(1.0 + d3*d3 + d8*d8);
238 f4 = PetscSqrtScalar(1.0 + d3*d3 + d2*d2);
239 f5 = PetscSqrtScalar(1.0 + d2*d2 + d5*d5);
240 f6 = PetscSqrtScalar(1.0 + d4*d4 + d6*d6);
241
242 df1dxc /= f1;
243 df2dxc /= f2;
244 df3dxc /= f3;
245 df4dxc /= f4;
246 df5dxc /= f5;
247 df6dxc /= f6;
248
249 g[row] = (df1dxc+df2dxc+df3dxc+df4dxc+df5dxc+df6dxc)/2.0;
250 }
251 }
252
253 /* Restore vectors */
254 ierr = VecRestoreArray(X, &x);CHKERRQ(ierr);
255 ierr = VecRestoreArray(G, &g);CHKERRQ(ierr);
256 ierr = PetscLogFlops(67*mx*my);CHKERRQ(ierr);
257 PetscFunctionReturn(0);
258 }
259
260 /* ------------------------------------------------------------------- */
261 /*
262 FormJacobian - Evaluates Jacobian matrix.
263
264 Input Parameters:
265 . tao - the TAO_APPLICATION context
266 . X - input vector
267 . ptr - optional user-defined context, as set by TaoSetJacobian()
268
269 Output Parameters:
270 . tH - Jacobian matrix
271
272 */
FormJacobian(Tao tao,Vec X,Mat H,Mat tHPre,void * ptr)273 PetscErrorCode FormJacobian(Tao tao, Vec X, Mat H, Mat tHPre, void *ptr)
274 {
275 AppCtx *user = (AppCtx *) ptr;
276 PetscErrorCode ierr;
277 PetscInt i,j,k,row;
278 PetscInt mx=user->mx, my=user->my;
279 PetscInt col[7];
280 PetscReal hx=1.0/(mx+1), hy=1.0/(my+1), hydhx=hy/hx, hxdhy=hx/hy;
281 PetscReal f1,f2,f3,f4,f5,f6,d1,d2,d3,d4,d5,d6,d7,d8,xc,xl,xr,xt,xb,xlt,xrb;
282 PetscReal hl,hr,ht,hb,hc,htl,hbr;
283 const PetscScalar *x;
284 PetscScalar v[7];
285 PetscBool assembled;
286
287 /* Set various matrix options */
288 ierr = MatSetOption(H,MAT_IGNORE_OFF_PROC_ENTRIES,PETSC_TRUE);CHKERRQ(ierr);
289 ierr = MatAssembled(H,&assembled);CHKERRQ(ierr);
290 if (assembled){ierr = MatZeroEntries(H);CHKERRQ(ierr);}
291
292 /* Get pointers to vector data */
293 ierr = VecGetArrayRead(X, &x);CHKERRQ(ierr);
294
295 /* Compute Jacobian over the locally owned part of the mesh */
296 for (i=0; i< mx; i++){
297 for (j=0; j<my; j++){
298 row= j*mx + i;
299
300 xc = x[row];
301 xlt=xrb=xl=xr=xb=xt=xc;
302
303 /* Left side */
304 if (i==0){
305 xl = user->left[j+1];
306 xlt = user->left[j+2];
307 } else {
308 xl = x[row-1];
309 }
310
311 if (j==0){
312 xb = user->bottom[i+1];
313 xrb = user->bottom[i+2];
314 } else {
315 xb = x[row-mx];
316 }
317
318 if (i+1 == mx){
319 xr = user->right[j+1];
320 xrb = user->right[j];
321 } else {
322 xr = x[row+1];
323 }
324
325 if (j+1==my){
326 xt = user->top[i+1];
327 xlt = user->top[i];
328 }else {
329 xt = x[row+mx];
330 }
331
332 if (i>0 && j+1<my){
333 xlt = x[row-1+mx];
334 }
335 if (j>0 && i+1<mx){
336 xrb = x[row+1-mx];
337 }
338
339
340 d1 = (xc-xl)/hx;
341 d2 = (xc-xr)/hx;
342 d3 = (xc-xt)/hy;
343 d4 = (xc-xb)/hy;
344 d5 = (xrb-xr)/hy;
345 d6 = (xrb-xb)/hx;
346 d7 = (xlt-xl)/hy;
347 d8 = (xlt-xt)/hx;
348
349 f1 = PetscSqrtScalar(1.0 + d1*d1 + d7*d7);
350 f2 = PetscSqrtScalar(1.0 + d1*d1 + d4*d4);
351 f3 = PetscSqrtScalar(1.0 + d3*d3 + d8*d8);
352 f4 = PetscSqrtScalar(1.0 + d3*d3 + d2*d2);
353 f5 = PetscSqrtScalar(1.0 + d2*d2 + d5*d5);
354 f6 = PetscSqrtScalar(1.0 + d4*d4 + d6*d6);
355
356
357 hl = (-hydhx*(1.0+d7*d7)+d1*d7)/(f1*f1*f1)+(-hydhx*(1.0+d4*d4)+d1*d4)/(f2*f2*f2);
358 hr = (-hydhx*(1.0+d5*d5)+d2*d5)/(f5*f5*f5)+(-hydhx*(1.0+d3*d3)+d2*d3)/(f4*f4*f4);
359 ht = (-hxdhy*(1.0+d8*d8)+d3*d8)/(f3*f3*f3)+(-hxdhy*(1.0+d2*d2)+d2*d3)/(f4*f4*f4);
360 hb = (-hxdhy*(1.0+d6*d6)+d4*d6)/(f6*f6*f6)+(-hxdhy*(1.0+d1*d1)+d1*d4)/(f2*f2*f2);
361
362 hbr = -d2*d5/(f5*f5*f5) - d4*d6/(f6*f6*f6);
363 htl = -d1*d7/(f1*f1*f1) - d3*d8/(f3*f3*f3);
364
365 hc = hydhx*(1.0+d7*d7)/(f1*f1*f1) + hxdhy*(1.0+d8*d8)/(f3*f3*f3) + hydhx*(1.0+d5*d5)/(f5*f5*f5) + hxdhy*(1.0+d6*d6)/(f6*f6*f6) +
366 (hxdhy*(1.0+d1*d1)+hydhx*(1.0+d4*d4)-2*d1*d4)/(f2*f2*f2) + (hxdhy*(1.0+d2*d2)+hydhx*(1.0+d3*d3)-2*d2*d3)/(f4*f4*f4);
367
368 hl/=2.0; hr/=2.0; ht/=2.0; hb/=2.0; hbr/=2.0; htl/=2.0; hc/=2.0;
369
370 k=0;
371 if (j>0){
372 v[k]=hb; col[k]=row - mx; k++;
373 }
374
375 if (j>0 && i < mx -1){
376 v[k]=hbr; col[k]=row - mx+1; k++;
377 }
378
379 if (i>0){
380 v[k]= hl; col[k]=row - 1; k++;
381 }
382
383 v[k]= hc; col[k]=row; k++;
384
385 if (i < mx-1){
386 v[k]= hr; col[k]=row+1; k++;
387 }
388
389 if (i>0 && j < my-1){
390 v[k]= htl; col[k] = row+mx-1; k++;
391 }
392
393 if (j < my-1){
394 v[k]= ht; col[k] = row+mx; k++;
395 }
396
397 /*
398 Set matrix values using local numbering, which was defined
399 earlier, in the main routine.
400 */
401 ierr = MatSetValues(H,1,&row,k,col,v,INSERT_VALUES);CHKERRQ(ierr);
402 }
403 }
404
405 /* Restore vectors */
406 ierr = VecRestoreArrayRead(X,&x);CHKERRQ(ierr);
407
408 /* Assemble the matrix */
409 ierr = MatAssemblyBegin(H,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
410 ierr = MatAssemblyEnd(H,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
411 ierr = PetscLogFlops(199*mx*my);CHKERRQ(ierr);
412 PetscFunctionReturn(0);
413 }
414
415 /* ------------------------------------------------------------------- */
416 /*
417 MSA_BoundaryConditions - Calculates the boundary conditions for
418 the region.
419
420 Input Parameter:
421 . user - user-defined application context
422
423 Output Parameter:
424 . user - user-defined application context
425 */
MSA_BoundaryConditions(AppCtx * user)426 static PetscErrorCode MSA_BoundaryConditions(AppCtx * user)
427 {
428 PetscErrorCode ierr;
429 PetscInt i,j,k,limit=0,maxits=5;
430 PetscInt mx=user->mx,my=user->my;
431 PetscInt bsize=0, lsize=0, tsize=0, rsize=0;
432 PetscReal one=1.0, two=2.0, three=3.0, tol=1e-10;
433 PetscReal fnorm,det,hx,hy,xt=0,yt=0;
434 PetscReal u1,u2,nf1,nf2,njac11,njac12,njac21,njac22;
435 PetscReal b=-0.5, t=0.5, l=-0.5, r=0.5;
436 PetscReal *boundary;
437
438 PetscFunctionBegin;
439 bsize=mx+2; lsize=my+2; rsize=my+2; tsize=mx+2;
440
441 ierr = PetscMalloc1(bsize, &user->bottom);CHKERRQ(ierr);
442 ierr = PetscMalloc1(tsize, &user->top);CHKERRQ(ierr);
443 ierr = PetscMalloc1(lsize, &user->left);CHKERRQ(ierr);
444 ierr = PetscMalloc1(rsize, &user->right);CHKERRQ(ierr);
445
446 hx= (r-l)/(mx+1); hy=(t-b)/(my+1);
447
448 for (j=0; j<4; j++){
449 if (j==0){
450 yt=b;
451 xt=l;
452 limit=bsize;
453 boundary=user->bottom;
454 } else if (j==1){
455 yt=t;
456 xt=l;
457 limit=tsize;
458 boundary=user->top;
459 } else if (j==2){
460 yt=b;
461 xt=l;
462 limit=lsize;
463 boundary=user->left;
464 } else { /* if (j==3) */
465 yt=b;
466 xt=r;
467 limit=rsize;
468 boundary=user->right;
469 }
470
471 for (i=0; i<limit; i++){
472 u1=xt;
473 u2=-yt;
474 for (k=0; k<maxits; k++){
475 nf1=u1 + u1*u2*u2 - u1*u1*u1/three-xt;
476 nf2=-u2 - u1*u1*u2 + u2*u2*u2/three-yt;
477 fnorm=PetscSqrtScalar(nf1*nf1+nf2*nf2);
478 if (fnorm <= tol) break;
479 njac11=one+u2*u2-u1*u1;
480 njac12=two*u1*u2;
481 njac21=-two*u1*u2;
482 njac22=-one - u1*u1 + u2*u2;
483 det = njac11*njac22-njac21*njac12;
484 u1 = u1-(njac22*nf1-njac12*nf2)/det;
485 u2 = u2-(njac11*nf2-njac21*nf1)/det;
486 }
487
488 boundary[i]=u1*u1-u2*u2;
489 if (j==0 || j==1) {
490 xt=xt+hx;
491 } else { /* if (j==2 || j==3) */
492 yt=yt+hy;
493 }
494 }
495 }
496 PetscFunctionReturn(0);
497 }
498
499 /* ------------------------------------------------------------------- */
500 /*
501 MSA_InitialPoint - Calculates the initial guess in one of three ways.
502
503 Input Parameters:
504 . user - user-defined application context
505 . X - vector for initial guess
506
507 Output Parameters:
508 . X - newly computed initial guess
509 */
MSA_InitialPoint(AppCtx * user,Vec X)510 static PetscErrorCode MSA_InitialPoint(AppCtx * user, Vec X)
511 {
512 PetscErrorCode ierr;
513 PetscInt start=-1,i,j;
514 PetscScalar zero=0.0;
515 PetscBool flg;
516
517 PetscFunctionBegin;
518 ierr = PetscOptionsGetInt(NULL,NULL,"-start",&start,&flg);CHKERRQ(ierr);
519
520 if (flg && start==0){ /* The zero vector is reasonable */
521 ierr = VecSet(X, zero);CHKERRQ(ierr);
522 } else { /* Take an average of the boundary conditions */
523 PetscInt row;
524 PetscInt mx=user->mx,my=user->my;
525 PetscScalar *x;
526
527 /* Get pointers to vector data */
528 ierr = VecGetArray(X,&x);CHKERRQ(ierr);
529
530 /* Perform local computations */
531 for (j=0; j<my; j++){
532 for (i=0; i< mx; i++){
533 row=(j)*mx + (i);
534 x[row] = (((j+1)*user->bottom[i+1]+(my-j+1)*user->top[i+1])/(my+2)+ ((i+1)*user->left[j+1]+(mx-i+1)*user->right[j+1])/(mx+2))/2.0;
535 }
536 }
537
538 /* Restore vectors */
539 ierr = VecRestoreArray(X,&x);CHKERRQ(ierr);
540 }
541 PetscFunctionReturn(0);
542 }
543
544
545 /*TEST
546
547 build:
548 requires: !complex
549
550 test:
551 args: -tao_monitor -tao_view -tao_type ssils -tao_gttol 1.e-5
552 requires: !single
553
554 test:
555 suffix: 2
556 args: -tao_monitor -tao_view -tao_type ssfls -tao_gttol 1.e-5
557
558 TEST*/
559