Actual source code: minsurf1.c
1: #include <petsctao.h>
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";
20: /*T
21: Concepts: TAO^Solving a complementarity problem
22: Routines: TaoCreate(); TaoDestroy();
24: Processors: 1
25: T*/
27: /*
28: User-defined application context - contains data needed by the
29: application-provided call-back routines, FormFunctionGradient(),
30: FormHessian().
31: */
32: typedef struct {
33: PetscInt mx, my;
34: PetscReal *bottom, *top, *left, *right;
35: } AppCtx;
37: /* -------- User-defined Routines --------- */
39: static PetscErrorCode MSA_BoundaryConditions(AppCtx *);
40: static PetscErrorCode MSA_InitialPoint(AppCtx *, Vec);
41: PetscErrorCode FormConstraints(Tao, Vec, Vec, void *);
42: PetscErrorCode FormJacobian(Tao, Vec, Mat, Mat, void *);
44: int main(int argc, char **argv)
45: {
46: Vec x; /* solution vector */
47: Vec c; /* Constraints function vector */
48: Vec xl,xu; /* Bounds on the variables */
49: PetscBool flg; /* A return variable when checking for user options */
50: Tao tao; /* TAO solver context */
51: Mat J; /* Jacobian matrix */
52: PetscInt N; /* Number of elements in vector */
53: PetscScalar lb = PETSC_NINFINITY; /* lower bound constant */
54: PetscScalar ub = PETSC_INFINITY; /* upper bound constant */
55: AppCtx user; /* user-defined work context */
57: /* Initialize PETSc, TAO */
58: PetscInitialize(&argc, &argv, (char *)0, help);
60: /* Specify default dimension of the problem */
61: user.mx = 4; user.my = 4;
63: /* Check for any command line arguments that override defaults */
64: PetscOptionsGetInt(NULL,NULL, "-mx", &user.mx, &flg);
65: PetscOptionsGetInt(NULL,NULL, "-my", &user.my, &flg);
67: /* Calculate any derived values from parameters */
68: N = user.mx*user.my;
70: PetscPrintf(PETSC_COMM_SELF,"\n---- Minimum Surface Area Problem -----\n");
71: PetscPrintf(PETSC_COMM_SELF,"mx:%D, my:%D\n", user.mx,user.my);
73: /* Create appropriate vectors and matrices */
74: VecCreateSeq(MPI_COMM_SELF, N, &x);
75: VecDuplicate(x, &c);
76: MatCreateSeqAIJ(MPI_COMM_SELF, N, N, 7, NULL, &J);
78: /* The TAO code begins here */
80: /* Create TAO solver and set desired solution method */
81: TaoCreate(PETSC_COMM_SELF,&tao);
82: TaoSetType(tao,TAOSSILS);
84: /* Set data structure */
85: TaoSetSolution(tao, x);
87: /* Set routines for constraints function and Jacobian evaluation */
88: TaoSetConstraintsRoutine(tao, c, FormConstraints, (void *)&user);
89: TaoSetJacobianRoutine(tao, J, J, FormJacobian, (void *)&user);
91: /* Set the variable bounds */
92: MSA_BoundaryConditions(&user);
94: /* Set initial solution guess */
95: MSA_InitialPoint(&user, x);
97: /* Set Bounds on variables */
98: VecDuplicate(x, &xl);
99: VecDuplicate(x, &xu);
100: VecSet(xl, lb);
101: VecSet(xu, ub);
102: TaoSetVariableBounds(tao,xl,xu);
104: /* Check for any tao command line options */
105: TaoSetFromOptions(tao);
107: /* Solve the application */
108: TaoSolve(tao);
110: /* Free Tao data structures */
111: TaoDestroy(&tao);
113: /* Free PETSc data structures */
114: VecDestroy(&x);
115: VecDestroy(&xl);
116: VecDestroy(&xu);
117: VecDestroy(&c);
118: MatDestroy(&J);
120: /* Free user-created data structures */
121: PetscFree(user.bottom);
122: PetscFree(user.top);
123: PetscFree(user.left);
124: PetscFree(user.right);
126: PetscFinalize();
127: return 0;
128: }
130: /* -------------------------------------------------------------------- */
132: /* FormConstraints - Evaluates gradient of f.
134: Input Parameters:
135: . tao - the TAO_APPLICATION context
136: . X - input vector
137: . ptr - optional user-defined context, as set by TaoSetConstraintsRoutine()
139: Output Parameters:
140: . G - vector containing the newly evaluated gradient
141: */
142: PetscErrorCode FormConstraints(Tao tao, Vec X, Vec G, void *ptr)
143: {
144: AppCtx *user = (AppCtx *) ptr;
145: PetscInt i,j,row;
146: PetscInt mx=user->mx, my=user->my;
147: PetscReal hx=1.0/(mx+1),hy=1.0/(my+1), hydhx=hy/hx, hxdhy=hx/hy;
148: PetscReal f1,f2,f3,f4,f5,f6,d1,d2,d3,d4,d5,d6,d7,d8,xc,xl,xr,xt,xb,xlt,xrb;
149: PetscReal df1dxc,df2dxc,df3dxc,df4dxc,df5dxc,df6dxc;
150: PetscScalar zero=0.0;
151: PetscScalar *g, *x;
153: /* Initialize vector to zero */
154: VecSet(G, zero);
156: /* Get pointers to vector data */
157: VecGetArray(X, &x);
158: VecGetArray(G, &g);
160: /* Compute function over the locally owned part of the mesh */
161: for (j=0; j<my; j++) {
162: for (i=0; i< mx; i++) {
163: row= j*mx + i;
165: xc = x[row];
166: xlt=xrb=xl=xr=xb=xt=xc;
168: if (i==0) { /* left side */
169: xl= user->left[j+1];
170: xlt = user->left[j+2];
171: } else {
172: xl = x[row-1];
173: }
175: if (j==0) { /* bottom side */
176: xb=user->bottom[i+1];
177: xrb = user->bottom[i+2];
178: } else {
179: xb = x[row-mx];
180: }
182: if (i+1 == mx) { /* right side */
183: xr=user->right[j+1];
184: xrb = user->right[j];
185: } else {
186: xr = x[row+1];
187: }
189: if (j+1==0+my) { /* top side */
190: xt=user->top[i+1];
191: xlt = user->top[i];
192: }else {
193: xt = x[row+mx];
194: }
196: if (i>0 && j+1<my) {
197: xlt = x[row-1+mx];
198: }
199: if (j>0 && i+1<mx) {
200: xrb = x[row+1-mx];
201: }
203: d1 = (xc-xl);
204: d2 = (xc-xr);
205: d3 = (xc-xt);
206: d4 = (xc-xb);
207: d5 = (xr-xrb);
208: d6 = (xrb-xb);
209: d7 = (xlt-xl);
210: d8 = (xt-xlt);
212: df1dxc = d1*hydhx;
213: df2dxc = (d1*hydhx + d4*hxdhy);
214: df3dxc = d3*hxdhy;
215: df4dxc = (d2*hydhx + d3*hxdhy);
216: df5dxc = d2*hydhx;
217: df6dxc = d4*hxdhy;
219: d1 /= hx;
220: d2 /= hx;
221: d3 /= hy;
222: d4 /= hy;
223: d5 /= hy;
224: d6 /= hx;
225: d7 /= hy;
226: d8 /= hx;
228: f1 = PetscSqrtScalar(1.0 + d1*d1 + d7*d7);
229: f2 = PetscSqrtScalar(1.0 + d1*d1 + d4*d4);
230: f3 = PetscSqrtScalar(1.0 + d3*d3 + d8*d8);
231: f4 = PetscSqrtScalar(1.0 + d3*d3 + d2*d2);
232: f5 = PetscSqrtScalar(1.0 + d2*d2 + d5*d5);
233: f6 = PetscSqrtScalar(1.0 + d4*d4 + d6*d6);
235: df1dxc /= f1;
236: df2dxc /= f2;
237: df3dxc /= f3;
238: df4dxc /= f4;
239: df5dxc /= f5;
240: df6dxc /= f6;
242: g[row] = (df1dxc+df2dxc+df3dxc+df4dxc+df5dxc+df6dxc)/2.0;
243: }
244: }
246: /* Restore vectors */
247: VecRestoreArray(X, &x);
248: VecRestoreArray(G, &g);
249: PetscLogFlops(67*mx*my);
250: return 0;
251: }
253: /* ------------------------------------------------------------------- */
254: /*
255: FormJacobian - Evaluates Jacobian matrix.
257: Input Parameters:
258: . tao - the TAO_APPLICATION context
259: . X - input vector
260: . ptr - optional user-defined context, as set by TaoSetJacobian()
262: Output Parameters:
263: . tH - Jacobian matrix
265: */
266: PetscErrorCode FormJacobian(Tao tao, Vec X, Mat H, Mat tHPre, void *ptr)
267: {
268: AppCtx *user = (AppCtx *) ptr;
269: PetscInt i,j,k,row;
270: PetscInt mx=user->mx, my=user->my;
271: PetscInt col[7];
272: PetscReal hx=1.0/(mx+1), hy=1.0/(my+1), hydhx=hy/hx, hxdhy=hx/hy;
273: PetscReal f1,f2,f3,f4,f5,f6,d1,d2,d3,d4,d5,d6,d7,d8,xc,xl,xr,xt,xb,xlt,xrb;
274: PetscReal hl,hr,ht,hb,hc,htl,hbr;
275: const PetscScalar *x;
276: PetscScalar v[7];
277: PetscBool assembled;
279: /* Set various matrix options */
280: MatSetOption(H,MAT_IGNORE_OFF_PROC_ENTRIES,PETSC_TRUE);
281: MatAssembled(H,&assembled);
282: if (assembled) MatZeroEntries(H);
284: /* Get pointers to vector data */
285: VecGetArrayRead(X, &x);
287: /* Compute Jacobian over the locally owned part of the mesh */
288: for (i=0; i< mx; i++) {
289: for (j=0; j<my; j++) {
290: row= j*mx + i;
292: xc = x[row];
293: xlt=xrb=xl=xr=xb=xt=xc;
295: /* Left side */
296: if (i==0) {
297: xl = user->left[j+1];
298: xlt = user->left[j+2];
299: } else {
300: xl = x[row-1];
301: }
303: if (j==0) {
304: xb = user->bottom[i+1];
305: xrb = user->bottom[i+2];
306: } else {
307: xb = x[row-mx];
308: }
310: if (i+1 == mx) {
311: xr = user->right[j+1];
312: xrb = user->right[j];
313: } else {
314: xr = x[row+1];
315: }
317: if (j+1==my) {
318: xt = user->top[i+1];
319: xlt = user->top[i];
320: }else {
321: xt = x[row+mx];
322: }
324: if (i>0 && j+1<my) {
325: xlt = x[row-1+mx];
326: }
327: if (j>0 && i+1<mx) {
328: xrb = x[row+1-mx];
329: }
331: d1 = (xc-xl)/hx;
332: d2 = (xc-xr)/hx;
333: d3 = (xc-xt)/hy;
334: d4 = (xc-xb)/hy;
335: d5 = (xrb-xr)/hy;
336: d6 = (xrb-xb)/hx;
337: d7 = (xlt-xl)/hy;
338: d8 = (xlt-xt)/hx;
340: f1 = PetscSqrtScalar(1.0 + d1*d1 + d7*d7);
341: f2 = PetscSqrtScalar(1.0 + d1*d1 + d4*d4);
342: f3 = PetscSqrtScalar(1.0 + d3*d3 + d8*d8);
343: f4 = PetscSqrtScalar(1.0 + d3*d3 + d2*d2);
344: f5 = PetscSqrtScalar(1.0 + d2*d2 + d5*d5);
345: f6 = PetscSqrtScalar(1.0 + d4*d4 + d6*d6);
347: hl = (-hydhx*(1.0+d7*d7)+d1*d7)/(f1*f1*f1)+(-hydhx*(1.0+d4*d4)+d1*d4)/(f2*f2*f2);
348: hr = (-hydhx*(1.0+d5*d5)+d2*d5)/(f5*f5*f5)+(-hydhx*(1.0+d3*d3)+d2*d3)/(f4*f4*f4);
349: ht = (-hxdhy*(1.0+d8*d8)+d3*d8)/(f3*f3*f3)+(-hxdhy*(1.0+d2*d2)+d2*d3)/(f4*f4*f4);
350: hb = (-hxdhy*(1.0+d6*d6)+d4*d6)/(f6*f6*f6)+(-hxdhy*(1.0+d1*d1)+d1*d4)/(f2*f2*f2);
352: hbr = -d2*d5/(f5*f5*f5) - d4*d6/(f6*f6*f6);
353: htl = -d1*d7/(f1*f1*f1) - d3*d8/(f3*f3*f3);
355: 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) +
356: (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);
358: hl/=2.0; hr/=2.0; ht/=2.0; hb/=2.0; hbr/=2.0; htl/=2.0; hc/=2.0;
360: k=0;
361: if (j>0) {
362: v[k]=hb; col[k]=row - mx; k++;
363: }
365: if (j>0 && i < mx -1) {
366: v[k]=hbr; col[k]=row - mx+1; k++;
367: }
369: if (i>0) {
370: v[k]= hl; col[k]=row - 1; k++;
371: }
373: v[k]= hc; col[k]=row; k++;
375: if (i < mx-1) {
376: v[k]= hr; col[k]=row+1; k++;
377: }
379: if (i>0 && j < my-1) {
380: v[k]= htl; col[k] = row+mx-1; k++;
381: }
383: if (j < my-1) {
384: v[k]= ht; col[k] = row+mx; k++;
385: }
387: /*
388: Set matrix values using local numbering, which was defined
389: earlier, in the main routine.
390: */
391: MatSetValues(H,1,&row,k,col,v,INSERT_VALUES);
392: }
393: }
395: /* Restore vectors */
396: VecRestoreArrayRead(X,&x);
398: /* Assemble the matrix */
399: MatAssemblyBegin(H,MAT_FINAL_ASSEMBLY);
400: MatAssemblyEnd(H,MAT_FINAL_ASSEMBLY);
401: PetscLogFlops(199*mx*my);
402: return 0;
403: }
405: /* ------------------------------------------------------------------- */
406: /*
407: MSA_BoundaryConditions - Calculates the boundary conditions for
408: the region.
410: Input Parameter:
411: . user - user-defined application context
413: Output Parameter:
414: . user - user-defined application context
415: */
416: static PetscErrorCode MSA_BoundaryConditions(AppCtx * user)
417: {
418: PetscInt i,j,k,limit=0,maxits=5;
419: PetscInt mx=user->mx,my=user->my;
420: PetscInt bsize=0, lsize=0, tsize=0, rsize=0;
421: PetscReal one=1.0, two=2.0, three=3.0, tol=1e-10;
422: PetscReal fnorm,det,hx,hy,xt=0,yt=0;
423: PetscReal u1,u2,nf1,nf2,njac11,njac12,njac21,njac22;
424: PetscReal b=-0.5, t=0.5, l=-0.5, r=0.5;
425: PetscReal *boundary;
427: bsize=mx+2; lsize=my+2; rsize=my+2; tsize=mx+2;
429: PetscMalloc1(bsize, &user->bottom);
430: PetscMalloc1(tsize, &user->top);
431: PetscMalloc1(lsize, &user->left);
432: PetscMalloc1(rsize, &user->right);
434: hx= (r-l)/(mx+1); hy=(t-b)/(my+1);
436: for (j=0; j<4; j++) {
437: if (j==0) {
438: yt=b;
439: xt=l;
440: limit=bsize;
441: boundary=user->bottom;
442: } else if (j==1) {
443: yt=t;
444: xt=l;
445: limit=tsize;
446: boundary=user->top;
447: } else if (j==2) {
448: yt=b;
449: xt=l;
450: limit=lsize;
451: boundary=user->left;
452: } else { /* if (j==3) */
453: yt=b;
454: xt=r;
455: limit=rsize;
456: boundary=user->right;
457: }
459: for (i=0; i<limit; i++) {
460: u1=xt;
461: u2=-yt;
462: for (k=0; k<maxits; k++) {
463: nf1=u1 + u1*u2*u2 - u1*u1*u1/three-xt;
464: nf2=-u2 - u1*u1*u2 + u2*u2*u2/three-yt;
465: fnorm=PetscSqrtScalar(nf1*nf1+nf2*nf2);
466: if (fnorm <= tol) break;
467: njac11=one+u2*u2-u1*u1;
468: njac12=two*u1*u2;
469: njac21=-two*u1*u2;
470: njac22=-one - u1*u1 + u2*u2;
471: det = njac11*njac22-njac21*njac12;
472: u1 = u1-(njac22*nf1-njac12*nf2)/det;
473: u2 = u2-(njac11*nf2-njac21*nf1)/det;
474: }
476: boundary[i]=u1*u1-u2*u2;
477: if (j==0 || j==1) {
478: xt=xt+hx;
479: } else { /* if (j==2 || j==3) */
480: yt=yt+hy;
481: }
482: }
483: }
484: return 0;
485: }
487: /* ------------------------------------------------------------------- */
488: /*
489: MSA_InitialPoint - Calculates the initial guess in one of three ways.
491: Input Parameters:
492: . user - user-defined application context
493: . X - vector for initial guess
495: Output Parameters:
496: . X - newly computed initial guess
497: */
498: static PetscErrorCode MSA_InitialPoint(AppCtx * user, Vec X)
499: {
500: PetscInt start=-1,i,j;
501: PetscScalar zero=0.0;
502: PetscBool flg;
504: PetscOptionsGetInt(NULL,NULL,"-start",&start,&flg);
506: if (flg && start==0) { /* The zero vector is reasonable */
507: VecSet(X, zero);
508: } else { /* Take an average of the boundary conditions */
509: PetscInt row;
510: PetscInt mx=user->mx,my=user->my;
511: PetscScalar *x;
513: /* Get pointers to vector data */
514: VecGetArray(X,&x);
516: /* Perform local computations */
517: for (j=0; j<my; j++) {
518: for (i=0; i< mx; i++) {
519: row=(j)*mx + (i);
520: 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;
521: }
522: }
524: /* Restore vectors */
525: VecRestoreArray(X,&x);
526: }
527: return 0;
528: }
530: /*TEST
532: build:
533: requires: !complex
535: test:
536: args: -tao_monitor -tao_view -tao_type ssils -tao_gttol 1.e-5
537: requires: !single
539: test:
540: suffix: 2
541: args: -tao_monitor -tao_view -tao_type ssfls -tao_gttol 1.e-5
543: TEST*/