Actual source code: minsurf1.c
petsc-3.8.4 2018-03-24
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*/
28: /*
29: User-defined application context - contains data needed by the
30: application-provided call-back routines, FormFunctionGradient(),
31: FormHessian().
32: */
33: typedef struct {
34: PetscInt mx, my;
35: PetscReal *bottom, *top, *left, *right;
36: } AppCtx;
39: /* -------- User-defined Routines --------- */
41: static PetscErrorCode MSA_BoundaryConditions(AppCtx *);
42: static PetscErrorCode MSA_InitialPoint(AppCtx *, Vec);
43: PetscErrorCode FormConstraints(Tao, Vec, Vec, void *);
44: PetscErrorCode FormJacobian(Tao, Vec, Mat, Mat, void *);
46: int main(int argc, char **argv)
47: {
49: Vec x; /* solution vector */
50: Vec c; /* Constraints function vector */
51: Vec xl,xu; /* Bounds on the variables */
52: PetscBool flg; /* A return variable when checking for user options */
53: Tao tao; /* TAO solver context */
54: Mat J; /* Jacobian matrix */
55: PetscInt N; /* Number of elements in vector */
56: PetscScalar lb = PETSC_NINFINITY; /* lower bound constant */
57: PetscScalar ub = PETSC_INFINITY; /* upper bound constant */
58: AppCtx user; /* user-defined work context */
60: /* Initialize PETSc, TAO */
61: PetscInitialize(&argc, &argv, (char *)0, help );
63: /* Specify default dimension of the problem */
64: user.mx = 4; user.my = 4;
66: /* Check for any command line arguments that override defaults */
67: PetscOptionsGetInt(NULL,NULL, "-mx", &user.mx, &flg);
68: PetscOptionsGetInt(NULL,NULL, "-my", &user.my, &flg);
70: /* Calculate any derived values from parameters */
71: N = user.mx*user.my;
73: PetscPrintf(PETSC_COMM_SELF,"\n---- Minimum Surface Area Problem -----\n");
74: PetscPrintf(PETSC_COMM_SELF,"mx:%D, my:%D\n", user.mx,user.my);
76: /* Create appropriate vectors and matrices */
77: VecCreateSeq(MPI_COMM_SELF, N, &x);
78: VecDuplicate(x, &c);
79: MatCreateSeqAIJ(MPI_COMM_SELF, N, N, 7, NULL, &J);
81: /* The TAO code begins here */
83: /* Create TAO solver and set desired solution method */
84: TaoCreate(PETSC_COMM_SELF,&tao);
85: TaoSetType(tao,TAOSSILS);
87: /* Set data structure */
88: TaoSetInitialVector(tao, x);
90: /* Set routines for constraints function and Jacobian evaluation */
91: TaoSetConstraintsRoutine(tao, c, FormConstraints, (void *)&user);
92: TaoSetJacobianRoutine(tao, J, J, FormJacobian, (void *)&user);
94: /* Set the variable bounds */
95: MSA_BoundaryConditions(&user);
97: /* Set initial solution guess */
98: MSA_InitialPoint(&user, x);
100: /* Set Bounds on variables */
101: VecDuplicate(x, &xl);
102: VecDuplicate(x, &xu);
103: VecSet(xl, lb);
104: VecSet(xu, ub);
105: TaoSetVariableBounds(tao,xl,xu);
107: /* Check for any tao command line options */
108: TaoSetFromOptions(tao);
110: /* Solve the application */
111: TaoSolve(tao);
113: /* Free Tao data structures */
114: TaoDestroy(&tao);
116: /* Free PETSc data structures */
117: VecDestroy(&x);
118: VecDestroy(&xl);
119: VecDestroy(&xu);
120: VecDestroy(&c);
121: MatDestroy(&J);
123: /* Free user-created data structures */
124: PetscFree(user.bottom);
125: PetscFree(user.top);
126: PetscFree(user.left);
127: PetscFree(user.right);
129: PetscFinalize();
130: return ierr;
131: }
133: /* -------------------------------------------------------------------- */
135: /* FormConstraints - Evaluates gradient of f.
137: Input Parameters:
138: . tao - the TAO_APPLICATION context
139: . X - input vector
140: . ptr - optional user-defined context, as set by TaoSetConstraintsRoutine()
142: Output Parameters:
143: . G - vector containing the newly evaluated gradient
144: */
145: PetscErrorCode FormConstraints(Tao tao, Vec X, Vec G, void *ptr)
146: {
147: AppCtx *user = (AppCtx *) ptr;
149: PetscInt i,j,row;
150: PetscInt mx=user->mx, my=user->my;
151: PetscReal hx=1.0/(mx+1),hy=1.0/(my+1), hydhx=hy/hx, hxdhy=hx/hy;
152: PetscReal f1,f2,f3,f4,f5,f6,d1,d2,d3,d4,d5,d6,d7,d8,xc,xl,xr,xt,xb,xlt,xrb;
153: PetscReal df1dxc,df2dxc,df3dxc,df4dxc,df5dxc,df6dxc;
154: PetscScalar zero=0.0;
155: PetscScalar *g, *x;
158: /* Initialize vector to zero */
159: VecSet(G, zero);
161: /* Get pointers to vector data */
162: VecGetArray(X, &x);
163: VecGetArray(G, &g);
165: /* Compute function over the locally owned part of the mesh */
166: for (j=0; j<my; j++){
167: for (i=0; i< mx; i++){
168: row= j*mx + i;
170: xc = x[row];
171: xlt=xrb=xl=xr=xb=xt=xc;
173: if (i==0){ /* left side */
174: xl= user->left[j+1];
175: xlt = user->left[j+2];
176: } else {
177: xl = x[row-1];
178: }
180: if (j==0){ /* bottom side */
181: xb=user->bottom[i+1];
182: xrb = user->bottom[i+2];
183: } else {
184: xb = x[row-mx];
185: }
187: if (i+1 == mx){ /* right side */
188: xr=user->right[j+1];
189: xrb = user->right[j];
190: } else {
191: xr = x[row+1];
192: }
194: if (j+1==0+my){ /* top side */
195: xt=user->top[i+1];
196: xlt = user->top[i];
197: }else {
198: xt = x[row+mx];
199: }
201: if (i>0 && j+1<my){
202: xlt = x[row-1+mx];
203: }
204: if (j>0 && i+1<mx){
205: xrb = x[row+1-mx];
206: }
208: d1 = (xc-xl);
209: d2 = (xc-xr);
210: d3 = (xc-xt);
211: d4 = (xc-xb);
212: d5 = (xr-xrb);
213: d6 = (xrb-xb);
214: d7 = (xlt-xl);
215: d8 = (xt-xlt);
217: df1dxc = d1*hydhx;
218: df2dxc = ( d1*hydhx + d4*hxdhy );
219: df3dxc = d3*hxdhy;
220: df4dxc = ( d2*hydhx + d3*hxdhy );
221: df5dxc = d2*hydhx;
222: df6dxc = d4*hxdhy;
224: d1 /= hx;
225: d2 /= hx;
226: d3 /= hy;
227: d4 /= hy;
228: d5 /= hy;
229: d6 /= hx;
230: d7 /= hy;
231: d8 /= hx;
233: f1 = PetscSqrtScalar( 1.0 + d1*d1 + d7*d7);
234: f2 = PetscSqrtScalar( 1.0 + d1*d1 + d4*d4);
235: f3 = PetscSqrtScalar( 1.0 + d3*d3 + d8*d8);
236: f4 = PetscSqrtScalar( 1.0 + d3*d3 + d2*d2);
237: f5 = PetscSqrtScalar( 1.0 + d2*d2 + d5*d5);
238: f6 = PetscSqrtScalar( 1.0 + d4*d4 + d6*d6);
240: df1dxc /= f1;
241: df2dxc /= f2;
242: df3dxc /= f3;
243: df4dxc /= f4;
244: df5dxc /= f5;
245: df6dxc /= f6;
247: g[row] = (df1dxc+df2dxc+df3dxc+df4dxc+df5dxc+df6dxc )/2.0;
248: }
249: }
251: /* Restore vectors */
252: VecRestoreArray(X, &x);
253: VecRestoreArray(G, &g);
254: PetscLogFlops(67*mx*my);
255: return(0);
256: }
258: /* ------------------------------------------------------------------- */
259: /*
260: FormJacobian - Evaluates Jacobian matrix.
262: Input Parameters:
263: . tao - the TAO_APPLICATION context
264: . X - input vector
265: . ptr - optional user-defined context, as set by TaoSetJacobian()
267: Output Parameters:
268: . tH - Jacobian matrix
270: */
271: PetscErrorCode FormJacobian(Tao tao, Vec X, Mat H, Mat tHPre, void *ptr)
272: {
273: AppCtx *user = (AppCtx *) ptr;
275: PetscInt i,j,k,row;
276: PetscInt mx=user->mx, my=user->my;
277: PetscInt col[7];
278: PetscReal hx=1.0/(mx+1), hy=1.0/(my+1), hydhx=hy/hx, hxdhy=hx/hy;
279: PetscReal f1,f2,f3,f4,f5,f6,d1,d2,d3,d4,d5,d6,d7,d8,xc,xl,xr,xt,xb,xlt,xrb;
280: PetscReal hl,hr,ht,hb,hc,htl,hbr;
281: PetscScalar *x, v[7];
282: PetscBool assembled;
284: /* Set various matrix options */
285: MatSetOption(H,MAT_IGNORE_OFF_PROC_ENTRIES,PETSC_TRUE);
286: MatAssembled(H,&assembled);
287: if (assembled){MatZeroEntries(H); }
289: /* Get pointers to vector data */
290: VecGetArray(X, &x);
292: /* Compute Jacobian over the locally owned part of the mesh */
293: for (i=0; i< mx; i++){
294: for (j=0; j<my; j++){
295: row= j*mx + i;
297: xc = x[row];
298: xlt=xrb=xl=xr=xb=xt=xc;
300: /* Left side */
301: if (i==0){
302: xl= user->left[j+1];
303: xlt = user->left[j+2];
304: } else {
305: xl = x[row-1];
306: }
308: if (j==0){
309: xb=user->bottom[i+1];
310: xrb = user->bottom[i+2];
311: } else {
312: xb = x[row-mx];
313: }
315: if (i+1 == mx){
316: xr=user->right[j+1];
317: xrb = user->right[j];
318: } else {
319: xr = x[row+1];
320: }
322: if (j+1==my){
323: xt=user->top[i+1];
324: xlt = user->top[i];
325: }else {
326: xt = x[row+mx];
327: }
329: if (i>0 && j+1<my){
330: xlt = x[row-1+mx];
331: }
332: if (j>0 && i+1<mx){
333: xrb = x[row+1-mx];
334: }
337: d1 = (xc-xl)/hx;
338: d2 = (xc-xr)/hx;
339: d3 = (xc-xt)/hy;
340: d4 = (xc-xb)/hy;
341: d5 = (xrb-xr)/hy;
342: d6 = (xrb-xb)/hx;
343: d7 = (xlt-xl)/hy;
344: d8 = (xlt-xt)/hx;
346: f1 = PetscSqrtScalar( 1.0 + d1*d1 + d7*d7);
347: f2 = PetscSqrtScalar( 1.0 + d1*d1 + d4*d4);
348: f3 = PetscSqrtScalar( 1.0 + d3*d3 + d8*d8);
349: f4 = PetscSqrtScalar( 1.0 + d3*d3 + d2*d2);
350: f5 = PetscSqrtScalar( 1.0 + d2*d2 + d5*d5);
351: f6 = PetscSqrtScalar( 1.0 + d4*d4 + d6*d6);
354: hl = (-hydhx*(1.0+d7*d7)+d1*d7)/(f1*f1*f1)+(-hydhx*(1.0+d4*d4)+d1*d4)/(f2*f2*f2);
355: hr = (-hydhx*(1.0+d5*d5)+d2*d5)/(f5*f5*f5)+(-hydhx*(1.0+d3*d3)+d2*d3)/(f4*f4*f4);
356: ht = (-hxdhy*(1.0+d8*d8)+d3*d8)/(f3*f3*f3)+(-hxdhy*(1.0+d2*d2)+d2*d3)/(f4*f4*f4);
357: hb = (-hxdhy*(1.0+d6*d6)+d4*d6)/(f6*f6*f6)+(-hxdhy*(1.0+d1*d1)+d1*d4)/(f2*f2*f2);
359: hbr = -d2*d5/(f5*f5*f5) - d4*d6/(f6*f6*f6);
360: htl = -d1*d7/(f1*f1*f1) - d3*d8/(f3*f3*f3);
362: 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) +
363: (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);
365: hl/=2.0; hr/=2.0; ht/=2.0; hb/=2.0; hbr/=2.0; htl/=2.0; hc/=2.0;
367: k=0;
368: if (j>0){
369: v[k]=hb; col[k]=row - mx; k++;
370: }
372: if (j>0 && i < mx -1){
373: v[k]=hbr; col[k]=row - mx+1; k++;
374: }
376: if (i>0){
377: v[k]= hl; col[k]=row - 1; k++;
378: }
380: v[k]= hc; col[k]=row; k++;
382: if (i < mx-1 ){
383: v[k]= hr; col[k]=row+1; k++;
384: }
386: if (i>0 && j < my-1 ){
387: v[k]= htl; col[k] = row+mx-1; k++;
388: }
390: if (j < my-1 ){
391: v[k]= ht; col[k] = row+mx; k++;
392: }
394: /*
395: Set matrix values using local numbering, which was defined
396: earlier, in the main routine.
397: */
398: MatSetValues(H,1,&row,k,col,v,INSERT_VALUES);
399: }
400: }
402: /* Restore vectors */
403: VecRestoreArray(X,&x);
405: /* Assemble the matrix */
406: MatAssemblyBegin(H,MAT_FINAL_ASSEMBLY);
407: MatAssemblyEnd(H,MAT_FINAL_ASSEMBLY);
408: PetscLogFlops(199*mx*my);
409: return(0);
410: }
412: /* ------------------------------------------------------------------- */
413: /*
414: MSA_BoundaryConditions - Calculates the boundary conditions for
415: the region.
417: Input Parameter:
418: . user - user-defined application context
420: Output Parameter:
421: . user - user-defined application context
422: */
423: static PetscErrorCode MSA_BoundaryConditions(AppCtx * user)
424: {
425: PetscErrorCode ierr;
426: PetscInt i,j,k,limit=0,maxits=5;
427: PetscInt mx=user->mx,my=user->my;
428: PetscInt bsize=0, lsize=0, tsize=0, rsize=0;
429: PetscReal one=1.0, two=2.0, three=3.0, tol=1e-10;
430: PetscReal fnorm,det,hx,hy,xt=0,yt=0;
431: PetscReal u1,u2,nf1,nf2,njac11,njac12,njac21,njac22;
432: PetscReal b=-0.5, t=0.5, l=-0.5, r=0.5;
433: PetscReal *boundary;
436: bsize=mx+2; lsize=my+2; rsize=my+2; tsize=mx+2;
438: PetscMalloc1(bsize, &user->bottom);
439: PetscMalloc1(tsize, &user->top);
440: PetscMalloc1(lsize, &user->left);
441: PetscMalloc1(rsize, &user->right);
443: hx= (r-l)/(mx+1); hy=(t-b)/(my+1);
445: for (j=0; j<4; j++){
446: if (j==0){
447: yt=b;
448: xt=l;
449: limit=bsize;
450: boundary=user->bottom;
451: } else if (j==1){
452: yt=t;
453: xt=l;
454: limit=tsize;
455: boundary=user->top;
456: } else if (j==2){
457: yt=b;
458: xt=l;
459: limit=lsize;
460: boundary=user->left;
461: } else { /* if (j==3) */
462: yt=b;
463: xt=r;
464: limit=rsize;
465: boundary=user->right;
466: }
468: for (i=0; i<limit; i++){
469: u1=xt;
470: u2=-yt;
471: for (k=0; k<maxits; k++){
472: nf1=u1 + u1*u2*u2 - u1*u1*u1/three-xt;
473: nf2=-u2 - u1*u1*u2 + u2*u2*u2/three-yt;
474: fnorm=PetscSqrtScalar(nf1*nf1+nf2*nf2);
475: if (fnorm <= tol) break;
476: njac11=one+u2*u2-u1*u1;
477: njac12=two*u1*u2;
478: njac21=-two*u1*u2;
479: njac22=-one - u1*u1 + u2*u2;
480: det = njac11*njac22-njac21*njac12;
481: u1 = u1-(njac22*nf1-njac12*nf2)/det;
482: u2 = u2-(njac11*nf2-njac21*nf1)/det;
483: }
485: boundary[i]=u1*u1-u2*u2;
486: if (j==0 || j==1) {
487: xt=xt+hx;
488: } else { /* if (j==2 || j==3) */
489: yt=yt+hy;
490: }
491: }
492: }
493: return(0);
494: }
496: /* ------------------------------------------------------------------- */
497: /*
498: MSA_InitialPoint - Calculates the initial guess in one of three ways.
500: Input Parameters:
501: . user - user-defined application context
502: . X - vector for initial guess
504: Output Parameters:
505: . X - newly computed initial guess
506: */
507: static PetscErrorCode MSA_InitialPoint(AppCtx * user, Vec X)
508: {
510: PetscInt start=-1,i,j;
511: PetscScalar zero=0.0;
512: PetscBool flg;
515: PetscOptionsGetInt(NULL,NULL,"-start",&start,&flg);
517: if (flg && start==0){ /* The zero vector is reasonable */
518: VecSet(X, zero);
519: } else { /* Take an average of the boundary conditions */
520: PetscInt row;
521: PetscInt mx=user->mx,my=user->my;
522: PetscScalar *x;
524: /* Get pointers to vector data */
525: VecGetArray(X,&x);
527: /* Perform local computations */
528: for (j=0; j<my; j++){
529: for (i=0; i< mx; i++){
530: row=(j)*mx + (i);
531: 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;
532: }
533: }
535: /* Restore vectors */
536: VecRestoreArray(X,&x);
537: }
538: return(0);
539: }