Actual source code: minsurf1.c
petsc-3.7.5 2017-01-01
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 *);
48: int main(int argc, char **argv)
49: {
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 */
62: /* Initialize PETSc, TAO */
63: PetscInitialize(&argc, &argv, (char *)0, help );
65: /* Specify default dimension of the problem */
66: user.mx = 4; user.my = 4;
68: /* Check for any command line arguments that override defaults */
69: PetscOptionsGetInt(NULL,NULL, "-mx", &user.mx, &flg);
70: PetscOptionsGetInt(NULL,NULL, "-my", &user.my, &flg);
72: /* Calculate any derived values from parameters */
73: N = user.mx*user.my;
75: PetscPrintf(PETSC_COMM_SELF,"\n---- Minimum Surface Area Problem -----\n");
76: PetscPrintf(PETSC_COMM_SELF,"mx:%D, my:%D\n", user.mx,user.my);
78: /* Create appropriate vectors and matrices */
79: VecCreateSeq(MPI_COMM_SELF, N, &x);
80: VecDuplicate(x, &c);
81: MatCreateSeqAIJ(MPI_COMM_SELF, N, N, 7, NULL, &J);
83: /* The TAO code begins here */
85: /* Create TAO solver and set desired solution method */
86: TaoCreate(PETSC_COMM_SELF,&tao);
87: TaoSetType(tao,TAOSSILS);
89: /* Set data structure */
90: TaoSetInitialVector(tao, x);
92: /* Set routines for constraints function and Jacobian evaluation */
93: TaoSetConstraintsRoutine(tao, c, FormConstraints, (void *)&user);
94: TaoSetJacobianRoutine(tao, J, J, FormJacobian, (void *)&user);
96: /* Set the variable bounds */
97: MSA_BoundaryConditions(&user);
99: /* Set initial solution guess */
100: MSA_InitialPoint(&user, x);
102: /* Set Bounds on variables */
103: VecDuplicate(x, &xl);
104: VecDuplicate(x, &xu);
105: VecSet(xl, lb);
106: VecSet(xu, ub);
107: TaoSetVariableBounds(tao,xl,xu);
109: /* Check for any tao command line options */
110: TaoSetFromOptions(tao);
112: /* Solve the application */
113: TaoSolve(tao);
115: /* Free Tao data structures */
116: TaoDestroy(&tao);
118: /* Free PETSc data structures */
119: VecDestroy(&x);
120: VecDestroy(&xl);
121: VecDestroy(&xu);
122: VecDestroy(&c);
123: MatDestroy(&J);
125: /* Free user-created data structures */
126: PetscFree(user.bottom);
127: PetscFree(user.top);
128: PetscFree(user.left);
129: PetscFree(user.right);
131: PetscFinalize();
132: return 0;
133: }
135: /* -------------------------------------------------------------------- */
139: /* FormConstraints - Evaluates gradient of f.
141: Input Parameters:
142: . tao - the TAO_APPLICATION context
143: . X - input vector
144: . ptr - optional user-defined context, as set by TaoSetConstraintsRoutine()
146: Output Parameters:
147: . G - vector containing the newly evaluated gradient
148: */
149: PetscErrorCode FormConstraints(Tao tao, Vec X, Vec G, void *ptr)
150: {
151: AppCtx *user = (AppCtx *) ptr;
153: PetscInt i,j,row;
154: PetscInt mx=user->mx, my=user->my;
155: PetscReal hx=1.0/(mx+1),hy=1.0/(my+1), hydhx=hy/hx, hxdhy=hx/hy;
156: PetscReal f1,f2,f3,f4,f5,f6,d1,d2,d3,d4,d5,d6,d7,d8,xc,xl,xr,xt,xb,xlt,xrb;
157: PetscReal df1dxc,df2dxc,df3dxc,df4dxc,df5dxc,df6dxc;
158: PetscScalar zero=0.0;
159: PetscScalar *g, *x;
162: /* Initialize vector to zero */
163: VecSet(G, zero);
165: /* Get pointers to vector data */
166: VecGetArray(X, &x);
167: VecGetArray(G, &g);
169: /* Compute function over the locally owned part of the mesh */
170: for (j=0; j<my; j++){
171: for (i=0; i< mx; i++){
172: row= j*mx + i;
174: xc = x[row];
175: xlt=xrb=xl=xr=xb=xt=xc;
177: if (i==0){ /* left side */
178: xl= user->left[j+1];
179: xlt = user->left[j+2];
180: } else {
181: xl = x[row-1];
182: }
184: if (j==0){ /* bottom side */
185: xb=user->bottom[i+1];
186: xrb = user->bottom[i+2];
187: } else {
188: xb = x[row-mx];
189: }
191: if (i+1 == mx){ /* right side */
192: xr=user->right[j+1];
193: xrb = user->right[j];
194: } else {
195: xr = x[row+1];
196: }
198: if (j+1==0+my){ /* top side */
199: xt=user->top[i+1];
200: xlt = user->top[i];
201: }else {
202: xt = x[row+mx];
203: }
205: if (i>0 && j+1<my){
206: xlt = x[row-1+mx];
207: }
208: if (j>0 && i+1<mx){
209: xrb = x[row+1-mx];
210: }
212: d1 = (xc-xl);
213: d2 = (xc-xr);
214: d3 = (xc-xt);
215: d4 = (xc-xb);
216: d5 = (xr-xrb);
217: d6 = (xrb-xb);
218: d7 = (xlt-xl);
219: d8 = (xt-xlt);
221: df1dxc = d1*hydhx;
222: df2dxc = ( d1*hydhx + d4*hxdhy );
223: df3dxc = d3*hxdhy;
224: df4dxc = ( d2*hydhx + d3*hxdhy );
225: df5dxc = d2*hydhx;
226: df6dxc = d4*hxdhy;
228: d1 /= hx;
229: d2 /= hx;
230: d3 /= hy;
231: d4 /= hy;
232: d5 /= hy;
233: d6 /= hx;
234: d7 /= hy;
235: d8 /= hx;
237: f1 = PetscSqrtScalar( 1.0 + d1*d1 + d7*d7);
238: f2 = PetscSqrtScalar( 1.0 + d1*d1 + d4*d4);
239: f3 = PetscSqrtScalar( 1.0 + d3*d3 + d8*d8);
240: f4 = PetscSqrtScalar( 1.0 + d3*d3 + d2*d2);
241: f5 = PetscSqrtScalar( 1.0 + d2*d2 + d5*d5);
242: f6 = PetscSqrtScalar( 1.0 + d4*d4 + d6*d6);
244: df1dxc /= f1;
245: df2dxc /= f2;
246: df3dxc /= f3;
247: df4dxc /= f4;
248: df5dxc /= f5;
249: df6dxc /= f6;
251: g[row] = (df1dxc+df2dxc+df3dxc+df4dxc+df5dxc+df6dxc )/2.0;
252: }
253: }
255: /* Restore vectors */
256: VecRestoreArray(X, &x);
257: VecRestoreArray(G, &g);
258: PetscLogFlops(67*mx*my);
259: return(0);
260: }
262: /* ------------------------------------------------------------------- */
265: /*
266: FormJacobian - Evaluates Jacobian matrix.
268: Input Parameters:
269: . tao - the TAO_APPLICATION context
270: . X - input vector
271: . ptr - optional user-defined context, as set by TaoSetJacobian()
273: Output Parameters:
274: . tH - Jacobian matrix
276: */
277: PetscErrorCode FormJacobian(Tao tao, Vec X, Mat H, Mat tHPre, void *ptr)
278: {
279: AppCtx *user = (AppCtx *) ptr;
281: PetscInt i,j,k,row;
282: PetscInt mx=user->mx, my=user->my;
283: PetscInt col[7];
284: PetscReal hx=1.0/(mx+1), hy=1.0/(my+1), hydhx=hy/hx, hxdhy=hx/hy;
285: PetscReal f1,f2,f3,f4,f5,f6,d1,d2,d3,d4,d5,d6,d7,d8,xc,xl,xr,xt,xb,xlt,xrb;
286: PetscReal hl,hr,ht,hb,hc,htl,hbr;
287: PetscScalar *x, v[7];
288: PetscBool assembled;
290: /* Set various matrix options */
291: MatSetOption(H,MAT_IGNORE_OFF_PROC_ENTRIES,PETSC_TRUE);
292: MatAssembled(H,&assembled);
293: if (assembled){MatZeroEntries(H); }
295: /* Get pointers to vector data */
296: VecGetArray(X, &x);
298: /* Compute Jacobian over the locally owned part of the mesh */
299: for (i=0; i< mx; i++){
300: for (j=0; j<my; j++){
301: row= j*mx + i;
303: xc = x[row];
304: xlt=xrb=xl=xr=xb=xt=xc;
306: /* Left side */
307: if (i==0){
308: xl= user->left[j+1];
309: xlt = user->left[j+2];
310: } else {
311: xl = x[row-1];
312: }
314: if (j==0){
315: xb=user->bottom[i+1];
316: xrb = user->bottom[i+2];
317: } else {
318: xb = x[row-mx];
319: }
321: if (i+1 == mx){
322: xr=user->right[j+1];
323: xrb = user->right[j];
324: } else {
325: xr = x[row+1];
326: }
328: if (j+1==my){
329: xt=user->top[i+1];
330: xlt = user->top[i];
331: }else {
332: xt = x[row+mx];
333: }
335: if (i>0 && j+1<my){
336: xlt = x[row-1+mx];
337: }
338: if (j>0 && i+1<mx){
339: xrb = x[row+1-mx];
340: }
343: d1 = (xc-xl)/hx;
344: d2 = (xc-xr)/hx;
345: d3 = (xc-xt)/hy;
346: d4 = (xc-xb)/hy;
347: d5 = (xrb-xr)/hy;
348: d6 = (xrb-xb)/hx;
349: d7 = (xlt-xl)/hy;
350: d8 = (xlt-xt)/hx;
352: f1 = PetscSqrtScalar( 1.0 + d1*d1 + d7*d7);
353: f2 = PetscSqrtScalar( 1.0 + d1*d1 + d4*d4);
354: f3 = PetscSqrtScalar( 1.0 + d3*d3 + d8*d8);
355: f4 = PetscSqrtScalar( 1.0 + d3*d3 + d2*d2);
356: f5 = PetscSqrtScalar( 1.0 + d2*d2 + d5*d5);
357: f6 = PetscSqrtScalar( 1.0 + d4*d4 + d6*d6);
360: hl = (-hydhx*(1.0+d7*d7)+d1*d7)/(f1*f1*f1)+(-hydhx*(1.0+d4*d4)+d1*d4)/(f2*f2*f2);
361: hr = (-hydhx*(1.0+d5*d5)+d2*d5)/(f5*f5*f5)+(-hydhx*(1.0+d3*d3)+d2*d3)/(f4*f4*f4);
362: ht = (-hxdhy*(1.0+d8*d8)+d3*d8)/(f3*f3*f3)+(-hxdhy*(1.0+d2*d2)+d2*d3)/(f4*f4*f4);
363: hb = (-hxdhy*(1.0+d6*d6)+d4*d6)/(f6*f6*f6)+(-hxdhy*(1.0+d1*d1)+d1*d4)/(f2*f2*f2);
365: hbr = -d2*d5/(f5*f5*f5) - d4*d6/(f6*f6*f6);
366: htl = -d1*d7/(f1*f1*f1) - d3*d8/(f3*f3*f3);
368: 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) +
369: (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);
371: hl/=2.0; hr/=2.0; ht/=2.0; hb/=2.0; hbr/=2.0; htl/=2.0; hc/=2.0;
373: k=0;
374: if (j>0){
375: v[k]=hb; col[k]=row - mx; k++;
376: }
378: if (j>0 && i < mx -1){
379: v[k]=hbr; col[k]=row - mx+1; k++;
380: }
382: if (i>0){
383: v[k]= hl; col[k]=row - 1; k++;
384: }
386: v[k]= hc; col[k]=row; k++;
388: if (i < mx-1 ){
389: v[k]= hr; col[k]=row+1; k++;
390: }
392: if (i>0 && j < my-1 ){
393: v[k]= htl; col[k] = row+mx-1; k++;
394: }
396: if (j < my-1 ){
397: v[k]= ht; col[k] = row+mx; k++;
398: }
400: /*
401: Set matrix values using local numbering, which was defined
402: earlier, in the main routine.
403: */
404: MatSetValues(H,1,&row,k,col,v,INSERT_VALUES);
405: }
406: }
408: /* Restore vectors */
409: VecRestoreArray(X,&x);
411: /* Assemble the matrix */
412: MatAssemblyBegin(H,MAT_FINAL_ASSEMBLY);
413: MatAssemblyEnd(H,MAT_FINAL_ASSEMBLY);
414: PetscLogFlops(199*mx*my);
415: return(0);
416: }
418: /* ------------------------------------------------------------------- */
421: /*
422: MSA_BoundaryConditions - Calculates the boundary conditions for
423: the region.
425: Input Parameter:
426: . user - user-defined application context
428: Output Parameter:
429: . user - user-defined application context
430: */
431: static PetscErrorCode MSA_BoundaryConditions(AppCtx * user)
432: {
433: PetscErrorCode ierr;
434: PetscInt i,j,k,limit=0,maxits=5;
435: PetscInt mx=user->mx,my=user->my;
436: PetscInt bsize=0, lsize=0, tsize=0, rsize=0;
437: PetscReal one=1.0, two=2.0, three=3.0, tol=1e-10;
438: PetscReal fnorm,det,hx,hy,xt=0,yt=0;
439: PetscReal u1,u2,nf1,nf2,njac11,njac12,njac21,njac22;
440: PetscReal b=-0.5, t=0.5, l=-0.5, r=0.5;
441: PetscReal *boundary;
444: bsize=mx+2; lsize=my+2; rsize=my+2; tsize=mx+2;
446: PetscMalloc1(bsize, &user->bottom);
447: PetscMalloc1(tsize, &user->top);
448: PetscMalloc1(lsize, &user->left);
449: PetscMalloc1(rsize, &user->right);
451: hx= (r-l)/(mx+1); hy=(t-b)/(my+1);
453: for (j=0; j<4; j++){
454: if (j==0){
455: yt=b;
456: xt=l;
457: limit=bsize;
458: boundary=user->bottom;
459: } else if (j==1){
460: yt=t;
461: xt=l;
462: limit=tsize;
463: boundary=user->top;
464: } else if (j==2){
465: yt=b;
466: xt=l;
467: limit=lsize;
468: boundary=user->left;
469: } else { /* if (j==3) */
470: yt=b;
471: xt=r;
472: limit=rsize;
473: boundary=user->right;
474: }
476: for (i=0; i<limit; i++){
477: u1=xt;
478: u2=-yt;
479: for (k=0; k<maxits; k++){
480: nf1=u1 + u1*u2*u2 - u1*u1*u1/three-xt;
481: nf2=-u2 - u1*u1*u2 + u2*u2*u2/three-yt;
482: fnorm=PetscSqrtScalar(nf1*nf1+nf2*nf2);
483: if (fnorm <= tol) break;
484: njac11=one+u2*u2-u1*u1;
485: njac12=two*u1*u2;
486: njac21=-two*u1*u2;
487: njac22=-one - u1*u1 + u2*u2;
488: det = njac11*njac22-njac21*njac12;
489: u1 = u1-(njac22*nf1-njac12*nf2)/det;
490: u2 = u2-(njac11*nf2-njac21*nf1)/det;
491: }
493: boundary[i]=u1*u1-u2*u2;
494: if (j==0 || j==1) {
495: xt=xt+hx;
496: } else { /* if (j==2 || j==3) */
497: yt=yt+hy;
498: }
499: }
500: }
501: return(0);
502: }
504: /* ------------------------------------------------------------------- */
507: /*
508: MSA_InitialPoint - Calculates the initial guess in one of three ways.
510: Input Parameters:
511: . user - user-defined application context
512: . X - vector for initial guess
514: Output Parameters:
515: . X - newly computed initial guess
516: */
517: static PetscErrorCode MSA_InitialPoint(AppCtx * user, Vec X)
518: {
520: PetscInt start=-1,i,j;
521: PetscScalar zero=0.0;
522: PetscBool flg;
525: PetscOptionsGetInt(NULL,NULL,"-start",&start,&flg);
527: if (flg && start==0){ /* The zero vector is reasonable */
528: VecSet(X, zero);
529: } else { /* Take an average of the boundary conditions */
530: PetscInt row;
531: PetscInt mx=user->mx,my=user->my;
532: PetscScalar *x;
534: /* Get pointers to vector data */
535: VecGetArray(X,&x);
537: /* Perform local computations */
538: for (j=0; j<my; j++){
539: for (i=0; i< mx; i++){
540: row=(j)*mx + (i);
541: 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;
542: }
543: }
545: /* Restore vectors */
546: VecRestoreArray(X,&x);
547: }
548: return(0);
549: }