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: /*
21: User-defined application context - contains data needed by the
22: application-provided call-back routines, FormFunctionGradient(),
23: FormHessian().
24: */
25: typedef struct {
26: PetscInt mx, my;
27: PetscReal *bottom, *top, *left, *right;
28: } AppCtx;
30: /* -------- User-defined Routines --------- */
32: static PetscErrorCode MSA_BoundaryConditions(AppCtx *);
33: static PetscErrorCode MSA_InitialPoint(AppCtx *, Vec);
34: PetscErrorCode FormConstraints(Tao, Vec, Vec, void *);
35: PetscErrorCode FormJacobian(Tao, Vec, Mat, Mat, void *);
37: int main(int argc, char **argv)
38: {
39: Vec x; /* solution vector */
40: Vec c; /* Constraints function vector */
41: Vec xl,xu; /* Bounds on the variables */
42: PetscBool flg; /* A return variable when checking for user options */
43: Tao tao; /* TAO solver context */
44: Mat J; /* Jacobian matrix */
45: PetscInt N; /* Number of elements in vector */
46: PetscScalar lb = PETSC_NINFINITY; /* lower bound constant */
47: PetscScalar ub = PETSC_INFINITY; /* upper bound constant */
48: AppCtx user; /* user-defined work context */
50: /* Initialize PETSc, TAO */
51: PetscInitialize(&argc, &argv, (char *)0, help);
53: /* Specify default dimension of the problem */
54: user.mx = 4; user.my = 4;
56: /* Check for any command line arguments that override defaults */
57: PetscOptionsGetInt(NULL,NULL, "-mx", &user.mx, &flg);
58: PetscOptionsGetInt(NULL,NULL, "-my", &user.my, &flg);
60: /* Calculate any derived values from parameters */
61: N = user.mx*user.my;
63: PetscPrintf(PETSC_COMM_SELF,"\n---- Minimum Surface Area Problem -----\n");
64: PetscPrintf(PETSC_COMM_SELF,"mx:%D, my:%D\n", user.mx,user.my);
66: /* Create appropriate vectors and matrices */
67: VecCreateSeq(MPI_COMM_SELF, N, &x);
68: VecDuplicate(x, &c);
69: MatCreateSeqAIJ(MPI_COMM_SELF, N, N, 7, NULL, &J);
71: /* The TAO code begins here */
73: /* Create TAO solver and set desired solution method */
74: TaoCreate(PETSC_COMM_SELF,&tao);
75: TaoSetType(tao,TAOSSILS);
77: /* Set data structure */
78: TaoSetSolution(tao, x);
80: /* Set routines for constraints function and Jacobian evaluation */
81: TaoSetConstraintsRoutine(tao, c, FormConstraints, (void *)&user);
82: TaoSetJacobianRoutine(tao, J, J, FormJacobian, (void *)&user);
84: /* Set the variable bounds */
85: MSA_BoundaryConditions(&user);
87: /* Set initial solution guess */
88: MSA_InitialPoint(&user, x);
90: /* Set Bounds on variables */
91: VecDuplicate(x, &xl);
92: VecDuplicate(x, &xu);
93: VecSet(xl, lb);
94: VecSet(xu, ub);
95: TaoSetVariableBounds(tao,xl,xu);
97: /* Check for any tao command line options */
98: TaoSetFromOptions(tao);
100: /* Solve the application */
101: TaoSolve(tao);
103: /* Free Tao data structures */
104: TaoDestroy(&tao);
106: /* Free PETSc data structures */
107: VecDestroy(&x);
108: VecDestroy(&xl);
109: VecDestroy(&xu);
110: VecDestroy(&c);
111: MatDestroy(&J);
113: /* Free user-created data structures */
114: PetscFree(user.bottom);
115: PetscFree(user.top);
116: PetscFree(user.left);
117: PetscFree(user.right);
119: PetscFinalize();
120: return 0;
121: }
123: /* -------------------------------------------------------------------- */
125: /* FormConstraints - Evaluates gradient of f.
127: Input Parameters:
128: . tao - the TAO_APPLICATION context
129: . X - input vector
130: . ptr - optional user-defined context, as set by TaoSetConstraintsRoutine()
132: Output Parameters:
133: . G - vector containing the newly evaluated gradient
134: */
135: PetscErrorCode FormConstraints(Tao tao, Vec X, Vec G, void *ptr)
136: {
137: AppCtx *user = (AppCtx *) ptr;
138: PetscInt i,j,row;
139: PetscInt mx=user->mx, my=user->my;
140: PetscReal hx=1.0/(mx+1),hy=1.0/(my+1), hydhx=hy/hx, hxdhy=hx/hy;
141: PetscReal f1,f2,f3,f4,f5,f6,d1,d2,d3,d4,d5,d6,d7,d8,xc,xl,xr,xt,xb,xlt,xrb;
142: PetscReal df1dxc,df2dxc,df3dxc,df4dxc,df5dxc,df6dxc;
143: PetscScalar zero=0.0;
144: PetscScalar *g, *x;
146: /* Initialize vector to zero */
147: VecSet(G, zero);
149: /* Get pointers to vector data */
150: VecGetArray(X, &x);
151: VecGetArray(G, &g);
153: /* Compute function over the locally owned part of the mesh */
154: for (j=0; j<my; j++) {
155: for (i=0; i< mx; i++) {
156: row= j*mx + i;
158: xc = x[row];
159: xlt=xrb=xl=xr=xb=xt=xc;
161: if (i==0) { /* left side */
162: xl= user->left[j+1];
163: xlt = user->left[j+2];
164: } else {
165: xl = x[row-1];
166: }
168: if (j==0) { /* bottom side */
169: xb=user->bottom[i+1];
170: xrb = user->bottom[i+2];
171: } else {
172: xb = x[row-mx];
173: }
175: if (i+1 == mx) { /* right side */
176: xr=user->right[j+1];
177: xrb = user->right[j];
178: } else {
179: xr = x[row+1];
180: }
182: if (j+1==0+my) { /* top side */
183: xt=user->top[i+1];
184: xlt = user->top[i];
185: }else {
186: xt = x[row+mx];
187: }
189: if (i>0 && j+1<my) {
190: xlt = x[row-1+mx];
191: }
192: if (j>0 && i+1<mx) {
193: xrb = x[row+1-mx];
194: }
196: d1 = (xc-xl);
197: d2 = (xc-xr);
198: d3 = (xc-xt);
199: d4 = (xc-xb);
200: d5 = (xr-xrb);
201: d6 = (xrb-xb);
202: d7 = (xlt-xl);
203: d8 = (xt-xlt);
205: df1dxc = d1*hydhx;
206: df2dxc = (d1*hydhx + d4*hxdhy);
207: df3dxc = d3*hxdhy;
208: df4dxc = (d2*hydhx + d3*hxdhy);
209: df5dxc = d2*hydhx;
210: df6dxc = d4*hxdhy;
212: d1 /= hx;
213: d2 /= hx;
214: d3 /= hy;
215: d4 /= hy;
216: d5 /= hy;
217: d6 /= hx;
218: d7 /= hy;
219: d8 /= hx;
221: f1 = PetscSqrtScalar(1.0 + d1*d1 + d7*d7);
222: f2 = PetscSqrtScalar(1.0 + d1*d1 + d4*d4);
223: f3 = PetscSqrtScalar(1.0 + d3*d3 + d8*d8);
224: f4 = PetscSqrtScalar(1.0 + d3*d3 + d2*d2);
225: f5 = PetscSqrtScalar(1.0 + d2*d2 + d5*d5);
226: f6 = PetscSqrtScalar(1.0 + d4*d4 + d6*d6);
228: df1dxc /= f1;
229: df2dxc /= f2;
230: df3dxc /= f3;
231: df4dxc /= f4;
232: df5dxc /= f5;
233: df6dxc /= f6;
235: g[row] = (df1dxc+df2dxc+df3dxc+df4dxc+df5dxc+df6dxc)/2.0;
236: }
237: }
239: /* Restore vectors */
240: VecRestoreArray(X, &x);
241: VecRestoreArray(G, &g);
242: PetscLogFlops(67*mx*my);
243: return 0;
244: }
246: /* ------------------------------------------------------------------- */
247: /*
248: FormJacobian - Evaluates Jacobian matrix.
250: Input Parameters:
251: . tao - the TAO_APPLICATION context
252: . X - input vector
253: . ptr - optional user-defined context, as set by TaoSetJacobian()
255: Output Parameters:
256: . tH - Jacobian matrix
258: */
259: PetscErrorCode FormJacobian(Tao tao, Vec X, Mat H, Mat tHPre, void *ptr)
260: {
261: AppCtx *user = (AppCtx *) ptr;
262: PetscInt i,j,k,row;
263: PetscInt mx=user->mx, my=user->my;
264: PetscInt col[7];
265: PetscReal hx=1.0/(mx+1), hy=1.0/(my+1), hydhx=hy/hx, hxdhy=hx/hy;
266: PetscReal f1,f2,f3,f4,f5,f6,d1,d2,d3,d4,d5,d6,d7,d8,xc,xl,xr,xt,xb,xlt,xrb;
267: PetscReal hl,hr,ht,hb,hc,htl,hbr;
268: const PetscScalar *x;
269: PetscScalar v[7];
270: PetscBool assembled;
272: /* Set various matrix options */
273: MatSetOption(H,MAT_IGNORE_OFF_PROC_ENTRIES,PETSC_TRUE);
274: MatAssembled(H,&assembled);
275: if (assembled) MatZeroEntries(H);
277: /* Get pointers to vector data */
278: VecGetArrayRead(X, &x);
280: /* Compute Jacobian over the locally owned part of the mesh */
281: for (i=0; i< mx; i++) {
282: for (j=0; j<my; j++) {
283: row= j*mx + i;
285: xc = x[row];
286: xlt=xrb=xl=xr=xb=xt=xc;
288: /* Left side */
289: if (i==0) {
290: xl = user->left[j+1];
291: xlt = user->left[j+2];
292: } else {
293: xl = x[row-1];
294: }
296: if (j==0) {
297: xb = user->bottom[i+1];
298: xrb = user->bottom[i+2];
299: } else {
300: xb = x[row-mx];
301: }
303: if (i+1 == mx) {
304: xr = user->right[j+1];
305: xrb = user->right[j];
306: } else {
307: xr = x[row+1];
308: }
310: if (j+1==my) {
311: xt = user->top[i+1];
312: xlt = user->top[i];
313: }else {
314: xt = x[row+mx];
315: }
317: if (i>0 && j+1<my) {
318: xlt = x[row-1+mx];
319: }
320: if (j>0 && i+1<mx) {
321: xrb = x[row+1-mx];
322: }
324: d1 = (xc-xl)/hx;
325: d2 = (xc-xr)/hx;
326: d3 = (xc-xt)/hy;
327: d4 = (xc-xb)/hy;
328: d5 = (xrb-xr)/hy;
329: d6 = (xrb-xb)/hx;
330: d7 = (xlt-xl)/hy;
331: d8 = (xlt-xt)/hx;
333: f1 = PetscSqrtScalar(1.0 + d1*d1 + d7*d7);
334: f2 = PetscSqrtScalar(1.0 + d1*d1 + d4*d4);
335: f3 = PetscSqrtScalar(1.0 + d3*d3 + d8*d8);
336: f4 = PetscSqrtScalar(1.0 + d3*d3 + d2*d2);
337: f5 = PetscSqrtScalar(1.0 + d2*d2 + d5*d5);
338: f6 = PetscSqrtScalar(1.0 + d4*d4 + d6*d6);
340: hl = (-hydhx*(1.0+d7*d7)+d1*d7)/(f1*f1*f1)+(-hydhx*(1.0+d4*d4)+d1*d4)/(f2*f2*f2);
341: hr = (-hydhx*(1.0+d5*d5)+d2*d5)/(f5*f5*f5)+(-hydhx*(1.0+d3*d3)+d2*d3)/(f4*f4*f4);
342: ht = (-hxdhy*(1.0+d8*d8)+d3*d8)/(f3*f3*f3)+(-hxdhy*(1.0+d2*d2)+d2*d3)/(f4*f4*f4);
343: hb = (-hxdhy*(1.0+d6*d6)+d4*d6)/(f6*f6*f6)+(-hxdhy*(1.0+d1*d1)+d1*d4)/(f2*f2*f2);
345: hbr = -d2*d5/(f5*f5*f5) - d4*d6/(f6*f6*f6);
346: htl = -d1*d7/(f1*f1*f1) - d3*d8/(f3*f3*f3);
348: 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) +
349: (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);
351: hl/=2.0; hr/=2.0; ht/=2.0; hb/=2.0; hbr/=2.0; htl/=2.0; hc/=2.0;
353: k=0;
354: if (j>0) {
355: v[k]=hb; col[k]=row - mx; k++;
356: }
358: if (j>0 && i < mx -1) {
359: v[k]=hbr; col[k]=row - mx+1; k++;
360: }
362: if (i>0) {
363: v[k]= hl; col[k]=row - 1; k++;
364: }
366: v[k]= hc; col[k]=row; k++;
368: if (i < mx-1) {
369: v[k]= hr; col[k]=row+1; k++;
370: }
372: if (i>0 && j < my-1) {
373: v[k]= htl; col[k] = row+mx-1; k++;
374: }
376: if (j < my-1) {
377: v[k]= ht; col[k] = row+mx; k++;
378: }
380: /*
381: Set matrix values using local numbering, which was defined
382: earlier, in the main routine.
383: */
384: MatSetValues(H,1,&row,k,col,v,INSERT_VALUES);
385: }
386: }
388: /* Restore vectors */
389: VecRestoreArrayRead(X,&x);
391: /* Assemble the matrix */
392: MatAssemblyBegin(H,MAT_FINAL_ASSEMBLY);
393: MatAssemblyEnd(H,MAT_FINAL_ASSEMBLY);
394: PetscLogFlops(199*mx*my);
395: return 0;
396: }
398: /* ------------------------------------------------------------------- */
399: /*
400: MSA_BoundaryConditions - Calculates the boundary conditions for
401: the region.
403: Input Parameter:
404: . user - user-defined application context
406: Output Parameter:
407: . user - user-defined application context
408: */
409: static PetscErrorCode MSA_BoundaryConditions(AppCtx * user)
410: {
411: PetscInt i,j,k,limit=0,maxits=5;
412: PetscInt mx=user->mx,my=user->my;
413: PetscInt bsize=0, lsize=0, tsize=0, rsize=0;
414: PetscReal one=1.0, two=2.0, three=3.0, tol=1e-10;
415: PetscReal fnorm,det,hx,hy,xt=0,yt=0;
416: PetscReal u1,u2,nf1,nf2,njac11,njac12,njac21,njac22;
417: PetscReal b=-0.5, t=0.5, l=-0.5, r=0.5;
418: PetscReal *boundary;
420: bsize=mx+2; lsize=my+2; rsize=my+2; tsize=mx+2;
422: PetscMalloc1(bsize, &user->bottom);
423: PetscMalloc1(tsize, &user->top);
424: PetscMalloc1(lsize, &user->left);
425: PetscMalloc1(rsize, &user->right);
427: hx= (r-l)/(mx+1); hy=(t-b)/(my+1);
429: for (j=0; j<4; j++) {
430: if (j==0) {
431: yt=b;
432: xt=l;
433: limit=bsize;
434: boundary=user->bottom;
435: } else if (j==1) {
436: yt=t;
437: xt=l;
438: limit=tsize;
439: boundary=user->top;
440: } else if (j==2) {
441: yt=b;
442: xt=l;
443: limit=lsize;
444: boundary=user->left;
445: } else { /* if (j==3) */
446: yt=b;
447: xt=r;
448: limit=rsize;
449: boundary=user->right;
450: }
452: for (i=0; i<limit; i++) {
453: u1=xt;
454: u2=-yt;
455: for (k=0; k<maxits; k++) {
456: nf1=u1 + u1*u2*u2 - u1*u1*u1/three-xt;
457: nf2=-u2 - u1*u1*u2 + u2*u2*u2/three-yt;
458: fnorm=PetscSqrtScalar(nf1*nf1+nf2*nf2);
459: if (fnorm <= tol) break;
460: njac11=one+u2*u2-u1*u1;
461: njac12=two*u1*u2;
462: njac21=-two*u1*u2;
463: njac22=-one - u1*u1 + u2*u2;
464: det = njac11*njac22-njac21*njac12;
465: u1 = u1-(njac22*nf1-njac12*nf2)/det;
466: u2 = u2-(njac11*nf2-njac21*nf1)/det;
467: }
469: boundary[i]=u1*u1-u2*u2;
470: if (j==0 || j==1) {
471: xt=xt+hx;
472: } else { /* if (j==2 || j==3) */
473: yt=yt+hy;
474: }
475: }
476: }
477: return 0;
478: }
480: /* ------------------------------------------------------------------- */
481: /*
482: MSA_InitialPoint - Calculates the initial guess in one of three ways.
484: Input Parameters:
485: . user - user-defined application context
486: . X - vector for initial guess
488: Output Parameters:
489: . X - newly computed initial guess
490: */
491: static PetscErrorCode MSA_InitialPoint(AppCtx * user, Vec X)
492: {
493: PetscInt start=-1,i,j;
494: PetscScalar zero=0.0;
495: PetscBool flg;
497: PetscOptionsGetInt(NULL,NULL,"-start",&start,&flg);
499: if (flg && start==0) { /* The zero vector is reasonable */
500: VecSet(X, zero);
501: } else { /* Take an average of the boundary conditions */
502: PetscInt row;
503: PetscInt mx=user->mx,my=user->my;
504: PetscScalar *x;
506: /* Get pointers to vector data */
507: VecGetArray(X,&x);
509: /* Perform local computations */
510: for (j=0; j<my; j++) {
511: for (i=0; i< mx; i++) {
512: row=(j)*mx + (i);
513: 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;
514: }
515: }
517: /* Restore vectors */
518: VecRestoreArray(X,&x);
519: }
520: return 0;
521: }
523: /*TEST
525: build:
526: requires: !complex
528: test:
529: args: -tao_monitor -tao_view -tao_type ssils -tao_gttol 1.e-5
530: requires: !single
532: test:
533: suffix: 2
534: args: -tao_monitor -tao_view -tao_type ssfls -tao_gttol 1.e-5
536: TEST*/