Actual source code: minsurf1.c
1: #include <petsctao.h>
3: static char help[] = "This example demonstrates use of the TAO package to\n\
4: solve an unconstrained system of equations. This example is based on a\n\
5: problem from the MINPACK-2 test suite. Given a rectangular 2-D domain and\n\
6: boundary values along the edges of the domain, the objective is to find the\n\
7: surface with the minimal area that satisfies the boundary conditions.\n\
8: This application solves this problem using complimentarity -- We are actually\n\
9: solving the system (grad f)_i >= 0, if x_i == l_i \n\
10: (grad f)_i = 0, if l_i < x_i < u_i \n\
11: (grad f)_i <= 0, if x_i == u_i \n\
12: where f is the function to be minimized. \n\
13: \n\
14: The command line options are:\n\
15: -mx <xg>, where <xg> = number of grid points in the 1st coordinate direction\n\
16: -my <yg>, where <yg> = number of grid points in the 2nd coordinate direction\n\
17: -start <st>, where <st> =0 for zero vector, and an average of the boundary conditions otherwise \n\n";
19: /*
20: User-defined application context - contains data needed by the
21: application-provided call-back routines, FormFunctionGradient(),
22: FormHessian().
23: */
24: typedef struct {
25: PetscInt mx, my;
26: PetscReal *bottom, *top, *left, *right;
27: } AppCtx;
29: /* -------- User-defined Routines --------- */
31: static PetscErrorCode MSA_BoundaryConditions(AppCtx *);
32: static PetscErrorCode MSA_InitialPoint(AppCtx *, Vec);
33: PetscErrorCode FormConstraints(Tao, Vec, Vec, void *);
34: PetscErrorCode FormJacobian(Tao, Vec, Mat, Mat, void *);
36: int main(int argc, char **argv)
37: {
38: Vec x; /* solution vector */
39: Vec c; /* Constraints function vector */
40: Vec xl, xu; /* Bounds on the variables */
41: PetscBool flg; /* A return variable when checking for user options */
42: Tao tao; /* TAO solver context */
43: Mat J; /* Jacobian matrix */
44: PetscInt N; /* Number of elements in vector */
45: PetscScalar lb = PETSC_NINFINITY; /* lower bound constant */
46: PetscScalar ub = PETSC_INFINITY; /* upper bound constant */
47: AppCtx user; /* user-defined work context */
49: /* Initialize PETSc, TAO */
51: PetscInitialize(&argc, &argv, (char *)0, help);
53: /* Specify default dimension of the problem */
54: user.mx = 4;
55: user.my = 4;
57: /* Check for any command line arguments that override defaults */
58: PetscOptionsGetInt(NULL, NULL, "-mx", &user.mx, &flg);
59: PetscOptionsGetInt(NULL, NULL, "-my", &user.my, &flg);
61: /* Calculate any derived values from parameters */
62: N = user.mx * user.my;
64: PetscPrintf(PETSC_COMM_SELF, "\n---- Minimum Surface Area Problem -----\n");
65: PetscPrintf(PETSC_COMM_SELF, "mx:%" PetscInt_FMT ", my:%" PetscInt_FMT "\n", user.mx, user.my);
67: /* Create appropriate vectors and matrices */
68: VecCreateSeq(MPI_COMM_SELF, N, &x);
69: VecDuplicate(x, &c);
70: MatCreateSeqAIJ(MPI_COMM_SELF, N, N, 7, NULL, &J);
72: /* The TAO code begins here */
74: /* Create TAO solver and set desired solution method */
75: TaoCreate(PETSC_COMM_SELF, &tao);
76: TaoSetType(tao, TAOSSILS);
78: /* Set data structure */
79: TaoSetSolution(tao, x);
81: /* Set routines for constraints function and Jacobian evaluation */
82: TaoSetConstraintsRoutine(tao, c, FormConstraints, (void *)&user);
83: TaoSetJacobianRoutine(tao, J, J, FormJacobian, (void *)&user);
85: /* Set the variable bounds */
86: MSA_BoundaryConditions(&user);
88: /* Set initial solution guess */
89: MSA_InitialPoint(&user, x);
91: /* Set Bounds on variables */
92: VecDuplicate(x, &xl);
93: VecDuplicate(x, &xu);
94: VecSet(xl, lb);
95: VecSet(xu, ub);
96: TaoSetVariableBounds(tao, xl, xu);
98: /* Check for any tao command line options */
99: TaoSetFromOptions(tao);
101: /* Solve the application */
102: TaoSolve(tao);
104: /* Free Tao data structures */
105: TaoDestroy(&tao);
107: /* Free PETSc data structures */
108: VecDestroy(&x);
109: VecDestroy(&xl);
110: VecDestroy(&xu);
111: VecDestroy(&c);
112: MatDestroy(&J);
114: /* Free user-created data structures */
115: PetscFree(user.bottom);
116: PetscFree(user.top);
117: PetscFree(user.left);
118: PetscFree(user.right);
120: PetscFinalize();
121: return 0;
122: }
124: /* -------------------------------------------------------------------- */
126: /* FormConstraints - Evaluates gradient of f.
128: Input Parameters:
129: . tao - the TAO_APPLICATION context
130: . X - input vector
131: . ptr - optional user-defined context, as set by TaoSetConstraintsRoutine()
133: Output Parameters:
134: . G - vector containing the newly evaluated gradient
135: */
136: PetscErrorCode FormConstraints(Tao tao, Vec X, Vec G, void *ptr)
137: {
138: AppCtx *user = (AppCtx *)ptr;
139: PetscInt i, j, row;
140: PetscInt mx = user->mx, my = user->my;
141: PetscReal hx = 1.0 / (mx + 1), hy = 1.0 / (my + 1), hydhx = hy / hx, hxdhy = hx / hy;
142: PetscReal f1, f2, f3, f4, f5, f6, d1, d2, d3, d4, d5, d6, d7, d8, xc, xl, xr, xt, xb, xlt, xrb;
143: PetscReal df1dxc, df2dxc, df3dxc, df4dxc, df5dxc, df6dxc;
144: PetscScalar zero = 0.0;
145: PetscScalar *g, *x;
147: /* Initialize vector to zero */
148: VecSet(G, zero);
150: /* Get pointers to vector data */
151: VecGetArray(X, &x);
152: VecGetArray(G, &g);
154: /* Compute function over the locally owned part of the mesh */
155: for (j = 0; j < my; j++) {
156: for (i = 0; i < mx; i++) {
157: row = j * mx + i;
159: xc = x[row];
160: xlt = xrb = xl = xr = xb = xt = xc;
162: if (i == 0) { /* left side */
163: xl = user->left[j + 1];
164: xlt = user->left[j + 2];
165: } else {
166: xl = x[row - 1];
167: }
169: if (j == 0) { /* bottom side */
170: xb = user->bottom[i + 1];
171: xrb = user->bottom[i + 2];
172: } else {
173: xb = x[row - mx];
174: }
176: if (i + 1 == mx) { /* right side */
177: xr = user->right[j + 1];
178: xrb = user->right[j];
179: } else {
180: xr = x[row + 1];
181: }
183: if (j + 1 == 0 + my) { /* top side */
184: xt = user->top[i + 1];
185: xlt = user->top[i];
186: } else {
187: xt = x[row + mx];
188: }
190: if (i > 0 && j + 1 < my) xlt = x[row - 1 + mx];
191: if (j > 0 && i + 1 < mx) xrb = x[row + 1 - mx];
193: d1 = (xc - xl);
194: d2 = (xc - xr);
195: d3 = (xc - xt);
196: d4 = (xc - xb);
197: d5 = (xr - xrb);
198: d6 = (xrb - xb);
199: d7 = (xlt - xl);
200: d8 = (xt - xlt);
202: df1dxc = d1 * hydhx;
203: df2dxc = (d1 * hydhx + d4 * hxdhy);
204: df3dxc = d3 * hxdhy;
205: df4dxc = (d2 * hydhx + d3 * hxdhy);
206: df5dxc = d2 * hydhx;
207: df6dxc = d4 * hxdhy;
209: d1 /= hx;
210: d2 /= hx;
211: d3 /= hy;
212: d4 /= hy;
213: d5 /= hy;
214: d6 /= hx;
215: d7 /= hy;
216: d8 /= hx;
218: f1 = PetscSqrtScalar(1.0 + d1 * d1 + d7 * d7);
219: f2 = PetscSqrtScalar(1.0 + d1 * d1 + d4 * d4);
220: f3 = PetscSqrtScalar(1.0 + d3 * d3 + d8 * d8);
221: f4 = PetscSqrtScalar(1.0 + d3 * d3 + d2 * d2);
222: f5 = PetscSqrtScalar(1.0 + d2 * d2 + d5 * d5);
223: f6 = PetscSqrtScalar(1.0 + d4 * d4 + d6 * d6);
225: df1dxc /= f1;
226: df2dxc /= f2;
227: df3dxc /= f3;
228: df4dxc /= f4;
229: df5dxc /= f5;
230: df6dxc /= f6;
232: g[row] = (df1dxc + df2dxc + df3dxc + df4dxc + df5dxc + df6dxc) / 2.0;
233: }
234: }
236: /* Restore vectors */
237: VecRestoreArray(X, &x);
238: VecRestoreArray(G, &g);
239: PetscLogFlops(67 * mx * my);
240: return 0;
241: }
243: /* ------------------------------------------------------------------- */
244: /*
245: FormJacobian - Evaluates Jacobian matrix.
247: Input Parameters:
248: . tao - the TAO_APPLICATION context
249: . X - input vector
250: . ptr - optional user-defined context, as set by TaoSetJacobian()
252: Output Parameters:
253: . tH - Jacobian matrix
255: */
256: PetscErrorCode FormJacobian(Tao tao, Vec X, Mat H, Mat tHPre, void *ptr)
257: {
258: AppCtx *user = (AppCtx *)ptr;
259: PetscInt i, j, k, row;
260: PetscInt mx = user->mx, my = user->my;
261: PetscInt col[7];
262: PetscReal hx = 1.0 / (mx + 1), hy = 1.0 / (my + 1), hydhx = hy / hx, hxdhy = hx / hy;
263: PetscReal f1, f2, f3, f4, f5, f6, d1, d2, d3, d4, d5, d6, d7, d8, xc, xl, xr, xt, xb, xlt, xrb;
264: PetscReal hl, hr, ht, hb, hc, htl, hbr;
265: const PetscScalar *x;
266: PetscScalar v[7];
267: PetscBool assembled;
269: /* Set various matrix options */
270: MatSetOption(H, MAT_IGNORE_OFF_PROC_ENTRIES, PETSC_TRUE);
271: MatAssembled(H, &assembled);
272: if (assembled) MatZeroEntries(H);
274: /* Get pointers to vector data */
275: VecGetArrayRead(X, &x);
277: /* Compute Jacobian over the locally owned part of the mesh */
278: for (i = 0; i < mx; i++) {
279: for (j = 0; j < my; j++) {
280: row = j * mx + i;
282: xc = x[row];
283: xlt = xrb = xl = xr = xb = xt = xc;
285: /* Left side */
286: if (i == 0) {
287: xl = user->left[j + 1];
288: xlt = user->left[j + 2];
289: } else {
290: xl = x[row - 1];
291: }
293: if (j == 0) {
294: xb = user->bottom[i + 1];
295: xrb = user->bottom[i + 2];
296: } else {
297: xb = x[row - mx];
298: }
300: if (i + 1 == mx) {
301: xr = user->right[j + 1];
302: xrb = user->right[j];
303: } else {
304: xr = x[row + 1];
305: }
307: if (j + 1 == my) {
308: xt = user->top[i + 1];
309: xlt = user->top[i];
310: } else {
311: xt = x[row + mx];
312: }
314: if (i > 0 && j + 1 < my) xlt = x[row - 1 + mx];
315: if (j > 0 && i + 1 < mx) xrb = x[row + 1 - mx];
317: d1 = (xc - xl) / hx;
318: d2 = (xc - xr) / hx;
319: d3 = (xc - xt) / hy;
320: d4 = (xc - xb) / hy;
321: d5 = (xrb - xr) / hy;
322: d6 = (xrb - xb) / hx;
323: d7 = (xlt - xl) / hy;
324: d8 = (xlt - xt) / hx;
326: f1 = PetscSqrtScalar(1.0 + d1 * d1 + d7 * d7);
327: f2 = PetscSqrtScalar(1.0 + d1 * d1 + d4 * d4);
328: f3 = PetscSqrtScalar(1.0 + d3 * d3 + d8 * d8);
329: f4 = PetscSqrtScalar(1.0 + d3 * d3 + d2 * d2);
330: f5 = PetscSqrtScalar(1.0 + d2 * d2 + d5 * d5);
331: f6 = PetscSqrtScalar(1.0 + d4 * d4 + d6 * d6);
333: hl = (-hydhx * (1.0 + d7 * d7) + d1 * d7) / (f1 * f1 * f1) + (-hydhx * (1.0 + d4 * d4) + d1 * d4) / (f2 * f2 * f2);
334: hr = (-hydhx * (1.0 + d5 * d5) + d2 * d5) / (f5 * f5 * f5) + (-hydhx * (1.0 + d3 * d3) + d2 * d3) / (f4 * f4 * f4);
335: ht = (-hxdhy * (1.0 + d8 * d8) + d3 * d8) / (f3 * f3 * f3) + (-hxdhy * (1.0 + d2 * d2) + d2 * d3) / (f4 * f4 * f4);
336: hb = (-hxdhy * (1.0 + d6 * d6) + d4 * d6) / (f6 * f6 * f6) + (-hxdhy * (1.0 + d1 * d1) + d1 * d4) / (f2 * f2 * f2);
338: hbr = -d2 * d5 / (f5 * f5 * f5) - d4 * d6 / (f6 * f6 * f6);
339: htl = -d1 * d7 / (f1 * f1 * f1) - d3 * d8 / (f3 * f3 * f3);
341: 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) + (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);
343: hl /= 2.0;
344: hr /= 2.0;
345: ht /= 2.0;
346: hb /= 2.0;
347: hbr /= 2.0;
348: htl /= 2.0;
349: hc /= 2.0;
351: k = 0;
352: if (j > 0) {
353: v[k] = hb;
354: col[k] = row - mx;
355: k++;
356: }
358: if (j > 0 && i < mx - 1) {
359: v[k] = hbr;
360: col[k] = row - mx + 1;
361: k++;
362: }
364: if (i > 0) {
365: v[k] = hl;
366: col[k] = row - 1;
367: k++;
368: }
370: v[k] = hc;
371: col[k] = row;
372: k++;
374: if (i < mx - 1) {
375: v[k] = hr;
376: col[k] = row + 1;
377: k++;
378: }
380: if (i > 0 && j < my - 1) {
381: v[k] = htl;
382: col[k] = row + mx - 1;
383: k++;
384: }
386: if (j < my - 1) {
387: v[k] = ht;
388: col[k] = row + mx;
389: k++;
390: }
392: /*
393: Set matrix values using local numbering, which was defined
394: earlier, in the main routine.
395: */
396: MatSetValues(H, 1, &row, k, col, v, INSERT_VALUES);
397: }
398: }
400: /* Restore vectors */
401: VecRestoreArrayRead(X, &x);
403: /* Assemble the matrix */
404: MatAssemblyBegin(H, MAT_FINAL_ASSEMBLY);
405: MatAssemblyEnd(H, MAT_FINAL_ASSEMBLY);
406: PetscLogFlops(199 * mx * my);
407: return 0;
408: }
410: /* ------------------------------------------------------------------- */
411: /*
412: MSA_BoundaryConditions - Calculates the boundary conditions for
413: the region.
415: Input Parameter:
416: . user - user-defined application context
418: Output Parameter:
419: . user - user-defined application context
420: */
421: static PetscErrorCode MSA_BoundaryConditions(AppCtx *user)
422: {
423: PetscInt i, j, k, limit = 0, maxits = 5;
424: PetscInt mx = user->mx, my = user->my;
425: PetscInt bsize = 0, lsize = 0, tsize = 0, rsize = 0;
426: PetscReal one = 1.0, two = 2.0, three = 3.0, tol = 1e-10;
427: PetscReal fnorm, det, hx, hy, xt = 0, yt = 0;
428: PetscReal u1, u2, nf1, nf2, njac11, njac12, njac21, njac22;
429: PetscReal b = -0.5, t = 0.5, l = -0.5, r = 0.5;
430: PetscReal *boundary;
432: bsize = mx + 2;
433: lsize = my + 2;
434: rsize = my + 2;
435: tsize = mx + 2;
437: PetscMalloc1(bsize, &user->bottom);
438: PetscMalloc1(tsize, &user->top);
439: PetscMalloc1(lsize, &user->left);
440: PetscMalloc1(rsize, &user->right);
442: hx = (r - l) / (mx + 1);
443: 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: {
509: PetscInt start = -1, i, j;
510: PetscScalar zero = 0.0;
511: PetscBool flg;
513: PetscOptionsGetInt(NULL, NULL, "-start", &start, &flg);
515: if (flg && start == 0) { /* The zero vector is reasonable */
516: VecSet(X, zero);
517: } else { /* Take an average of the boundary conditions */
518: PetscInt row;
519: PetscInt mx = user->mx, my = user->my;
520: PetscScalar *x;
522: /* Get pointers to vector data */
523: VecGetArray(X, &x);
525: /* Perform local computations */
526: for (j = 0; j < my; j++) {
527: for (i = 0; i < mx; i++) {
528: row = (j)*mx + (i);
529: 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;
530: }
531: }
533: /* Restore vectors */
534: VecRestoreArray(X, &x);
535: }
536: return 0;
537: }
539: /*TEST
541: build:
542: requires: !complex
544: test:
545: args: -tao_monitor -tao_view -tao_type ssils -tao_gttol 1.e-5
546: requires: !single
548: test:
549: suffix: 2
550: args: -tao_monitor -tao_view -tao_type ssfls -tao_gttol 1.e-5
552: TEST*/