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 */
50: PetscFunctionBeginUser;
51: PetscCall(PetscInitialize(&argc, &argv, NULL, 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: PetscCall(PetscOptionsGetInt(NULL, NULL, "-mx", &user.mx, &flg));
59: PetscCall(PetscOptionsGetInt(NULL, NULL, "-my", &user.my, &flg));
61: /* Calculate any derived values from parameters */
62: N = user.mx * user.my;
64: PetscCall(PetscPrintf(PETSC_COMM_SELF, "\n---- Minimum Surface Area Problem -----\n"));
65: PetscCall(PetscPrintf(PETSC_COMM_SELF, "mx:%" PetscInt_FMT ", my:%" PetscInt_FMT "\n", user.mx, user.my));
67: /* Create appropriate vectors and matrices */
68: PetscCall(VecCreateSeq(MPI_COMM_SELF, N, &x));
69: PetscCall(VecDuplicate(x, &c));
70: PetscCall(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: PetscCall(TaoCreate(PETSC_COMM_SELF, &tao));
76: PetscCall(TaoSetType(tao, TAOSSILS));
78: /* Set data structure */
79: PetscCall(TaoSetSolution(tao, x));
81: /* Set routines for constraints function and Jacobian evaluation */
82: PetscCall(TaoSetConstraintsRoutine(tao, c, FormConstraints, (void *)&user));
83: PetscCall(TaoSetJacobianRoutine(tao, J, J, FormJacobian, (void *)&user));
85: /* Set the variable bounds */
86: PetscCall(MSA_BoundaryConditions(&user));
88: /* Set initial solution guess */
89: PetscCall(MSA_InitialPoint(&user, x));
91: /* Set Bounds on variables */
92: PetscCall(VecDuplicate(x, &xl));
93: PetscCall(VecDuplicate(x, &xu));
94: PetscCall(VecSet(xl, lb));
95: PetscCall(VecSet(xu, ub));
96: PetscCall(TaoSetVariableBounds(tao, xl, xu));
98: /* Check for any tao command line options */
99: PetscCall(TaoSetFromOptions(tao));
101: /* Solve the application */
102: PetscCall(TaoSolve(tao));
104: /* Free Tao data structures */
105: PetscCall(TaoDestroy(&tao));
107: /* Free PETSc data structures */
108: PetscCall(VecDestroy(&x));
109: PetscCall(VecDestroy(&xl));
110: PetscCall(VecDestroy(&xu));
111: PetscCall(VecDestroy(&c));
112: PetscCall(MatDestroy(&J));
114: /* Free user-created data structures */
115: PetscCall(PetscFree(user.bottom));
116: PetscCall(PetscFree(user.top));
117: PetscCall(PetscFree(user.left));
118: PetscCall(PetscFree(user.right));
120: PetscCall(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: PetscFunctionBegin;
148: /* Initialize vector to zero */
149: PetscCall(VecSet(G, zero));
151: /* Get pointers to vector data */
152: PetscCall(VecGetArray(X, &x));
153: PetscCall(VecGetArray(G, &g));
155: /* Compute function over the locally owned part of the mesh */
156: for (j = 0; j < my; j++) {
157: for (i = 0; i < mx; i++) {
158: row = j * mx + i;
160: xc = x[row];
161: xlt = xrb = xl = xr = xb = xt = xc;
163: if (i == 0) { /* left side */
164: xl = user->left[j + 1];
165: xlt = user->left[j + 2];
166: } else {
167: xl = x[row - 1];
168: }
170: if (j == 0) { /* bottom side */
171: xb = user->bottom[i + 1];
172: xrb = user->bottom[i + 2];
173: } else {
174: xb = x[row - mx];
175: }
177: if (i + 1 == mx) { /* right side */
178: xr = user->right[j + 1];
179: xrb = user->right[j];
180: } else {
181: xr = x[row + 1];
182: }
184: if (j + 1 == 0 + my) { /* top side */
185: xt = user->top[i + 1];
186: xlt = user->top[i];
187: } else {
188: xt = x[row + mx];
189: }
191: if (i > 0 && j + 1 < my) xlt = x[row - 1 + mx];
192: if (j > 0 && i + 1 < mx) xrb = x[row + 1 - mx];
194: d1 = (xc - xl);
195: d2 = (xc - xr);
196: d3 = (xc - xt);
197: d4 = (xc - xb);
198: d5 = (xr - xrb);
199: d6 = (xrb - xb);
200: d7 = (xlt - xl);
201: d8 = (xt - xlt);
203: df1dxc = d1 * hydhx;
204: df2dxc = (d1 * hydhx + d4 * hxdhy);
205: df3dxc = d3 * hxdhy;
206: df4dxc = (d2 * hydhx + d3 * hxdhy);
207: df5dxc = d2 * hydhx;
208: df6dxc = d4 * hxdhy;
210: d1 /= hx;
211: d2 /= hx;
212: d3 /= hy;
213: d4 /= hy;
214: d5 /= hy;
215: d6 /= hx;
216: d7 /= hy;
217: d8 /= hx;
219: f1 = PetscSqrtScalar(1.0 + d1 * d1 + d7 * d7);
220: f2 = PetscSqrtScalar(1.0 + d1 * d1 + d4 * d4);
221: f3 = PetscSqrtScalar(1.0 + d3 * d3 + d8 * d8);
222: f4 = PetscSqrtScalar(1.0 + d3 * d3 + d2 * d2);
223: f5 = PetscSqrtScalar(1.0 + d2 * d2 + d5 * d5);
224: f6 = PetscSqrtScalar(1.0 + d4 * d4 + d6 * d6);
226: df1dxc /= f1;
227: df2dxc /= f2;
228: df3dxc /= f3;
229: df4dxc /= f4;
230: df5dxc /= f5;
231: df6dxc /= f6;
233: g[row] = (df1dxc + df2dxc + df3dxc + df4dxc + df5dxc + df6dxc) / 2.0;
234: }
235: }
237: /* Restore vectors */
238: PetscCall(VecRestoreArray(X, &x));
239: PetscCall(VecRestoreArray(G, &g));
240: PetscCall(PetscLogFlops(67 * mx * my));
241: PetscFunctionReturn(PETSC_SUCCESS);
242: }
244: /* ------------------------------------------------------------------- */
245: /*
246: FormJacobian - Evaluates Jacobian matrix.
248: Input Parameters:
249: . tao - the TAO_APPLICATION context
250: . X - input vector
251: . ptr - optional user-defined context, as set by TaoSetJacobian()
253: Output Parameters:
254: . tH - Jacobian matrix
256: */
257: PetscErrorCode FormJacobian(Tao tao, Vec X, Mat H, Mat tHPre, void *ptr)
258: {
259: AppCtx *user = (AppCtx *)ptr;
260: PetscInt i, j, k, row;
261: PetscInt mx = user->mx, my = user->my;
262: PetscInt col[7];
263: PetscReal hx = 1.0 / (mx + 1), hy = 1.0 / (my + 1), hydhx = hy / hx, hxdhy = hx / hy;
264: PetscReal f1, f2, f3, f4, f5, f6, d1, d2, d3, d4, d5, d6, d7, d8, xc, xl, xr, xt, xb, xlt, xrb;
265: PetscReal hl, hr, ht, hb, hc, htl, hbr;
266: const PetscScalar *x;
267: PetscScalar v[7];
268: PetscBool assembled;
270: /* Set various matrix options */
271: PetscFunctionBegin;
272: PetscCall(MatSetOption(H, MAT_IGNORE_OFF_PROC_ENTRIES, PETSC_TRUE));
273: PetscCall(MatAssembled(H, &assembled));
274: if (assembled) PetscCall(MatZeroEntries(H));
276: /* Get pointers to vector data */
277: PetscCall(VecGetArrayRead(X, &x));
279: /* Compute Jacobian over the locally owned part of the mesh */
280: for (i = 0; i < mx; i++) {
281: for (j = 0; j < my; j++) {
282: row = j * mx + i;
284: xc = x[row];
285: xlt = xrb = xl = xr = xb = xt = xc;
287: /* Left side */
288: if (i == 0) {
289: xl = user->left[j + 1];
290: xlt = user->left[j + 2];
291: } else {
292: xl = x[row - 1];
293: }
295: if (j == 0) {
296: xb = user->bottom[i + 1];
297: xrb = user->bottom[i + 2];
298: } else {
299: xb = x[row - mx];
300: }
302: if (i + 1 == mx) {
303: xr = user->right[j + 1];
304: xrb = user->right[j];
305: } else {
306: xr = x[row + 1];
307: }
309: if (j + 1 == my) {
310: xt = user->top[i + 1];
311: xlt = user->top[i];
312: } else {
313: xt = x[row + mx];
314: }
316: if (i > 0 && j + 1 < my) xlt = x[row - 1 + mx];
317: if (j > 0 && i + 1 < mx) xrb = x[row + 1 - mx];
319: d1 = (xc - xl) / hx;
320: d2 = (xc - xr) / hx;
321: d3 = (xc - xt) / hy;
322: d4 = (xc - xb) / hy;
323: d5 = (xrb - xr) / hy;
324: d6 = (xrb - xb) / hx;
325: d7 = (xlt - xl) / hy;
326: d8 = (xlt - xt) / hx;
328: f1 = PetscSqrtScalar(1.0 + d1 * d1 + d7 * d7);
329: f2 = PetscSqrtScalar(1.0 + d1 * d1 + d4 * d4);
330: f3 = PetscSqrtScalar(1.0 + d3 * d3 + d8 * d8);
331: f4 = PetscSqrtScalar(1.0 + d3 * d3 + d2 * d2);
332: f5 = PetscSqrtScalar(1.0 + d2 * d2 + d5 * d5);
333: f6 = PetscSqrtScalar(1.0 + d4 * d4 + d6 * d6);
335: hl = (-hydhx * (1.0 + d7 * d7) + d1 * d7) / (f1 * f1 * f1) + (-hydhx * (1.0 + d4 * d4) + d1 * d4) / (f2 * f2 * f2);
336: hr = (-hydhx * (1.0 + d5 * d5) + d2 * d5) / (f5 * f5 * f5) + (-hydhx * (1.0 + d3 * d3) + d2 * d3) / (f4 * f4 * f4);
337: ht = (-hxdhy * (1.0 + d8 * d8) + d3 * d8) / (f3 * f3 * f3) + (-hxdhy * (1.0 + d2 * d2) + d2 * d3) / (f4 * f4 * f4);
338: hb = (-hxdhy * (1.0 + d6 * d6) + d4 * d6) / (f6 * f6 * f6) + (-hxdhy * (1.0 + d1 * d1) + d1 * d4) / (f2 * f2 * f2);
340: hbr = -d2 * d5 / (f5 * f5 * f5) - d4 * d6 / (f6 * f6 * f6);
341: htl = -d1 * d7 / (f1 * f1 * f1) - d3 * d8 / (f3 * f3 * f3);
343: 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);
345: hl /= 2.0;
346: hr /= 2.0;
347: ht /= 2.0;
348: hb /= 2.0;
349: hbr /= 2.0;
350: htl /= 2.0;
351: hc /= 2.0;
353: k = 0;
354: if (j > 0) {
355: v[k] = hb;
356: col[k] = row - mx;
357: k++;
358: }
360: if (j > 0 && i < mx - 1) {
361: v[k] = hbr;
362: col[k] = row - mx + 1;
363: k++;
364: }
366: if (i > 0) {
367: v[k] = hl;
368: col[k] = row - 1;
369: k++;
370: }
372: v[k] = hc;
373: col[k] = row;
374: k++;
376: if (i < mx - 1) {
377: v[k] = hr;
378: col[k] = row + 1;
379: k++;
380: }
382: if (i > 0 && j < my - 1) {
383: v[k] = htl;
384: col[k] = row + mx - 1;
385: k++;
386: }
388: if (j < my - 1) {
389: v[k] = ht;
390: col[k] = row + mx;
391: k++;
392: }
394: /*
395: Set matrix values using local numbering, which was defined
396: earlier, in the main routine.
397: */
398: PetscCall(MatSetValues(H, 1, &row, k, col, v, INSERT_VALUES));
399: }
400: }
402: /* Restore vectors */
403: PetscCall(VecRestoreArrayRead(X, &x));
405: /* Assemble the matrix */
406: PetscCall(MatAssemblyBegin(H, MAT_FINAL_ASSEMBLY));
407: PetscCall(MatAssemblyEnd(H, MAT_FINAL_ASSEMBLY));
408: PetscCall(PetscLogFlops(199 * mx * my));
409: PetscFunctionReturn(PETSC_SUCCESS);
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: PetscInt i, j, k, limit = 0, maxits = 5;
426: PetscInt mx = user->mx, my = user->my;
427: PetscInt bsize = 0, lsize = 0, tsize = 0, rsize = 0;
428: PetscReal one = 1.0, two = 2.0, three = 3.0, tol = 1e-10;
429: PetscReal fnorm, det, hx, hy, xt = 0, yt = 0;
430: PetscReal u1, u2, nf1, nf2, njac11, njac12, njac21, njac22;
431: PetscReal b = -0.5, t = 0.5, l = -0.5, r = 0.5;
432: PetscReal *boundary;
434: PetscFunctionBegin;
435: bsize = mx + 2;
436: lsize = my + 2;
437: rsize = my + 2;
438: tsize = mx + 2;
440: PetscCall(PetscMalloc1(bsize, &user->bottom));
441: PetscCall(PetscMalloc1(tsize, &user->top));
442: PetscCall(PetscMalloc1(lsize, &user->left));
443: PetscCall(PetscMalloc1(rsize, &user->right));
445: hx = (r - l) / (mx + 1);
446: hy = (t - b) / (my + 1);
448: for (j = 0; j < 4; j++) {
449: if (j == 0) {
450: yt = b;
451: xt = l;
452: limit = bsize;
453: boundary = user->bottom;
454: } else if (j == 1) {
455: yt = t;
456: xt = l;
457: limit = tsize;
458: boundary = user->top;
459: } else if (j == 2) {
460: yt = b;
461: xt = l;
462: limit = lsize;
463: boundary = user->left;
464: } else { /* if (j==3) */
465: yt = b;
466: xt = r;
467: limit = rsize;
468: boundary = user->right;
469: }
471: for (i = 0; i < limit; i++) {
472: u1 = xt;
473: u2 = -yt;
474: for (k = 0; k < maxits; k++) {
475: nf1 = u1 + u1 * u2 * u2 - u1 * u1 * u1 / three - xt;
476: nf2 = -u2 - u1 * u1 * u2 + u2 * u2 * u2 / three - yt;
477: fnorm = PetscSqrtScalar(nf1 * nf1 + nf2 * nf2);
478: if (fnorm <= tol) break;
479: njac11 = one + u2 * u2 - u1 * u1;
480: njac12 = two * u1 * u2;
481: njac21 = -two * u1 * u2;
482: njac22 = -one - u1 * u1 + u2 * u2;
483: det = njac11 * njac22 - njac21 * njac12;
484: u1 = u1 - (njac22 * nf1 - njac12 * nf2) / det;
485: u2 = u2 - (njac11 * nf2 - njac21 * nf1) / det;
486: }
488: boundary[i] = u1 * u1 - u2 * u2;
489: if (j == 0 || j == 1) {
490: xt = xt + hx;
491: } else { /* if (j==2 || j==3) */
492: yt = yt + hy;
493: }
494: }
495: }
496: PetscFunctionReturn(PETSC_SUCCESS);
497: }
499: /* ------------------------------------------------------------------- */
500: /*
501: MSA_InitialPoint - Calculates the initial guess in one of three ways.
503: Input Parameters:
504: . user - user-defined application context
505: . X - vector for initial guess
507: Output Parameters:
508: . X - newly computed initial guess
509: */
510: static PetscErrorCode MSA_InitialPoint(AppCtx *user, Vec X)
511: {
512: PetscInt start = -1, i, j;
513: PetscScalar zero = 0.0;
514: PetscBool flg;
516: PetscFunctionBegin;
517: PetscCall(PetscOptionsGetInt(NULL, NULL, "-start", &start, &flg));
519: if (flg && start == 0) { /* The zero vector is reasonable */
520: PetscCall(VecSet(X, zero));
521: } else { /* Take an average of the boundary conditions */
522: PetscInt row;
523: PetscInt mx = user->mx, my = user->my;
524: PetscScalar *x;
526: /* Get pointers to vector data */
527: PetscCall(VecGetArray(X, &x));
529: /* Perform local computations */
530: for (j = 0; j < my; j++) {
531: for (i = 0; i < mx; i++) {
532: row = (j)*mx + (i);
533: 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;
534: }
535: }
537: /* Restore vectors */
538: PetscCall(VecRestoreArray(X, &x));
539: }
540: PetscFunctionReturn(PETSC_SUCCESS);
541: }
543: /*TEST
545: build:
546: requires: !complex
548: test:
549: args: -tao_monitor -tao_view -tao_type ssils -tao_gttol 1.e-5
550: requires: !single
552: test:
553: suffix: 2
554: args: -tao_monitor -tao_view -tao_type ssfls -tao_gttol 1.e-5
556: TEST*/