1 #include <petsctao.h> 2 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"; 19 20 /*T 21 Concepts: TAO^Solving a complementarity problem 22 Routines: TaoCreate(); TaoDestroy(); 23 24 Processors: 1 25 T*/ 26 27 28 29 30 /* 31 User-defined application context - contains data needed by the 32 application-provided call-back routines, FormFunctionGradient(), 33 FormHessian(). 34 */ 35 typedef struct { 36 PetscInt mx, my; 37 PetscReal *bottom, *top, *left, *right; 38 } AppCtx; 39 40 41 /* -------- User-defined Routines --------- */ 42 43 static PetscErrorCode MSA_BoundaryConditions(AppCtx *); 44 static PetscErrorCode MSA_InitialPoint(AppCtx *, Vec); 45 PetscErrorCode FormConstraints(Tao, Vec, Vec, void *); 46 PetscErrorCode FormJacobian(Tao, Vec, Mat, Mat, void *); 47 48 int main(int argc, char **argv) 49 { 50 PetscErrorCode ierr; /* used to check for functions returning nonzeros */ 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 */ 61 62 /* Initialize PETSc, TAO */ 63 ierr = PetscInitialize(&argc, &argv, (char *)0, help);if (ierr) return ierr; 64 65 /* Specify default dimension of the problem */ 66 user.mx = 4; user.my = 4; 67 68 /* Check for any command line arguments that override defaults */ 69 ierr = PetscOptionsGetInt(NULL,NULL, "-mx", &user.mx, &flg);CHKERRQ(ierr); 70 ierr = PetscOptionsGetInt(NULL,NULL, "-my", &user.my, &flg);CHKERRQ(ierr); 71 72 /* Calculate any derived values from parameters */ 73 N = user.mx*user.my; 74 75 ierr = PetscPrintf(PETSC_COMM_SELF,"\n---- Minimum Surface Area Problem -----\n");CHKERRQ(ierr); 76 ierr = PetscPrintf(PETSC_COMM_SELF,"mx:%D, my:%D\n", user.mx,user.my);CHKERRQ(ierr); 77 78 /* Create appropriate vectors and matrices */ 79 ierr = VecCreateSeq(MPI_COMM_SELF, N, &x);CHKERRQ(ierr); 80 ierr = VecDuplicate(x, &c);CHKERRQ(ierr); 81 ierr = MatCreateSeqAIJ(MPI_COMM_SELF, N, N, 7, NULL, &J);CHKERRQ(ierr); 82 83 /* The TAO code begins here */ 84 85 /* Create TAO solver and set desired solution method */ 86 ierr = TaoCreate(PETSC_COMM_SELF,&tao);CHKERRQ(ierr); 87 ierr = TaoSetType(tao,TAOSSILS);CHKERRQ(ierr); 88 89 /* Set data structure */ 90 ierr = TaoSetInitialVector(tao, x);CHKERRQ(ierr); 91 92 /* Set routines for constraints function and Jacobian evaluation */ 93 ierr = TaoSetConstraintsRoutine(tao, c, FormConstraints, (void *)&user);CHKERRQ(ierr); 94 ierr = TaoSetJacobianRoutine(tao, J, J, FormJacobian, (void *)&user);CHKERRQ(ierr); 95 96 /* Set the variable bounds */ 97 ierr = MSA_BoundaryConditions(&user);CHKERRQ(ierr); 98 99 /* Set initial solution guess */ 100 ierr = MSA_InitialPoint(&user, x);CHKERRQ(ierr); 101 102 /* Set Bounds on variables */ 103 ierr = VecDuplicate(x, &xl);CHKERRQ(ierr); 104 ierr = VecDuplicate(x, &xu);CHKERRQ(ierr); 105 ierr = VecSet(xl, lb);CHKERRQ(ierr); 106 ierr = VecSet(xu, ub);CHKERRQ(ierr); 107 ierr = TaoSetVariableBounds(tao,xl,xu);CHKERRQ(ierr); 108 109 /* Check for any tao command line options */ 110 ierr = TaoSetFromOptions(tao);CHKERRQ(ierr); 111 112 /* Solve the application */ 113 ierr = TaoSolve(tao);CHKERRQ(ierr); 114 115 /* Free Tao data structures */ 116 ierr = TaoDestroy(&tao);CHKERRQ(ierr); 117 118 /* Free PETSc data structures */ 119 ierr = VecDestroy(&x);CHKERRQ(ierr); 120 ierr = VecDestroy(&xl);CHKERRQ(ierr); 121 ierr = VecDestroy(&xu);CHKERRQ(ierr); 122 ierr = VecDestroy(&c);CHKERRQ(ierr); 123 ierr = MatDestroy(&J);CHKERRQ(ierr); 124 125 /* Free user-created data structures */ 126 ierr = PetscFree(user.bottom);CHKERRQ(ierr); 127 ierr = PetscFree(user.top);CHKERRQ(ierr); 128 ierr = PetscFree(user.left);CHKERRQ(ierr); 129 ierr = PetscFree(user.right);CHKERRQ(ierr); 130 131 ierr = PetscFinalize(); 132 return ierr; 133 } 134 135 /* -------------------------------------------------------------------- */ 136 137 /* FormConstraints - Evaluates gradient of f. 138 139 Input Parameters: 140 . tao - the TAO_APPLICATION context 141 . X - input vector 142 . ptr - optional user-defined context, as set by TaoSetConstraintsRoutine() 143 144 Output Parameters: 145 . G - vector containing the newly evaluated gradient 146 */ 147 PetscErrorCode FormConstraints(Tao tao, Vec X, Vec G, void *ptr) 148 { 149 AppCtx *user = (AppCtx *) ptr; 150 PetscErrorCode ierr; 151 PetscInt i,j,row; 152 PetscInt mx=user->mx, my=user->my; 153 PetscReal hx=1.0/(mx+1),hy=1.0/(my+1), hydhx=hy/hx, hxdhy=hx/hy; 154 PetscReal f1,f2,f3,f4,f5,f6,d1,d2,d3,d4,d5,d6,d7,d8,xc,xl,xr,xt,xb,xlt,xrb; 155 PetscReal df1dxc,df2dxc,df3dxc,df4dxc,df5dxc,df6dxc; 156 PetscScalar zero=0.0; 157 PetscScalar *g, *x; 158 159 PetscFunctionBegin; 160 /* Initialize vector to zero */ 161 ierr = VecSet(G, zero);CHKERRQ(ierr); 162 163 /* Get pointers to vector data */ 164 ierr = VecGetArray(X, &x);CHKERRQ(ierr); 165 ierr = VecGetArray(G, &g);CHKERRQ(ierr); 166 167 /* Compute function over the locally owned part of the mesh */ 168 for (j=0; j<my; j++){ 169 for (i=0; i< mx; i++){ 170 row= j*mx + i; 171 172 xc = x[row]; 173 xlt=xrb=xl=xr=xb=xt=xc; 174 175 if (i==0){ /* left side */ 176 xl= user->left[j+1]; 177 xlt = user->left[j+2]; 178 } else { 179 xl = x[row-1]; 180 } 181 182 if (j==0){ /* bottom side */ 183 xb=user->bottom[i+1]; 184 xrb = user->bottom[i+2]; 185 } else { 186 xb = x[row-mx]; 187 } 188 189 if (i+1 == mx){ /* right side */ 190 xr=user->right[j+1]; 191 xrb = user->right[j]; 192 } else { 193 xr = x[row+1]; 194 } 195 196 if (j+1==0+my){ /* top side */ 197 xt=user->top[i+1]; 198 xlt = user->top[i]; 199 }else { 200 xt = x[row+mx]; 201 } 202 203 if (i>0 && j+1<my){ 204 xlt = x[row-1+mx]; 205 } 206 if (j>0 && i+1<mx){ 207 xrb = x[row+1-mx]; 208 } 209 210 d1 = (xc-xl); 211 d2 = (xc-xr); 212 d3 = (xc-xt); 213 d4 = (xc-xb); 214 d5 = (xr-xrb); 215 d6 = (xrb-xb); 216 d7 = (xlt-xl); 217 d8 = (xt-xlt); 218 219 df1dxc = d1*hydhx; 220 df2dxc = (d1*hydhx + d4*hxdhy); 221 df3dxc = d3*hxdhy; 222 df4dxc = (d2*hydhx + d3*hxdhy); 223 df5dxc = d2*hydhx; 224 df6dxc = d4*hxdhy; 225 226 d1 /= hx; 227 d2 /= hx; 228 d3 /= hy; 229 d4 /= hy; 230 d5 /= hy; 231 d6 /= hx; 232 d7 /= hy; 233 d8 /= hx; 234 235 f1 = PetscSqrtScalar(1.0 + d1*d1 + d7*d7); 236 f2 = PetscSqrtScalar(1.0 + d1*d1 + d4*d4); 237 f3 = PetscSqrtScalar(1.0 + d3*d3 + d8*d8); 238 f4 = PetscSqrtScalar(1.0 + d3*d3 + d2*d2); 239 f5 = PetscSqrtScalar(1.0 + d2*d2 + d5*d5); 240 f6 = PetscSqrtScalar(1.0 + d4*d4 + d6*d6); 241 242 df1dxc /= f1; 243 df2dxc /= f2; 244 df3dxc /= f3; 245 df4dxc /= f4; 246 df5dxc /= f5; 247 df6dxc /= f6; 248 249 g[row] = (df1dxc+df2dxc+df3dxc+df4dxc+df5dxc+df6dxc)/2.0; 250 } 251 } 252 253 /* Restore vectors */ 254 ierr = VecRestoreArray(X, &x);CHKERRQ(ierr); 255 ierr = VecRestoreArray(G, &g);CHKERRQ(ierr); 256 ierr = PetscLogFlops(67*mx*my);CHKERRQ(ierr); 257 PetscFunctionReturn(0); 258 } 259 260 /* ------------------------------------------------------------------- */ 261 /* 262 FormJacobian - Evaluates Jacobian matrix. 263 264 Input Parameters: 265 . tao - the TAO_APPLICATION context 266 . X - input vector 267 . ptr - optional user-defined context, as set by TaoSetJacobian() 268 269 Output Parameters: 270 . tH - Jacobian matrix 271 272 */ 273 PetscErrorCode FormJacobian(Tao tao, Vec X, Mat H, Mat tHPre, void *ptr) 274 { 275 AppCtx *user = (AppCtx *) ptr; 276 PetscErrorCode ierr; 277 PetscInt i,j,k,row; 278 PetscInt mx=user->mx, my=user->my; 279 PetscInt col[7]; 280 PetscReal hx=1.0/(mx+1), hy=1.0/(my+1), hydhx=hy/hx, hxdhy=hx/hy; 281 PetscReal f1,f2,f3,f4,f5,f6,d1,d2,d3,d4,d5,d6,d7,d8,xc,xl,xr,xt,xb,xlt,xrb; 282 PetscReal hl,hr,ht,hb,hc,htl,hbr; 283 const PetscScalar *x; 284 PetscScalar v[7]; 285 PetscBool assembled; 286 287 /* Set various matrix options */ 288 ierr = MatSetOption(H,MAT_IGNORE_OFF_PROC_ENTRIES,PETSC_TRUE);CHKERRQ(ierr); 289 ierr = MatAssembled(H,&assembled);CHKERRQ(ierr); 290 if (assembled){ierr = MatZeroEntries(H);CHKERRQ(ierr);} 291 292 /* Get pointers to vector data */ 293 ierr = VecGetArrayRead(X, &x);CHKERRQ(ierr); 294 295 /* Compute Jacobian over the locally owned part of the mesh */ 296 for (i=0; i< mx; i++){ 297 for (j=0; j<my; j++){ 298 row= j*mx + i; 299 300 xc = x[row]; 301 xlt=xrb=xl=xr=xb=xt=xc; 302 303 /* Left side */ 304 if (i==0){ 305 xl = user->left[j+1]; 306 xlt = user->left[j+2]; 307 } else { 308 xl = x[row-1]; 309 } 310 311 if (j==0){ 312 xb = user->bottom[i+1]; 313 xrb = user->bottom[i+2]; 314 } else { 315 xb = x[row-mx]; 316 } 317 318 if (i+1 == mx){ 319 xr = user->right[j+1]; 320 xrb = user->right[j]; 321 } else { 322 xr = x[row+1]; 323 } 324 325 if (j+1==my){ 326 xt = user->top[i+1]; 327 xlt = user->top[i]; 328 }else { 329 xt = x[row+mx]; 330 } 331 332 if (i>0 && j+1<my){ 333 xlt = x[row-1+mx]; 334 } 335 if (j>0 && i+1<mx){ 336 xrb = x[row+1-mx]; 337 } 338 339 340 d1 = (xc-xl)/hx; 341 d2 = (xc-xr)/hx; 342 d3 = (xc-xt)/hy; 343 d4 = (xc-xb)/hy; 344 d5 = (xrb-xr)/hy; 345 d6 = (xrb-xb)/hx; 346 d7 = (xlt-xl)/hy; 347 d8 = (xlt-xt)/hx; 348 349 f1 = PetscSqrtScalar(1.0 + d1*d1 + d7*d7); 350 f2 = PetscSqrtScalar(1.0 + d1*d1 + d4*d4); 351 f3 = PetscSqrtScalar(1.0 + d3*d3 + d8*d8); 352 f4 = PetscSqrtScalar(1.0 + d3*d3 + d2*d2); 353 f5 = PetscSqrtScalar(1.0 + d2*d2 + d5*d5); 354 f6 = PetscSqrtScalar(1.0 + d4*d4 + d6*d6); 355 356 357 hl = (-hydhx*(1.0+d7*d7)+d1*d7)/(f1*f1*f1)+(-hydhx*(1.0+d4*d4)+d1*d4)/(f2*f2*f2); 358 hr = (-hydhx*(1.0+d5*d5)+d2*d5)/(f5*f5*f5)+(-hydhx*(1.0+d3*d3)+d2*d3)/(f4*f4*f4); 359 ht = (-hxdhy*(1.0+d8*d8)+d3*d8)/(f3*f3*f3)+(-hxdhy*(1.0+d2*d2)+d2*d3)/(f4*f4*f4); 360 hb = (-hxdhy*(1.0+d6*d6)+d4*d6)/(f6*f6*f6)+(-hxdhy*(1.0+d1*d1)+d1*d4)/(f2*f2*f2); 361 362 hbr = -d2*d5/(f5*f5*f5) - d4*d6/(f6*f6*f6); 363 htl = -d1*d7/(f1*f1*f1) - d3*d8/(f3*f3*f3); 364 365 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) + 366 (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); 367 368 hl/=2.0; hr/=2.0; ht/=2.0; hb/=2.0; hbr/=2.0; htl/=2.0; hc/=2.0; 369 370 k=0; 371 if (j>0){ 372 v[k]=hb; col[k]=row - mx; k++; 373 } 374 375 if (j>0 && i < mx -1){ 376 v[k]=hbr; col[k]=row - mx+1; k++; 377 } 378 379 if (i>0){ 380 v[k]= hl; col[k]=row - 1; k++; 381 } 382 383 v[k]= hc; col[k]=row; k++; 384 385 if (i < mx-1){ 386 v[k]= hr; col[k]=row+1; k++; 387 } 388 389 if (i>0 && j < my-1){ 390 v[k]= htl; col[k] = row+mx-1; k++; 391 } 392 393 if (j < my-1){ 394 v[k]= ht; col[k] = row+mx; k++; 395 } 396 397 /* 398 Set matrix values using local numbering, which was defined 399 earlier, in the main routine. 400 */ 401 ierr = MatSetValues(H,1,&row,k,col,v,INSERT_VALUES);CHKERRQ(ierr); 402 } 403 } 404 405 /* Restore vectors */ 406 ierr = VecRestoreArrayRead(X,&x);CHKERRQ(ierr); 407 408 /* Assemble the matrix */ 409 ierr = MatAssemblyBegin(H,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr); 410 ierr = MatAssemblyEnd(H,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr); 411 ierr = PetscLogFlops(199*mx*my);CHKERRQ(ierr); 412 PetscFunctionReturn(0); 413 } 414 415 /* ------------------------------------------------------------------- */ 416 /* 417 MSA_BoundaryConditions - Calculates the boundary conditions for 418 the region. 419 420 Input Parameter: 421 . user - user-defined application context 422 423 Output Parameter: 424 . user - user-defined application context 425 */ 426 static PetscErrorCode MSA_BoundaryConditions(AppCtx * user) 427 { 428 PetscErrorCode ierr; 429 PetscInt i,j,k,limit=0,maxits=5; 430 PetscInt mx=user->mx,my=user->my; 431 PetscInt bsize=0, lsize=0, tsize=0, rsize=0; 432 PetscReal one=1.0, two=2.0, three=3.0, tol=1e-10; 433 PetscReal fnorm,det,hx,hy,xt=0,yt=0; 434 PetscReal u1,u2,nf1,nf2,njac11,njac12,njac21,njac22; 435 PetscReal b=-0.5, t=0.5, l=-0.5, r=0.5; 436 PetscReal *boundary; 437 438 PetscFunctionBegin; 439 bsize=mx+2; lsize=my+2; rsize=my+2; tsize=mx+2; 440 441 ierr = PetscMalloc1(bsize, &user->bottom);CHKERRQ(ierr); 442 ierr = PetscMalloc1(tsize, &user->top);CHKERRQ(ierr); 443 ierr = PetscMalloc1(lsize, &user->left);CHKERRQ(ierr); 444 ierr = PetscMalloc1(rsize, &user->right);CHKERRQ(ierr); 445 446 hx= (r-l)/(mx+1); hy=(t-b)/(my+1); 447 448 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 } 470 471 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 } 487 488 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(0); 497 } 498 499 /* ------------------------------------------------------------------- */ 500 /* 501 MSA_InitialPoint - Calculates the initial guess in one of three ways. 502 503 Input Parameters: 504 . user - user-defined application context 505 . X - vector for initial guess 506 507 Output Parameters: 508 . X - newly computed initial guess 509 */ 510 static PetscErrorCode MSA_InitialPoint(AppCtx * user, Vec X) 511 { 512 PetscErrorCode ierr; 513 PetscInt start=-1,i,j; 514 PetscScalar zero=0.0; 515 PetscBool flg; 516 517 PetscFunctionBegin; 518 ierr = PetscOptionsGetInt(NULL,NULL,"-start",&start,&flg);CHKERRQ(ierr); 519 520 if (flg && start==0){ /* The zero vector is reasonable */ 521 ierr = VecSet(X, zero);CHKERRQ(ierr); 522 } else { /* Take an average of the boundary conditions */ 523 PetscInt row; 524 PetscInt mx=user->mx,my=user->my; 525 PetscScalar *x; 526 527 /* Get pointers to vector data */ 528 ierr = VecGetArray(X,&x);CHKERRQ(ierr); 529 530 /* Perform local computations */ 531 for (j=0; j<my; j++){ 532 for (i=0; i< mx; i++){ 533 row=(j)*mx + (i); 534 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; 535 } 536 } 537 538 /* Restore vectors */ 539 ierr = VecRestoreArray(X,&x);CHKERRQ(ierr); 540 } 541 PetscFunctionReturn(0); 542 } 543 544 545 /*TEST 546 547 build: 548 requires: !complex 549 550 test: 551 args: -tao_monitor -tao_view -tao_type ssils -tao_gttol 1.e-5 552 requires: !single 553 554 test: 555 suffix: 2 556 args: -tao_monitor -tao_view -tao_type ssfls -tao_gttol 1.e-5 557 558 TEST*/ 559