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 User-defined application context - contains data needed by the 29 application-provided call-back routines, FormFunctionGradient(), 30 FormHessian(). 31 */ 32 typedef struct { 33 PetscInt mx, my; 34 PetscReal *bottom, *top, *left, *right; 35 } AppCtx; 36 37 /* -------- User-defined Routines --------- */ 38 39 static PetscErrorCode MSA_BoundaryConditions(AppCtx *); 40 static PetscErrorCode MSA_InitialPoint(AppCtx *, Vec); 41 PetscErrorCode FormConstraints(Tao, Vec, Vec, void *); 42 PetscErrorCode FormJacobian(Tao, Vec, Mat, Mat, void *); 43 44 int main(int argc, char **argv) 45 { 46 Vec x; /* solution vector */ 47 Vec c; /* Constraints function vector */ 48 Vec xl,xu; /* Bounds on the variables */ 49 PetscBool flg; /* A return variable when checking for user options */ 50 Tao tao; /* TAO solver context */ 51 Mat J; /* Jacobian matrix */ 52 PetscInt N; /* Number of elements in vector */ 53 PetscScalar lb = PETSC_NINFINITY; /* lower bound constant */ 54 PetscScalar ub = PETSC_INFINITY; /* upper bound constant */ 55 AppCtx user; /* user-defined work context */ 56 57 /* Initialize PETSc, TAO */ 58 CHKERRQ(PetscInitialize(&argc, &argv, (char *)0, help)); 59 60 /* Specify default dimension of the problem */ 61 user.mx = 4; user.my = 4; 62 63 /* Check for any command line arguments that override defaults */ 64 CHKERRQ(PetscOptionsGetInt(NULL,NULL, "-mx", &user.mx, &flg)); 65 CHKERRQ(PetscOptionsGetInt(NULL,NULL, "-my", &user.my, &flg)); 66 67 /* Calculate any derived values from parameters */ 68 N = user.mx*user.my; 69 70 CHKERRQ(PetscPrintf(PETSC_COMM_SELF,"\n---- Minimum Surface Area Problem -----\n")); 71 CHKERRQ(PetscPrintf(PETSC_COMM_SELF,"mx:%D, my:%D\n", user.mx,user.my)); 72 73 /* Create appropriate vectors and matrices */ 74 CHKERRQ(VecCreateSeq(MPI_COMM_SELF, N, &x)); 75 CHKERRQ(VecDuplicate(x, &c)); 76 CHKERRQ(MatCreateSeqAIJ(MPI_COMM_SELF, N, N, 7, NULL, &J)); 77 78 /* The TAO code begins here */ 79 80 /* Create TAO solver and set desired solution method */ 81 CHKERRQ(TaoCreate(PETSC_COMM_SELF,&tao)); 82 CHKERRQ(TaoSetType(tao,TAOSSILS)); 83 84 /* Set data structure */ 85 CHKERRQ(TaoSetSolution(tao, x)); 86 87 /* Set routines for constraints function and Jacobian evaluation */ 88 CHKERRQ(TaoSetConstraintsRoutine(tao, c, FormConstraints, (void *)&user)); 89 CHKERRQ(TaoSetJacobianRoutine(tao, J, J, FormJacobian, (void *)&user)); 90 91 /* Set the variable bounds */ 92 CHKERRQ(MSA_BoundaryConditions(&user)); 93 94 /* Set initial solution guess */ 95 CHKERRQ(MSA_InitialPoint(&user, x)); 96 97 /* Set Bounds on variables */ 98 CHKERRQ(VecDuplicate(x, &xl)); 99 CHKERRQ(VecDuplicate(x, &xu)); 100 CHKERRQ(VecSet(xl, lb)); 101 CHKERRQ(VecSet(xu, ub)); 102 CHKERRQ(TaoSetVariableBounds(tao,xl,xu)); 103 104 /* Check for any tao command line options */ 105 CHKERRQ(TaoSetFromOptions(tao)); 106 107 /* Solve the application */ 108 CHKERRQ(TaoSolve(tao)); 109 110 /* Free Tao data structures */ 111 CHKERRQ(TaoDestroy(&tao)); 112 113 /* Free PETSc data structures */ 114 CHKERRQ(VecDestroy(&x)); 115 CHKERRQ(VecDestroy(&xl)); 116 CHKERRQ(VecDestroy(&xu)); 117 CHKERRQ(VecDestroy(&c)); 118 CHKERRQ(MatDestroy(&J)); 119 120 /* Free user-created data structures */ 121 CHKERRQ(PetscFree(user.bottom)); 122 CHKERRQ(PetscFree(user.top)); 123 CHKERRQ(PetscFree(user.left)); 124 CHKERRQ(PetscFree(user.right)); 125 126 CHKERRQ(PetscFinalize()); 127 return 0; 128 } 129 130 /* -------------------------------------------------------------------- */ 131 132 /* FormConstraints - Evaluates gradient of f. 133 134 Input Parameters: 135 . tao - the TAO_APPLICATION context 136 . X - input vector 137 . ptr - optional user-defined context, as set by TaoSetConstraintsRoutine() 138 139 Output Parameters: 140 . G - vector containing the newly evaluated gradient 141 */ 142 PetscErrorCode FormConstraints(Tao tao, Vec X, Vec G, void *ptr) 143 { 144 AppCtx *user = (AppCtx *) ptr; 145 PetscInt i,j,row; 146 PetscInt mx=user->mx, my=user->my; 147 PetscReal hx=1.0/(mx+1),hy=1.0/(my+1), hydhx=hy/hx, hxdhy=hx/hy; 148 PetscReal f1,f2,f3,f4,f5,f6,d1,d2,d3,d4,d5,d6,d7,d8,xc,xl,xr,xt,xb,xlt,xrb; 149 PetscReal df1dxc,df2dxc,df3dxc,df4dxc,df5dxc,df6dxc; 150 PetscScalar zero=0.0; 151 PetscScalar *g, *x; 152 153 PetscFunctionBegin; 154 /* Initialize vector to zero */ 155 CHKERRQ(VecSet(G, zero)); 156 157 /* Get pointers to vector data */ 158 CHKERRQ(VecGetArray(X, &x)); 159 CHKERRQ(VecGetArray(G, &g)); 160 161 /* Compute function over the locally owned part of the mesh */ 162 for (j=0; j<my; j++) { 163 for (i=0; i< mx; i++) { 164 row= j*mx + i; 165 166 xc = x[row]; 167 xlt=xrb=xl=xr=xb=xt=xc; 168 169 if (i==0) { /* left side */ 170 xl= user->left[j+1]; 171 xlt = user->left[j+2]; 172 } else { 173 xl = x[row-1]; 174 } 175 176 if (j==0) { /* bottom side */ 177 xb=user->bottom[i+1]; 178 xrb = user->bottom[i+2]; 179 } else { 180 xb = x[row-mx]; 181 } 182 183 if (i+1 == mx) { /* right side */ 184 xr=user->right[j+1]; 185 xrb = user->right[j]; 186 } else { 187 xr = x[row+1]; 188 } 189 190 if (j+1==0+my) { /* top side */ 191 xt=user->top[i+1]; 192 xlt = user->top[i]; 193 }else { 194 xt = x[row+mx]; 195 } 196 197 if (i>0 && j+1<my) { 198 xlt = x[row-1+mx]; 199 } 200 if (j>0 && i+1<mx) { 201 xrb = x[row+1-mx]; 202 } 203 204 d1 = (xc-xl); 205 d2 = (xc-xr); 206 d3 = (xc-xt); 207 d4 = (xc-xb); 208 d5 = (xr-xrb); 209 d6 = (xrb-xb); 210 d7 = (xlt-xl); 211 d8 = (xt-xlt); 212 213 df1dxc = d1*hydhx; 214 df2dxc = (d1*hydhx + d4*hxdhy); 215 df3dxc = d3*hxdhy; 216 df4dxc = (d2*hydhx + d3*hxdhy); 217 df5dxc = d2*hydhx; 218 df6dxc = d4*hxdhy; 219 220 d1 /= hx; 221 d2 /= hx; 222 d3 /= hy; 223 d4 /= hy; 224 d5 /= hy; 225 d6 /= hx; 226 d7 /= hy; 227 d8 /= hx; 228 229 f1 = PetscSqrtScalar(1.0 + d1*d1 + d7*d7); 230 f2 = PetscSqrtScalar(1.0 + d1*d1 + d4*d4); 231 f3 = PetscSqrtScalar(1.0 + d3*d3 + d8*d8); 232 f4 = PetscSqrtScalar(1.0 + d3*d3 + d2*d2); 233 f5 = PetscSqrtScalar(1.0 + d2*d2 + d5*d5); 234 f6 = PetscSqrtScalar(1.0 + d4*d4 + d6*d6); 235 236 df1dxc /= f1; 237 df2dxc /= f2; 238 df3dxc /= f3; 239 df4dxc /= f4; 240 df5dxc /= f5; 241 df6dxc /= f6; 242 243 g[row] = (df1dxc+df2dxc+df3dxc+df4dxc+df5dxc+df6dxc)/2.0; 244 } 245 } 246 247 /* Restore vectors */ 248 CHKERRQ(VecRestoreArray(X, &x)); 249 CHKERRQ(VecRestoreArray(G, &g)); 250 CHKERRQ(PetscLogFlops(67*mx*my)); 251 PetscFunctionReturn(0); 252 } 253 254 /* ------------------------------------------------------------------- */ 255 /* 256 FormJacobian - Evaluates Jacobian matrix. 257 258 Input Parameters: 259 . tao - the TAO_APPLICATION context 260 . X - input vector 261 . ptr - optional user-defined context, as set by TaoSetJacobian() 262 263 Output Parameters: 264 . tH - Jacobian matrix 265 266 */ 267 PetscErrorCode FormJacobian(Tao tao, Vec X, Mat H, Mat tHPre, void *ptr) 268 { 269 AppCtx *user = (AppCtx *) ptr; 270 PetscInt i,j,k,row; 271 PetscInt mx=user->mx, my=user->my; 272 PetscInt col[7]; 273 PetscReal hx=1.0/(mx+1), hy=1.0/(my+1), hydhx=hy/hx, hxdhy=hx/hy; 274 PetscReal f1,f2,f3,f4,f5,f6,d1,d2,d3,d4,d5,d6,d7,d8,xc,xl,xr,xt,xb,xlt,xrb; 275 PetscReal hl,hr,ht,hb,hc,htl,hbr; 276 const PetscScalar *x; 277 PetscScalar v[7]; 278 PetscBool assembled; 279 280 /* Set various matrix options */ 281 PetscFunctionBegin; 282 CHKERRQ(MatSetOption(H,MAT_IGNORE_OFF_PROC_ENTRIES,PETSC_TRUE)); 283 CHKERRQ(MatAssembled(H,&assembled)); 284 if (assembled) CHKERRQ(MatZeroEntries(H)); 285 286 /* Get pointers to vector data */ 287 CHKERRQ(VecGetArrayRead(X, &x)); 288 289 /* Compute Jacobian over the locally owned part of the mesh */ 290 for (i=0; i< mx; i++) { 291 for (j=0; j<my; j++) { 292 row= j*mx + i; 293 294 xc = x[row]; 295 xlt=xrb=xl=xr=xb=xt=xc; 296 297 /* Left side */ 298 if (i==0) { 299 xl = user->left[j+1]; 300 xlt = user->left[j+2]; 301 } else { 302 xl = x[row-1]; 303 } 304 305 if (j==0) { 306 xb = user->bottom[i+1]; 307 xrb = user->bottom[i+2]; 308 } else { 309 xb = x[row-mx]; 310 } 311 312 if (i+1 == mx) { 313 xr = user->right[j+1]; 314 xrb = user->right[j]; 315 } else { 316 xr = x[row+1]; 317 } 318 319 if (j+1==my) { 320 xt = user->top[i+1]; 321 xlt = user->top[i]; 322 }else { 323 xt = x[row+mx]; 324 } 325 326 if (i>0 && j+1<my) { 327 xlt = x[row-1+mx]; 328 } 329 if (j>0 && i+1<mx) { 330 xrb = x[row+1-mx]; 331 } 332 333 d1 = (xc-xl)/hx; 334 d2 = (xc-xr)/hx; 335 d3 = (xc-xt)/hy; 336 d4 = (xc-xb)/hy; 337 d5 = (xrb-xr)/hy; 338 d6 = (xrb-xb)/hx; 339 d7 = (xlt-xl)/hy; 340 d8 = (xlt-xt)/hx; 341 342 f1 = PetscSqrtScalar(1.0 + d1*d1 + d7*d7); 343 f2 = PetscSqrtScalar(1.0 + d1*d1 + d4*d4); 344 f3 = PetscSqrtScalar(1.0 + d3*d3 + d8*d8); 345 f4 = PetscSqrtScalar(1.0 + d3*d3 + d2*d2); 346 f5 = PetscSqrtScalar(1.0 + d2*d2 + d5*d5); 347 f6 = PetscSqrtScalar(1.0 + d4*d4 + d6*d6); 348 349 hl = (-hydhx*(1.0+d7*d7)+d1*d7)/(f1*f1*f1)+(-hydhx*(1.0+d4*d4)+d1*d4)/(f2*f2*f2); 350 hr = (-hydhx*(1.0+d5*d5)+d2*d5)/(f5*f5*f5)+(-hydhx*(1.0+d3*d3)+d2*d3)/(f4*f4*f4); 351 ht = (-hxdhy*(1.0+d8*d8)+d3*d8)/(f3*f3*f3)+(-hxdhy*(1.0+d2*d2)+d2*d3)/(f4*f4*f4); 352 hb = (-hxdhy*(1.0+d6*d6)+d4*d6)/(f6*f6*f6)+(-hxdhy*(1.0+d1*d1)+d1*d4)/(f2*f2*f2); 353 354 hbr = -d2*d5/(f5*f5*f5) - d4*d6/(f6*f6*f6); 355 htl = -d1*d7/(f1*f1*f1) - d3*d8/(f3*f3*f3); 356 357 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) + 358 (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); 359 360 hl/=2.0; hr/=2.0; ht/=2.0; hb/=2.0; hbr/=2.0; htl/=2.0; hc/=2.0; 361 362 k=0; 363 if (j>0) { 364 v[k]=hb; col[k]=row - mx; k++; 365 } 366 367 if (j>0 && i < mx -1) { 368 v[k]=hbr; col[k]=row - mx+1; k++; 369 } 370 371 if (i>0) { 372 v[k]= hl; col[k]=row - 1; k++; 373 } 374 375 v[k]= hc; col[k]=row; k++; 376 377 if (i < mx-1) { 378 v[k]= hr; col[k]=row+1; k++; 379 } 380 381 if (i>0 && j < my-1) { 382 v[k]= htl; col[k] = row+mx-1; k++; 383 } 384 385 if (j < my-1) { 386 v[k]= ht; col[k] = row+mx; k++; 387 } 388 389 /* 390 Set matrix values using local numbering, which was defined 391 earlier, in the main routine. 392 */ 393 CHKERRQ(MatSetValues(H,1,&row,k,col,v,INSERT_VALUES)); 394 } 395 } 396 397 /* Restore vectors */ 398 CHKERRQ(VecRestoreArrayRead(X,&x)); 399 400 /* Assemble the matrix */ 401 CHKERRQ(MatAssemblyBegin(H,MAT_FINAL_ASSEMBLY)); 402 CHKERRQ(MatAssemblyEnd(H,MAT_FINAL_ASSEMBLY)); 403 CHKERRQ(PetscLogFlops(199*mx*my)); 404 PetscFunctionReturn(0); 405 } 406 407 /* ------------------------------------------------------------------- */ 408 /* 409 MSA_BoundaryConditions - Calculates the boundary conditions for 410 the region. 411 412 Input Parameter: 413 . user - user-defined application context 414 415 Output Parameter: 416 . user - user-defined application context 417 */ 418 static PetscErrorCode MSA_BoundaryConditions(AppCtx * user) 419 { 420 PetscInt i,j,k,limit=0,maxits=5; 421 PetscInt mx=user->mx,my=user->my; 422 PetscInt bsize=0, lsize=0, tsize=0, rsize=0; 423 PetscReal one=1.0, two=2.0, three=3.0, tol=1e-10; 424 PetscReal fnorm,det,hx,hy,xt=0,yt=0; 425 PetscReal u1,u2,nf1,nf2,njac11,njac12,njac21,njac22; 426 PetscReal b=-0.5, t=0.5, l=-0.5, r=0.5; 427 PetscReal *boundary; 428 429 PetscFunctionBegin; 430 bsize=mx+2; lsize=my+2; rsize=my+2; tsize=mx+2; 431 432 CHKERRQ(PetscMalloc1(bsize, &user->bottom)); 433 CHKERRQ(PetscMalloc1(tsize, &user->top)); 434 CHKERRQ(PetscMalloc1(lsize, &user->left)); 435 CHKERRQ(PetscMalloc1(rsize, &user->right)); 436 437 hx= (r-l)/(mx+1); hy=(t-b)/(my+1); 438 439 for (j=0; j<4; j++) { 440 if (j==0) { 441 yt=b; 442 xt=l; 443 limit=bsize; 444 boundary=user->bottom; 445 } else if (j==1) { 446 yt=t; 447 xt=l; 448 limit=tsize; 449 boundary=user->top; 450 } else if (j==2) { 451 yt=b; 452 xt=l; 453 limit=lsize; 454 boundary=user->left; 455 } else { /* if (j==3) */ 456 yt=b; 457 xt=r; 458 limit=rsize; 459 boundary=user->right; 460 } 461 462 for (i=0; i<limit; i++) { 463 u1=xt; 464 u2=-yt; 465 for (k=0; k<maxits; k++) { 466 nf1=u1 + u1*u2*u2 - u1*u1*u1/three-xt; 467 nf2=-u2 - u1*u1*u2 + u2*u2*u2/three-yt; 468 fnorm=PetscSqrtScalar(nf1*nf1+nf2*nf2); 469 if (fnorm <= tol) break; 470 njac11=one+u2*u2-u1*u1; 471 njac12=two*u1*u2; 472 njac21=-two*u1*u2; 473 njac22=-one - u1*u1 + u2*u2; 474 det = njac11*njac22-njac21*njac12; 475 u1 = u1-(njac22*nf1-njac12*nf2)/det; 476 u2 = u2-(njac11*nf2-njac21*nf1)/det; 477 } 478 479 boundary[i]=u1*u1-u2*u2; 480 if (j==0 || j==1) { 481 xt=xt+hx; 482 } else { /* if (j==2 || j==3) */ 483 yt=yt+hy; 484 } 485 } 486 } 487 PetscFunctionReturn(0); 488 } 489 490 /* ------------------------------------------------------------------- */ 491 /* 492 MSA_InitialPoint - Calculates the initial guess in one of three ways. 493 494 Input Parameters: 495 . user - user-defined application context 496 . X - vector for initial guess 497 498 Output Parameters: 499 . X - newly computed initial guess 500 */ 501 static PetscErrorCode MSA_InitialPoint(AppCtx * user, Vec X) 502 { 503 PetscInt start=-1,i,j; 504 PetscScalar zero=0.0; 505 PetscBool flg; 506 507 PetscFunctionBegin; 508 CHKERRQ(PetscOptionsGetInt(NULL,NULL,"-start",&start,&flg)); 509 510 if (flg && start==0) { /* The zero vector is reasonable */ 511 CHKERRQ(VecSet(X, zero)); 512 } else { /* Take an average of the boundary conditions */ 513 PetscInt row; 514 PetscInt mx=user->mx,my=user->my; 515 PetscScalar *x; 516 517 /* Get pointers to vector data */ 518 CHKERRQ(VecGetArray(X,&x)); 519 520 /* Perform local computations */ 521 for (j=0; j<my; j++) { 522 for (i=0; i< mx; i++) { 523 row=(j)*mx + (i); 524 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; 525 } 526 } 527 528 /* Restore vectors */ 529 CHKERRQ(VecRestoreArray(X,&x)); 530 } 531 PetscFunctionReturn(0); 532 } 533 534 /*TEST 535 536 build: 537 requires: !complex 538 539 test: 540 args: -tao_monitor -tao_view -tao_type ssils -tao_gttol 1.e-5 541 requires: !single 542 543 test: 544 suffix: 2 545 args: -tao_monitor -tao_view -tao_type ssfls -tao_gttol 1.e-5 546 547 TEST*/ 548