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