1 static char help[] = "Simple example to test separable objective optimizers.\n"; 2 3 #include <petsc.h> 4 #include <petsctao.h> 5 #include <petscvec.h> 6 #include <petscmath.h> 7 8 #define NWORKLEFT 4 9 #define NWORKRIGHT 12 10 11 typedef struct _UserCtx { 12 PetscInt m; /* The row dimension of F */ 13 PetscInt n; /* The column dimension of F */ 14 PetscInt matops; /* Matrix format. 0 for stencil, 1 for random */ 15 PetscInt iter; /* Number of iterations for ADMM */ 16 PetscReal hStart; /* Starting point for Taylor test */ 17 PetscReal hFactor; /* Taylor test step factor */ 18 PetscReal hMin; /* Taylor test end goal */ 19 PetscReal alpha; /* regularization constant applied to || x ||_p */ 20 PetscReal eps; /* small constant for approximating gradient of || x ||_1 */ 21 PetscReal mu; /* the augmented Lagrangian term in ADMM */ 22 PetscReal abstol; 23 PetscReal reltol; 24 Mat F; /* matrix in least squares component $(1/2) * || F x - d ||_2^2$ */ 25 Mat W; /* Workspace matrix. ATA */ 26 Mat Hm; /* Hessian Misfit*/ 27 Mat Hr; /* Hessian Reg*/ 28 Vec d; /* RHS in least squares component $(1/2) * || F x - d ||_2^2$ */ 29 Vec workLeft[NWORKLEFT]; /* Workspace for temporary vec */ 30 Vec workRight[NWORKRIGHT]; /* Workspace for temporary vec */ 31 NormType p; 32 PetscRandom rctx; 33 PetscBool taylor; /* Flag to determine whether to run Taylor test or not */ 34 PetscBool use_admm; /* Flag to determine whether to run Taylor test or not */ 35 } *UserCtx; 36 37 static PetscErrorCode CreateRHS(UserCtx ctx) 38 { 39 PetscFunctionBegin; 40 /* build the rhs d in ctx */ 41 PetscCall(VecCreate(PETSC_COMM_WORLD, &(ctx->d))); 42 PetscCall(VecSetSizes(ctx->d, PETSC_DECIDE, ctx->m)); 43 PetscCall(VecSetFromOptions(ctx->d)); 44 PetscCall(VecSetRandom(ctx->d, ctx->rctx)); 45 PetscFunctionReturn(PETSC_SUCCESS); 46 } 47 48 static PetscErrorCode CreateMatrix(UserCtx ctx) 49 { 50 PetscInt Istart, Iend, i, j, Ii, gridN, I_n, I_s, I_e, I_w; 51 PetscLogStage stage; 52 53 PetscFunctionBegin; 54 /* build the matrix F in ctx */ 55 PetscCall(MatCreate(PETSC_COMM_WORLD, &(ctx->F))); 56 PetscCall(MatSetSizes(ctx->F, PETSC_DECIDE, PETSC_DECIDE, ctx->m, ctx->n)); 57 PetscCall(MatSetType(ctx->F, MATAIJ)); /* TODO: Decide specific SetType other than dummy*/ 58 PetscCall(MatMPIAIJSetPreallocation(ctx->F, 5, NULL, 5, NULL)); /*TODO: some number other than 5?*/ 59 PetscCall(MatSeqAIJSetPreallocation(ctx->F, 5, NULL)); 60 PetscCall(MatSetUp(ctx->F)); 61 PetscCall(MatGetOwnershipRange(ctx->F, &Istart, &Iend)); 62 PetscCall(PetscLogStageRegister("Assembly", &stage)); 63 PetscCall(PetscLogStagePush(stage)); 64 65 /* Set matrix elements in 2-D five point stencil format. */ 66 if (!(ctx->matops)) { 67 PetscCheck(ctx->m == ctx->n, PETSC_COMM_WORLD, PETSC_ERR_ARG_SIZ, "Stencil matrix must be square"); 68 gridN = (PetscInt)PetscSqrtReal((PetscReal)ctx->m); 69 PetscCheck(gridN * gridN == ctx->m, PETSC_COMM_WORLD, PETSC_ERR_ARG_SIZ, "Number of rows must be square"); 70 for (Ii = Istart; Ii < Iend; Ii++) { 71 i = Ii / gridN; 72 j = Ii % gridN; 73 I_n = i * gridN + j + 1; 74 if (j + 1 >= gridN) I_n = -1; 75 I_s = i * gridN + j - 1; 76 if (j - 1 < 0) I_s = -1; 77 I_e = (i + 1) * gridN + j; 78 if (i + 1 >= gridN) I_e = -1; 79 I_w = (i - 1) * gridN + j; 80 if (i - 1 < 0) I_w = -1; 81 PetscCall(MatSetValue(ctx->F, Ii, Ii, 4., INSERT_VALUES)); 82 PetscCall(MatSetValue(ctx->F, Ii, I_n, -1., INSERT_VALUES)); 83 PetscCall(MatSetValue(ctx->F, Ii, I_s, -1., INSERT_VALUES)); 84 PetscCall(MatSetValue(ctx->F, Ii, I_e, -1., INSERT_VALUES)); 85 PetscCall(MatSetValue(ctx->F, Ii, I_w, -1., INSERT_VALUES)); 86 } 87 } else PetscCall(MatSetRandom(ctx->F, ctx->rctx)); 88 PetscCall(MatAssemblyBegin(ctx->F, MAT_FINAL_ASSEMBLY)); 89 PetscCall(MatAssemblyEnd(ctx->F, MAT_FINAL_ASSEMBLY)); 90 PetscCall(PetscLogStagePop()); 91 /* Stencil matrix is symmetric. Setting symmetric flag for ICC/Cholesky preconditioner */ 92 if (!(ctx->matops)) PetscCall(MatSetOption(ctx->F, MAT_SYMMETRIC, PETSC_TRUE)); 93 PetscCall(MatTransposeMatMult(ctx->F, ctx->F, MAT_INITIAL_MATRIX, PETSC_DEFAULT, &(ctx->W))); 94 /* Setup Hessian Workspace in same shape as W */ 95 PetscCall(MatDuplicate(ctx->W, MAT_DO_NOT_COPY_VALUES, &(ctx->Hm))); 96 PetscCall(MatDuplicate(ctx->W, MAT_DO_NOT_COPY_VALUES, &(ctx->Hr))); 97 PetscFunctionReturn(PETSC_SUCCESS); 98 } 99 100 static PetscErrorCode SetupWorkspace(UserCtx ctx) 101 { 102 PetscInt i; 103 104 PetscFunctionBegin; 105 PetscCall(MatCreateVecs(ctx->F, &ctx->workLeft[0], &ctx->workRight[0])); 106 for (i = 1; i < NWORKLEFT; i++) PetscCall(VecDuplicate(ctx->workLeft[0], &(ctx->workLeft[i]))); 107 for (i = 1; i < NWORKRIGHT; i++) PetscCall(VecDuplicate(ctx->workRight[0], &(ctx->workRight[i]))); 108 PetscFunctionReturn(PETSC_SUCCESS); 109 } 110 111 static PetscErrorCode ConfigureContext(UserCtx ctx) 112 { 113 PetscFunctionBegin; 114 ctx->m = 16; 115 ctx->n = 16; 116 ctx->eps = 1.e-3; 117 ctx->abstol = 1.e-4; 118 ctx->reltol = 1.e-2; 119 ctx->hStart = 1.; 120 ctx->hMin = 1.e-3; 121 ctx->hFactor = 0.5; 122 ctx->alpha = 1.; 123 ctx->mu = 1.0; 124 ctx->matops = 0; 125 ctx->iter = 10; 126 ctx->p = NORM_2; 127 ctx->taylor = PETSC_TRUE; 128 ctx->use_admm = PETSC_FALSE; 129 PetscOptionsBegin(PETSC_COMM_WORLD, NULL, "Configure separable objection example", "ex4.c"); 130 PetscCall(PetscOptionsInt("-m", "The row dimension of matrix F", "ex4.c", ctx->m, &(ctx->m), NULL)); 131 PetscCall(PetscOptionsInt("-n", "The column dimension of matrix F", "ex4.c", ctx->n, &(ctx->n), NULL)); 132 PetscCall(PetscOptionsInt("-matrix_format", "Decide format of F matrix. 0 for stencil, 1 for random", "ex4.c", ctx->matops, &(ctx->matops), NULL)); 133 PetscCall(PetscOptionsInt("-iter", "Iteration number ADMM", "ex4.c", ctx->iter, &(ctx->iter), NULL)); 134 PetscCall(PetscOptionsReal("-alpha", "The regularization multiplier. 1 default", "ex4.c", ctx->alpha, &(ctx->alpha), NULL)); 135 PetscCall(PetscOptionsReal("-epsilon", "The small constant added to |x_i| in the denominator to approximate the gradient of ||x||_1", "ex4.c", ctx->eps, &(ctx->eps), NULL)); 136 PetscCall(PetscOptionsReal("-mu", "The augmented lagrangian multiplier in ADMM", "ex4.c", ctx->mu, &(ctx->mu), NULL)); 137 PetscCall(PetscOptionsReal("-hStart", "Taylor test starting point. 1 default.", "ex4.c", ctx->hStart, &(ctx->hStart), NULL)); 138 PetscCall(PetscOptionsReal("-hFactor", "Taylor test multiplier factor. 0.5 default", "ex4.c", ctx->hFactor, &(ctx->hFactor), NULL)); 139 PetscCall(PetscOptionsReal("-hMin", "Taylor test ending condition. 1.e-3 default", "ex4.c", ctx->hMin, &(ctx->hMin), NULL)); 140 PetscCall(PetscOptionsReal("-abstol", "Absolute stopping criterion for ADMM", "ex4.c", ctx->abstol, &(ctx->abstol), NULL)); 141 PetscCall(PetscOptionsReal("-reltol", "Relative stopping criterion for ADMM", "ex4.c", ctx->reltol, &(ctx->reltol), NULL)); 142 PetscCall(PetscOptionsBool("-taylor", "Flag for Taylor test. Default is true.", "ex4.c", ctx->taylor, &(ctx->taylor), NULL)); 143 PetscCall(PetscOptionsBool("-use_admm", "Use the ADMM solver in this example.", "ex4.c", ctx->use_admm, &(ctx->use_admm), NULL)); 144 PetscCall(PetscOptionsEnum("-p", "Norm type.", "ex4.c", NormTypes, (PetscEnum)ctx->p, (PetscEnum *)&(ctx->p), NULL)); 145 PetscOptionsEnd(); 146 /* Creating random ctx */ 147 PetscCall(PetscRandomCreate(PETSC_COMM_WORLD, &(ctx->rctx))); 148 PetscCall(PetscRandomSetFromOptions(ctx->rctx)); 149 PetscCall(CreateMatrix(ctx)); 150 PetscCall(CreateRHS(ctx)); 151 PetscCall(SetupWorkspace(ctx)); 152 PetscFunctionReturn(PETSC_SUCCESS); 153 } 154 155 static PetscErrorCode DestroyContext(UserCtx *ctx) 156 { 157 PetscInt i; 158 159 PetscFunctionBegin; 160 PetscCall(MatDestroy(&((*ctx)->F))); 161 PetscCall(MatDestroy(&((*ctx)->W))); 162 PetscCall(MatDestroy(&((*ctx)->Hm))); 163 PetscCall(MatDestroy(&((*ctx)->Hr))); 164 PetscCall(VecDestroy(&((*ctx)->d))); 165 for (i = 0; i < NWORKLEFT; i++) PetscCall(VecDestroy(&((*ctx)->workLeft[i]))); 166 for (i = 0; i < NWORKRIGHT; i++) PetscCall(VecDestroy(&((*ctx)->workRight[i]))); 167 PetscCall(PetscRandomDestroy(&((*ctx)->rctx))); 168 PetscCall(PetscFree(*ctx)); 169 PetscFunctionReturn(PETSC_SUCCESS); 170 } 171 172 /* compute (1/2) * ||F x - d||^2 */ 173 static PetscErrorCode ObjectiveMisfit(Tao tao, Vec x, PetscReal *J, void *_ctx) 174 { 175 UserCtx ctx = (UserCtx)_ctx; 176 Vec y; 177 178 PetscFunctionBegin; 179 y = ctx->workLeft[0]; 180 PetscCall(MatMult(ctx->F, x, y)); 181 PetscCall(VecAXPY(y, -1., ctx->d)); 182 PetscCall(VecDot(y, y, J)); 183 *J *= 0.5; 184 PetscFunctionReturn(PETSC_SUCCESS); 185 } 186 187 /* compute V = FTFx - FTd */ 188 static PetscErrorCode GradientMisfit(Tao tao, Vec x, Vec V, void *_ctx) 189 { 190 UserCtx ctx = (UserCtx)_ctx; 191 Vec FTFx, FTd; 192 193 PetscFunctionBegin; 194 /* work1 is A^T Ax, work2 is Ab, W is A^T A*/ 195 FTFx = ctx->workRight[0]; 196 FTd = ctx->workRight[1]; 197 PetscCall(MatMult(ctx->W, x, FTFx)); 198 PetscCall(MatMultTranspose(ctx->F, ctx->d, FTd)); 199 PetscCall(VecWAXPY(V, -1., FTd, FTFx)); 200 PetscFunctionReturn(PETSC_SUCCESS); 201 } 202 203 /* returns FTF */ 204 static PetscErrorCode HessianMisfit(Tao tao, Vec x, Mat H, Mat Hpre, void *_ctx) 205 { 206 UserCtx ctx = (UserCtx)_ctx; 207 208 PetscFunctionBegin; 209 if (H != ctx->W) PetscCall(MatCopy(ctx->W, H, DIFFERENT_NONZERO_PATTERN)); 210 if (Hpre != ctx->W) PetscCall(MatCopy(ctx->W, Hpre, DIFFERENT_NONZERO_PATTERN)); 211 PetscFunctionReturn(PETSC_SUCCESS); 212 } 213 214 /* computes augment Lagrangian objective (with scaled dual): 215 * 0.5 * ||F x - d||^2 + 0.5 * mu ||x - z + u||^2 */ 216 static PetscErrorCode ObjectiveMisfitADMM(Tao tao, Vec x, PetscReal *J, void *_ctx) 217 { 218 UserCtx ctx = (UserCtx)_ctx; 219 PetscReal mu, workNorm, misfit; 220 Vec z, u, temp; 221 222 PetscFunctionBegin; 223 mu = ctx->mu; 224 z = ctx->workRight[5]; 225 u = ctx->workRight[6]; 226 temp = ctx->workRight[10]; 227 /* misfit = f(x) */ 228 PetscCall(ObjectiveMisfit(tao, x, &misfit, _ctx)); 229 PetscCall(VecCopy(x, temp)); 230 /* temp = x - z + u */ 231 PetscCall(VecAXPBYPCZ(temp, -1., 1., 1., z, u)); 232 /* workNorm = ||x - z + u||^2 */ 233 PetscCall(VecDot(temp, temp, &workNorm)); 234 /* augment Lagrangian objective (with scaled dual): f(x) + 0.5 * mu ||x -z + u||^2 */ 235 *J = misfit + 0.5 * mu * workNorm; 236 PetscFunctionReturn(PETSC_SUCCESS); 237 } 238 239 /* computes FTFx - FTd mu*(x - z + u) */ 240 static PetscErrorCode GradientMisfitADMM(Tao tao, Vec x, Vec V, void *_ctx) 241 { 242 UserCtx ctx = (UserCtx)_ctx; 243 PetscReal mu; 244 Vec z, u, temp; 245 246 PetscFunctionBegin; 247 mu = ctx->mu; 248 z = ctx->workRight[5]; 249 u = ctx->workRight[6]; 250 temp = ctx->workRight[10]; 251 PetscCall(GradientMisfit(tao, x, V, _ctx)); 252 PetscCall(VecCopy(x, temp)); 253 /* temp = x - z + u */ 254 PetscCall(VecAXPBYPCZ(temp, -1., 1., 1., z, u)); 255 /* V = FTFx - FTd mu*(x - z + u) */ 256 PetscCall(VecAXPY(V, mu, temp)); 257 PetscFunctionReturn(PETSC_SUCCESS); 258 } 259 260 /* returns FTF + diag(mu) */ 261 static PetscErrorCode HessianMisfitADMM(Tao tao, Vec x, Mat H, Mat Hpre, void *_ctx) 262 { 263 UserCtx ctx = (UserCtx)_ctx; 264 265 PetscFunctionBegin; 266 PetscCall(MatCopy(ctx->W, H, DIFFERENT_NONZERO_PATTERN)); 267 PetscCall(MatShift(H, ctx->mu)); 268 if (Hpre != H) PetscCall(MatCopy(H, Hpre, DIFFERENT_NONZERO_PATTERN)); 269 PetscFunctionReturn(PETSC_SUCCESS); 270 } 271 272 /* computes || x ||_p (mult by 0.5 in case of NORM_2) */ 273 static PetscErrorCode ObjectiveRegularization(Tao tao, Vec x, PetscReal *J, void *_ctx) 274 { 275 UserCtx ctx = (UserCtx)_ctx; 276 PetscReal norm; 277 278 PetscFunctionBegin; 279 *J = 0; 280 PetscCall(VecNorm(x, ctx->p, &norm)); 281 if (ctx->p == NORM_2) norm = 0.5 * norm * norm; 282 *J = ctx->alpha * norm; 283 PetscFunctionReturn(PETSC_SUCCESS); 284 } 285 286 /* NORM_2 Case: return x 287 * NORM_1 Case: x/(|x| + eps) 288 * Else: TODO */ 289 static PetscErrorCode GradientRegularization(Tao tao, Vec x, Vec V, void *_ctx) 290 { 291 UserCtx ctx = (UserCtx)_ctx; 292 PetscReal eps = ctx->eps; 293 294 PetscFunctionBegin; 295 if (ctx->p == NORM_2) { 296 PetscCall(VecCopy(x, V)); 297 } else if (ctx->p == NORM_1) { 298 PetscCall(VecCopy(x, ctx->workRight[1])); 299 PetscCall(VecAbs(ctx->workRight[1])); 300 PetscCall(VecShift(ctx->workRight[1], eps)); 301 PetscCall(VecPointwiseDivide(V, x, ctx->workRight[1])); 302 } else SETERRQ(PetscObjectComm((PetscObject)tao), PETSC_ERR_ARG_OUTOFRANGE, "Example only works for NORM_1 and NORM_2"); 303 PetscFunctionReturn(PETSC_SUCCESS); 304 } 305 306 /* NORM_2 Case: returns diag(mu) 307 * NORM_1 Case: diag(mu* 1/sqrt(x_i^2 + eps) * (1 - x_i^2/ABS(x_i^2+eps))) */ 308 static PetscErrorCode HessianRegularization(Tao tao, Vec x, Mat H, Mat Hpre, void *_ctx) 309 { 310 UserCtx ctx = (UserCtx)_ctx; 311 PetscReal eps = ctx->eps; 312 Vec copy1, copy2, copy3; 313 314 PetscFunctionBegin; 315 if (ctx->p == NORM_2) { 316 /* Identity matrix scaled by mu */ 317 PetscCall(MatZeroEntries(H)); 318 PetscCall(MatShift(H, ctx->mu)); 319 if (Hpre != H) { 320 PetscCall(MatZeroEntries(Hpre)); 321 PetscCall(MatShift(Hpre, ctx->mu)); 322 } 323 } else if (ctx->p == NORM_1) { 324 /* 1/sqrt(x_i^2 + eps) * (1 - x_i^2/ABS(x_i^2+eps)) */ 325 copy1 = ctx->workRight[1]; 326 copy2 = ctx->workRight[2]; 327 copy3 = ctx->workRight[3]; 328 /* copy1 : 1/sqrt(x_i^2 + eps) */ 329 PetscCall(VecCopy(x, copy1)); 330 PetscCall(VecPow(copy1, 2)); 331 PetscCall(VecShift(copy1, eps)); 332 PetscCall(VecSqrtAbs(copy1)); 333 PetscCall(VecReciprocal(copy1)); 334 /* copy2: x_i^2.*/ 335 PetscCall(VecCopy(x, copy2)); 336 PetscCall(VecPow(copy2, 2)); 337 /* copy3: abs(x_i^2 + eps) */ 338 PetscCall(VecCopy(x, copy3)); 339 PetscCall(VecPow(copy3, 2)); 340 PetscCall(VecShift(copy3, eps)); 341 PetscCall(VecAbs(copy3)); 342 /* copy2: 1 - x_i^2/abs(x_i^2 + eps) */ 343 PetscCall(VecPointwiseDivide(copy2, copy2, copy3)); 344 PetscCall(VecScale(copy2, -1.)); 345 PetscCall(VecShift(copy2, 1.)); 346 PetscCall(VecAXPY(copy1, 1., copy2)); 347 PetscCall(VecScale(copy1, ctx->mu)); 348 PetscCall(MatZeroEntries(H)); 349 PetscCall(MatDiagonalSet(H, copy1, INSERT_VALUES)); 350 if (Hpre != H) { 351 PetscCall(MatZeroEntries(Hpre)); 352 PetscCall(MatDiagonalSet(Hpre, copy1, INSERT_VALUES)); 353 } 354 } else SETERRQ(PetscObjectComm((PetscObject)tao), PETSC_ERR_ARG_OUTOFRANGE, "Example only works for NORM_1 and NORM_2"); 355 PetscFunctionReturn(PETSC_SUCCESS); 356 } 357 358 /* NORM_2 Case: 0.5 || x ||_2 + 0.5 * mu * ||x + u - z||^2 359 * Else : || x ||_2 + 0.5 * mu * ||x + u - z||^2 */ 360 static PetscErrorCode ObjectiveRegularizationADMM(Tao tao, Vec z, PetscReal *J, void *_ctx) 361 { 362 UserCtx ctx = (UserCtx)_ctx; 363 PetscReal mu, workNorm, reg; 364 Vec x, u, temp; 365 366 PetscFunctionBegin; 367 mu = ctx->mu; 368 x = ctx->workRight[4]; 369 u = ctx->workRight[6]; 370 temp = ctx->workRight[10]; 371 PetscCall(ObjectiveRegularization(tao, z, ®, _ctx)); 372 PetscCall(VecCopy(z, temp)); 373 /* temp = x + u -z */ 374 PetscCall(VecAXPBYPCZ(temp, 1., 1., -1., x, u)); 375 /* workNorm = ||x + u - z ||^2 */ 376 PetscCall(VecDot(temp, temp, &workNorm)); 377 *J = reg + 0.5 * mu * workNorm; 378 PetscFunctionReturn(PETSC_SUCCESS); 379 } 380 381 /* NORM_2 Case: x - mu*(x + u - z) 382 * NORM_1 Case: x/(|x| + eps) - mu*(x + u - z) 383 * Else: TODO */ 384 static PetscErrorCode GradientRegularizationADMM(Tao tao, Vec z, Vec V, void *_ctx) 385 { 386 UserCtx ctx = (UserCtx)_ctx; 387 PetscReal mu; 388 Vec x, u, temp; 389 390 PetscFunctionBegin; 391 mu = ctx->mu; 392 x = ctx->workRight[4]; 393 u = ctx->workRight[6]; 394 temp = ctx->workRight[10]; 395 PetscCall(GradientRegularization(tao, z, V, _ctx)); 396 PetscCall(VecCopy(z, temp)); 397 /* temp = x + u -z */ 398 PetscCall(VecAXPBYPCZ(temp, 1., 1., -1., x, u)); 399 PetscCall(VecAXPY(V, -mu, temp)); 400 PetscFunctionReturn(PETSC_SUCCESS); 401 } 402 403 /* NORM_2 Case: returns diag(mu) 404 * NORM_1 Case: FTF + diag(mu) */ 405 static PetscErrorCode HessianRegularizationADMM(Tao tao, Vec x, Mat H, Mat Hpre, void *_ctx) 406 { 407 UserCtx ctx = (UserCtx)_ctx; 408 409 PetscFunctionBegin; 410 if (ctx->p == NORM_2) { 411 /* Identity matrix scaled by mu */ 412 PetscCall(MatZeroEntries(H)); 413 PetscCall(MatShift(H, ctx->mu)); 414 if (Hpre != H) { 415 PetscCall(MatZeroEntries(Hpre)); 416 PetscCall(MatShift(Hpre, ctx->mu)); 417 } 418 } else if (ctx->p == NORM_1) { 419 PetscCall(HessianMisfit(tao, x, H, Hpre, (void *)ctx)); 420 PetscCall(MatShift(H, ctx->mu)); 421 if (Hpre != H) PetscCall(MatShift(Hpre, ctx->mu)); 422 } else SETERRQ(PetscObjectComm((PetscObject)tao), PETSC_ERR_ARG_OUTOFRANGE, "Example only works for NORM_1 and NORM_2"); 423 PetscFunctionReturn(PETSC_SUCCESS); 424 } 425 426 /* NORM_2 Case : (1/2) * ||F x - d||^2 + 0.5 * || x ||_p 427 * NORM_1 Case : (1/2) * ||F x - d||^2 + || x ||_p */ 428 static PetscErrorCode ObjectiveComplete(Tao tao, Vec x, PetscReal *J, void *ctx) 429 { 430 PetscReal Jm, Jr; 431 432 PetscFunctionBegin; 433 PetscCall(ObjectiveMisfit(tao, x, &Jm, ctx)); 434 PetscCall(ObjectiveRegularization(tao, x, &Jr, ctx)); 435 *J = Jm + Jr; 436 PetscFunctionReturn(PETSC_SUCCESS); 437 } 438 439 /* NORM_2 Case: FTFx - FTd + x 440 * NORM_1 Case: FTFx - FTd + x/(|x| + eps) */ 441 static PetscErrorCode GradientComplete(Tao tao, Vec x, Vec V, void *ctx) 442 { 443 UserCtx cntx = (UserCtx)ctx; 444 445 PetscFunctionBegin; 446 PetscCall(GradientMisfit(tao, x, cntx->workRight[2], ctx)); 447 PetscCall(GradientRegularization(tao, x, cntx->workRight[3], ctx)); 448 PetscCall(VecWAXPY(V, 1, cntx->workRight[2], cntx->workRight[3])); 449 PetscFunctionReturn(PETSC_SUCCESS); 450 } 451 452 /* NORM_2 Case: diag(mu) + FTF 453 * NORM_1 Case: diag(mu* 1/sqrt(x_i^2 + eps) * (1 - x_i^2/ABS(x_i^2+eps))) + FTF */ 454 static PetscErrorCode HessianComplete(Tao tao, Vec x, Mat H, Mat Hpre, void *ctx) 455 { 456 Mat tempH; 457 458 PetscFunctionBegin; 459 PetscCall(MatDuplicate(H, MAT_SHARE_NONZERO_PATTERN, &tempH)); 460 PetscCall(HessianMisfit(tao, x, H, H, ctx)); 461 PetscCall(HessianRegularization(tao, x, tempH, tempH, ctx)); 462 PetscCall(MatAXPY(H, 1., tempH, DIFFERENT_NONZERO_PATTERN)); 463 if (Hpre != H) PetscCall(MatCopy(H, Hpre, DIFFERENT_NONZERO_PATTERN)); 464 PetscCall(MatDestroy(&tempH)); 465 PetscFunctionReturn(PETSC_SUCCESS); 466 } 467 468 static PetscErrorCode TaoSolveADMM(UserCtx ctx, Vec x) 469 { 470 PetscInt i; 471 PetscReal u_norm, r_norm, s_norm, primal, dual, x_norm, z_norm; 472 Tao tao1, tao2; 473 Vec xk, z, u, diff, zold, zdiff, temp; 474 PetscReal mu; 475 476 PetscFunctionBegin; 477 xk = ctx->workRight[4]; 478 z = ctx->workRight[5]; 479 u = ctx->workRight[6]; 480 diff = ctx->workRight[7]; 481 zold = ctx->workRight[8]; 482 zdiff = ctx->workRight[9]; 483 temp = ctx->workRight[11]; 484 mu = ctx->mu; 485 PetscCall(VecSet(u, 0.)); 486 PetscCall(TaoCreate(PETSC_COMM_WORLD, &tao1)); 487 PetscCall(TaoSetType(tao1, TAONLS)); 488 PetscCall(TaoSetObjective(tao1, ObjectiveMisfitADMM, (void *)ctx)); 489 PetscCall(TaoSetGradient(tao1, NULL, GradientMisfitADMM, (void *)ctx)); 490 PetscCall(TaoSetHessian(tao1, ctx->Hm, ctx->Hm, HessianMisfitADMM, (void *)ctx)); 491 PetscCall(VecSet(xk, 0.)); 492 PetscCall(TaoSetSolution(tao1, xk)); 493 PetscCall(TaoSetOptionsPrefix(tao1, "misfit_")); 494 PetscCall(TaoSetFromOptions(tao1)); 495 PetscCall(TaoCreate(PETSC_COMM_WORLD, &tao2)); 496 if (ctx->p == NORM_2) { 497 PetscCall(TaoSetType(tao2, TAONLS)); 498 PetscCall(TaoSetObjective(tao2, ObjectiveRegularizationADMM, (void *)ctx)); 499 PetscCall(TaoSetGradient(tao2, NULL, GradientRegularizationADMM, (void *)ctx)); 500 PetscCall(TaoSetHessian(tao2, ctx->Hr, ctx->Hr, HessianRegularizationADMM, (void *)ctx)); 501 } 502 PetscCall(VecSet(z, 0.)); 503 PetscCall(TaoSetSolution(tao2, z)); 504 PetscCall(TaoSetOptionsPrefix(tao2, "reg_")); 505 PetscCall(TaoSetFromOptions(tao2)); 506 507 for (i = 0; i < ctx->iter; i++) { 508 PetscCall(VecCopy(z, zold)); 509 PetscCall(TaoSolve(tao1)); /* Updates xk */ 510 if (ctx->p == NORM_1) { 511 PetscCall(VecWAXPY(temp, 1., xk, u)); 512 PetscCall(TaoSoftThreshold(temp, -ctx->alpha / mu, ctx->alpha / mu, z)); 513 } else { 514 PetscCall(TaoSolve(tao2)); /* Update zk */ 515 } 516 /* u = u + xk -z */ 517 PetscCall(VecAXPBYPCZ(u, 1., -1., 1., xk, z)); 518 /* r_norm : norm(x-z) */ 519 PetscCall(VecWAXPY(diff, -1., z, xk)); 520 PetscCall(VecNorm(diff, NORM_2, &r_norm)); 521 /* s_norm : norm(-mu(z-zold)) */ 522 PetscCall(VecWAXPY(zdiff, -1., zold, z)); 523 PetscCall(VecNorm(zdiff, NORM_2, &s_norm)); 524 s_norm = s_norm * mu; 525 /* primal : sqrt(n)*ABSTOL + RELTOL*max(norm(x), norm(-z))*/ 526 PetscCall(VecNorm(xk, NORM_2, &x_norm)); 527 PetscCall(VecNorm(z, NORM_2, &z_norm)); 528 primal = PetscSqrtReal(ctx->n) * ctx->abstol + ctx->reltol * PetscMax(x_norm, z_norm); 529 /* Duality : sqrt(n)*ABSTOL + RELTOL*norm(mu*u)*/ 530 PetscCall(VecNorm(u, NORM_2, &u_norm)); 531 dual = PetscSqrtReal(ctx->n) * ctx->abstol + ctx->reltol * u_norm * mu; 532 PetscCall(PetscPrintf(PetscObjectComm((PetscObject)tao1), "Iter %" PetscInt_FMT " : ||x-z||: %g, mu*||z-zold||: %g\n", i, (double)r_norm, (double)s_norm)); 533 if (r_norm < primal && s_norm < dual) break; 534 } 535 PetscCall(VecCopy(xk, x)); 536 PetscCall(TaoDestroy(&tao1)); 537 PetscCall(TaoDestroy(&tao2)); 538 PetscFunctionReturn(PETSC_SUCCESS); 539 } 540 541 /* Second order Taylor remainder convergence test */ 542 static PetscErrorCode TaylorTest(UserCtx ctx, Tao tao, Vec x, PetscReal *C) 543 { 544 PetscReal h, J, temp; 545 PetscInt i, j; 546 PetscInt numValues; 547 PetscReal Jx, Jxhat_comp, Jxhat_pred; 548 PetscReal *Js, *hs; 549 PetscReal gdotdx; 550 PetscReal minrate = PETSC_MAX_REAL; 551 MPI_Comm comm = PetscObjectComm((PetscObject)x); 552 Vec g, dx, xhat; 553 554 PetscFunctionBegin; 555 PetscCall(VecDuplicate(x, &g)); 556 PetscCall(VecDuplicate(x, &xhat)); 557 /* choose a perturbation direction */ 558 PetscCall(VecDuplicate(x, &dx)); 559 PetscCall(VecSetRandom(dx, ctx->rctx)); 560 /* evaluate objective at x: J(x) */ 561 PetscCall(TaoComputeObjective(tao, x, &Jx)); 562 /* evaluate gradient at x, save in vector g */ 563 PetscCall(TaoComputeGradient(tao, x, g)); 564 PetscCall(VecDot(g, dx, &gdotdx)); 565 566 for (numValues = 0, h = ctx->hStart; h >= ctx->hMin; h *= ctx->hFactor) numValues++; 567 PetscCall(PetscCalloc2(numValues, &Js, numValues, &hs)); 568 for (i = 0, h = ctx->hStart; h >= ctx->hMin; h *= ctx->hFactor, i++) { 569 PetscCall(VecWAXPY(xhat, h, dx, x)); 570 PetscCall(TaoComputeObjective(tao, xhat, &Jxhat_comp)); 571 /* J(\hat(x)) \approx J(x) + g^T (xhat - x) = J(x) + h * g^T dx */ 572 Jxhat_pred = Jx + h * gdotdx; 573 /* Vector to dJdm scalar? Dot?*/ 574 J = PetscAbsReal(Jxhat_comp - Jxhat_pred); 575 PetscCall(PetscPrintf(comm, "J(xhat): %g, predicted: %g, diff %g\n", (double)Jxhat_comp, (double)Jxhat_pred, (double)J)); 576 Js[i] = J; 577 hs[i] = h; 578 } 579 for (j = 1; j < numValues; j++) { 580 temp = PetscLogReal(Js[j] / Js[j - 1]) / PetscLogReal(hs[j] / hs[j - 1]); 581 PetscCall(PetscPrintf(comm, "Convergence rate step %" PetscInt_FMT ": %g\n", j - 1, (double)temp)); 582 minrate = PetscMin(minrate, temp); 583 } 584 /* If O is not ~2, then the test is wrong */ 585 PetscCall(PetscFree2(Js, hs)); 586 *C = minrate; 587 PetscCall(VecDestroy(&dx)); 588 PetscCall(VecDestroy(&xhat)); 589 PetscCall(VecDestroy(&g)); 590 PetscFunctionReturn(PETSC_SUCCESS); 591 } 592 593 int main(int argc, char **argv) 594 { 595 UserCtx ctx; 596 Tao tao; 597 Vec x; 598 Mat H; 599 600 PetscFunctionBeginUser; 601 PetscCall(PetscInitialize(&argc, &argv, NULL, help)); 602 PetscCall(PetscNew(&ctx)); 603 PetscCall(ConfigureContext(ctx)); 604 /* Define two functions that could pass as objectives to TaoSetObjective(): one 605 * for the misfit component, and one for the regularization component */ 606 /* ObjectiveMisfit() and ObjectiveRegularization() */ 607 608 /* Define a single function that calls both components adds them together: the complete objective, 609 * in the absence of a Tao implementation that handles separability */ 610 /* ObjectiveComplete() */ 611 PetscCall(TaoCreate(PETSC_COMM_WORLD, &tao)); 612 PetscCall(TaoSetType(tao, TAONM)); 613 PetscCall(TaoSetObjective(tao, ObjectiveComplete, (void *)ctx)); 614 PetscCall(TaoSetGradient(tao, NULL, GradientComplete, (void *)ctx)); 615 PetscCall(MatDuplicate(ctx->W, MAT_SHARE_NONZERO_PATTERN, &H)); 616 PetscCall(TaoSetHessian(tao, H, H, HessianComplete, (void *)ctx)); 617 PetscCall(MatCreateVecs(ctx->F, NULL, &x)); 618 PetscCall(VecSet(x, 0.)); 619 PetscCall(TaoSetSolution(tao, x)); 620 PetscCall(TaoSetFromOptions(tao)); 621 if (ctx->use_admm) PetscCall(TaoSolveADMM(ctx, x)); 622 else PetscCall(TaoSolve(tao)); 623 /* examine solution */ 624 PetscCall(VecViewFromOptions(x, NULL, "-view_sol")); 625 if (ctx->taylor) { 626 PetscReal rate; 627 PetscCall(TaylorTest(ctx, tao, x, &rate)); 628 } 629 PetscCall(MatDestroy(&H)); 630 PetscCall(TaoDestroy(&tao)); 631 PetscCall(VecDestroy(&x)); 632 PetscCall(DestroyContext(&ctx)); 633 PetscCall(PetscFinalize()); 634 return 0; 635 } 636 637 /*TEST 638 639 build: 640 requires: !complex 641 642 test: 643 suffix: 0 644 args: 645 646 test: 647 suffix: l1_1 648 args: -p 1 -tao_type lmvm -alpha 1. -epsilon 1.e-7 -m 64 -n 64 -view_sol -matrix_format 1 649 650 test: 651 suffix: hessian_1 652 args: -matrix_format 1 -m 100 -n 100 -tao_monitor -p 1 -tao_type nls 653 654 test: 655 suffix: hessian_2 656 args: -matrix_format 1 -m 100 -n 100 -tao_monitor -p 2 -tao_type nls 657 658 test: 659 suffix: nm_1 660 args: -matrix_format 1 -m 100 -n 100 -tao_monitor -p 1 -tao_type nm -tao_max_it 50 661 662 test: 663 suffix: nm_2 664 args: -matrix_format 1 -m 100 -n 100 -tao_monitor -p 2 -tao_type nm -tao_max_it 50 665 666 test: 667 suffix: lmvm_1 668 args: -matrix_format 1 -m 100 -n 100 -tao_monitor -p 1 -tao_type lmvm -tao_max_it 40 669 670 test: 671 suffix: lmvm_2 672 args: -matrix_format 1 -m 100 -n 100 -tao_monitor -p 2 -tao_type lmvm -tao_max_it 15 673 674 test: 675 suffix: soft_threshold_admm_1 676 args: -matrix_format 1 -m 100 -n 100 -tao_monitor -p 1 -use_admm 677 678 test: 679 suffix: hessian_admm_1 680 args: -matrix_format 1 -m 100 -n 100 -tao_monitor -p 1 -use_admm -reg_tao_type nls -misfit_tao_type nls 681 682 test: 683 suffix: hessian_admm_2 684 args: -matrix_format 1 -m 100 -n 100 -tao_monitor -p 2 -use_admm -reg_tao_type nls -misfit_tao_type nls 685 686 test: 687 suffix: nm_admm_1 688 args: -matrix_format 1 -m 100 -n 100 -tao_monitor -p 1 -use_admm -reg_tao_type nm -misfit_tao_type nm 689 690 test: 691 suffix: nm_admm_2 692 args: -matrix_format 1 -m 100 -n 100 -tao_monitor -p 2 -use_admm -reg_tao_type nm -misfit_tao_type nm -iter 7 693 694 test: 695 suffix: lmvm_admm_1 696 args: -matrix_format 1 -m 100 -n 100 -tao_monitor -p 1 -use_admm -reg_tao_type lmvm -misfit_tao_type lmvm 697 698 test: 699 suffix: lmvm_admm_2 700 args: -matrix_format 1 -m 100 -n 100 -tao_monitor -p 2 -use_admm -reg_tao_type lmvm -misfit_tao_type lmvm 701 702 TEST*/ 703