Actual source code: ex4.c

  1: static char help[] = "Simple example to test separable objective optimizers.\n";

  3: #include <petsc.h>
  4: #include <petsctao.h>
  5: #include <petscvec.h>
  6: #include <petscmath.h>

  8: #define NWORKLEFT  4
  9: #define NWORKRIGHT 12

 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;    /* Numer 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;

 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: }

 48: static PetscErrorCode CreateMatrix(UserCtx ctx)
 49: {
 50:   PetscInt Istart, Iend, i, j, Ii, gridN, I_n, I_s, I_e, I_w;
 51: #if defined(PETSC_USE_LOG)
 52:   PetscLogStage stage;
 53: #endif

 55:   PetscFunctionBegin;
 56:   /* build the matrix F in ctx */
 57:   PetscCall(MatCreate(PETSC_COMM_WORLD, &(ctx->F)));
 58:   PetscCall(MatSetSizes(ctx->F, PETSC_DECIDE, PETSC_DECIDE, ctx->m, ctx->n));
 59:   PetscCall(MatSetType(ctx->F, MATAIJ));                          /* TODO: Decide specific SetType other than dummy*/
 60:   PetscCall(MatMPIAIJSetPreallocation(ctx->F, 5, NULL, 5, NULL)); /*TODO: some number other than 5?*/
 61:   PetscCall(MatSeqAIJSetPreallocation(ctx->F, 5, NULL));
 62:   PetscCall(MatSetUp(ctx->F));
 63:   PetscCall(MatGetOwnershipRange(ctx->F, &Istart, &Iend));
 64:   PetscCall(PetscLogStageRegister("Assembly", &stage));
 65:   PetscCall(PetscLogStagePush(stage));

 67:   /* Set matrix elements in  2-D five point stencil format. */
 68:   if (!(ctx->matops)) {
 69:     PetscCheck(ctx->m == ctx->n, PETSC_COMM_WORLD, PETSC_ERR_ARG_SIZ, "Stencil matrix must be square");
 70:     gridN = (PetscInt)PetscSqrtReal((PetscReal)ctx->m);
 71:     PetscCheck(gridN * gridN == ctx->m, PETSC_COMM_WORLD, PETSC_ERR_ARG_SIZ, "Number of rows must be square");
 72:     for (Ii = Istart; Ii < Iend; Ii++) {
 73:       i   = Ii / gridN;
 74:       j   = Ii % gridN;
 75:       I_n = i * gridN + j + 1;
 76:       if (j + 1 >= gridN) I_n = -1;
 77:       I_s = i * gridN + j - 1;
 78:       if (j - 1 < 0) I_s = -1;
 79:       I_e = (i + 1) * gridN + j;
 80:       if (i + 1 >= gridN) I_e = -1;
 81:       I_w = (i - 1) * gridN + j;
 82:       if (i - 1 < 0) I_w = -1;
 83:       PetscCall(MatSetValue(ctx->F, Ii, Ii, 4., INSERT_VALUES));
 84:       PetscCall(MatSetValue(ctx->F, Ii, I_n, -1., INSERT_VALUES));
 85:       PetscCall(MatSetValue(ctx->F, Ii, I_s, -1., INSERT_VALUES));
 86:       PetscCall(MatSetValue(ctx->F, Ii, I_e, -1., INSERT_VALUES));
 87:       PetscCall(MatSetValue(ctx->F, Ii, I_w, -1., INSERT_VALUES));
 88:     }
 89:   } else PetscCall(MatSetRandom(ctx->F, ctx->rctx));
 90:   PetscCall(MatAssemblyBegin(ctx->F, MAT_FINAL_ASSEMBLY));
 91:   PetscCall(MatAssemblyEnd(ctx->F, MAT_FINAL_ASSEMBLY));
 92:   PetscCall(PetscLogStagePop());
 93:   /* Stencil matrix is symmetric. Setting symmetric flag for ICC/Cholesky preconditioner */
 94:   if (!(ctx->matops)) PetscCall(MatSetOption(ctx->F, MAT_SYMMETRIC, PETSC_TRUE));
 95:   PetscCall(MatTransposeMatMult(ctx->F, ctx->F, MAT_INITIAL_MATRIX, PETSC_DEFAULT, &(ctx->W)));
 96:   /* Setup Hessian Workspace in same shape as W */
 97:   PetscCall(MatDuplicate(ctx->W, MAT_DO_NOT_COPY_VALUES, &(ctx->Hm)));
 98:   PetscCall(MatDuplicate(ctx->W, MAT_DO_NOT_COPY_VALUES, &(ctx->Hr)));
 99:   PetscFunctionReturn(PETSC_SUCCESS);
100: }

102: static PetscErrorCode SetupWorkspace(UserCtx ctx)
103: {
104:   PetscInt i;

106:   PetscFunctionBegin;
107:   PetscCall(MatCreateVecs(ctx->F, &ctx->workLeft[0], &ctx->workRight[0]));
108:   for (i = 1; i < NWORKLEFT; i++) PetscCall(VecDuplicate(ctx->workLeft[0], &(ctx->workLeft[i])));
109:   for (i = 1; i < NWORKRIGHT; i++) PetscCall(VecDuplicate(ctx->workRight[0], &(ctx->workRight[i])));
110:   PetscFunctionReturn(PETSC_SUCCESS);
111: }

113: static PetscErrorCode ConfigureContext(UserCtx ctx)
114: {
115:   PetscFunctionBegin;
116:   ctx->m        = 16;
117:   ctx->n        = 16;
118:   ctx->eps      = 1.e-3;
119:   ctx->abstol   = 1.e-4;
120:   ctx->reltol   = 1.e-2;
121:   ctx->hStart   = 1.;
122:   ctx->hMin     = 1.e-3;
123:   ctx->hFactor  = 0.5;
124:   ctx->alpha    = 1.;
125:   ctx->mu       = 1.0;
126:   ctx->matops   = 0;
127:   ctx->iter     = 10;
128:   ctx->p        = NORM_2;
129:   ctx->taylor   = PETSC_TRUE;
130:   ctx->use_admm = PETSC_FALSE;
131:   PetscOptionsBegin(PETSC_COMM_WORLD, NULL, "Configure separable objection example", "ex4.c");
132:   PetscCall(PetscOptionsInt("-m", "The row dimension of matrix F", "ex4.c", ctx->m, &(ctx->m), NULL));
133:   PetscCall(PetscOptionsInt("-n", "The column dimension of matrix F", "ex4.c", ctx->n, &(ctx->n), NULL));
134:   PetscCall(PetscOptionsInt("-matrix_format", "Decide format of F matrix. 0 for stencil, 1 for random", "ex4.c", ctx->matops, &(ctx->matops), NULL));
135:   PetscCall(PetscOptionsInt("-iter", "Iteration number ADMM", "ex4.c", ctx->iter, &(ctx->iter), NULL));
136:   PetscCall(PetscOptionsReal("-alpha", "The regularization multiplier. 1 default", "ex4.c", ctx->alpha, &(ctx->alpha), NULL));
137:   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));
138:   PetscCall(PetscOptionsReal("-mu", "The augmented lagrangian multiplier in ADMM", "ex4.c", ctx->mu, &(ctx->mu), NULL));
139:   PetscCall(PetscOptionsReal("-hStart", "Taylor test starting point. 1 default.", "ex4.c", ctx->hStart, &(ctx->hStart), NULL));
140:   PetscCall(PetscOptionsReal("-hFactor", "Taylor test multiplier factor. 0.5 default", "ex4.c", ctx->hFactor, &(ctx->hFactor), NULL));
141:   PetscCall(PetscOptionsReal("-hMin", "Taylor test ending condition. 1.e-3 default", "ex4.c", ctx->hMin, &(ctx->hMin), NULL));
142:   PetscCall(PetscOptionsReal("-abstol", "Absolute stopping criterion for ADMM", "ex4.c", ctx->abstol, &(ctx->abstol), NULL));
143:   PetscCall(PetscOptionsReal("-reltol", "Relative stopping criterion for ADMM", "ex4.c", ctx->reltol, &(ctx->reltol), NULL));
144:   PetscCall(PetscOptionsBool("-taylor", "Flag for Taylor test. Default is true.", "ex4.c", ctx->taylor, &(ctx->taylor), NULL));
145:   PetscCall(PetscOptionsBool("-use_admm", "Use the ADMM solver in this example.", "ex4.c", ctx->use_admm, &(ctx->use_admm), NULL));
146:   PetscCall(PetscOptionsEnum("-p", "Norm type.", "ex4.c", NormTypes, (PetscEnum)ctx->p, (PetscEnum *)&(ctx->p), NULL));
147:   PetscOptionsEnd();
148:   /* Creating random ctx */
149:   PetscCall(PetscRandomCreate(PETSC_COMM_WORLD, &(ctx->rctx)));
150:   PetscCall(PetscRandomSetFromOptions(ctx->rctx));
151:   PetscCall(CreateMatrix(ctx));
152:   PetscCall(CreateRHS(ctx));
153:   PetscCall(SetupWorkspace(ctx));
154:   PetscFunctionReturn(PETSC_SUCCESS);
155: }

157: static PetscErrorCode DestroyContext(UserCtx *ctx)
158: {
159:   PetscInt i;

161:   PetscFunctionBegin;
162:   PetscCall(MatDestroy(&((*ctx)->F)));
163:   PetscCall(MatDestroy(&((*ctx)->W)));
164:   PetscCall(MatDestroy(&((*ctx)->Hm)));
165:   PetscCall(MatDestroy(&((*ctx)->Hr)));
166:   PetscCall(VecDestroy(&((*ctx)->d)));
167:   for (i = 0; i < NWORKLEFT; i++) PetscCall(VecDestroy(&((*ctx)->workLeft[i])));
168:   for (i = 0; i < NWORKRIGHT; i++) PetscCall(VecDestroy(&((*ctx)->workRight[i])));
169:   PetscCall(PetscRandomDestroy(&((*ctx)->rctx)));
170:   PetscCall(PetscFree(*ctx));
171:   PetscFunctionReturn(PETSC_SUCCESS);
172: }

174: /* compute (1/2) * ||F x - d||^2 */
175: static PetscErrorCode ObjectiveMisfit(Tao tao, Vec x, PetscReal *J, void *_ctx)
176: {
177:   UserCtx ctx = (UserCtx)_ctx;
178:   Vec     y;

180:   PetscFunctionBegin;
181:   y = ctx->workLeft[0];
182:   PetscCall(MatMult(ctx->F, x, y));
183:   PetscCall(VecAXPY(y, -1., ctx->d));
184:   PetscCall(VecDot(y, y, J));
185:   *J *= 0.5;
186:   PetscFunctionReturn(PETSC_SUCCESS);
187: }

189: /* compute V = FTFx - FTd */
190: static PetscErrorCode GradientMisfit(Tao tao, Vec x, Vec V, void *_ctx)
191: {
192:   UserCtx ctx = (UserCtx)_ctx;
193:   Vec     FTFx, FTd;

195:   PetscFunctionBegin;
196:   /* work1 is A^T Ax, work2 is Ab, W is A^T A*/
197:   FTFx = ctx->workRight[0];
198:   FTd  = ctx->workRight[1];
199:   PetscCall(MatMult(ctx->W, x, FTFx));
200:   PetscCall(MatMultTranspose(ctx->F, ctx->d, FTd));
201:   PetscCall(VecWAXPY(V, -1., FTd, FTFx));
202:   PetscFunctionReturn(PETSC_SUCCESS);
203: }

205: /* returns FTF */
206: static PetscErrorCode HessianMisfit(Tao tao, Vec x, Mat H, Mat Hpre, void *_ctx)
207: {
208:   UserCtx ctx = (UserCtx)_ctx;

210:   PetscFunctionBegin;
211:   if (H != ctx->W) PetscCall(MatCopy(ctx->W, H, DIFFERENT_NONZERO_PATTERN));
212:   if (Hpre != ctx->W) PetscCall(MatCopy(ctx->W, Hpre, DIFFERENT_NONZERO_PATTERN));
213:   PetscFunctionReturn(PETSC_SUCCESS);
214: }

216: /* computes augment Lagrangian objective (with scaled dual):
217:  * 0.5 * ||F x - d||^2  + 0.5 * mu ||x - z + u||^2 */
218: static PetscErrorCode ObjectiveMisfitADMM(Tao tao, Vec x, PetscReal *J, void *_ctx)
219: {
220:   UserCtx   ctx = (UserCtx)_ctx;
221:   PetscReal mu, workNorm, misfit;
222:   Vec       z, u, temp;

224:   PetscFunctionBegin;
225:   mu   = ctx->mu;
226:   z    = ctx->workRight[5];
227:   u    = ctx->workRight[6];
228:   temp = ctx->workRight[10];
229:   /* misfit = f(x) */
230:   PetscCall(ObjectiveMisfit(tao, x, &misfit, _ctx));
231:   PetscCall(VecCopy(x, temp));
232:   /* temp = x - z + u */
233:   PetscCall(VecAXPBYPCZ(temp, -1., 1., 1., z, u));
234:   /* workNorm = ||x - z + u||^2 */
235:   PetscCall(VecDot(temp, temp, &workNorm));
236:   /* augment Lagrangian objective (with scaled dual): f(x) + 0.5 * mu ||x -z + u||^2 */
237:   *J = misfit + 0.5 * mu * workNorm;
238:   PetscFunctionReturn(PETSC_SUCCESS);
239: }

241: /* computes FTFx - FTd  mu*(x - z + u) */
242: static PetscErrorCode GradientMisfitADMM(Tao tao, Vec x, Vec V, void *_ctx)
243: {
244:   UserCtx   ctx = (UserCtx)_ctx;
245:   PetscReal mu;
246:   Vec       z, u, temp;

248:   PetscFunctionBegin;
249:   mu   = ctx->mu;
250:   z    = ctx->workRight[5];
251:   u    = ctx->workRight[6];
252:   temp = ctx->workRight[10];
253:   PetscCall(GradientMisfit(tao, x, V, _ctx));
254:   PetscCall(VecCopy(x, temp));
255:   /* temp = x - z + u */
256:   PetscCall(VecAXPBYPCZ(temp, -1., 1., 1., z, u));
257:   /* V =  FTFx - FTd  mu*(x - z + u) */
258:   PetscCall(VecAXPY(V, mu, temp));
259:   PetscFunctionReturn(PETSC_SUCCESS);
260: }

262: /* returns FTF + diag(mu) */
263: static PetscErrorCode HessianMisfitADMM(Tao tao, Vec x, Mat H, Mat Hpre, void *_ctx)
264: {
265:   UserCtx ctx = (UserCtx)_ctx;

267:   PetscFunctionBegin;
268:   PetscCall(MatCopy(ctx->W, H, DIFFERENT_NONZERO_PATTERN));
269:   PetscCall(MatShift(H, ctx->mu));
270:   if (Hpre != H) PetscCall(MatCopy(H, Hpre, DIFFERENT_NONZERO_PATTERN));
271:   PetscFunctionReturn(PETSC_SUCCESS);
272: }

274: /* computes || x ||_p (mult by 0.5 in case of NORM_2) */
275: static PetscErrorCode ObjectiveRegularization(Tao tao, Vec x, PetscReal *J, void *_ctx)
276: {
277:   UserCtx   ctx = (UserCtx)_ctx;
278:   PetscReal norm;

280:   PetscFunctionBegin;
281:   *J = 0;
282:   PetscCall(VecNorm(x, ctx->p, &norm));
283:   if (ctx->p == NORM_2) norm = 0.5 * norm * norm;
284:   *J = ctx->alpha * norm;
285:   PetscFunctionReturn(PETSC_SUCCESS);
286: }

288: /* NORM_2 Case: return x
289:  * NORM_1 Case: x/(|x| + eps)
290:  * Else: TODO */
291: static PetscErrorCode GradientRegularization(Tao tao, Vec x, Vec V, void *_ctx)
292: {
293:   UserCtx   ctx = (UserCtx)_ctx;
294:   PetscReal eps = ctx->eps;

296:   PetscFunctionBegin;
297:   if (ctx->p == NORM_2) {
298:     PetscCall(VecCopy(x, V));
299:   } else if (ctx->p == NORM_1) {
300:     PetscCall(VecCopy(x, ctx->workRight[1]));
301:     PetscCall(VecAbs(ctx->workRight[1]));
302:     PetscCall(VecShift(ctx->workRight[1], eps));
303:     PetscCall(VecPointwiseDivide(V, x, ctx->workRight[1]));
304:   } else SETERRQ(PetscObjectComm((PetscObject)tao), PETSC_ERR_ARG_OUTOFRANGE, "Example only works for NORM_1 and NORM_2");
305:   PetscFunctionReturn(PETSC_SUCCESS);
306: }

308: /* NORM_2 Case: returns diag(mu)
309:  * NORM_1 Case: diag(mu* 1/sqrt(x_i^2 + eps) * (1 - x_i^2/ABS(x_i^2+eps)))  */
310: static PetscErrorCode HessianRegularization(Tao tao, Vec x, Mat H, Mat Hpre, void *_ctx)
311: {
312:   UserCtx   ctx = (UserCtx)_ctx;
313:   PetscReal eps = ctx->eps;
314:   Vec       copy1, copy2, copy3;

316:   PetscFunctionBegin;
317:   if (ctx->p == NORM_2) {
318:     /* Identity matrix scaled by mu */
319:     PetscCall(MatZeroEntries(H));
320:     PetscCall(MatShift(H, ctx->mu));
321:     if (Hpre != H) {
322:       PetscCall(MatZeroEntries(Hpre));
323:       PetscCall(MatShift(Hpre, ctx->mu));
324:     }
325:   } else if (ctx->p == NORM_1) {
326:     /* 1/sqrt(x_i^2 + eps) * (1 - x_i^2/ABS(x_i^2+eps)) */
327:     copy1 = ctx->workRight[1];
328:     copy2 = ctx->workRight[2];
329:     copy3 = ctx->workRight[3];
330:     /* copy1 : 1/sqrt(x_i^2 + eps) */
331:     PetscCall(VecCopy(x, copy1));
332:     PetscCall(VecPow(copy1, 2));
333:     PetscCall(VecShift(copy1, eps));
334:     PetscCall(VecSqrtAbs(copy1));
335:     PetscCall(VecReciprocal(copy1));
336:     /* copy2:  x_i^2.*/
337:     PetscCall(VecCopy(x, copy2));
338:     PetscCall(VecPow(copy2, 2));
339:     /* copy3: abs(x_i^2 + eps) */
340:     PetscCall(VecCopy(x, copy3));
341:     PetscCall(VecPow(copy3, 2));
342:     PetscCall(VecShift(copy3, eps));
343:     PetscCall(VecAbs(copy3));
344:     /* copy2: 1 - x_i^2/abs(x_i^2 + eps) */
345:     PetscCall(VecPointwiseDivide(copy2, copy2, copy3));
346:     PetscCall(VecScale(copy2, -1.));
347:     PetscCall(VecShift(copy2, 1.));
348:     PetscCall(VecAXPY(copy1, 1., copy2));
349:     PetscCall(VecScale(copy1, ctx->mu));
350:     PetscCall(MatZeroEntries(H));
351:     PetscCall(MatDiagonalSet(H, copy1, INSERT_VALUES));
352:     if (Hpre != H) {
353:       PetscCall(MatZeroEntries(Hpre));
354:       PetscCall(MatDiagonalSet(Hpre, copy1, INSERT_VALUES));
355:     }
356:   } else SETERRQ(PetscObjectComm((PetscObject)tao), PETSC_ERR_ARG_OUTOFRANGE, "Example only works for NORM_1 and NORM_2");
357:   PetscFunctionReturn(PETSC_SUCCESS);
358: }

360: /* NORM_2 Case: 0.5 || x ||_2 + 0.5 * mu * ||x + u - z||^2
361:  * Else : || x ||_2 + 0.5 * mu * ||x + u - z||^2 */
362: static PetscErrorCode ObjectiveRegularizationADMM(Tao tao, Vec z, PetscReal *J, void *_ctx)
363: {
364:   UserCtx   ctx = (UserCtx)_ctx;
365:   PetscReal mu, workNorm, reg;
366:   Vec       x, u, temp;

368:   PetscFunctionBegin;
369:   mu   = ctx->mu;
370:   x    = ctx->workRight[4];
371:   u    = ctx->workRight[6];
372:   temp = ctx->workRight[10];
373:   PetscCall(ObjectiveRegularization(tao, z, &reg, _ctx));
374:   PetscCall(VecCopy(z, temp));
375:   /* temp = x + u -z */
376:   PetscCall(VecAXPBYPCZ(temp, 1., 1., -1., x, u));
377:   /* workNorm = ||x + u - z ||^2 */
378:   PetscCall(VecDot(temp, temp, &workNorm));
379:   *J = reg + 0.5 * mu * workNorm;
380:   PetscFunctionReturn(PETSC_SUCCESS);
381: }

383: /* NORM_2 Case: x - mu*(x + u - z)
384:  * NORM_1 Case: x/(|x| + eps) - mu*(x + u - z)
385:  * Else: TODO */
386: static PetscErrorCode GradientRegularizationADMM(Tao tao, Vec z, Vec V, void *_ctx)
387: {
388:   UserCtx   ctx = (UserCtx)_ctx;
389:   PetscReal mu;
390:   Vec       x, u, temp;

392:   PetscFunctionBegin;
393:   mu   = ctx->mu;
394:   x    = ctx->workRight[4];
395:   u    = ctx->workRight[6];
396:   temp = ctx->workRight[10];
397:   PetscCall(GradientRegularization(tao, z, V, _ctx));
398:   PetscCall(VecCopy(z, temp));
399:   /* temp = x + u -z */
400:   PetscCall(VecAXPBYPCZ(temp, 1., 1., -1., x, u));
401:   PetscCall(VecAXPY(V, -mu, temp));
402:   PetscFunctionReturn(PETSC_SUCCESS);
403: }

405: /* NORM_2 Case: returns diag(mu)
406:  * NORM_1 Case: FTF + diag(mu) */
407: static PetscErrorCode HessianRegularizationADMM(Tao tao, Vec x, Mat H, Mat Hpre, void *_ctx)
408: {
409:   UserCtx ctx = (UserCtx)_ctx;

411:   PetscFunctionBegin;
412:   if (ctx->p == NORM_2) {
413:     /* Identity matrix scaled by mu */
414:     PetscCall(MatZeroEntries(H));
415:     PetscCall(MatShift(H, ctx->mu));
416:     if (Hpre != H) {
417:       PetscCall(MatZeroEntries(Hpre));
418:       PetscCall(MatShift(Hpre, ctx->mu));
419:     }
420:   } else if (ctx->p == NORM_1) {
421:     PetscCall(HessianMisfit(tao, x, H, Hpre, (void *)ctx));
422:     PetscCall(MatShift(H, ctx->mu));
423:     if (Hpre != H) PetscCall(MatShift(Hpre, ctx->mu));
424:   } else SETERRQ(PetscObjectComm((PetscObject)tao), PETSC_ERR_ARG_OUTOFRANGE, "Example only works for NORM_1 and NORM_2");
425:   PetscFunctionReturn(PETSC_SUCCESS);
426: }

428: /* NORM_2 Case : (1/2) * ||F x - d||^2 + 0.5 * || x ||_p
429: *  NORM_1 Case : (1/2) * ||F x - d||^2 + || x ||_p */
430: static PetscErrorCode ObjectiveComplete(Tao tao, Vec x, PetscReal *J, void *ctx)
431: {
432:   PetscReal Jm, Jr;

434:   PetscFunctionBegin;
435:   PetscCall(ObjectiveMisfit(tao, x, &Jm, ctx));
436:   PetscCall(ObjectiveRegularization(tao, x, &Jr, ctx));
437:   *J = Jm + Jr;
438:   PetscFunctionReturn(PETSC_SUCCESS);
439: }

441: /* NORM_2 Case: FTFx - FTd + x
442:  * NORM_1 Case: FTFx - FTd + x/(|x| + eps) */
443: static PetscErrorCode GradientComplete(Tao tao, Vec x, Vec V, void *ctx)
444: {
445:   UserCtx cntx = (UserCtx)ctx;

447:   PetscFunctionBegin;
448:   PetscCall(GradientMisfit(tao, x, cntx->workRight[2], ctx));
449:   PetscCall(GradientRegularization(tao, x, cntx->workRight[3], ctx));
450:   PetscCall(VecWAXPY(V, 1, cntx->workRight[2], cntx->workRight[3]));
451:   PetscFunctionReturn(PETSC_SUCCESS);
452: }

454: /* NORM_2 Case: diag(mu) + FTF
455:  * NORM_1 Case: diag(mu* 1/sqrt(x_i^2 + eps) * (1 - x_i^2/ABS(x_i^2+eps))) + FTF  */
456: static PetscErrorCode HessianComplete(Tao tao, Vec x, Mat H, Mat Hpre, void *ctx)
457: {
458:   Mat tempH;

460:   PetscFunctionBegin;
461:   PetscCall(MatDuplicate(H, MAT_SHARE_NONZERO_PATTERN, &tempH));
462:   PetscCall(HessianMisfit(tao, x, H, H, ctx));
463:   PetscCall(HessianRegularization(tao, x, tempH, tempH, ctx));
464:   PetscCall(MatAXPY(H, 1., tempH, DIFFERENT_NONZERO_PATTERN));
465:   if (Hpre != H) PetscCall(MatCopy(H, Hpre, DIFFERENT_NONZERO_PATTERN));
466:   PetscCall(MatDestroy(&tempH));
467:   PetscFunctionReturn(PETSC_SUCCESS);
468: }

470: static PetscErrorCode TaoSolveADMM(UserCtx ctx, Vec x)
471: {
472:   PetscInt  i;
473:   PetscReal u_norm, r_norm, s_norm, primal, dual, x_norm, z_norm;
474:   Tao       tao1, tao2;
475:   Vec       xk, z, u, diff, zold, zdiff, temp;
476:   PetscReal mu;

478:   PetscFunctionBegin;
479:   xk    = ctx->workRight[4];
480:   z     = ctx->workRight[5];
481:   u     = ctx->workRight[6];
482:   diff  = ctx->workRight[7];
483:   zold  = ctx->workRight[8];
484:   zdiff = ctx->workRight[9];
485:   temp  = ctx->workRight[11];
486:   mu    = ctx->mu;
487:   PetscCall(VecSet(u, 0.));
488:   PetscCall(TaoCreate(PETSC_COMM_WORLD, &tao1));
489:   PetscCall(TaoSetType(tao1, TAONLS));
490:   PetscCall(TaoSetObjective(tao1, ObjectiveMisfitADMM, (void *)ctx));
491:   PetscCall(TaoSetGradient(tao1, NULL, GradientMisfitADMM, (void *)ctx));
492:   PetscCall(TaoSetHessian(tao1, ctx->Hm, ctx->Hm, HessianMisfitADMM, (void *)ctx));
493:   PetscCall(VecSet(xk, 0.));
494:   PetscCall(TaoSetSolution(tao1, xk));
495:   PetscCall(TaoSetOptionsPrefix(tao1, "misfit_"));
496:   PetscCall(TaoSetFromOptions(tao1));
497:   PetscCall(TaoCreate(PETSC_COMM_WORLD, &tao2));
498:   if (ctx->p == NORM_2) {
499:     PetscCall(TaoSetType(tao2, TAONLS));
500:     PetscCall(TaoSetObjective(tao2, ObjectiveRegularizationADMM, (void *)ctx));
501:     PetscCall(TaoSetGradient(tao2, NULL, GradientRegularizationADMM, (void *)ctx));
502:     PetscCall(TaoSetHessian(tao2, ctx->Hr, ctx->Hr, HessianRegularizationADMM, (void *)ctx));
503:   }
504:   PetscCall(VecSet(z, 0.));
505:   PetscCall(TaoSetSolution(tao2, z));
506:   PetscCall(TaoSetOptionsPrefix(tao2, "reg_"));
507:   PetscCall(TaoSetFromOptions(tao2));

509:   for (i = 0; i < ctx->iter; i++) {
510:     PetscCall(VecCopy(z, zold));
511:     PetscCall(TaoSolve(tao1)); /* Updates xk */
512:     if (ctx->p == NORM_1) {
513:       PetscCall(VecWAXPY(temp, 1., xk, u));
514:       PetscCall(TaoSoftThreshold(temp, -ctx->alpha / mu, ctx->alpha / mu, z));
515:     } else {
516:       PetscCall(TaoSolve(tao2)); /* Update zk */
517:     }
518:     /* u = u + xk -z */
519:     PetscCall(VecAXPBYPCZ(u, 1., -1., 1., xk, z));
520:     /* r_norm : norm(x-z) */
521:     PetscCall(VecWAXPY(diff, -1., z, xk));
522:     PetscCall(VecNorm(diff, NORM_2, &r_norm));
523:     /* s_norm : norm(-mu(z-zold)) */
524:     PetscCall(VecWAXPY(zdiff, -1., zold, z));
525:     PetscCall(VecNorm(zdiff, NORM_2, &s_norm));
526:     s_norm = s_norm * mu;
527:     /* primal : sqrt(n)*ABSTOL + RELTOL*max(norm(x), norm(-z))*/
528:     PetscCall(VecNorm(xk, NORM_2, &x_norm));
529:     PetscCall(VecNorm(z, NORM_2, &z_norm));
530:     primal = PetscSqrtReal(ctx->n) * ctx->abstol + ctx->reltol * PetscMax(x_norm, z_norm);
531:     /* Duality : sqrt(n)*ABSTOL + RELTOL*norm(mu*u)*/
532:     PetscCall(VecNorm(u, NORM_2, &u_norm));
533:     dual = PetscSqrtReal(ctx->n) * ctx->abstol + ctx->reltol * u_norm * mu;
534:     PetscCall(PetscPrintf(PetscObjectComm((PetscObject)tao1), "Iter %" PetscInt_FMT " : ||x-z||: %g, mu*||z-zold||: %g\n", i, (double)r_norm, (double)s_norm));
535:     if (r_norm < primal && s_norm < dual) break;
536:   }
537:   PetscCall(VecCopy(xk, x));
538:   PetscCall(TaoDestroy(&tao1));
539:   PetscCall(TaoDestroy(&tao2));
540:   PetscFunctionReturn(PETSC_SUCCESS);
541: }

543: /* Second order Taylor remainder convergence test */
544: static PetscErrorCode TaylorTest(UserCtx ctx, Tao tao, Vec x, PetscReal *C)
545: {
546:   PetscReal  h, J, temp;
547:   PetscInt   i, j;
548:   PetscInt   numValues;
549:   PetscReal  Jx, Jxhat_comp, Jxhat_pred;
550:   PetscReal *Js, *hs;
551:   PetscReal  gdotdx;
552:   PetscReal  minrate = PETSC_MAX_REAL;
553:   MPI_Comm   comm    = PetscObjectComm((PetscObject)x);
554:   Vec        g, dx, xhat;

556:   PetscFunctionBegin;
557:   PetscCall(VecDuplicate(x, &g));
558:   PetscCall(VecDuplicate(x, &xhat));
559:   /* choose a perturbation direction */
560:   PetscCall(VecDuplicate(x, &dx));
561:   PetscCall(VecSetRandom(dx, ctx->rctx));
562:   /* evaluate objective at x: J(x) */
563:   PetscCall(TaoComputeObjective(tao, x, &Jx));
564:   /* evaluate gradient at x, save in vector g */
565:   PetscCall(TaoComputeGradient(tao, x, g));
566:   PetscCall(VecDot(g, dx, &gdotdx));

568:   for (numValues = 0, h = ctx->hStart; h >= ctx->hMin; h *= ctx->hFactor) numValues++;
569:   PetscCall(PetscCalloc2(numValues, &Js, numValues, &hs));
570:   for (i = 0, h = ctx->hStart; h >= ctx->hMin; h *= ctx->hFactor, i++) {
571:     PetscCall(VecWAXPY(xhat, h, dx, x));
572:     PetscCall(TaoComputeObjective(tao, xhat, &Jxhat_comp));
573:     /* J(\hat(x)) \approx J(x) + g^T (xhat - x) = J(x) + h * g^T dx */
574:     Jxhat_pred = Jx + h * gdotdx;
575:     /* Vector to dJdm scalar? Dot?*/
576:     J = PetscAbsReal(Jxhat_comp - Jxhat_pred);
577:     PetscCall(PetscPrintf(comm, "J(xhat): %g, predicted: %g, diff %g\n", (double)Jxhat_comp, (double)Jxhat_pred, (double)J));
578:     Js[i] = J;
579:     hs[i] = h;
580:   }
581:   for (j = 1; j < numValues; j++) {
582:     temp = PetscLogReal(Js[j] / Js[j - 1]) / PetscLogReal(hs[j] / hs[j - 1]);
583:     PetscCall(PetscPrintf(comm, "Convergence rate step %" PetscInt_FMT ": %g\n", j - 1, (double)temp));
584:     minrate = PetscMin(minrate, temp);
585:   }
586:   /* If O is not ~2, then the test is wrong */
587:   PetscCall(PetscFree2(Js, hs));
588:   *C = minrate;
589:   PetscCall(VecDestroy(&dx));
590:   PetscCall(VecDestroy(&xhat));
591:   PetscCall(VecDestroy(&g));
592:   PetscFunctionReturn(PETSC_SUCCESS);
593: }

595: int main(int argc, char **argv)
596: {
597:   UserCtx ctx;
598:   Tao     tao;
599:   Vec     x;
600:   Mat     H;

602:   PetscFunctionBeginUser;
603:   PetscCall(PetscInitialize(&argc, &argv, NULL, help));
604:   PetscCall(PetscNew(&ctx));
605:   PetscCall(ConfigureContext(ctx));
606:   /* Define two functions that could pass as objectives to TaoSetObjective(): one
607:    * for the misfit component, and one for the regularization component */
608:   /* ObjectiveMisfit() and ObjectiveRegularization() */

610:   /* Define a single function that calls both components adds them together: the complete objective,
611:    * in the absence of a Tao implementation that handles separability */
612:   /* ObjectiveComplete() */
613:   PetscCall(TaoCreate(PETSC_COMM_WORLD, &tao));
614:   PetscCall(TaoSetType(tao, TAONM));
615:   PetscCall(TaoSetObjective(tao, ObjectiveComplete, (void *)ctx));
616:   PetscCall(TaoSetGradient(tao, NULL, GradientComplete, (void *)ctx));
617:   PetscCall(MatDuplicate(ctx->W, MAT_SHARE_NONZERO_PATTERN, &H));
618:   PetscCall(TaoSetHessian(tao, H, H, HessianComplete, (void *)ctx));
619:   PetscCall(MatCreateVecs(ctx->F, NULL, &x));
620:   PetscCall(VecSet(x, 0.));
621:   PetscCall(TaoSetSolution(tao, x));
622:   PetscCall(TaoSetFromOptions(tao));
623:   if (ctx->use_admm) PetscCall(TaoSolveADMM(ctx, x));
624:   else PetscCall(TaoSolve(tao));
625:   /* examine solution */
626:   PetscCall(VecViewFromOptions(x, NULL, "-view_sol"));
627:   if (ctx->taylor) {
628:     PetscReal rate;
629:     PetscCall(TaylorTest(ctx, tao, x, &rate));
630:   }
631:   PetscCall(MatDestroy(&H));
632:   PetscCall(TaoDestroy(&tao));
633:   PetscCall(VecDestroy(&x));
634:   PetscCall(DestroyContext(&ctx));
635:   PetscCall(PetscFinalize());
636:   return 0;
637: }

639: /*TEST

641:   build:
642:     requires: !complex

644:   test:
645:     suffix: 0
646:     args:

648:   test:
649:     suffix: l1_1
650:     args: -p 1 -tao_type lmvm -alpha 1. -epsilon 1.e-7 -m 64 -n 64 -view_sol -matrix_format 1

652:   test:
653:     suffix: hessian_1
654:     args: -matrix_format 1 -m 100 -n 100 -tao_monitor -p 1 -tao_type nls

656:   test:
657:     suffix: hessian_2
658:     args: -matrix_format 1 -m 100 -n 100 -tao_monitor -p 2 -tao_type nls

660:   test:
661:     suffix: nm_1
662:     args: -matrix_format 1 -m 100 -n 100 -tao_monitor -p 1 -tao_type nm -tao_max_it 50

664:   test:
665:     suffix: nm_2
666:     args: -matrix_format 1 -m 100 -n 100 -tao_monitor -p 2 -tao_type nm -tao_max_it 50

668:   test:
669:     suffix: lmvm_1
670:     args: -matrix_format 1 -m 100 -n 100 -tao_monitor -p 1 -tao_type lmvm -tao_max_it 40

672:   test:
673:     suffix: lmvm_2
674:     args: -matrix_format 1 -m 100 -n 100 -tao_monitor -p 2 -tao_type lmvm -tao_max_it 15

676:   test:
677:     suffix: soft_threshold_admm_1
678:     args: -matrix_format 1 -m 100 -n 100 -tao_monitor -p 1 -use_admm

680:   test:
681:     suffix: hessian_admm_1
682:     args: -matrix_format 1 -m 100 -n 100 -tao_monitor -p 1 -use_admm -reg_tao_type nls -misfit_tao_type nls

684:   test:
685:     suffix: hessian_admm_2
686:     args: -matrix_format 1 -m 100 -n 100 -tao_monitor -p 2 -use_admm -reg_tao_type nls -misfit_tao_type nls

688:   test:
689:     suffix: nm_admm_1
690:     args: -matrix_format 1 -m 100 -n 100 -tao_monitor -p 1 -use_admm -reg_tao_type nm -misfit_tao_type nm

692:   test:
693:     suffix: nm_admm_2
694:     args: -matrix_format 1 -m 100 -n 100 -tao_monitor -p 2 -use_admm -reg_tao_type nm -misfit_tao_type nm -iter 7

696:   test:
697:     suffix: lmvm_admm_1
698:     args: -matrix_format 1 -m 100 -n 100 -tao_monitor -p 1 -use_admm -reg_tao_type lmvm -misfit_tao_type lmvm

700:   test:
701:     suffix: lmvm_admm_2
702:     args: -matrix_format 1 -m 100 -n 100 -tao_monitor -p 2 -use_admm -reg_tao_type lmvm -misfit_tao_type lmvm

704: TEST*/