Actual source code: pgmres.c
2: /*
3: This file implements PGMRES (a Pipelined Generalized Minimal Residual method)
4: */
6: #include <../src/ksp/ksp/impls/gmres/pgmres/pgmresimpl.h>
7: #define PGMRES_DELTA_DIRECTIONS 10
8: #define PGMRES_DEFAULT_MAXK 30
10: static PetscErrorCode KSPPGMRESUpdateHessenberg(KSP,PetscInt,PetscBool*,PetscReal*);
11: static PetscErrorCode KSPPGMRESBuildSoln(PetscScalar*,Vec,Vec,KSP,PetscInt);
13: /*
15: KSPSetUp_PGMRES - Sets up the workspace needed by pgmres.
17: This is called once, usually automatically by KSPSolve() or KSPSetUp(),
18: but can be called directly by KSPSetUp().
20: */
21: static PetscErrorCode KSPSetUp_PGMRES(KSP ksp)
22: {
23: KSPSetUp_GMRES(ksp);
24: return 0;
25: }
27: /*
29: KSPPGMRESCycle - Run pgmres, possibly with restart. Return residual
30: history if requested.
32: input parameters:
33: . pgmres - structure containing parameters and work areas
35: output parameters:
36: . itcount - number of iterations used. If null, ignored.
37: . converged - 0 if not converged
39: Notes:
40: On entry, the value in vector VEC_VV(0) should be
41: the initial residual.
43: */
44: static PetscErrorCode KSPPGMRESCycle(PetscInt *itcount,KSP ksp)
45: {
46: KSP_PGMRES *pgmres = (KSP_PGMRES*)(ksp->data);
47: PetscReal res_norm,res,newnorm;
48: PetscInt it = 0,j,k;
49: PetscBool hapend = PETSC_FALSE;
51: if (itcount) *itcount = 0;
52: VecNormalize(VEC_VV(0),&res_norm);
53: KSPCheckNorm(ksp,res_norm);
54: res = res_norm;
55: *RS(0) = res_norm;
57: /* check for the convergence */
58: PetscObjectSAWsTakeAccess((PetscObject)ksp);
59: if (ksp->normtype != KSP_NORM_NONE) ksp->rnorm = res;
60: else ksp->rnorm = 0;
61: PetscObjectSAWsGrantAccess((PetscObject)ksp);
62: pgmres->it = it-2;
63: KSPLogResidualHistory(ksp,ksp->rnorm);
64: KSPMonitor(ksp,ksp->its,ksp->rnorm);
65: if (!res) {
66: ksp->reason = KSP_CONVERGED_ATOL;
67: PetscInfo(ksp,"Converged due to zero residual norm on entry\n");
68: return 0;
69: }
71: (*ksp->converged)(ksp,ksp->its,ksp->rnorm,&ksp->reason,ksp->cnvP);
72: for (; !ksp->reason; it++) {
73: Vec Zcur,Znext;
74: if (pgmres->vv_allocated <= it + VEC_OFFSET + 1) {
75: KSPGMRESGetNewVectors(ksp,it+1);
76: }
77: /* VEC_VV(it-1) is orthogonal, it will be normalized once the VecNorm arrives. */
78: Zcur = VEC_VV(it); /* Zcur is not yet orthogonal, but the VecMDot to orthogonalize it has been started. */
79: Znext = VEC_VV(it+1); /* This iteration will compute Znext, update with a deferred correction once we know how
80: * Zcur relates to the previous vectors, and start the reduction to orthogonalize it. */
82: if (it < pgmres->max_k+1 && ksp->its+1 < PetscMax(2,ksp->max_it)) { /* We don't know whether what we have computed is enough, so apply the matrix. */
83: KSP_PCApplyBAorAB(ksp,Zcur,Znext,VEC_TEMP_MATOP);
84: }
86: if (it > 1) { /* Complete the pending reduction */
87: VecNormEnd(VEC_VV(it-1),NORM_2,&newnorm);
88: *HH(it-1,it-2) = newnorm;
89: }
90: if (it > 0) { /* Finish the reduction computing the latest column of H */
91: VecMDotEnd(Zcur,it,&(VEC_VV(0)),HH(0,it-1));
92: }
94: if (it > 1) {
95: /* normalize the base vector from two iterations ago, basis is complete up to here */
96: VecScale(VEC_VV(it-1),1./ *HH(it-1,it-2));
98: KSPPGMRESUpdateHessenberg(ksp,it-2,&hapend,&res);
99: pgmres->it = it-2;
100: ksp->its++;
101: if (ksp->normtype != KSP_NORM_NONE) ksp->rnorm = res;
102: else ksp->rnorm = 0;
104: (*ksp->converged)(ksp,ksp->its,ksp->rnorm,&ksp->reason,ksp->cnvP);
105: if (it < pgmres->max_k+1 || ksp->reason || ksp->its == ksp->max_it) { /* Monitor if we are done or still iterating, but not before a restart. */
106: KSPLogResidualHistory(ksp,ksp->rnorm);
107: KSPMonitor(ksp,ksp->its,ksp->rnorm);
108: }
109: if (ksp->reason) break;
110: /* Catch error in happy breakdown and signal convergence and break from loop */
111: if (hapend) {
113: else {
114: ksp->reason = KSP_DIVERGED_BREAKDOWN;
115: break;
116: }
117: }
119: if (!(it < pgmres->max_k+1 && ksp->its < ksp->max_it)) break;
121: /* The it-2 column of H was not scaled when we computed Zcur, apply correction */
122: VecScale(Zcur,1./ *HH(it-1,it-2));
123: /* And Znext computed in this iteration was computed using the under-scaled Zcur */
124: VecScale(Znext,1./ *HH(it-1,it-2));
126: /* In the previous iteration, we projected an unnormalized Zcur against the Krylov basis, so we need to fix the column of H resulting from that projection. */
127: for (k=0; k<it; k++) *HH(k,it-1) /= *HH(it-1,it-2);
128: /* When Zcur was projected against the Krylov basis, VV(it-1) was still not normalized, so fix that too. This
129: * column is complete except for HH(it,it-1) which we won't know until the next iteration. */
130: *HH(it-1,it-1) /= *HH(it-1,it-2);
131: }
133: if (it > 0) {
134: PetscScalar *work;
135: if (!pgmres->orthogwork) PetscMalloc1(pgmres->max_k + 2,&pgmres->orthogwork);
136: work = pgmres->orthogwork;
137: /* Apply correction computed by the VecMDot in the last iteration to Znext. The original form is
138: *
139: * Znext -= sum_{j=0}^{i-1} Z[j+1] * H[j,i-1]
140: *
141: * where
142: *
143: * Z[j] = sum_{k=0}^j V[k] * H[k,j-1]
144: *
145: * substituting
146: *
147: * Znext -= sum_{j=0}^{i-1} sum_{k=0}^{j+1} V[k] * H[k,j] * H[j,i-1]
148: *
149: * rearranging the iteration space from row-column to column-row
150: *
151: * Znext -= sum_{k=0}^i sum_{j=k-1}^{i-1} V[k] * H[k,j] * H[j,i-1]
152: *
153: * Note that column it-1 of HH is correct. For all previous columns, we must look at HES because HH has already
154: * been transformed to upper triangular form.
155: */
156: for (k=0; k<it+1; k++) {
157: work[k] = 0;
158: for (j=PetscMax(0,k-1); j<it-1; j++) work[k] -= *HES(k,j) * *HH(j,it-1);
159: }
160: VecMAXPY(Znext,it+1,work,&VEC_VV(0));
161: VecAXPY(Znext,-*HH(it-1,it-1),Zcur);
163: /* Orthogonalize Zcur against existing basis vectors. */
164: for (k=0; k<it; k++) work[k] = -*HH(k,it-1);
165: VecMAXPY(Zcur,it,work,&VEC_VV(0));
166: /* Zcur is now orthogonal, and will be referred to as VEC_VV(it) again, though it is still not normalized. */
167: /* Begin computing the norm of the new vector, will be normalized after the MatMult in the next iteration. */
168: VecNormBegin(VEC_VV(it),NORM_2,&newnorm);
169: }
171: /* Compute column of H (to the diagonal, but not the subdiagonal) to be able to orthogonalize the newest vector. */
172: VecMDotBegin(Znext,it+1,&VEC_VV(0),HH(0,it));
174: /* Start an asynchronous split-mode reduction, the result of the MDot and Norm will be collected on the next iteration. */
175: PetscCommSplitReductionBegin(PetscObjectComm((PetscObject)Znext));
176: }
178: if (itcount) *itcount = it-1; /* Number of iterations actually completed. */
180: /*
181: Down here we have to solve for the "best" coefficients of the Krylov
182: columns, add the solution values together, and possibly unwind the
183: preconditioning from the solution
184: */
185: /* Form the solution (or the solution so far) */
186: KSPPGMRESBuildSoln(RS(0),ksp->vec_sol,ksp->vec_sol,ksp,it-2);
187: return 0;
188: }
190: /*
191: KSPSolve_PGMRES - This routine applies the PGMRES method.
193: Input Parameter:
194: . ksp - the Krylov space object that was set to use pgmres
196: Output Parameter:
197: . outits - number of iterations used
199: */
200: static PetscErrorCode KSPSolve_PGMRES(KSP ksp)
201: {
202: PetscInt its,itcount;
203: KSP_PGMRES *pgmres = (KSP_PGMRES*)ksp->data;
204: PetscBool guess_zero = ksp->guess_zero;
207: PetscObjectSAWsTakeAccess((PetscObject)ksp);
208: ksp->its = 0;
209: PetscObjectSAWsGrantAccess((PetscObject)ksp);
211: itcount = 0;
212: ksp->reason = KSP_CONVERGED_ITERATING;
213: while (!ksp->reason) {
214: KSPInitialResidual(ksp,ksp->vec_sol,VEC_TEMP,VEC_TEMP_MATOP,VEC_VV(0),ksp->vec_rhs);
215: KSPPGMRESCycle(&its,ksp);
216: itcount += its;
217: if (itcount >= ksp->max_it) {
218: if (!ksp->reason) ksp->reason = KSP_DIVERGED_ITS;
219: break;
220: }
221: ksp->guess_zero = PETSC_FALSE; /* every future call to KSPInitialResidual() will have nonzero guess */
222: }
223: ksp->guess_zero = guess_zero; /* restore if user provided nonzero initial guess */
224: return 0;
225: }
227: static PetscErrorCode KSPDestroy_PGMRES(KSP ksp)
228: {
229: KSPDestroy_GMRES(ksp);
230: return 0;
231: }
233: /*
234: KSPPGMRESBuildSoln - create the solution from the starting vector and the
235: current iterates.
237: Input parameters:
238: nrs - work area of size it + 1.
239: vguess - index of initial guess
240: vdest - index of result. Note that vguess may == vdest (replace
241: guess with the solution).
242: it - HH upper triangular part is a block of size (it+1) x (it+1)
244: This is an internal routine that knows about the PGMRES internals.
245: */
246: static PetscErrorCode KSPPGMRESBuildSoln(PetscScalar *nrs,Vec vguess,Vec vdest,KSP ksp,PetscInt it)
247: {
248: PetscScalar tt;
249: PetscInt k,j;
250: KSP_PGMRES *pgmres = (KSP_PGMRES*)(ksp->data);
252: /* Solve for solution vector that minimizes the residual */
254: if (it < 0) { /* no pgmres steps have been performed */
255: VecCopy(vguess,vdest); /* VecCopy() is smart, exits immediately if vguess == vdest */
256: return 0;
257: }
259: /* solve the upper triangular system - RS is the right side and HH is
260: the upper triangular matrix - put soln in nrs */
261: if (*HH(it,it) != 0.0) nrs[it] = *RS(it) / *HH(it,it);
262: else nrs[it] = 0.0;
264: for (k=it-1; k>=0; k--) {
265: tt = *RS(k);
266: for (j=k+1; j<=it; j++) tt -= *HH(k,j) * nrs[j];
267: nrs[k] = tt / *HH(k,k);
268: }
270: /* Accumulate the correction to the solution of the preconditioned problem in TEMP */
271: VecZeroEntries(VEC_TEMP);
272: VecMAXPY(VEC_TEMP,it+1,nrs,&VEC_VV(0));
273: KSPUnwindPreconditioner(ksp,VEC_TEMP,VEC_TEMP_MATOP);
274: /* add solution to previous solution */
275: if (vdest == vguess) {
276: VecAXPY(vdest,1.0,VEC_TEMP);
277: } else {
278: VecWAXPY(vdest,1.0,VEC_TEMP,vguess);
279: }
280: return 0;
281: }
283: /*
285: KSPPGMRESUpdateHessenberg - Do the scalar work for the orthogonalization.
286: Return new residual.
288: input parameters:
290: . ksp - Krylov space object
291: . it - plane rotations are applied to the (it+1)th column of the
292: modified hessenberg (i.e. HH(:,it))
293: . hapend - PETSC_FALSE not happy breakdown ending.
295: output parameters:
296: . res - the new residual
298: */
299: /*
300: . it - column of the Hessenberg that is complete, PGMRES is actually computing two columns ahead of this
301: */
302: static PetscErrorCode KSPPGMRESUpdateHessenberg(KSP ksp,PetscInt it,PetscBool *hapend,PetscReal *res)
303: {
304: PetscScalar *hh,*cc,*ss,*rs;
305: PetscInt j;
306: PetscReal hapbnd;
307: KSP_PGMRES *pgmres = (KSP_PGMRES*)(ksp->data);
309: hh = HH(0,it); /* pointer to beginning of column to update */
310: cc = CC(0); /* beginning of cosine rotations */
311: ss = SS(0); /* beginning of sine rotations */
312: rs = RS(0); /* right hand side of least squares system */
314: /* The Hessenberg matrix is now correct through column it, save that form for possible spectral analysis */
315: for (j=0; j<=it+1; j++) *HES(j,it) = hh[j];
317: /* check for the happy breakdown */
318: hapbnd = PetscMin(PetscAbsScalar(hh[it+1] / rs[it]),pgmres->haptol);
319: if (PetscAbsScalar(hh[it+1]) < hapbnd) {
320: PetscInfo(ksp,"Detected happy breakdown, current hapbnd = %14.12e H(%D,%D) = %14.12e\n",(double)hapbnd,it+1,it,(double)PetscAbsScalar(*HH(it+1,it)));
321: *hapend = PETSC_TRUE;
322: }
324: /* Apply all the previously computed plane rotations to the new column
325: of the Hessenberg matrix */
326: /* Note: this uses the rotation [conj(c) s ; -s c], c= cos(theta), s= sin(theta),
327: and some refs have [c s ; -conj(s) c] (don't be confused!) */
329: for (j=0; j<it; j++) {
330: PetscScalar hhj = hh[j];
331: hh[j] = PetscConj(cc[j])*hhj + ss[j]*hh[j+1];
332: hh[j+1] = -ss[j] *hhj + cc[j]*hh[j+1];
333: }
335: /*
336: compute the new plane rotation, and apply it to:
337: 1) the right-hand-side of the Hessenberg system (RS)
338: note: it affects RS(it) and RS(it+1)
339: 2) the new column of the Hessenberg matrix
340: note: it affects HH(it,it) which is currently pointed to
341: by hh and HH(it+1, it) (*(hh+1))
342: thus obtaining the updated value of the residual...
343: */
345: /* compute new plane rotation */
347: if (!*hapend) {
348: PetscReal delta = PetscSqrtReal(PetscSqr(PetscAbsScalar(hh[it])) + PetscSqr(PetscAbsScalar(hh[it+1])));
349: if (delta == 0.0) {
350: ksp->reason = KSP_DIVERGED_NULL;
351: return 0;
352: }
354: cc[it] = hh[it] / delta; /* new cosine value */
355: ss[it] = hh[it+1] / delta; /* new sine value */
357: hh[it] = PetscConj(cc[it])*hh[it] + ss[it]*hh[it+1];
358: rs[it+1] = -ss[it]*rs[it];
359: rs[it] = PetscConj(cc[it])*rs[it];
360: *res = PetscAbsScalar(rs[it+1]);
361: } else { /* happy breakdown: HH(it+1, it) = 0, therefore we don't need to apply
362: another rotation matrix (so RH doesn't change). The new residual is
363: always the new sine term times the residual from last time (RS(it)),
364: but now the new sine rotation would be zero...so the residual should
365: be zero...so we will multiply "zero" by the last residual. This might
366: not be exactly what we want to do here -could just return "zero". */
368: *res = 0.0;
369: }
370: return 0;
371: }
373: /*
374: KSPBuildSolution_PGMRES
376: Input Parameter:
377: . ksp - the Krylov space object
378: . ptr-
380: Output Parameter:
381: . result - the solution
383: Note: this calls KSPPGMRESBuildSoln - the same function that KSPPGMRESCycle
384: calls directly.
386: */
387: PetscErrorCode KSPBuildSolution_PGMRES(KSP ksp,Vec ptr,Vec *result)
388: {
389: KSP_PGMRES *pgmres = (KSP_PGMRES*)ksp->data;
391: if (!ptr) {
392: if (!pgmres->sol_temp) {
393: VecDuplicate(ksp->vec_sol,&pgmres->sol_temp);
394: PetscLogObjectParent((PetscObject)ksp,(PetscObject)pgmres->sol_temp);
395: }
396: ptr = pgmres->sol_temp;
397: }
398: if (!pgmres->nrs) {
399: /* allocate the work area */
400: PetscMalloc1(pgmres->max_k,&pgmres->nrs);
401: PetscLogObjectMemory((PetscObject)ksp,pgmres->max_k*sizeof(PetscScalar));
402: }
404: KSPPGMRESBuildSoln(pgmres->nrs,ksp->vec_sol,ptr,ksp,pgmres->it);
405: if (result) *result = ptr;
406: return 0;
407: }
409: PetscErrorCode KSPSetFromOptions_PGMRES(PetscOptionItems *PetscOptionsObject,KSP ksp)
410: {
411: KSPSetFromOptions_GMRES(PetscOptionsObject,ksp);
412: PetscOptionsHead(PetscOptionsObject,"KSP pipelined GMRES Options");
413: PetscOptionsTail();
414: return 0;
415: }
417: PetscErrorCode KSPReset_PGMRES(KSP ksp)
418: {
419: KSPReset_GMRES(ksp);
420: return 0;
421: }
423: /*MC
424: KSPPGMRES - Implements the Pipelined Generalized Minimal Residual method.
426: Options Database Keys:
427: + -ksp_gmres_restart <restart> - the number of Krylov directions to orthogonalize against
428: . -ksp_gmres_haptol <tol> - sets the tolerance for "happy ending" (exact convergence)
429: . -ksp_gmres_preallocate - preallocate all the Krylov search directions initially (otherwise groups of
430: vectors are allocated as needed)
431: . -ksp_gmres_classicalgramschmidt - use classical (unmodified) Gram-Schmidt to orthogonalize against the Krylov space (fast) (the default)
432: . -ksp_gmres_modifiedgramschmidt - use modified Gram-Schmidt in the orthogonalization (more stable, but slower)
433: . -ksp_gmres_cgs_refinement_type <refine_never,refine_ifneeded,refine_always> - determine if iterative refinement is used to increase the
434: stability of the classical Gram-Schmidt orthogonalization.
435: - -ksp_gmres_krylov_monitor - plot the Krylov space generated
437: Level: beginner
439: Notes:
440: MPI configuration may be necessary for reductions to make asynchronous progress, which is important for performance of pipelined methods.
441: See the FAQ on the PETSc website for details.
443: Reference:
444: Ghysels, Ashby, Meerbergen, Vanroose, Hiding global communication latencies in the GMRES algorithm on massively parallel machines, 2012.
446: Developer Notes:
447: This object is subclassed off of KSPGMRES
449: .seealso: KSPCreate(), KSPSetType(), KSPType (for list of available types), KSP, KSPGMRES, KSPLGMRES, KSPPIPECG, KSPPIPECR,
450: KSPGMRESSetRestart(), KSPGMRESSetHapTol(), KSPGMRESSetPreAllocateVectors(), KSPGMRESSetOrthogonalization(), KSPGMRESGetOrthogonalization(),
451: KSPGMRESClassicalGramSchmidtOrthogonalization(), KSPGMRESModifiedGramSchmidtOrthogonalization(),
452: KSPGMRESCGSRefinementType, KSPGMRESSetCGSRefinementType(), KSPGMRESGetCGSRefinementType(), KSPGMRESMonitorKrylov()
453: M*/
455: PETSC_EXTERN PetscErrorCode KSPCreate_PGMRES(KSP ksp)
456: {
457: KSP_PGMRES *pgmres;
459: PetscNewLog(ksp,&pgmres);
461: ksp->data = (void*)pgmres;
462: ksp->ops->buildsolution = KSPBuildSolution_PGMRES;
463: ksp->ops->setup = KSPSetUp_PGMRES;
464: ksp->ops->solve = KSPSolve_PGMRES;
465: ksp->ops->reset = KSPReset_PGMRES;
466: ksp->ops->destroy = KSPDestroy_PGMRES;
467: ksp->ops->view = KSPView_GMRES;
468: ksp->ops->setfromoptions = KSPSetFromOptions_PGMRES;
469: ksp->ops->computeextremesingularvalues = KSPComputeExtremeSingularValues_GMRES;
470: ksp->ops->computeeigenvalues = KSPComputeEigenvalues_GMRES;
472: KSPSetSupportedNorm(ksp,KSP_NORM_PRECONDITIONED,PC_LEFT,3);
473: KSPSetSupportedNorm(ksp,KSP_NORM_UNPRECONDITIONED,PC_RIGHT,2);
474: KSPSetSupportedNorm(ksp,KSP_NORM_NONE,PC_RIGHT,1);
476: PetscObjectComposeFunction((PetscObject)ksp,"KSPGMRESSetPreAllocateVectors_C",KSPGMRESSetPreAllocateVectors_GMRES);
477: PetscObjectComposeFunction((PetscObject)ksp,"KSPGMRESSetOrthogonalization_C",KSPGMRESSetOrthogonalization_GMRES);
478: PetscObjectComposeFunction((PetscObject)ksp,"KSPGMRESGetOrthogonalization_C",KSPGMRESGetOrthogonalization_GMRES);
479: PetscObjectComposeFunction((PetscObject)ksp,"KSPGMRESSetRestart_C",KSPGMRESSetRestart_GMRES);
480: PetscObjectComposeFunction((PetscObject)ksp,"KSPGMRESGetRestart_C",KSPGMRESGetRestart_GMRES);
481: PetscObjectComposeFunction((PetscObject)ksp,"KSPGMRESSetCGSRefinementType_C",KSPGMRESSetCGSRefinementType_GMRES);
482: PetscObjectComposeFunction((PetscObject)ksp,"KSPGMRESGetCGSRefinementType_C",KSPGMRESGetCGSRefinementType_GMRES);
484: pgmres->nextra_vecs = 1;
485: pgmres->haptol = 1.0e-30;
486: pgmres->q_preallocate = 0;
487: pgmres->delta_allocate = PGMRES_DELTA_DIRECTIONS;
488: pgmres->orthog = KSPGMRESClassicalGramSchmidtOrthogonalization;
489: pgmres->nrs = NULL;
490: pgmres->sol_temp = NULL;
491: pgmres->max_k = PGMRES_DEFAULT_MAXK;
492: pgmres->Rsvd = NULL;
493: pgmres->orthogwork = NULL;
494: pgmres->cgstype = KSP_GMRES_CGS_REFINE_NEVER;
495: return 0;
496: }