Actual source code: lmekrylov.c

slepc-3.15.2 2021-09-20
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  1: /*
  2:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  3:    SLEPc - Scalable Library for Eigenvalue Problem Computations
  4:    Copyright (c) 2002-2021, Universitat Politecnica de Valencia, Spain

  6:    This file is part of SLEPc.
  7:    SLEPc is distributed under a 2-clause BSD license (see LICENSE).
  8:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  9: */
 10: /*
 11:    SLEPc matrix equation solver: "krylov"

 13:    Method: Arnoldi with Eiermann-Ernst restart

 15:    Algorithm:

 17:        Project the equation onto the Arnoldi basis and solve the compressed
 18:        equation the Hessenberg matrix H, restart by discarding the Krylov
 19:        basis but keeping H.

 21:    References:

 23:        [1] Y. Saad, "Numerical solution of large Lyapunov equations", in
 24:            Signal processing, scattering and operator theory, and numerical
 25:            methods, vol. 5 of Progr. Systems Control Theory, pages 503-511,
 26:            1990.

 28:        [2] D. Kressner, "Memory-efficient Krylov subspace techniques for
 29:            solving large-scale Lyapunov equations", in 2008 IEEE Int. Conf.
 30:            Computer-Aided Control Systems, pages 613-618, 2008.
 31: */

 33: #include <slepc/private/lmeimpl.h>
 34: #include <slepcblaslapack.h>

 36: PetscErrorCode LMESetUp_Krylov(LME lme)
 37: {
 39:   PetscInt       N;

 42:   MatGetSize(lme->A,&N,NULL);
 43:   if (lme->ncv==PETSC_DEFAULT) lme->ncv = PetscMin(30,N);
 44:   if (lme->max_it==PETSC_DEFAULT) lme->max_it = 100;
 45:   LMEAllocateSolution(lme,1);
 46:   return(0);
 47: }

 49: PetscErrorCode LMESolve_Krylov_Lyapunov_Vec(LME lme,Vec b,PetscBool fixed,PetscInt rrank,BV C1,BV *X1,PetscInt *col,PetscBool *fail,PetscInt *totalits)
 50: {
 52:   PetscInt       n=0,m,ldh,ldg=0,i,j,rank=0,lrank,pass,nouter=0,its;
 53:   PetscReal      bnorm,beta,errest;
 54:   PetscBool      breakdown;
 55:   PetscScalar    *H,*G=NULL,*Gnew=NULL,*L,*U,*r,*Qarray,sone=1.0,zero=0.0;
 56:   PetscBLASInt   n_,m_,rk_;
 57:   Mat            Q;

 60:   *fail = PETSC_FALSE;
 61:   its = 0;
 62:   m  = lme->ncv;
 63:   ldh = m+1;
 64:   PetscCalloc1(ldh*m,&H);

 66:   VecNorm(b,NORM_2,&bnorm);
 67:   if (!bnorm) SETERRQ(PetscObjectComm((PetscObject)lme),PETSC_ERR_ARG_WRONG,"Cannot process a zero vector in the right-hand side");

 69:   for (pass=0;pass<2;pass++) {

 71:     /* set initial vector to b/||b|| */
 72:     BVInsertVec(lme->V,0,b);
 73:     BVScaleColumn(lme->V,0,1.0/bnorm);

 75:     /* Restart loop */
 76:     while ((pass==0 && !*fail) || (pass==1 && its+1<nouter)) {
 77:       its++;

 79:       /* compute Arnoldi factorization */
 80:       BVMatArnoldi(lme->V,lme->A,H,ldh,0,&m,&beta,&breakdown);

 82:       if (pass==0) {
 83:         /* glue together the previous H and the new H obtained with Arnoldi */
 84:         ldg = n+m+1;
 85:         PetscCalloc1(ldg*(n+m),&Gnew);
 86:         for (j=0;j<m;j++) {
 87:           PetscArraycpy(Gnew+n+(j+n)*ldg,H+j*ldh,m);
 88:         }
 89:         Gnew[n+m+(n+m-1)*ldg] = beta;
 90:         if (G) {
 91:           for (j=0;j<n;j++) {
 92:             PetscArraycpy(Gnew+j*ldg,G+j*(n+1),n+1);
 93:           }
 94:           PetscFree(G);
 95:         }
 96:         G = Gnew;
 97:         n += m;
 98:       } else {
 99:         /* update Z = Z + V(:,1:m)*Q    with   Q=U(blk,:)*P(1:nrk,:)'  */
100:         MatCreateSeqDense(PETSC_COMM_SELF,m,*col+rank,NULL,&Q);
101:         MatDenseGetArray(Q,&Qarray);
102:         PetscBLASIntCast(m,&m_);
103:         PetscBLASIntCast(n,&n_);
104:         PetscBLASIntCast(rank,&rk_);
105:         PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&m_,&rk_,&rk_,&sone,U+its*m,&n_,L,&n_,&zero,Qarray+(*col)*m,&m_));
106:         MatDenseRestoreArray(Q,&Qarray);
107:         BVSetActiveColumns(*X1,*col,*col+rank);
108:         BVMult(*X1,1.0,1.0,lme->V,Q);
109:         MatDestroy(&Q);
110:       }

112:       if (pass==0) {
113:         /* solve compressed Lyapunov equation */
114:         PetscCalloc1(n,&r);
115:         PetscCalloc1(n*n,&L);
116:         r[0] = bnorm;
117:         errest = PetscAbsScalar(G[n+(n-1)*ldg]);
118:         LMEDenseHessLyapunovChol(lme,n,G,ldg,1,r,n,L,n,&errest);
119:         LMEMonitor(lme,*totalits+its,errest);
120:         PetscFree(r);

122:         /* check convergence */
123:         if (errest<lme->tol) {
124:           lme->errest += errest;
125:           PetscMalloc1(n*n,&U);
126:           /* transpose L */
127:           for (j=0;j<n;j++) {
128:             for (i=j+1;i<n;i++) {
129:               L[i+j*n] = PetscConj(L[j+i*n]);
130:               L[j+i*n] = 0.0;
131:             }
132:           }
133:           LMEDenseRankSVD(lme,n,L,n,U,n,&lrank);
134:           PetscInfo1(lme,"Rank of the Cholesky factor = %D\n",lrank);
135:           nouter = its;
136:           its = -1;
137:           if (!fixed) {  /* X1 was not set by user, allocate it with rank columns */
138:             rank = lrank;
139:             if (*col) {
140:               BVResize(*X1,*col+rank,PETSC_TRUE);
141:             } else {
142:               BVDuplicateResize(C1,rank,X1);
143:             }
144:           } else rank = PetscMin(lrank,rrank);
145:           PetscFree(G);
146:           break;
147:         } else {
148:           PetscFree(L);
149:           if (*totalits+its>=lme->max_it) *fail = PETSC_TRUE;
150:         }
151:       }

153:       /* restart with vector v_{m+1} */
154:       if (!*fail) {
155:         BVCopyColumn(lme->V,m,0);
156:       }
157:     }
158:   }

160:   *col += rank;
161:   *totalits += its+1;
162:   PetscFree(H);
163:   if (L) { PetscFree(L); }
164:   if (U) { PetscFree(U); }
165:   return(0);
166: }

168: PetscErrorCode LMESolve_Krylov_Lyapunov(LME lme)
169: {
171:   PetscBool      fail,fixed = lme->X? PETSC_TRUE: PETSC_FALSE;
172:   PetscInt       i,k,rank=0,col=0;
173:   Vec            b;
174:   BV             X1=NULL,C1;
175:   Mat            X1m,X1t,C1m;

178:   MatLRCGetMats(lme->C,NULL,&C1m,NULL,NULL);
179:   BVCreateFromMat(C1m,&C1);
180:   BVSetFromOptions(C1);
181:   BVGetActiveColumns(C1,NULL,&k);
182:   if (fixed) {
183:     MatLRCGetMats(lme->X,NULL,&X1m,NULL,NULL);
184:     BVCreateFromMat(X1m,&X1);
185:     BVSetFromOptions(X1);
186:     BVGetActiveColumns(X1,NULL,&rank);
187:     rank = rank/k;
188:   }
189:   for (i=0;i<k;i++) {
190:     BVGetColumn(C1,i,&b);
191:     LMESolve_Krylov_Lyapunov_Vec(lme,b,fixed,rank,C1,&X1,&col,&fail,&lme->its);
192:     BVRestoreColumn(C1,i,&b);
193:     if (fail) {
194:       lme->reason = LME_DIVERGED_ITS;
195:       break;
196:     }
197:   }
198:   if (lme->reason==LME_CONVERGED_ITERATING) lme->reason = LME_CONVERGED_TOL;
199:   BVCreateMat(X1,&X1t);
200:   if (fixed) {
201:     MatCopy(X1t,X1m,SAME_NONZERO_PATTERN);
202:   } else {
203:     MatCreateLRC(NULL,X1t,NULL,NULL,&lme->X);
204:   }
205:   MatDestroy(&X1t);
206:   BVDestroy(&C1);
207:   BVDestroy(&X1);
208:   return(0);
209: }

211: SLEPC_EXTERN PetscErrorCode LMECreate_Krylov(LME lme)
212: {
214:   lme->ops->solve[LME_LYAPUNOV]      = LMESolve_Krylov_Lyapunov;
215:   lme->ops->setup                    = LMESetUp_Krylov;
216:   return(0);
217: }