Actual source code: mfnexpokit.c

slepc-3.11.2 2019-07-30
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  1: /*
  2:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  3:    SLEPc - Scalable Library for Eigenvalue Problem Computations
  4:    Copyright (c) 2002-2019, 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 function solver: "expokit"

 13:    Method: Arnoldi method tailored for the matrix exponential

 15:    Algorithm:

 17:        Uses Arnoldi relations to compute exp(t_step*A)*v_last for
 18:        several time steps.

 20:    References:

 22:        [1] R. Sidje, "Expokit: a software package for computing matrix
 23:            exponentials", ACM Trans. Math. Softw. 24(1):130-156, 1998.
 24: */

 26: #include <slepc/private/mfnimpl.h>

 28: PetscErrorCode MFNSetUp_Expokit(MFN mfn)
 29: {
 31:   PetscInt       N;
 32:   PetscBool      isexp;

 35:   MatGetSize(mfn->A,&N,NULL);
 36:   if (!mfn->ncv) mfn->ncv = PetscMin(30,N);
 37:   if (!mfn->max_it) mfn->max_it = 100;
 38:   MFNAllocateSolution(mfn,2);

 40:   PetscObjectTypeCompare((PetscObject)mfn->fn,FNEXP,&isexp);
 41:   if (!isexp) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"This solver only supports the exponential function");
 42:   return(0);
 43: }

 45: PetscErrorCode MFNSolve_Expokit(MFN mfn,Vec b,Vec x)
 46: {
 48:   PetscInt       mxstep,mxrej,m,mb,ld,i,j,ireject,mx,k1;
 49:   Vec            v,r;
 50:   Mat            M=NULL,K=NULL;
 51:   FN             fn;
 52:   PetscScalar    *H,*B,*F,*betaF,t,sgn,sfactor;
 53:   PetscReal      anorm,avnorm,tol,err_loc,rndoff;
 54:   PetscReal      t_out,t_new,t_now,t_step;
 55:   PetscReal      xm,fact,s,p1,p2;
 56:   PetscReal      beta,beta2,gamma,delta;
 57:   PetscBool      breakdown;

 60:   m   = mfn->ncv;
 61:   tol = mfn->tol;
 62:   mxstep = mfn->max_it;
 63:   mxrej = 10;
 64:   gamma = 0.9;
 65:   delta = 1.2;
 66:   mb    = m;
 67:   FNGetScale(mfn->fn,&t,&sfactor);
 68:   FNDuplicate(mfn->fn,PetscObjectComm((PetscObject)mfn->fn),&fn);
 69:   FNSetScale(fn,1.0,1.0);
 70:   t_out = PetscAbsScalar(t);
 71:   t_now = 0.0;
 72:   MatNorm(mfn->A,NORM_INFINITY,&anorm);
 73:   rndoff = anorm*PETSC_MACHINE_EPSILON;

 75:   k1 = 2;
 76:   xm = 1.0/(PetscReal)m;
 77:   beta = mfn->bnorm;
 78:   fact = PetscPowRealInt((m+1)/2.72,m+1)*PetscSqrtReal(2*PETSC_PI*(m+1));
 79:   t_new = (1.0/anorm)*PetscPowReal((fact*tol)/(4.0*beta*anorm),xm);
 80:   s = PetscPowReal(10.0,PetscFloorReal(PetscLog10Real(t_new))-1);
 81:   t_new = PetscCeilReal(t_new/s)*s;
 82:   sgn = t/PetscAbsScalar(t);

 84:   VecCopy(b,x);
 85:   ld = m+2;
 86:   PetscCalloc3(m+1,&betaF,ld*ld,&H,ld*ld,&B);

 88:   while (mfn->reason == MFN_CONVERGED_ITERATING) {
 89:     mfn->its++;
 90:     if (PetscIsInfOrNanReal(t_new)) t_new = PETSC_MAX_REAL;
 91:     t_step = PetscMin(t_out-t_now,t_new);
 92:     BVInsertVec(mfn->V,0,x);
 93:     BVScaleColumn(mfn->V,0,1.0/beta);
 94:     MFNBasicArnoldi(mfn,H,ld,0,&mb,&beta2,&breakdown);
 95:     if (breakdown) {
 96:       k1 = 0;
 97:       t_step = t_out-t_now;
 98:     }
 99:     if (k1!=0) {
100:       H[m+1+ld*m] = 1.0;
101:       BVGetColumn(mfn->V,m,&v);
102:       BVGetColumn(mfn->V,m+1,&r);
103:       MatMult(mfn->A,v,r);
104:       BVRestoreColumn(mfn->V,m,&v);
105:       BVRestoreColumn(mfn->V,m+1,&r);
106:       BVNormColumn(mfn->V,m+1,NORM_2,&avnorm);
107:     }
108:     PetscMemcpy(B,H,ld*ld*sizeof(PetscScalar));

110:     ireject = 0;
111:     while (ireject <= mxrej) {
112:       mx = mb + k1;
113:       for (i=0;i<mx;i++) {
114:         for (j=0;j<mx;j++) {
115:           H[i+j*ld] = sgn*B[i+j*ld]*t_step;
116:         }
117:       }
118:       MFN_CreateDenseMat(mx,&M);
119:       MFN_CreateDenseMat(mx,&K);
120:       MatDenseGetArray(M,&F);
121:       for (j=0;j<mx;j++) {
122:         PetscMemcpy(F+j*mx,H+j*ld,mx*sizeof(PetscScalar));
123:       }
124:       MatDenseRestoreArray(M,&F);
125:       FNEvaluateFunctionMat(fn,M,K);

127:       if (k1==0) {
128:         err_loc = tol;
129:         break;
130:       } else {
131:         MatDenseGetArray(K,&F);
132:         p1 = PetscAbsScalar(beta*F[m]);
133:         p2 = PetscAbsScalar(beta*F[m+1]*avnorm);
134:         MatDenseRestoreArray(K,&F);
135:         if (p1 > 10*p2) {
136:           err_loc = p2;
137:           xm = 1.0/(PetscReal)m;
138:         } else if (p1 > p2) {
139:           err_loc = (p1*p2)/(p1-p2);
140:           xm = 1.0/(PetscReal)m;
141:         } else {
142:           err_loc = p1;
143:           xm = 1.0/(PetscReal)(m-1);
144:         }
145:       }
146:       if (err_loc <= delta*t_step*tol) break;
147:       else {
148:         t_step = gamma*t_step*PetscPowReal(t_step*tol/err_loc,xm);
149:         s = PetscPowReal(10.0,PetscFloorReal(PetscLog10Real(t_step))-1);
150:         t_step = PetscCeilReal(t_step/s)*s;
151:         ireject = ireject+1;
152:       }
153:     }

155:     mx = mb + PetscMax(0,k1-1);
156:     MatDenseGetArray(K,&F);
157:     for (j=0;j<mx;j++) betaF[j] = beta*F[j];
158:     MatDenseRestoreArray(K,&F);
159:     BVSetActiveColumns(mfn->V,0,mx);
160:     BVMultVec(mfn->V,1.0,0.0,x,betaF);
161:     VecNorm(x,NORM_2,&beta);

163:     t_now = t_now+t_step;
164:     if (t_now>=t_out) mfn->reason = MFN_CONVERGED_TOL;
165:     else {
166:       t_new = gamma*t_step*PetscPowReal((t_step*tol)/err_loc,xm);
167:       s = PetscPowReal(10.0,PetscFloorReal(PetscLog10Real(t_new))-1);
168:       t_new = PetscCeilReal(t_new/s)*s;
169:     }
170:     err_loc = PetscMax(err_loc,rndoff);
171:     if (mfn->its==mxstep) mfn->reason = MFN_DIVERGED_ITS;
172:     MFNMonitor(mfn,mfn->its,err_loc);
173:   }
174:   VecScale(x,sfactor);

176:   MatDestroy(&M);
177:   MatDestroy(&K);
178:   FNDestroy(&fn);
179:   PetscFree3(betaF,H,B);
180:   return(0);
181: }

183: SLEPC_EXTERN PetscErrorCode MFNCreate_Expokit(MFN mfn)
184: {
186:   mfn->ops->solve          = MFNSolve_Expokit;
187:   mfn->ops->setup          = MFNSetUp_Expokit;
188:   return(0);
189: }