Actual source code: mfnexpokit.c
slepc-3.10.1 2018-10-23
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      1: /*
      2:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
      3:    SLEPc - Scalable Library for Eigenvalue Problem Computations
      4:    Copyright (c) 2002-2018, 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: PETSC_EXTERN PetscErrorCode MFNCreate_Expokit(MFN mfn)
    184: {
    186:   mfn->ops->solve          = MFNSolve_Expokit;
    187:   mfn->ops->setup          = MFNSetUp_Expokit;
    188:   return(0);
    189: }