Date: 2021-09-20T09:44:12+00:00

Actual source code: mfnexpokit.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 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==PETSC_DEFAULT) mfn->ncv = PetscMin(30,N);
     37:   if (mfn->max_it==PETSC_DEFAULT) 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: {
     47:   PetscErrorCode    ierr;
     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:   const PetscScalar *pK;
     54:   PetscReal         anorm,avnorm,tol,err_loc,rndoff,t_out,t_new,t_now,t_step;
     55:   PetscReal         xm,fact,s,p1,p2,beta,beta2,gamma,delta;
     56:   PetscBool         breakdown;

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

     74:   k1 = 2;
     75:   xm = 1.0/(PetscReal)m;
     76:   beta = mfn->bnorm;
     77:   fact = PetscPowRealInt((m+1)/2.72,m+1)*PetscSqrtReal(2.0*PETSC_PI*(m+1));
     78:   t_new = (1.0/anorm)*PetscPowReal((fact*tol)/(4.0*beta*anorm),xm);
     79:   s = PetscPowReal(10.0,PetscFloorReal(PetscLog10Real(t_new))-1);
     80:   t_new = PetscCeilReal(t_new/s)*s;
     81:   sgn = t/PetscAbsScalar(t);

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

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

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

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

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

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

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

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