Actual source code: mfnexpokit.c

slepc-3.18.3 2023-03-24
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  1: /*
  2:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  3:    SLEPc - Scalable Library for Eigenvalue Problem Computations
  4:    Copyright (c) 2002-, 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: {
 30:   PetscInt       N;
 31:   PetscBool      isexp;

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

 38:   PetscObjectTypeCompare((PetscObject)mfn->fn,FNEXP,&isexp);
 40:   return 0;
 41: }

 43: PetscErrorCode MFNSolve_Expokit(MFN mfn,Vec b,Vec x)
 44: {
 45:   PetscInt          mxstep,mxrej,m,mb,ld,i,j,ireject,mx,k1;
 46:   Vec               v,r;
 47:   Mat               H,M=NULL,K=NULL;
 48:   FN                fn;
 49:   PetscScalar       *Harray,*B,*F,*betaF,t,sgn,sfactor;
 50:   const PetscScalar *pK;
 51:   PetscReal         anorm,avnorm,tol,err_loc,rndoff,t_out,t_new,t_now,t_step;
 52:   PetscReal         xm,fact,s,p1,p2,beta,beta2,gamma,delta;
 53:   PetscBool         breakdown;

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

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

 79:   VecCopy(b,x);
 80:   ld = m+2;
 81:   PetscCalloc2(m+1,&betaF,ld*ld,&B);
 82:   MatCreateSeqDense(PETSC_COMM_SELF,ld,ld,NULL,&H);
 83:   MatDenseGetArray(H,&Harray);

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

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

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

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

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

171:   MatDestroy(&M);
172:   MatDestroy(&K);
173:   FNDestroy(&fn);
174:   MatDenseRestoreArray(H,&Harray);
175:   MatDestroy(&H);
176:   PetscFree2(betaF,B);
177:   return 0;
178: }

180: SLEPC_EXTERN PetscErrorCode MFNCreate_Expokit(MFN mfn)
181: {
182:   mfn->ops->solve          = MFNSolve_Expokit;
183:   mfn->ops->setup          = MFNSetUp_Expokit;
184:   return 0;
185: }