Actual source code: agmresdeflation.c

      1: /*
      2:  *  This file computes data for the deflated restarting in the Newton-basis GMRES. At each restart (or at each detected stagnation in the adaptive strategy), a basis of an (approximated)invariant subspace corresponding to the smallest eigenvalues is extracted from the Krylov subspace. It is then used to augment the Newton basis.
      3:  *
      4:  * References : D. Nuentsa Wakam and J. Erhel, Parallelism and robustness in GMRES with the Newton basis and the deflation of eigenvalues. Research report INRIA RR-7787.
      5:  * Author: Desire NUENTSA WAKAM <desire.nuentsa_wakam@inria.fr>, 2011
      6:  */

      8: #include <../src/ksp/ksp/impls/gmres/agmres/agmresimpl.h>

     10: /* Quicksort algorithm to sort  the eigenvalues in increasing orders
     11:  * val_r - real part of eigenvalues, unchanged on exit.
     12:  * val_i - Imaginary part of eigenvalues unchanged on exit.
     13:  * size - Number of eigenvalues (with complex conjugates)
     14:  * perm - contains on exit the permutation vector to reorder the vectors val_r and val_i.
     15:  */
     16: #define  DEPTH  500
     17: static PetscErrorCode KSPAGMRESQuickSort(PetscScalar *val_r, PetscScalar *val_i, PetscInt size, PetscInt *perm)
     18: {

     20:   PetscInt    deb[DEPTH], fin[DEPTH];
     21:   PetscInt    ipivot;
     22:   PetscScalar pivot_r, pivot_i;
     23:   PetscInt    i, L, R, j;
     24:   PetscScalar abs_pivot;
     25:   PetscScalar abs_val;

     28:   /* initialize perm vector */
     29:   for (j = 0; j < size; j++) perm[j] = j;

     31:   deb[0] = 0;
     32:   fin[0] = size;
     33:   i      = 0;
     34:   while (i >= 0) {
     35:     L = deb[i];
     36:     R = fin[i] - 1;
     37:     if (L < R) {
     38:       pivot_r   = val_r[L];
     39:       pivot_i   = val_i[L];
     40:       abs_pivot = PetscSqrtReal(pivot_r * pivot_r + pivot_i * pivot_i);
     41:       ipivot    = perm[L];
     42:       if (i == DEPTH - 1) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_MEM, "Could cause stack overflow: Try to increase the value of DEPTH ");
     43:       while (L < R) {
     44:         abs_val = PetscSqrtReal(val_r[R] * val_r[R] + val_i[R] * val_i[R]);
     45:         while (abs_val >= abs_pivot && L < R) {
     46:           R--;
     47:           abs_val = PetscSqrtReal(val_r[R] * val_r[R] + val_i[R] * val_i[R]);
     48:         }
     49:         if (L < R) {
     50:           val_r[L] = val_r[R];
     51:           val_i[L] = val_i[R];
     52:           perm[L]  = perm[R];
     53:           L       += 1;
     54:         }
     55:         abs_val = PetscSqrtReal(val_r[L] * val_r[L] + val_i[L] * val_i[L]);
     56:         while (abs_val <= abs_pivot && L < R) {
     57:           L++;
     58:           abs_val = PetscSqrtReal(val_r[L] * val_r[L] + val_i[L] * val_i[L]);
     59:         }
     60:         if (L < R) {
     61:           val_r[R] = val_r[L];
     62:           val_i[R] = val_i[L];
     63:           perm[R]  = perm[L];
     64:           R       -= 1;
     65:         }
     66:       }
     67:       val_r[L] = pivot_r;
     68:       val_i[L] = pivot_i;
     69:       perm[L]  = ipivot;
     70:       deb[i+1] = L + 1;
     71:       fin[i+1] = fin[i];
     72:       fin[i]   = L;
     73:       i       += 1;
     74:       if (i == DEPTH - 1) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_MEM, "Could cause stack overflow: Try to increase the value of DEPTH ");
     75:     } else i--;
     76:   }
     77:   return(0);
     78: }

     80: /*
     81:  * Compute the Schur vectors from the generalized eigenvalue problem A.u =\lamba.B.u
     82:  * KspSize -  rank of the matrices A and B, size of the current Krylov basis
     83:  * A - Left matrix
     84:  * B - Right matrix
     85:  * ldA - first dimension of A as declared  in the calling program
     86:  * ldB - first dimension of B as declared  in the calling program
     87:  * IsReduced - specifies if the matrices are already in the reduced form,
     88:  * i.e A is a Hessenberg matrix and B is upper triangular.
     89:  * Sr - on exit, the extracted Schur vectors corresponding
     90:  * the smallest eigenvalues (with complex conjugates)
     91:  * CurNeig - Number of extracted eigenvalues
     92:  */
     93: static PetscErrorCode KSPAGMRESSchurForm(KSP ksp, PetscBLASInt KspSize, PetscScalar *A, PetscBLASInt ldA, PetscScalar *B, PetscBLASInt ldB, PetscBool IsReduced, PetscScalar *Sr, PetscInt *CurNeig)
     94: {
     95:   KSP_AGMRES     *agmres = (KSP_AGMRES*)ksp->data;
     96:   PetscInt       max_k   = agmres->max_k;
     97:   PetscBLASInt   r;
     98:   PetscInt       neig    = agmres->neig;
     99:   PetscScalar    *wr     = agmres->wr;
    100:   PetscScalar    *wi     = agmres->wi;
    101:   PetscScalar    *beta   = agmres->beta;
    102:   PetscScalar    *Q      = agmres->Q;
    103:   PetscScalar    *Z      = agmres->Z;
    104:   PetscScalar    *work   = agmres->work;
    105:   PetscBLASInt   *select = agmres->select;
    106:   PetscInt       *perm   = agmres->perm;
    107:   PetscBLASInt   sdim    = 0;
    108:   PetscInt       i,j;
    109:   PetscBLASInt   info;
    111:   PetscBLASInt   *iwork = agmres->iwork;
    112:   PetscBLASInt   N = MAXKSPSIZE;
    113:   PetscBLASInt   lwork,liwork;
    114:   PetscBLASInt   ilo,ihi;
    115:   PetscBLASInt   ijob,wantQ,wantZ;
    116:   PetscScalar    Dif[2];

    119:   ijob  = 2;
    120:   wantQ = 1;
    121:   wantZ = 1;
    122:   PetscBLASIntCast(PetscMax(8*N+16,4*neig*(N-neig)),&lwork);
    123:   PetscBLASIntCast(2*N*neig,&liwork);
    124:   ilo   = 1;
    125:   PetscBLASIntCast(KspSize,&ihi);

    127:   /* Compute the Schur form */
    128:   if (IsReduced) {                /* The eigenvalue problem is already in reduced form, meaning that A is upper Hessenberg and B is triangular */
    129:     PetscStackCallBLAS("LAPACKhgeqz",LAPACKhgeqz_("S", "I", "I", &KspSize, &ilo, &ihi, A, &ldA, B, &ldB, wr, wi, beta, Q, &N, Z, &N, work, &lwork, &info));
    130:     if (info) SETERRQ(PetscObjectComm((PetscObject)ksp),PETSC_ERR_PLIB, "Error while calling LAPACK routine xhgeqz_");
    131:   } else {
    132:     PetscStackCallBLAS("LAPACKgges",LAPACKgges_("V", "V", "N", NULL, &KspSize, A, &ldA, B, &ldB, &sdim, wr, wi, beta, Q, &N, Z, &N, work, &lwork, NULL, &info));
    133:     if (info) SETERRQ(PetscObjectComm((PetscObject)ksp),PETSC_ERR_PLIB, "Error while calling LAPACK routine xgges_");
    134:   }

    136:   /* We should avoid computing these ratio...  */
    137:   for (i = 0; i < KspSize; i++) {
    138:     if (beta[i] != 0.0) {
    139:       wr[i] /= beta[i];
    140:       wi[i] /= beta[i];
    141:     }
    142:   }

    144:   /* Sort the eigenvalues to extract the smallest ones */
    145:   KSPAGMRESQuickSort(wr, wi, KspSize, perm);

    147:   /* Count the number of extracted eigenvalues (with complex conjugates) */
    148:   r = 0;
    149:   while (r < neig) {
    150:     if (wi[r] != 0) r += 2;
    151:     else r += 1;
    152:   }
    153:   /* Reorder the Schur decomposition so that the cluster of smallest/largest eigenvalues appears in the leading diagonal blocks of A (and B)*/
    154:   PetscArrayzero(select, N);
    155:   if (!agmres->GreatestEig) {
    156:     for (j = 0; j < r; j++) select[perm[j]] = 1;
    157:   } else {
    158:     for (j = 0; j < r; j++) select[perm[KspSize-j-1]] = 1;
    159:   }
    160:   PetscStackCallBLAS("LAPACKtgsen",LAPACKtgsen_(&ijob, &wantQ, &wantZ, select, &KspSize, A, &ldA, B, &ldB, wr, wi, beta, Q, &N, Z, &N, &r, NULL, NULL, &(Dif[0]), work, &lwork, iwork, &liwork, &info));
    161:   if (info == 1) SETERRQ(PetscObjectComm((PetscObject)ksp),PETSC_ERR_PLIB, "UNABLE TO REORDER THE EIGENVALUES WITH THE LAPACK ROUTINE : ILL-CONDITIONED PROBLEM");
    162:   /* Extract the Schur vectors associated to the r smallest eigenvalues */
    163:   PetscArrayzero(Sr,(N+1)*r);
    164:   for (j = 0; j < r; j++) {
    165:     for (i = 0; i < KspSize; i++) {
    166:       Sr[j*(N+1)+i] = Z[j*N+i];
    167:     }
    168:   }

    170:   /* Broadcast Sr to all other processes to have consistent data;
    171:    * FIXME should investigate how to get unique Schur vectors (unique QR factorization, probably the sign of rotations) */
    172:   MPI_Bcast(Sr, (N+1)*r, MPIU_SCALAR, agmres->First, PetscObjectComm((PetscObject)ksp));
    173:   /* Update the Shift values for the Newton basis. This is surely necessary when applying the DeflationPrecond */
    174:   if (agmres->DeflPrecond) {
    175:     KSPAGMRESLejaOrdering(wr, wi, agmres->Rshift, agmres->Ishift, max_k);
    176:   }
    177:   *CurNeig = r; /* Number of extracted eigenvalues */
    178:   return(0);

    180: }

    182: /*
    183:  * This function form the matrices for the generalized eigenvalue problem,
    184:  * it then compute the Schur vectors needed to augment the Newton basis.
    185:  */
    186: PetscErrorCode KSPAGMRESComputeDeflationData(KSP ksp)
    187: {
    188:   KSP_AGMRES     *agmres  = (KSP_AGMRES*)ksp->data;
    189:   Vec            *U       = agmres->U;
    190:   Vec            *TmpU    = agmres->TmpU;
    191:   PetscScalar    *MatEigL = agmres->MatEigL;
    192:   PetscScalar    *MatEigR = agmres->MatEigR;
    193:   PetscScalar    *Sr      = agmres->Sr;
    194:   PetscScalar    alpha, beta;
    195:   PetscInt       i,j;
    197:   PetscInt       max_k = agmres->max_k;     /* size of the non - augmented subspace */
    198:   PetscInt       CurNeig;       /* Current number of extracted eigenvalues */
    199:   PetscInt       N        = MAXKSPSIZE;
    200:   PetscBLASInt   bN;
    201:   PetscInt       lC       = N + 1;
    202:   PetscInt       KspSize  = KSPSIZE;
    203:   PetscBLASInt   blC,bKspSize;
    204:   PetscInt       PrevNeig = agmres->r;

    207:   PetscLogEventBegin(KSP_AGMRESComputeDeflationData, ksp, 0,0,0);
    208:   if (agmres->neig <= 1) return(0);
    209:   /* Explicitly form MatEigL = H^T*H, It can also be formed as H^T+h_{N+1,N}H^-1e^T */
    210:   alpha = 1.0;
    211:   beta  = 0.0;
    212:   PetscBLASIntCast(KspSize,&bKspSize);
    213:   PetscBLASIntCast(lC,&blC);
    214:   PetscBLASIntCast(N,&bN);
    215:   PetscStackCallBLAS("BLASgemm",BLASgemm_("T", "N", &bKspSize, &bKspSize, &blC, &alpha, agmres->hes_origin, &blC, agmres->hes_origin, &blC, &beta, MatEigL, &bN));
    216:   if (!agmres->ritz) {
    217:     /* Form TmpU = V*H where V is the Newton basis orthogonalized  with roddec*/
    218:     for (j = 0; j < KspSize; j++) {
    219:       /* Apply the elementary reflectors (stored in Qloc) on H */
    220:       KSPAGMRESRodvec(ksp, KspSize+1, &agmres->hes_origin[j*lC], TmpU[j]);
    221:     }
    222:     /* Now form MatEigR = TmpU^T*W where W is [VEC_V(1:max_k); U] */
    223:     for (j = 0; j < max_k; j++) {
    224:       VecMDot(VEC_V(j), KspSize, TmpU, &MatEigR[j*N]);
    225:     }
    226:     for (j = max_k; j < KspSize; j++) {
    227:       VecMDot(U[j-max_k], KspSize, TmpU, &MatEigR[j*N]);
    228:     }
    229:   } else { /* Form H^T */
    230:     for (j = 0; j < N; j++) {
    231:       for (i = 0; i < N; i++) {
    232:         MatEigR[j*N+i] = agmres->hes_origin[i*lC+j];
    233:       }
    234:     }
    235:   }
    236:   /* Obtain the Schur form of  the generalized eigenvalue problem MatEigL*y = \lambda*MatEigR*y */
    237:   if (agmres->DeflPrecond) {
    238:     KSPAGMRESSchurForm(ksp, KspSize, agmres->hes_origin, lC, agmres->Rloc, lC, PETSC_TRUE, Sr, &CurNeig);
    239:   } else {
    240:     KSPAGMRESSchurForm(ksp, KspSize, MatEigL, N, MatEigR, N, PETSC_FALSE, Sr, &CurNeig);
    241:   }

    243:   if (agmres->DeflPrecond) { /* Switch to DGMRES to improve the basis of the invariant subspace associated to the deflation */
    244:     agmres->HasSchur = PETSC_TRUE;
    245:     KSPDGMRESComputeDeflationData(ksp, &CurNeig);
    246:     return(0);
    247:   }
    248:   /* Form the Schur vectors in the entire subspace: U = W * Sr where W = [VEC_V(1:max_k); U]*/
    249:   for (j = 0; j < PrevNeig; j++) { /* First, copy U to a temporary place */
    250:     VecCopy(U[j], TmpU[j]);
    251:   }

    253:   for (j = 0; j < CurNeig; j++) {
    254:     VecZeroEntries(U[j]);
    255:     VecMAXPY(U[j], max_k, &Sr[j*(N+1)], &VEC_V(0));
    256:     VecMAXPY(U[j], PrevNeig, &Sr[j*(N+1)+max_k], TmpU);
    257:   }
    258:   agmres->r = CurNeig;
    259:   PetscLogEventEnd(KSP_AGMRESComputeDeflationData, ksp, 0,0,0);
    260:   return(0);
    261: }