Actual source code: agmresdeflation.c

  1: /*
  2:   This file computes data for the deflated restarting in the Newton-basis GMRES.
  3:   At each restart (or at each detected stagnation in the adaptive strategy), a basis of an
  4:   (approximated)invariant subspace corresponding to the smallest eigenvalues is extracted from the Krylov subspace.
  5:   It is then used to augment the Newton basis.
  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: {
 19:   PetscInt    deb[DEPTH], fin[DEPTH];
 20:   PetscInt    ipivot;
 21:   PetscScalar pivot_r, pivot_i;
 22:   PetscInt    i, L, R, j;
 23:   PetscScalar abs_pivot;
 24:   PetscScalar abs_val;

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

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

 79: /*
 80:  Compute the Schur vectors from the generalized eigenvalue problem A.u =\lambda.B.u
 81:  KspSize -  rank of the matrices A and B, size of the current Krylov basis
 82:  A - Left matrix
 83:  B - Right matrix
 84:  ldA - first dimension of A as declared  in the calling program
 85:  ldB - first dimension of B as declared  in the calling program
 86:  IsReduced - specifies if the matrices are already in the reduced form,
 87:  i.e A is a Hessenberg matrix and B is upper triangular.
 88:  Sr - on exit, the extracted Schur vectors corresponding
 89:  the smallest eigenvalues (with complex conjugates)
 90:  CurNeig - Number of extracted eigenvalues
 91: */
 92: static PetscErrorCode KSPAGMRESSchurForm(KSP ksp, PetscBLASInt KspSize, PetscScalar *A, PetscBLASInt ldA, PetscScalar *B, PetscBLASInt ldB, PetscBool IsReduced, PetscScalar *Sr, PetscInt *CurNeig)
 93: {
 94:   KSP_AGMRES   *agmres = (KSP_AGMRES *)ksp->data;
 95:   PetscInt      max_k  = agmres->max_k;
 96:   PetscBLASInt  r;
 97:   PetscInt      neig   = agmres->neig;
 98:   PetscScalar  *wr     = agmres->wr;
 99:   PetscScalar  *wi     = agmres->wi;
100:   PetscScalar  *beta   = agmres->beta;
101:   PetscScalar  *Q      = agmres->Q;
102:   PetscScalar  *Z      = agmres->Z;
103:   PetscScalar  *work   = agmres->work;
104:   PetscBLASInt *select = agmres->select;
105:   PetscInt     *perm   = agmres->perm;
106:   PetscBLASInt  sdim   = 0;
107:   PetscInt      i, j;
108:   PetscBLASInt  info;
109:   PetscBLASInt *iwork = agmres->iwork;
110:   PetscBLASInt  N     = MAXKSPSIZE;
111:   PetscBLASInt  lwork, liwork;
112:   PetscBLASInt  ilo, ihi;
113:   PetscBLASInt  ijob, wantQ, wantZ;
114:   PetscScalar   Dif[2];

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

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

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

142:   /* Sort the eigenvalues to extract the smallest ones */
143:   PetscCall(KSPAGMRESQuickSort(wr, wi, KspSize, perm));

145:   /* Count the number of extracted eigenvalues (with complex conjugates) */
146:   r = 0;
147:   while (r < neig) {
148:     if (wi[r] != 0) r += 2;
149:     else r += 1;
150:   }
151:   /* Reorder the Schur decomposition so that the cluster of smallest/largest eigenvalues appears in the leading diagonal blocks of A (and B)*/
152:   PetscCall(PetscArrayzero(select, N));
153:   if (!agmres->GreatestEig) {
154:     for (j = 0; j < r; j++) select[perm[j]] = 1;
155:   } else {
156:     for (j = 0; j < r; j++) select[perm[KspSize - j - 1]] = 1;
157:   }
158:   PetscCallBLAS("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));
159:   PetscCheck(info != 1, PetscObjectComm((PetscObject)ksp), PETSC_ERR_PLIB, "UNABLE TO REORDER THE EIGENVALUES WITH THE LAPACK ROUTINE : ILL-CONDITIONED PROBLEM");
160:   /* Extract the Schur vectors associated to the r smallest eigenvalues */
161:   PetscCall(PetscArrayzero(Sr, (N + 1) * r));
162:   for (j = 0; j < r; j++) {
163:     for (i = 0; i < KspSize; i++) Sr[j * (N + 1) + i] = Z[j * N + i];
164:   }

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

175: /*
176:   Forms the matrices for the generalized eigenvalue problem,
177:   it then compute the Schur vectors needed to augment the Newton basis.
178: */
179: PetscErrorCode KSPAGMRESComputeDeflationData(KSP ksp)
180: {
181:   KSP_AGMRES  *agmres  = (KSP_AGMRES *)ksp->data;
182:   Vec         *U       = agmres->U;
183:   Vec         *TmpU    = agmres->TmpU;
184:   PetscScalar *MatEigL = agmres->MatEigL;
185:   PetscScalar *MatEigR = agmres->MatEigR;
186:   PetscScalar *Sr      = agmres->Sr;
187:   PetscScalar  alpha, beta;
188:   PetscInt     i, j;
189:   PetscInt     max_k = agmres->max_k; /* size of the non - augmented subspace */
190:   PetscInt     CurNeig;               /* Current number of extracted eigenvalues */
191:   PetscInt     N = MAXKSPSIZE;
192:   PetscBLASInt bN;
193:   PetscInt     lC      = N + 1;
194:   PetscInt     KspSize = KSPSIZE;
195:   PetscBLASInt blC, bKspSize;
196:   PetscInt     PrevNeig = agmres->r;

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

229:   if (agmres->DeflPrecond) { /* Switch to DGMRES to improve the basis of the invariant subspace associated to the deflation */
230:     agmres->HasSchur = PETSC_TRUE;
231:     PetscCall(KSPDGMRESComputeDeflationData(ksp, &CurNeig));
232:     PetscFunctionReturn(PETSC_SUCCESS);
233:   }
234:   /* Form the Schur vectors in the entire subspace: U = W * Sr where W = [VEC_V(1:max_k); U]*/
235:   for (j = 0; j < PrevNeig; j++) { /* First, copy U to a temporary place */
236:     PetscCall(VecCopy(U[j], TmpU[j]));
237:   }

239:   for (j = 0; j < CurNeig; j++) {
240:     PetscCall(VecMAXPBY(U[j], max_k, &Sr[j * (N + 1)], 0, &VEC_V(0)));
241:     PetscCall(VecMAXPY(U[j], PrevNeig, &Sr[j * (N + 1) + max_k], TmpU));
242:   }
243:   agmres->r = CurNeig;
244:   PetscCall(PetscLogEventEnd(KSP_AGMRESComputeDeflationData, ksp, 0, 0, 0));
245:   PetscFunctionReturn(PETSC_SUCCESS);
246: }