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Adding (Harmonic) Block KS
This commit is contained in:
Chulwoo Jung
@@ -85,8 +85,12 @@ NAMESPACE_CHECK(multigrid);
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#include <Grid/algorithms/FFT.h>
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#include <Grid/algorithms/iterative/KrylovSchur.h>
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#include <Grid/algorithms/iterative/BlockedKrylovSchur.h>
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#include <Grid/algorithms/iterative/HarmonicBlockedKrylovSchur.h>
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#include <Grid/algorithms/iterative/Arnoldi.h>
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#include <Grid/algorithms/iterative/LanczosBidiagonalization.h>
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#include <Grid/algorithms/iterative/RestartedLanczosBidiagonalization.h>
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#include <Grid/algorithms/iterative/GCR.h>
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#include <Grid/algorithms/iterative/MultiSplittingPreconditionedCG.h>
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#endif
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@@ -0,0 +1,701 @@
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/*************************************************************************************
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Grid physics library, www.github.com/paboyle/Grid
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Source file: ./lib/algorithms/iterative/BlockKrylovSchur.h
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Copyright (C) 2015
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Author: Peter Boyle <paboyle@ph.ed.ac.uk>
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Author: Chulwoo Jung <chulwoo@bnl.gov>
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This program is free software; you can redistribute it and/or modify
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it under the terms of the GNU General Public License as published by
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the Free Software Foundation; either version 2 of the License, or
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(at your option) any later version.
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This program is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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GNU General Public License for more details.
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You should have received a copy of the GNU General Public License along
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with this program; if not, write to the Free Software Foundation, Inc.,
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51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA.
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See the full license in the file "LICENSE" in the top level distribution directory
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*************************************************************************************/
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/* END LEGAL */
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#ifndef GRID_BLOCKED_KRYLOV_SCHUR_H
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#define GRID_BLOCKED_KRYLOV_SCHUR_H
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#include <iomanip>
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NAMESPACE_BEGIN(Grid);
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/**
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* Block (block-Arnoldi) restarted Krylov-Schur eigensolver for
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* general non-Hermitian operators.
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*
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* Algorithm
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* ---------
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* Uses a block Arnoldi factorisation of block size Nblock:
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*
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* A V_k = V_k H_k + F_k B_k^dag
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*
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* where
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* V_k = Nm*Nblock orthonormal basis vectors (stored flat in basis[])
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* H_k = (Nm*Nblock) x (Nm*Nblock) upper block-Hessenberg Rayleigh quotient
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* F_k = Nblock residual vectors (the next block beyond V_k)
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* B_k = (Nm*Nblock) x Nblock coupling matrix (non-zero only in last Nblock rows)
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*
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* Each block Arnoldi step applies A to each of the Nblock vectors in the
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* current block, orthogonalises against all previous basis vectors, and
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* reduces the residual block to upper-triangular form via Householder QR
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* (implemented here as modified Gram-Schmidt within the block).
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*
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* The restart is a thick restart via the Schur decomposition of H_k:
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* H_k = Q^dag S Q
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* The leading Nk*Nblock Schur vectors (chosen by RitzFilter) are retained,
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* the basis and Rayleigh quotient are truncated, and block Arnoldi continues
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* from the Nk-th block.
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*
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* Parameters
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* ----------
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* Nblock : block size p
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* Nm : number of block steps (total Krylov dimension = Nm * Nblock)
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* Nk : number of block steps to keep after each restart (Nk < Nm)
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* Nstop : declare convergence when this many eigenpairs have converged
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* MaxIter : maximum number of outer (restart) iterations
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* Tolerance : relative convergence tolerance (||r|| < Tolerance * |lambda_max|)
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*/
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template<class Field>
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class BlockKrylovSchur {
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//--------------------------------------------------------------------
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// Types
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//--------------------------------------------------------------------
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typedef Eigen::MatrixXcd CMat;
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typedef Eigen::VectorXcd CVec;
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//--------------------------------------------------------------------
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// Parameters (set by operator())
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//--------------------------------------------------------------------
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int Nblock; // block size
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int Nm; // block steps (total dim = Nm * Nblock)
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int Nk; // blocks retained after restart
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int Nstop;
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int MaxIter;
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RealD Tolerance;
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//--------------------------------------------------------------------
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// Internal state
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//--------------------------------------------------------------------
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LinearOperatorBase<Field>& Linop;
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GridBase* Grid_;
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RitzFilter ritzFilter;
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// Flat storage: basis[s*Nblock + t] is the t-th vector of block s
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// After construction: basis has Nm*Nblock entries
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std::vector<Field> basis;
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// Rayleigh quotient (Nm*Nblock) x (Nm*Nblock)
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CMat H;
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// Residual block: Nblock vectors (the (Nm+1)-th block, unnormalised before
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// QR; normalised and orthogonalised as part of block Arnoldi)
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std::vector<Field> F;
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// Coupling matrix B: (Nm*Nblock) x Nblock.
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// In exact arithmetic only the last Nblock rows are non-zero:
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// B(Nm*Nblock - Nblock + t, s) = H_{Nm+1, Nm}(t, s) (the subdiagonal block)
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// We keep it as a full matrix for generality after restarts.
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CMat B;
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RealD beta_k; // Frobenius norm of the last subdiagonal block
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RealD rtol; // absolute tolerance = Tolerance * approxLambdaMax
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// Output
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CVec evals;
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CMat littleEvecs; // Nm*Nblock columns
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std::vector<RealD> ritzEstimates;
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public:
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std::vector<Field> evecs;
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//--------------------------------------------------------------------
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// Constructor
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//--------------------------------------------------------------------
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BlockKrylovSchur(LinearOperatorBase<Field>& _Linop, GridBase* _Grid,
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RealD _Tolerance, RitzFilter _rf = EvalReSmall)
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: Linop(_Linop), Grid_(_Grid), Tolerance(_Tolerance), ritzFilter(_rf),
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Nblock(-1), Nm(-1), Nk(-1), Nstop(-1), MaxIter(-1),
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beta_k(0.0), rtol(0.0)
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{}
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//--------------------------------------------------------------------
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// Main entry point
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//--------------------------------------------------------------------
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/**
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* Run the blocked Krylov-Schur algorithm.
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*
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* Parameters
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* ----------
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* v0 : block of Nblock starting vectors (size >= Nblock)
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* _maxIter : maximum outer (restart) iterations
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* _Nm : number of block steps per cycle
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* _Nk : number of block steps to keep after restart (Nk < Nm)
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* _Nstop : stop after _Nstop eigenvalues converged
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* _Nblock : block size
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*/
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void operator()(const std::vector<Field>& v0, int _maxIter, int _Nm, int _Nk,
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int _Nstop, int _Nblock = 1, bool doubleOrthog = true,
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bool doVerify = false)
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{
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MaxIter = _maxIter;
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Nm = _Nm;
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Nk = _Nk;
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Nstop = _Nstop;
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Nblock = _Nblock;
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assert((int)v0.size() >= Nblock);
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assert(Nk < Nm);
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int N = Nm * Nblock; // total Krylov dimension
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// Approximate largest eigenvalue for tolerance normalisation
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RealD approxLambdaMax = approxMaxEval(v0[0]);
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rtol = Tolerance * approxLambdaMax;
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std::cout << GridLogMessage << "BlockKrylovSchur: approx max eval = "
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<< approxLambdaMax << ", rtol = " << rtol << std::endl;
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// Initialise
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H = CMat::Zero(N, N);
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B = CMat::Zero(N, Nblock);
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int start = 0;
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std::vector<Field> startBlock(v0.begin(), v0.begin() + Nblock);
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for (int iter = 0; iter < MaxIter; iter++) {
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std::cout << GridLogMessage << "BlockKrylovSchur: restart iteration " << iter << std::endl;
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// ---- Block Arnoldi: extend from block start to block Nm ----
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blockArnoldiIteration(startBlock, Nm, start, doubleOrthog);
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// After first full cycle start from block Nk
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start = Nk;
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if (doVerify) {
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std::string lbl = "iter " + std::to_string(iter) + " after Arnoldi";
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verify(lbl);
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}
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// ---- Schur decompose H ----
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ComplexSchurDecomposition schur(H, false, ritzFilter);
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std::cout << GridLogMessage << "BlockKrylovSchur: Schur decomposed." << std::endl;
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// Reorder: bring wanted Nk*Nblock Schur values to top-left
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schur.schurReorder(Nk * Nblock);
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std::cout << GridLogMessage << "BlockKrylovSchur: Schur reordered." << std::endl;
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CMat Q = schur.getMatrixQ();
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CMat Qt = Q.adjoint();
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// Rotate Krylov basis: basis_new[i] = sum_j basis[j] * Qt(j,i)
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std::vector<Field> basis2;
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constructUR(basis2, basis, Qt, N);
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basis = basis2;
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// Update b and H
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B = Q * B;
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H = schur.getMatrixS();
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// ---- Truncate to Nk*Nblock ----
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int Nkeep = Nk * Nblock;
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CMat Htmp = H(Eigen::seqN(0, Nkeep), Eigen::seqN(0, Nkeep));
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H = CMat::Zero(N, N);
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H(Eigen::seqN(0, Nkeep), Eigen::seqN(0, Nkeep)) = Htmp;
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std::vector<Field> basisTmp(basis.begin(), basis.begin() + Nkeep);
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basis = basisTmp;
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CMat Btmp = B(Eigen::seqN(0, Nkeep), Eigen::all);
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B = CMat::Zero(N, Nblock);
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B(Eigen::seqN(0, Nkeep), Eigen::all) = Btmp;
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// beta_k = Frobenius norm of the effective coupling
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beta_k = Btmp.norm();
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std::cout << GridLogMessage << "BlockKrylovSchur: beta_k = " << beta_k << std::endl;
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// Restart: the new starting block is F (the residual block from Arnoldi)
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startBlock = F;
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if (doVerify) {
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std::string lbl = "iter " + std::to_string(iter) + " after restart+truncation";
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verify(lbl);
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}
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// ---- Compute eigensystem of truncated H for convergence check ----
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CMat Hk = H(Eigen::seqN(0, Nkeep), Eigen::seqN(0, Nkeep));
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computeEigensystem(Hk, Nkeep);
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int Nconv = converged(Nkeep);
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std::cout << GridLogMessage << "BlockKrylovSchur: converged " << Nconv
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<< " / " << Nstop << std::endl;
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if (Nconv >= Nstop || iter == MaxIter - 1) {
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std::cout << GridLogMessage << "BlockKrylovSchur: done after " << iter
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<< " restarts, " << Nconv << " converged." << std::endl;
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std::cout << GridLogMessage << "Eigenvalues: " << evals.transpose() << std::endl;
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return;
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}
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}
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}
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// Accessors
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std::vector<Field> getEvecs() { return evecs; }
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CVec getEvals() { return evals; }
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std::vector<RealD> getRitzEstimates() { return ritzEstimates; }
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//--------------------------------------------------------------------
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// Verification: print H and B, check A V = V H + F B^dag explicitly
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//--------------------------------------------------------------------
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/**
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* Checks the block Arnoldi / Krylov-Schur decomposition
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*
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* A V = V H + F B^dag (KS)
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*
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* by explicit operator applications. For each basis vector j:
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*
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* w_j = A basis[j]
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*
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* The nBasis × nBasis matrix M of inner products is computed:
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*
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* M[i, j] = <basis[i] | A basis[j]>
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*
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* and compared column-by-column against H. Separately, the nBasis × Nblock
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* residual coupling matrix R is computed:
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*
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* R[j, t] = <basis[j] | F[t]> * ||F[t]|| (scaled by F-block norms)
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*
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* but since F is already normalised, R[j,t] = <basis[j] | F[t]>.
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*
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* The KS relation for column j reads:
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* w_j = sum_i basis[i] H[i,j] + sum_t F[t] B[j,t]*
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* so the deviation in column j is
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* dev_j = w_j - sum_i basis[i] M[i,j] (should be zero for exact arithmetic)
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* augmented by the F B^dag term in the last block.
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*
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* Prints:
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* - H (current Rayleigh quotient, nBasis × nBasis)
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* - B (coupling matrix, nBasis × Nblock)
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* - M (explicit inner product matrix <V | A V>)
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* - max |H[i,j] - M[i,j]| (should be O(machine epsilon))
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* - for each basis column j: || A v_j - V H[:,j] - F B[j,:]^* ||
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*
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* Parameters
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* ----------
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* label : string printed at the start (e.g. "after restart 2")
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*/
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void verify(const std::string& label = "")
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{
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int nBasis = (int)basis.size();
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int nF = (int)F.size();
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if (nBasis == 0) {
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std::cout << GridLogMessage
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<< "BlockKrylovSchur::verify [" << label
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<< "]: basis is empty." << std::endl;
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return;
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}
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std::cout << GridLogMessage
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<< "======== BlockKrylovSchur::verify [" << label << "] ========" << std::endl;
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std::cout << GridLogMessage
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<< " nBasis = " << nBasis << " Nblock = " << Nblock
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<< " nF = " << nF << std::endl;
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// ---- Print H ----
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std::cout << GridLogMessage << "H (" << nBasis << " x " << nBasis << "):" << std::endl;
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for (int i = 0; i < nBasis; i++) {
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for (int j = 0; j < nBasis; j++)
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std::cout << " " << std::setw(14) << H(i, j);
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std::cout << std::endl;
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}
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// ---- Print B ----
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std::cout << GridLogMessage << "B (" << nBasis << " x " << nF << "):" << std::endl;
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for (int i = 0; i < nBasis; i++) {
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for (int t = 0; t < nF; t++)
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std::cout << " " << std::setw(14) << B(i, t);
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std::cout << std::endl;
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}
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// ---- Compute M[i,j] = <basis[i] | A basis[j]> ----
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CMat M = CMat::Zero(nBasis, nBasis);
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Field w(Grid_);
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for (int j = 0; j < nBasis; j++) {
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Linop.Op(basis[j], w);
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for (int i = 0; i < nBasis; i++)
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M(i, j) = toStdCmplx(innerProduct(basis[i], w));
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}
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std::cout << GridLogMessage << "M = <V|AV> (" << nBasis << " x " << nBasis << "):" << std::endl;
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for (int i = 0; i < nBasis; i++) {
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for (int j = 0; j < nBasis; j++)
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std::cout << " " << std::setw(14) << M(i, j);
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std::cout << std::endl;
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}
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// ---- max |H - M| ----
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RealD maxHM = 0.0;
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for (int i = 0; i < nBasis; i++)
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for (int j = 0; j < nBasis; j++)
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maxHM = std::max(maxHM, std::abs(H(i,j) - M(i,j)));
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std::cout << GridLogMessage
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<< " max |H[i,j] - M[i,j]| = " << maxHM << std::endl;
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// ---- Check orthonormality of basis ----
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CMat G = CMat::Zero(nBasis, nBasis);
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for (int i = 0; i < nBasis; i++)
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for (int j = 0; j < nBasis; j++)
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G(i, j) = toStdCmplx(innerProduct(basis[i], basis[j]));
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CMat Gerr = G - CMat::Identity(nBasis, nBasis);
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std::cout << GridLogMessage
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<< " max |<V_i|V_j> - delta_ij| = " << Gerr.cwiseAbs().maxCoeff() << std::endl;
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// ---- Per-column residual: || A v_j - V H[:,j] - F B[j,:]^* || ----
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// For each basis vector j, compute A v_j then subtract V H[:,j] and F B[j,:]^*
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RealD maxColDev = 0.0;
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for (int j = 0; j < nBasis; j++) {
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Linop.Op(basis[j], w);
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// subtract V H[:,j]
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for (int i = 0; i < nBasis; i++)
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w -= basis[i] * H(i, j);
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// subtract F B[j,:]^* (F[t] * conj(B[j,t]))
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for (int t = 0; t < nF; t++)
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w -= F[t] * std::conj(B(j, t));
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RealD dev = std::sqrt(norm2(w));
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std::cout << GridLogMessage
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<< " || A v[" << j << "] - V H[:,j] - F B[j,:]* || = " << dev << std::endl;
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maxColDev = std::max(maxColDev, dev);
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}
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std::cout << GridLogMessage
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<< " max column deviation = " << maxColDev << std::endl;
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// ---- Check F block orthogonality against basis ----
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if (nF > 0) {
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RealD maxFV = 0.0;
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for (int t = 0; t < nF; t++)
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for (int i = 0; i < nBasis; i++) {
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RealD ip = std::abs(toStdCmplx(innerProduct(basis[i], F[t])));
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maxFV = std::max(maxFV, ip);
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}
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std::cout << GridLogMessage
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<< " max |<V_i | F_t>| (should be ~0) = " << maxFV << std::endl;
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}
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std::cout << GridLogMessage
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<< "======== end verify ========" << std::endl;
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||||
}
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private:
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//--------------------------------------------------------------------
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// Block Arnoldi iteration
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||||
//--------------------------------------------------------------------
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||||
/**
|
||||
* Extends the block Arnoldi factorisation from block index 'start' to
|
||||
* block index 'Nm'.
|
||||
*
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||||
* On entry (start > 0): basis[0..start*Nblock-1] already set,
|
||||
* H[0..start*Nblock-1, 0..start*Nblock-1] already set,
|
||||
* B[start*Nblock-1, :] set (coupling from prior residual block).
|
||||
* startBlock = the normalised residual block F from the previous cycle.
|
||||
*
|
||||
* On entry (start == 0): initialises everything from startBlock.
|
||||
*/
|
||||
void blockArnoldiIteration(std::vector<Field>& startBlock, int endBlock,
|
||||
int startIdx, bool doubleOrthog)
|
||||
{
|
||||
int N = Nm * Nblock;
|
||||
|
||||
if (startIdx == 0) {
|
||||
basis.clear();
|
||||
F.clear();
|
||||
H = CMat::Zero(N, N);
|
||||
B = CMat::Zero(N, Nblock);
|
||||
|
||||
// Orthonormalise starting block via modified Gram-Schmidt
|
||||
std::vector<Field> V0 = startBlock;
|
||||
blockOrthonormalise(V0);
|
||||
for (auto& v : V0) basis.push_back(v);
|
||||
} else {
|
||||
// Append residual block (startBlock = F_old) to basis.
|
||||
// The truncated KS relation after restart is:
|
||||
//
|
||||
// A V_k = V_k S_k + F_old B_old^dag (*)
|
||||
//
|
||||
// where V_k = basis[0:Nkeep], S_k is stored in H[0:Nkeep,0:Nkeep],
|
||||
// B_old = B[0:Nkeep,:], F_old = startBlock.
|
||||
//
|
||||
// Once F_old is appended as basis[Nkeep:Nkeep+Nblock], (*) becomes
|
||||
// a statement about the extended H matrix:
|
||||
//
|
||||
// H[Nkeep+t, j] = (B_old^dag)[t,j] = conj(B_old[j,t])
|
||||
// for t=0..Nblock-1, j=0..Nkeep-1
|
||||
//
|
||||
// These entries are the "restart coupling rows" that connect the new
|
||||
// block to all retained Schur vectors and must be set before Arnoldi
|
||||
// continues, otherwise A V_k = V_k H[:,0:Nkeep] would be missing the
|
||||
// F_old B_old^dag term for those columns.
|
||||
|
||||
int Nkeep = startIdx * Nblock;
|
||||
for (auto& v : startBlock) basis.push_back(v);
|
||||
|
||||
// Fill restart coupling rows into H
|
||||
for (int t = 0; t < Nblock; t++)
|
||||
for (int j = 0; j < Nkeep; j++)
|
||||
H(Nkeep + t, j) = std::conj(B(j, t));
|
||||
|
||||
// Zero out B for the retained columns now that the coupling is in H
|
||||
for (int j = 0; j < Nkeep; j++)
|
||||
for (int t = 0; t < Nblock; t++)
|
||||
B(j, t) = 0.0;
|
||||
}
|
||||
|
||||
// Main block Arnoldi loop
|
||||
for (int k = startIdx; k < endBlock; k++) {
|
||||
blockArnoldiStep(k, doubleOrthog);
|
||||
}
|
||||
}
|
||||
|
||||
//--------------------------------------------------------------------
|
||||
/**
|
||||
* One block Arnoldi step: extends by one block (Nblock vectors).
|
||||
*
|
||||
* Computes block column k of H and the next basis block V_{k+1}.
|
||||
*
|
||||
* Layout of basis (flat):
|
||||
* basis[j*Nblock + t] = t-th vector of j-th block, j = 0..k
|
||||
*
|
||||
* After this call:
|
||||
* H[i, k*Nblock : (k+1)*Nblock] filled for i = 0..(k+1)*Nblock - 1
|
||||
* basis[k*Nblock .. (k+1)*Nblock - 1] normalised (already set on entry)
|
||||
* F = residual block (to become V_{k+1} after this step if k < Nm-1)
|
||||
*
|
||||
* If k < Nm-1, also:
|
||||
* H[(k+1)*Nblock : (k+2)*Nblock, k*Nblock : (k+1)*Nblock] = subdiag block (from QR of residual)
|
||||
* basis extended by Nblock (the normalised residual vectors)
|
||||
*/
|
||||
void blockArnoldiStep(int k, bool doubleOrthog)
|
||||
{
|
||||
int kBase = k * Nblock; // first flat index of current block
|
||||
int prevN = kBase + Nblock; // number of basis vectors so far after this step
|
||||
int N = Nm * Nblock;
|
||||
|
||||
// W[t] = A * basis[kBase + t]
|
||||
std::vector<Field> W(Nblock, Field(Grid_));
|
||||
for (int t = 0; t < Nblock; t++) {
|
||||
Linop.Op(basis[kBase + t], W[t]);
|
||||
}
|
||||
|
||||
// Orthogonalise W against all current basis vectors (full reorthogonalisation)
|
||||
// H[i, kBase + t] = <basis[i] | W[t]>
|
||||
for (int pass = 0; pass < (doubleOrthog ? 2 : 1); pass++) {
|
||||
for (int i = 0; i < prevN; i++) {
|
||||
for (int t = 0; t < Nblock; t++) {
|
||||
ComplexD coeff = innerProduct(basis[i], W[t]);
|
||||
if (pass == 0)
|
||||
H(i, kBase + t) = toStdCmplx(coeff);
|
||||
else
|
||||
H(i, kBase + t) += toStdCmplx(coeff);
|
||||
W[t] -= coeff * basis[i];
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
// Store residual block F
|
||||
F = W;
|
||||
|
||||
if (k == Nm - 1) {
|
||||
// Last block: compute coupling matrix B for KS decomp.
|
||||
//
|
||||
// blockQR modifies F in-place (F → Q orthonormal) and returns R
|
||||
// such that W_orig[t] = sum_s F[s] * R[s,t] (W_orig = F_after * R).
|
||||
//
|
||||
// The KS relation for column j = kBase+t requires the coefficient of F[s]
|
||||
// to be (B†)[s,j] = conj(B[j,s]). Matching with R[s,t]:
|
||||
// conj(B[kBase+t, s]) = R[s,t] → B[kBase+t, s] = conj(R[s,t])
|
||||
//
|
||||
// Equivalently the last Nblock rows of B are R^H (Hermitian conjugate of R).
|
||||
// Note: for Nblock=1, R is scalar real positive, so this reduces to B = R. ✓
|
||||
CMat R = blockQR(F); // F is modified in-place to become Q; returns R
|
||||
for (int t = 0; t < Nblock; t++)
|
||||
for (int s = 0; s < Nblock; s++)
|
||||
B(kBase + t, s) = std::conj(R(s, t)); // B_block = R^H
|
||||
|
||||
beta_k = R.norm();
|
||||
return;
|
||||
}
|
||||
|
||||
// Not last block: QR-decompose residual to get V_{k+1}
|
||||
CMat R = blockQR(F); // F orthonormalised in-place, R is upper triangular
|
||||
|
||||
// Subdiagonal block of H: H[(k+1)*Nblock : (k+2)*Nblock, kBase : kBase+Nblock] = R
|
||||
int nextBase = (k + 1) * Nblock;
|
||||
for (int i = 0; i < Nblock; i++)
|
||||
for (int j = 0; j < Nblock; j++)
|
||||
H(nextBase + i, kBase + j) = R(i, j);
|
||||
|
||||
// Append normalised residual block to basis
|
||||
for (int t = 0; t < Nblock; t++)
|
||||
basis.push_back(F[t]);
|
||||
}
|
||||
|
||||
//--------------------------------------------------------------------
|
||||
// Block QR via modified Gram-Schmidt within the block
|
||||
//--------------------------------------------------------------------
|
||||
/**
|
||||
* Given a block of Nblock vectors W (not necessarily orthonormal),
|
||||
* orthonormalises them in-place and returns the upper-triangular R
|
||||
* such that W_in = W_out * R.
|
||||
*
|
||||
* Handles (near-)linear dependence by zeroing vectors below threshold.
|
||||
*/
|
||||
CMat blockQR(std::vector<Field>& W)
|
||||
{
|
||||
CMat R = CMat::Zero(Nblock, Nblock);
|
||||
const RealD deflThresh = 1e-14;
|
||||
|
||||
for (int j = 0; j < Nblock; j++) {
|
||||
// Orthogonalise W[j] against W[0..j-1]
|
||||
for (int i = 0; i < j; i++) {
|
||||
ComplexD coeff = innerProduct(W[i], W[j]);
|
||||
R(i, j) = toStdCmplx(coeff);
|
||||
W[j] -= coeff * W[i];
|
||||
}
|
||||
RealD nrm = std::sqrt(norm2(W[j]));
|
||||
R(j, j) = nrm;
|
||||
if (nrm > deflThresh) {
|
||||
W[j] *= (1.0 / nrm);
|
||||
} else {
|
||||
// deflation: zero this vector
|
||||
W[j] = Zero();
|
||||
std::cout << GridLogMessage
|
||||
<< "BlockKrylovSchur: deflation at block column " << j
|
||||
<< " (norm = " << nrm << ")" << std::endl;
|
||||
}
|
||||
}
|
||||
return R;
|
||||
}
|
||||
|
||||
//--------------------------------------------------------------------
|
||||
// Orthonormalise a block against itself (no prior basis)
|
||||
//--------------------------------------------------------------------
|
||||
void blockOrthonormalise(std::vector<Field>& V)
|
||||
{
|
||||
for (int j = 0; j < (int)V.size(); j++) {
|
||||
for (int i = 0; i < j; i++) {
|
||||
ComplexD c = innerProduct(V[i], V[j]);
|
||||
V[j] -= c * V[i];
|
||||
}
|
||||
RealD nrm = std::sqrt(norm2(V[j]));
|
||||
assert(nrm > 1e-14);
|
||||
V[j] *= (1.0 / nrm);
|
||||
}
|
||||
}
|
||||
|
||||
//--------------------------------------------------------------------
|
||||
// Basis rotation: UR[i] = sum_j U[j] * R(j, i)
|
||||
//--------------------------------------------------------------------
|
||||
void constructUR(std::vector<Field>& UR, std::vector<Field>& U,
|
||||
CMat& R, int N)
|
||||
{
|
||||
UR.clear();
|
||||
Field tmp(Grid_);
|
||||
for (int i = 0; i < N; i++) {
|
||||
tmp = Zero();
|
||||
for (int j = 0; j < N; j++)
|
||||
tmp += U[j] * R(j, i);
|
||||
UR.push_back(tmp);
|
||||
}
|
||||
}
|
||||
|
||||
//--------------------------------------------------------------------
|
||||
// Eigensystem of the truncated Rayleigh quotient
|
||||
//--------------------------------------------------------------------
|
||||
void computeEigensystem(CMat& Hk, int Nkeep)
|
||||
{
|
||||
Eigen::ComplexEigenSolver<CMat> es;
|
||||
es.compute(Hk);
|
||||
evals = es.eigenvalues();
|
||||
littleEvecs = es.eigenvectors();
|
||||
|
||||
evecs.clear();
|
||||
for (int k = 0; k < Nkeep; k++) {
|
||||
CVec vec = littleEvecs.col(k);
|
||||
Field tmp(Grid_);
|
||||
tmp = Zero();
|
||||
for (int j = 0; j < (int)basis.size() && j < Nkeep; j++)
|
||||
tmp += vec[j] * basis[j];
|
||||
evecs.push_back(tmp);
|
||||
}
|
||||
}
|
||||
|
||||
//--------------------------------------------------------------------
|
||||
// Convergence check
|
||||
//--------------------------------------------------------------------
|
||||
/**
|
||||
* An eigenpair (lambda, y) is converged if the Ritz estimate
|
||||
* r = || B^dag y ||
|
||||
* satisfies r < rtol. Here B is the (Nkeep x Nblock) coupling matrix
|
||||
* and y is the little eigenvector (Nkeep-vector) of H.
|
||||
*/
|
||||
int converged(int Nkeep)
|
||||
{
|
||||
ritzEstimates.clear();
|
||||
int Nconv = 0;
|
||||
|
||||
CMat Bk = B(Eigen::seqN(0, Nkeep), Eigen::all); // Nkeep x Nblock
|
||||
|
||||
for (int k = 0; k < Nkeep; k++) {
|
||||
CVec yk = littleEvecs.col(k); // Nkeep-vector
|
||||
CVec Bty = Bk.adjoint() * yk; // Nblock-vector
|
||||
RealD res = Bty.norm();
|
||||
ritzEstimates.push_back(res);
|
||||
std::cout << GridLogMessage << "BlockKrylovSchur: Ritz estimate[" << k
|
||||
<< "] = " << res << " eval = " << evals[k] << std::endl;
|
||||
if (res < rtol) Nconv++;
|
||||
}
|
||||
return Nconv;
|
||||
}
|
||||
|
||||
//--------------------------------------------------------------------
|
||||
// Approximate maximum eigenvalue via power iteration
|
||||
//--------------------------------------------------------------------
|
||||
RealD approxMaxEval(const Field& v0, int MAX_ITER = 50)
|
||||
{
|
||||
assert(norm2(v0) > 1e-8);
|
||||
RealD lam = 0.0, denom = std::sqrt(norm2(v0));
|
||||
Field vcur(Grid_), vtmp(Grid_);
|
||||
vcur = v0;
|
||||
for (int i = 0; i < MAX_ITER; i++) {
|
||||
Linop.Op(vcur, vtmp);
|
||||
vcur = vtmp;
|
||||
RealD num = std::sqrt(norm2(vcur));
|
||||
lam = num / denom;
|
||||
denom = num;
|
||||
}
|
||||
return lam;
|
||||
}
|
||||
|
||||
};
|
||||
|
||||
NAMESPACE_END(Grid);
|
||||
|
||||
#endif // GRID_BLOCKED_KRYLOV_SCHUR_H
|
||||
@@ -0,0 +1,700 @@
|
||||
/*************************************************************************************
|
||||
|
||||
Grid physics library, www.github.com/paboyle/Grid
|
||||
|
||||
Source file: ./lib/algorithms/iterative/HarmonicBlockKrylovSchur.h
|
||||
|
||||
Copyright (C) 2015
|
||||
|
||||
Author: Peter Boyle <paboyle@ph.ed.ac.uk>
|
||||
Author: Chulwoo Jung <chulwoo@bnl.gov>
|
||||
|
||||
This program is free software; you can redistribute it and/or modify
|
||||
it under the terms of the GNU General Public License as published by
|
||||
the Free Software Foundation; either version 2 of the License, or
|
||||
(at your option) any later version.
|
||||
|
||||
This program is distributed in the hope that it will be useful,
|
||||
but WITHOUT ANY WARRANTY; without even the implied warranty of
|
||||
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
|
||||
GNU General Public License for more details.
|
||||
|
||||
You should have received a copy of the GNU General Public License along
|
||||
with this program; if not, write to the Free Software Foundation, Inc.,
|
||||
51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA.
|
||||
|
||||
See the full license in the file "LICENSE" in the top level distribution directory
|
||||
*************************************************************************************/
|
||||
/* END LEGAL */
|
||||
#ifndef GRID_HARMONIC_BLOCKED_KRYLOV_SCHUR_H
|
||||
#define GRID_HARMONIC_BLOCKED_KRYLOV_SCHUR_H
|
||||
|
||||
#include <iomanip>
|
||||
|
||||
NAMESPACE_BEGIN(Grid);
|
||||
|
||||
/**
|
||||
* Block harmonic restarted Krylov-Schur eigensolver.
|
||||
*
|
||||
* Harmonic Ritz values
|
||||
* --------------------
|
||||
* Standard Ritz values of A in a Krylov space K_m minimise the residual
|
||||
* in a Galerkin sense; they are good approximations to eigenvalues at the
|
||||
* *exterior* of the spectrum. For eigenvalues *near* a target shift σ
|
||||
* (e.g. the smallest eigenvalues when σ=0) harmonic Ritz values are
|
||||
* better-suited: they are obtained by a Petrov-Galerkin condition that
|
||||
* requires the residual to be orthogonal to (A-σI)K_m instead of K_m.
|
||||
*
|
||||
* Given the block Arnoldi factorisation
|
||||
*
|
||||
* A V = V H + F B^dag (1)
|
||||
*
|
||||
* with V orthonormal (Nm*Nblock columns), H the (Nm*Nblock)² block
|
||||
* upper-Hessenberg Rayleigh quotient, F the Nblock residual vectors and B
|
||||
* the (Nm*Nblock)×Nblock coupling matrix, the harmonic Rayleigh quotient
|
||||
* relative to shift σ is
|
||||
*
|
||||
* Hhat = H + (H - σI)^{-H} B B^H (2)
|
||||
*
|
||||
* Derivation: the harmonic Ritz condition (A-σI)Vy ⊥ (A-σI)V leads to
|
||||
*
|
||||
* [ (H-σI)^H (H-σI) + B B^H ] y = μ (H-σI)^H y
|
||||
*
|
||||
* Left-multiplying by (H-σI)^{-H} and setting θ = μ + σ gives the
|
||||
* standard eigenvalue problem Hhat y = θ y with Hhat as in (2).
|
||||
*
|
||||
* The harmonic Ritz values θ_j are eigenvalues of Hhat; among these,
|
||||
* the ones closest to σ (smallest |θ_j - σ|) are the best approximations
|
||||
* to the eigenvalues of A near σ.
|
||||
*
|
||||
* Thick restart
|
||||
* -------------
|
||||
* The Schur decomposition Hhat = Q^dag S Q is computed and the
|
||||
* leading Nk*Nblock Schur values (sorted by the RitzFilter) are kept.
|
||||
* The same unitary rotation Q is applied to both the Krylov basis and
|
||||
* to the *original* Rayleigh quotient H (not Hhat) for the restart:
|
||||
*
|
||||
* V_new = V Q^dag [first Nk*Nblock columns]
|
||||
* H_new = Q H Q^dag [truncated Nk*Nblock × Nk*Nblock]
|
||||
* B_new = Q B [truncated Nk*Nblock × Nblock]
|
||||
*
|
||||
* Block Arnoldi then resumes from block Nk, restoring H to full size
|
||||
* as new columns are appended.
|
||||
*
|
||||
* Convergence
|
||||
* -----------
|
||||
* For a harmonic Ritz pair (θ, y) the true Ritz residual bound is
|
||||
*
|
||||
* || (A - θI) V y || ≤ || B^H y ||
|
||||
*
|
||||
* (same as for standard Ritz, because B captures the full coupling).
|
||||
* Convergence is declared when || B^H y || < Tolerance * approxLambdaMax.
|
||||
*
|
||||
* Parameters
|
||||
* ----------
|
||||
* shift : target shift σ (default 0.0)
|
||||
* Nblock : block size p
|
||||
* Nm : number of block steps (total dim = Nm * Nblock)
|
||||
* Nk : blocks to retain after each restart (Nk < Nm)
|
||||
* Nstop : stop when this many eigenpairs converge
|
||||
* MaxIter : maximum outer (restart) iterations
|
||||
* Tolerance: relative convergence tolerance
|
||||
*
|
||||
* Usage
|
||||
* -----
|
||||
* HarmonicBlockKrylovSchur<Field> hbks(LinOp, Grid, tol, shift, EvalNormSmall);
|
||||
* std::vector<Field> v0(Nblock, Field(Grid));
|
||||
* // fill v0 with random starting vectors
|
||||
* hbks(v0, maxIter, Nm, Nk, Nstop, Nblock);
|
||||
* auto evals = hbks.getEvals();
|
||||
* auto evecs = hbks.getEvecs();
|
||||
*/
|
||||
template<class Field>
|
||||
class HarmonicBlockKrylovSchur {
|
||||
|
||||
typedef Eigen::MatrixXcd CMat;
|
||||
typedef Eigen::VectorXcd CVec;
|
||||
|
||||
//--------------------------------------------------------------------
|
||||
// Parameters
|
||||
//--------------------------------------------------------------------
|
||||
int Nblock;
|
||||
int Nm;
|
||||
int Nk;
|
||||
int Nstop;
|
||||
int MaxIter;
|
||||
RealD Tolerance;
|
||||
ComplexD shift; // target shift σ
|
||||
|
||||
//--------------------------------------------------------------------
|
||||
// Internal state
|
||||
//--------------------------------------------------------------------
|
||||
LinearOperatorBase<Field>& Linop;
|
||||
GridBase* Grid_;
|
||||
RitzFilter ritzFilter;
|
||||
|
||||
std::vector<Field> basis; // Nm*Nblock flat basis
|
||||
CMat H; // (Nm*Nblock)² block-Hessenberg Rayleigh quotient
|
||||
std::vector<Field> F; // Nblock residual vectors
|
||||
CMat B; // (Nm*Nblock) × Nblock coupling matrix
|
||||
RealD beta_k;
|
||||
RealD rtol;
|
||||
|
||||
CVec evals;
|
||||
CMat littleEvecs;
|
||||
std::vector<RealD> ritzEstimates;
|
||||
|
||||
public:
|
||||
std::vector<Field> evecs;
|
||||
|
||||
//--------------------------------------------------------------------
|
||||
// Constructor
|
||||
//--------------------------------------------------------------------
|
||||
HarmonicBlockKrylovSchur(LinearOperatorBase<Field>& _Linop, GridBase* _Grid,
|
||||
RealD _Tolerance, ComplexD _shift = 0.0,
|
||||
RitzFilter _rf = EvalNormSmall)
|
||||
: Linop(_Linop), Grid_(_Grid), Tolerance(_Tolerance), shift(_shift),
|
||||
ritzFilter(_rf),
|
||||
Nblock(-1), Nm(-1), Nk(-1), Nstop(-1), MaxIter(-1),
|
||||
beta_k(0.0), rtol(0.0)
|
||||
{}
|
||||
|
||||
//--------------------------------------------------------------------
|
||||
// Main entry point
|
||||
//--------------------------------------------------------------------
|
||||
void operator()(const std::vector<Field>& v0, int _maxIter, int _Nm, int _Nk,
|
||||
int _Nstop, int _Nblock = 1, bool doubleOrthog = true,
|
||||
bool doVerify = false)
|
||||
{
|
||||
MaxIter = _maxIter;
|
||||
Nm = _Nm;
|
||||
Nk = _Nk;
|
||||
Nstop = _Nstop;
|
||||
Nblock = _Nblock;
|
||||
|
||||
assert((int)v0.size() >= Nblock);
|
||||
assert(Nk < Nm);
|
||||
|
||||
int N = Nm * Nblock;
|
||||
|
||||
RealD approxLambdaMax = approxMaxEval(v0[0]);
|
||||
rtol = Tolerance * approxLambdaMax;
|
||||
std::cout << GridLogMessage
|
||||
<< "HarmonicBlockKrylovSchur: approx max eval = " << approxLambdaMax
|
||||
<< ", rtol = " << rtol
|
||||
<< ", shift = " << shift << std::endl;
|
||||
|
||||
H = CMat::Zero(N, N);
|
||||
B = CMat::Zero(N, Nblock);
|
||||
|
||||
int start = 0;
|
||||
std::vector<Field> startBlock(v0.begin(), v0.begin() + Nblock);
|
||||
|
||||
for (int iter = 0; iter < MaxIter; iter++) {
|
||||
std::cout << GridLogMessage
|
||||
<< "HarmonicBlockKrylovSchur: restart iteration " << iter << std::endl;
|
||||
|
||||
// ---- Block Arnoldi: extend from block 'start' to block Nm ----
|
||||
blockArnoldiIteration(startBlock, Nm, start, doubleOrthog);
|
||||
start = Nk;
|
||||
|
||||
if (doVerify) {
|
||||
std::string lbl = "iter " + std::to_string(iter) + " after Arnoldi";
|
||||
verify(lbl);
|
||||
}
|
||||
|
||||
// ---- Form harmonic Rayleigh quotient ----
|
||||
// Hhat = H + (H - σI)^{-H} * B * B^H
|
||||
CMat Hhat = harmonicRayleigh(H, B, N);
|
||||
|
||||
// ---- Schur decompose Hhat ----
|
||||
ComplexSchurDecomposition schur(Hhat, false, ritzFilter);
|
||||
schur.schurReorder(Nk * Nblock);
|
||||
|
||||
std::cout << GridLogMessage
|
||||
<< "HarmonicBlockKrylovSchur: harmonic Ritz values (first Nk*Nblock):" << std::endl;
|
||||
CMat S = schur.getMatrixS();
|
||||
for (int i = 0; i < Nk * Nblock; i++)
|
||||
std::cout << GridLogMessage << " [" << i << "] " << S(i, i) << std::endl;
|
||||
|
||||
CMat Q = schur.getMatrixQ();
|
||||
CMat Qt = Q.adjoint();
|
||||
|
||||
// ---- Rotate Krylov basis using Q from Hhat ----
|
||||
std::vector<Field> basis2;
|
||||
constructUR(basis2, basis, Qt, N);
|
||||
basis = basis2;
|
||||
|
||||
// ---- Update H and B (rotate H, not Hhat) ----
|
||||
H = Q * H * Qt;
|
||||
B = Q * B;
|
||||
|
||||
// ---- Truncate to Nk*Nblock ----
|
||||
int Nkeep = Nk * Nblock;
|
||||
|
||||
CMat Htmp = H(Eigen::seqN(0, Nkeep), Eigen::seqN(0, Nkeep));
|
||||
H = CMat::Zero(N, N);
|
||||
H(Eigen::seqN(0, Nkeep), Eigen::seqN(0, Nkeep)) = Htmp;
|
||||
|
||||
std::vector<Field> basisTmp(basis.begin(), basis.begin() + Nkeep);
|
||||
basis = basisTmp;
|
||||
|
||||
CMat Btmp = B(Eigen::seqN(0, Nkeep), Eigen::all);
|
||||
B = CMat::Zero(N, Nblock);
|
||||
B(Eigen::seqN(0, Nkeep), Eigen::all) = Btmp;
|
||||
|
||||
beta_k = Btmp.norm();
|
||||
std::cout << GridLogMessage
|
||||
<< "HarmonicBlockKrylovSchur: beta_k = " << beta_k << std::endl;
|
||||
|
||||
// Restart from the residual block F (unchanged from last Arnoldi step).
|
||||
// Note: for a Hermitian operator the correct H rows H[i,j] for i >= Nkeep+Nblock,
|
||||
// j < Nkeep are filled via Hermitian symmetry inside blockArnoldiStep.
|
||||
startBlock = F;
|
||||
|
||||
if (doVerify) {
|
||||
std::string lbl = "iter " + std::to_string(iter) + " after restart+truncation";
|
||||
verify(lbl);
|
||||
}
|
||||
|
||||
// ---- Eigensystem of truncated H for convergence ----
|
||||
CMat Hk = H(Eigen::seqN(0, Nkeep), Eigen::seqN(0, Nkeep));
|
||||
computeEigensystem(Hk, Nkeep);
|
||||
|
||||
int Nconv = converged(Nkeep);
|
||||
std::cout << GridLogMessage
|
||||
<< "HarmonicBlockKrylovSchur: converged " << Nconv
|
||||
<< " / " << Nstop << std::endl;
|
||||
|
||||
if (Nconv >= Nstop || iter == MaxIter - 1) {
|
||||
std::cout << GridLogMessage
|
||||
<< "HarmonicBlockKrylovSchur: done after " << iter
|
||||
<< " restarts, " << Nconv << " converged." << std::endl;
|
||||
std::cout << GridLogMessage << "Eigenvalues: " << evals.transpose() << std::endl;
|
||||
return;
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
// Accessors
|
||||
std::vector<Field> getEvecs() { return evecs; }
|
||||
CVec getEvals() { return evals; }
|
||||
std::vector<RealD> getRitzEstimates() { return ritzEstimates; }
|
||||
|
||||
//--------------------------------------------------------------------
|
||||
// Verification: check A V = V H + F B^dag explicitly
|
||||
//--------------------------------------------------------------------
|
||||
/**
|
||||
* Checks the block Arnoldi / Krylov-Schur decomposition
|
||||
*
|
||||
* A V = V H + F B^dag (KS)
|
||||
*
|
||||
* by explicit operator applications. H here is the standard Rayleigh
|
||||
* quotient (not Hhat), so the KS relation is the same as for
|
||||
* BlockKrylovSchur.
|
||||
*
|
||||
* Prints:
|
||||
* - H (current Rayleigh quotient, nBasis × nBasis)
|
||||
* - B (coupling matrix, nBasis × Nblock)
|
||||
* - M (explicit inner product matrix <V | A V>)
|
||||
* - max |H[i,j] - M[i,j]| (should be O(machine epsilon))
|
||||
* - max |<V_i|V_j> - delta_ij| (orthonormality check)
|
||||
* - for each basis column j: || A v_j - V H[:,j] - F B[j,:]^* ||
|
||||
* - max |<V_i | F_t>| (F orthogonal to basis)
|
||||
*/
|
||||
void verify(const std::string& label = "")
|
||||
{
|
||||
int nBasis = (int)basis.size();
|
||||
int nF = (int)F.size();
|
||||
|
||||
if (nBasis == 0) {
|
||||
std::cout << GridLogMessage
|
||||
<< "HarmonicBlockKrylovSchur::verify [" << label
|
||||
<< "]: basis is empty." << std::endl;
|
||||
return;
|
||||
}
|
||||
|
||||
std::cout << GridLogMessage
|
||||
<< "======== HarmonicBlockKrylovSchur::verify [" << label << "] ========" << std::endl;
|
||||
std::cout << GridLogMessage
|
||||
<< " nBasis = " << nBasis << " Nblock = " << Nblock
|
||||
<< " nF = " << nF << std::endl;
|
||||
|
||||
// ---- Print H ----
|
||||
std::cout << GridLogMessage << "H (" << nBasis << " x " << nBasis << "):" << std::endl;
|
||||
for (int i = 0; i < nBasis; i++) {
|
||||
for (int j = 0; j < nBasis; j++)
|
||||
std::cout << " " << std::setw(14) << H(i, j);
|
||||
std::cout << std::endl;
|
||||
}
|
||||
|
||||
// ---- Print B ----
|
||||
std::cout << GridLogMessage << "B (" << nBasis << " x " << nF << "):" << std::endl;
|
||||
for (int i = 0; i < nBasis; i++) {
|
||||
for (int t = 0; t < nF; t++)
|
||||
std::cout << " " << std::setw(14) << B(i, t);
|
||||
std::cout << std::endl;
|
||||
}
|
||||
|
||||
// ---- Compute M[i,j] = <basis[i] | A basis[j]> ----
|
||||
CMat M = CMat::Zero(nBasis, nBasis);
|
||||
Field w(Grid_);
|
||||
for (int j = 0; j < nBasis; j++) {
|
||||
Linop.Op(basis[j], w);
|
||||
for (int i = 0; i < nBasis; i++)
|
||||
M(i, j) = toStdCmplx(innerProduct(basis[i], w));
|
||||
}
|
||||
|
||||
std::cout << GridLogMessage << "M = <V|AV> (" << nBasis << " x " << nBasis << "):" << std::endl;
|
||||
for (int i = 0; i < nBasis; i++) {
|
||||
for (int j = 0; j < nBasis; j++)
|
||||
std::cout << " " << std::setw(14) << M(i, j);
|
||||
std::cout << std::endl;
|
||||
}
|
||||
|
||||
// ---- max |H - M| ----
|
||||
RealD maxHM = 0.0;
|
||||
for (int i = 0; i < nBasis; i++)
|
||||
for (int j = 0; j < nBasis; j++)
|
||||
maxHM = std::max(maxHM, std::abs(H(i,j) - M(i,j)));
|
||||
std::cout << GridLogMessage
|
||||
<< " max |H[i,j] - M[i,j]| = " << maxHM << std::endl;
|
||||
|
||||
// ---- Check orthonormality of basis ----
|
||||
CMat G = CMat::Zero(nBasis, nBasis);
|
||||
for (int i = 0; i < nBasis; i++)
|
||||
for (int j = 0; j < nBasis; j++)
|
||||
G(i, j) = toStdCmplx(innerProduct(basis[i], basis[j]));
|
||||
CMat Gerr = G - CMat::Identity(nBasis, nBasis);
|
||||
std::cout << GridLogMessage
|
||||
<< " max |<V_i|V_j> - delta_ij| = " << Gerr.cwiseAbs().maxCoeff() << std::endl;
|
||||
|
||||
// ---- Per-column residual: || A v_j - V H[:,j] - F B[j,:]^* || ----
|
||||
RealD maxColDev = 0.0;
|
||||
for (int j = 0; j < nBasis; j++) {
|
||||
Linop.Op(basis[j], w);
|
||||
|
||||
// subtract V H[:,j]
|
||||
for (int i = 0; i < nBasis; i++)
|
||||
w -= basis[i] * H(i, j);
|
||||
|
||||
// subtract F B[j,:]^* (F[t] * conj(B[j,t]))
|
||||
for (int t = 0; t < nF; t++)
|
||||
w -= F[t] * std::conj(B(j, t));
|
||||
|
||||
RealD dev = std::sqrt(norm2(w));
|
||||
std::cout << GridLogMessage
|
||||
<< " || A v[" << j << "] - V H[:,j] - F B[j,:]* || = " << dev << std::endl;
|
||||
maxColDev = std::max(maxColDev, dev);
|
||||
}
|
||||
std::cout << GridLogMessage
|
||||
<< " max column deviation = " << maxColDev << std::endl;
|
||||
|
||||
// ---- Check F block orthogonality against basis ----
|
||||
if (nF > 0) {
|
||||
RealD maxFV = 0.0;
|
||||
for (int t = 0; t < nF; t++)
|
||||
for (int i = 0; i < nBasis; i++) {
|
||||
RealD ip = std::abs(toStdCmplx(innerProduct(basis[i], F[t])));
|
||||
maxFV = std::max(maxFV, ip);
|
||||
}
|
||||
std::cout << GridLogMessage
|
||||
<< " max |<V_i | F_t>| (should be ~0) = " << maxFV << std::endl;
|
||||
}
|
||||
|
||||
std::cout << GridLogMessage
|
||||
<< "======== end verify ========" << std::endl;
|
||||
}
|
||||
|
||||
private:
|
||||
|
||||
//--------------------------------------------------------------------
|
||||
// Harmonic Rayleigh quotient
|
||||
//--------------------------------------------------------------------
|
||||
/**
|
||||
* Forms the harmonic Rayleigh quotient relative to shift σ:
|
||||
*
|
||||
* Hhat = H + (H - σI)^{-H} * B * B^H
|
||||
*
|
||||
* where H is the N×N block-Hessenberg, B is the N×Nblock coupling matrix.
|
||||
*
|
||||
* The N×N solve (H - σI)^H X = B B^H is done via Eigen's LU
|
||||
* factorisation. If H - σI is (nearly) singular the result is
|
||||
* ill-conditioned; in that case σ should be perturbed slightly.
|
||||
*/
|
||||
CMat harmonicRayleigh(const CMat& H_, const CMat& B_, int N)
|
||||
{
|
||||
CMat K = H_ - shift * CMat::Identity(N, N);
|
||||
CMat KH = K.adjoint(); // (H - σI)^H
|
||||
|
||||
// Solve KH * X = B B^H → X = KH^{-1} B B^H = (H-σI)^{-H} B B^H
|
||||
CMat BBH = B_ * B_.adjoint(); // N × N
|
||||
CMat X = KH.lu().solve(BBH); // N × N
|
||||
|
||||
return H_ + X;
|
||||
}
|
||||
|
||||
//--------------------------------------------------------------------
|
||||
// Block Arnoldi iteration
|
||||
//--------------------------------------------------------------------
|
||||
void blockArnoldiIteration(std::vector<Field>& startBlock, int endBlock,
|
||||
int startIdx, bool doubleOrthog)
|
||||
{
|
||||
int N = Nm * Nblock;
|
||||
|
||||
if (startIdx == 0) {
|
||||
basis.clear();
|
||||
F.clear();
|
||||
H = CMat::Zero(N, N);
|
||||
B = CMat::Zero(N, Nblock);
|
||||
|
||||
std::vector<Field> V0 = startBlock;
|
||||
blockOrthonormalise(V0);
|
||||
for (auto& v : V0) basis.push_back(v);
|
||||
} else {
|
||||
// Append residual block (startBlock = F_old) to basis.
|
||||
// The truncated KS relation after restart is:
|
||||
//
|
||||
// A V_k = V_k S_k + F_old B_old^dag (*)
|
||||
//
|
||||
// where V_k = basis[0:Nkeep], S_k is stored in H[0:Nkeep,0:Nkeep],
|
||||
// B_old = B[0:Nkeep,:], F_old = startBlock.
|
||||
//
|
||||
// Once F_old is appended as basis[Nkeep:Nkeep+Nblock], (*) becomes
|
||||
// a statement about the extended H matrix:
|
||||
//
|
||||
// H[Nkeep+t, j] = (B_old^dag)[t,j] = conj(B_old[j,t])
|
||||
// for t=0..Nblock-1, j=0..Nkeep-1
|
||||
//
|
||||
// These "restart coupling rows" must be set before Arnoldi continues.
|
||||
int Nkeep = startIdx * Nblock;
|
||||
for (auto& v : startBlock) basis.push_back(v);
|
||||
|
||||
// Fill restart coupling rows into H
|
||||
for (int t = 0; t < Nblock; t++)
|
||||
for (int j = 0; j < Nkeep; j++)
|
||||
H(Nkeep + t, j) = std::conj(B(j, t));
|
||||
|
||||
// Zero out B for the retained columns now that the coupling is in H
|
||||
for (int j = 0; j < Nkeep; j++)
|
||||
for (int t = 0; t < Nblock; t++)
|
||||
B(j, t) = 0.0;
|
||||
}
|
||||
|
||||
for (int k = startIdx; k < endBlock; k++)
|
||||
blockArnoldiStep(k, doubleOrthog);
|
||||
}
|
||||
|
||||
//--------------------------------------------------------------------
|
||||
// One block Arnoldi step
|
||||
//--------------------------------------------------------------------
|
||||
void blockArnoldiStep(int k, bool doubleOrthog)
|
||||
{
|
||||
int kBase = k * Nblock;
|
||||
int prevN = kBase + Nblock;
|
||||
int N = Nm * Nblock;
|
||||
|
||||
std::vector<Field> W(Nblock, Field(Grid_));
|
||||
for (int t = 0; t < Nblock; t++)
|
||||
Linop.Op(basis[kBase + t], W[t]);
|
||||
|
||||
// Full reorthogonalisation against all current basis vectors
|
||||
for (int pass = 0; pass < (doubleOrthog ? 2 : 1); pass++) {
|
||||
for (int i = 0; i < prevN; i++) {
|
||||
for (int t = 0; t < Nblock; t++) {
|
||||
ComplexD coeff = innerProduct(basis[i], W[t]);
|
||||
if (pass == 0)
|
||||
H(i, kBase + t) = toStdCmplx(coeff);
|
||||
else
|
||||
H(i, kBase + t) += toStdCmplx(coeff);
|
||||
W[t] -= coeff * basis[i];
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
F = W;
|
||||
|
||||
if (k == Nm - 1) {
|
||||
// Last block: record coupling in B as R^H (Hermitian conjugate of QR factor)
|
||||
// KS relation requires B[kBase+t, s] = conj(R[s,t])
|
||||
CMat R = blockQR(F);
|
||||
for (int t = 0; t < Nblock; t++)
|
||||
for (int s = 0; s < Nblock; s++)
|
||||
B(kBase + t, s) = std::conj(R(s, t)); // B_block = R^H
|
||||
beta_k = R.norm();
|
||||
// Hermitian symmetry fill for last block (same as non-last path below)
|
||||
for (int t = 0; t < Nblock; t++)
|
||||
for (int j = 0; j < kBase; j++)
|
||||
H(kBase + t, j) = std::conj(H(j, kBase + t));
|
||||
return;
|
||||
}
|
||||
|
||||
// Not last: QR the residual, extend basis
|
||||
CMat R = blockQR(F);
|
||||
|
||||
int nextBase = (k + 1) * Nblock;
|
||||
for (int i = 0; i < Nblock; i++)
|
||||
for (int j = 0; j < Nblock; j++)
|
||||
H(nextBase + i, kBase + j) = R(i, j);
|
||||
|
||||
for (int t = 0; t < Nblock; t++)
|
||||
basis.push_back(F[t]);
|
||||
|
||||
// Hermitian symmetry fill: H[kBase+t, j] = conj(H[j, kBase+t]) for j < kBase.
|
||||
//
|
||||
// In a fresh block Arnoldi the Krylov structure forces H[kBase+t, j] = 0 for
|
||||
// j < kBase-Nblock (sub-subdiagonal), so this is a no-op.
|
||||
//
|
||||
// After a non-Schur restart (e.g. harmonic restart where H_new = Q H Q^dag is
|
||||
// a full matrix), A v_k_j for j < Nkeep has components in ALL new extended
|
||||
// vectors, making these elements non-zero. The Arnoldi step fills column
|
||||
// kBase+t (H[j, kBase+t] for j < prevN) via inner products, but never fills
|
||||
// the corresponding row. For a Hermitian operator the two are related by
|
||||
// H[kBase+t, j] = <basis[kBase+t] | A basis[j]>
|
||||
// = conj(<basis[j] | A basis[kBase+t]>) = conj(H[j, kBase+t])
|
||||
// Filling these ensures H = H^dag and fixes the M != H discrepancy that
|
||||
// corrupts subsequent Arnoldi steps after a harmonic restart.
|
||||
for (int t = 0; t < Nblock; t++)
|
||||
for (int j = 0; j < kBase; j++)
|
||||
H(kBase + t, j) = std::conj(H(j, kBase + t));
|
||||
}
|
||||
|
||||
//--------------------------------------------------------------------
|
||||
// Block QR (modified Gram-Schmidt within the block)
|
||||
//--------------------------------------------------------------------
|
||||
CMat blockQR(std::vector<Field>& W)
|
||||
{
|
||||
CMat R = CMat::Zero(Nblock, Nblock);
|
||||
const RealD deflThresh = 1e-14;
|
||||
|
||||
for (int j = 0; j < Nblock; j++) {
|
||||
for (int i = 0; i < j; i++) {
|
||||
ComplexD coeff = innerProduct(W[i], W[j]);
|
||||
R(i, j) = toStdCmplx(coeff);
|
||||
W[j] -= coeff * W[i];
|
||||
}
|
||||
RealD nrm = std::sqrt(norm2(W[j]));
|
||||
R(j, j) = nrm;
|
||||
if (nrm > deflThresh) {
|
||||
W[j] *= (1.0 / nrm);
|
||||
} else {
|
||||
W[j] = Zero();
|
||||
std::cout << GridLogMessage
|
||||
<< "HarmonicBlockKrylovSchur: deflation at block column " << j
|
||||
<< " (norm = " << nrm << ")" << std::endl;
|
||||
}
|
||||
}
|
||||
return R;
|
||||
}
|
||||
|
||||
//--------------------------------------------------------------------
|
||||
// Orthonormalise starting block
|
||||
//--------------------------------------------------------------------
|
||||
void blockOrthonormalise(std::vector<Field>& V)
|
||||
{
|
||||
for (int j = 0; j < (int)V.size(); j++) {
|
||||
for (int i = 0; i < j; i++) {
|
||||
ComplexD c = innerProduct(V[i], V[j]);
|
||||
V[j] -= c * V[i];
|
||||
}
|
||||
RealD nrm = std::sqrt(norm2(V[j]));
|
||||
assert(nrm > 1e-14);
|
||||
V[j] *= (1.0 / nrm);
|
||||
}
|
||||
}
|
||||
|
||||
//--------------------------------------------------------------------
|
||||
// Basis rotation: UR[i] = sum_j U[j] * R(j,i)
|
||||
//--------------------------------------------------------------------
|
||||
void constructUR(std::vector<Field>& UR, std::vector<Field>& U,
|
||||
CMat& R, int N)
|
||||
{
|
||||
UR.clear();
|
||||
Field tmp(Grid_);
|
||||
for (int i = 0; i < N; i++) {
|
||||
tmp = Zero();
|
||||
for (int j = 0; j < N; j++)
|
||||
tmp += U[j] * R(j, i);
|
||||
UR.push_back(tmp);
|
||||
}
|
||||
}
|
||||
|
||||
//--------------------------------------------------------------------
|
||||
// Eigensystem of the truncated H (not Hhat)
|
||||
//--------------------------------------------------------------------
|
||||
/**
|
||||
* Eigenvalues of H_k are the standard Ritz values in the retained
|
||||
* subspace. After convergence has been declared via harmonic estimates,
|
||||
* the final reported eigenvalues and vectors come from H_k (not Hhat_k),
|
||||
* since H_k contains the true projected operator.
|
||||
*/
|
||||
void computeEigensystem(CMat& Hk, int Nkeep)
|
||||
{
|
||||
Eigen::ComplexEigenSolver<CMat> es;
|
||||
es.compute(Hk);
|
||||
evals = es.eigenvalues();
|
||||
littleEvecs = es.eigenvectors();
|
||||
|
||||
evecs.clear();
|
||||
for (int k = 0; k < Nkeep; k++) {
|
||||
CVec vec = littleEvecs.col(k);
|
||||
Field tmp(Grid_);
|
||||
tmp = Zero();
|
||||
for (int j = 0; j < (int)basis.size() && j < Nkeep; j++)
|
||||
tmp += vec[j] * basis[j];
|
||||
evecs.push_back(tmp);
|
||||
}
|
||||
}
|
||||
|
||||
//--------------------------------------------------------------------
|
||||
// Convergence check
|
||||
//--------------------------------------------------------------------
|
||||
/**
|
||||
* Ritz estimate for eigenpair k: || B^H y_k ||
|
||||
* where y_k is the k-th eigenvector of the truncated H.
|
||||
* The same bound applies whether using Ritz or harmonic Ritz restart.
|
||||
*/
|
||||
int converged(int Nkeep)
|
||||
{
|
||||
ritzEstimates.clear();
|
||||
int Nconv = 0;
|
||||
|
||||
CMat Bk = B(Eigen::seqN(0, Nkeep), Eigen::all);
|
||||
|
||||
for (int k = 0; k < Nkeep; k++) {
|
||||
CVec yk = littleEvecs.col(k);
|
||||
CVec Bty = Bk.adjoint() * yk;
|
||||
RealD res = Bty.norm();
|
||||
ritzEstimates.push_back(res);
|
||||
std::cout << GridLogMessage
|
||||
<< "HarmonicBlockKrylovSchur: Ritz estimate[" << k
|
||||
<< "] = " << res << " eval = " << evals[k] << std::endl;
|
||||
if (res < rtol) Nconv++;
|
||||
}
|
||||
return Nconv;
|
||||
}
|
||||
|
||||
//--------------------------------------------------------------------
|
||||
// Approximate maximum eigenvalue (power iteration)
|
||||
//--------------------------------------------------------------------
|
||||
RealD approxMaxEval(const Field& v0, int MAX_ITER = 50)
|
||||
{
|
||||
assert(norm2(v0) > 1e-8);
|
||||
RealD lam = 0.0, denom = std::sqrt(norm2(v0));
|
||||
Field vcur(Grid_), vtmp(Grid_);
|
||||
vcur = v0;
|
||||
for (int i = 0; i < MAX_ITER; i++) {
|
||||
Linop.Op(vcur, vtmp);
|
||||
vcur = vtmp;
|
||||
RealD num = std::sqrt(norm2(vcur));
|
||||
lam = num / denom;
|
||||
denom = num;
|
||||
}
|
||||
return lam;
|
||||
}
|
||||
|
||||
};
|
||||
|
||||
NAMESPACE_END(Grid);
|
||||
|
||||
#endif // GRID_HARMONIC_BLOCKED_KRYLOV_SCHUR_H
|
||||
Reference in New Issue
Block a user