Computing and Deflating Eigenvalues While Solving Multiple Right-Hand Side Linear Systems with an Application to Quantum Chromodynamics

Abstract

We present a new algorithm that computes eigenvalues and eigenvectors of a Hermitian positive definite matrix while solving a linear system of equations with conjugate gradient (CG). Traditionally, all the CG iteration vectors could be saved and recombined through the eigenvectors of the tridiagonal projection matrix, which is equivalent theoretically to unrestarted Lanczos. Our algorithm capitalizes on the iteration vectors produced by CG to update only a small window of vectors that approximate the eigenvectors. While this window is restarted in a locally optimal way, the CG algorithm for the linear system is unaffected. Yet, in all our experiments, if the window has more than a properly chosen but small number of vectors, it converges to the required eigenvectors at a rate identical to unrestarted Lanczos. After the solution of the linear system, eigenvectors that have not accurately converged can be improved in an incremental fashion by solving additional linear systems. In this case, eigenvectors identified in earlier systems can be used to deflate, and thus accelerate, the convergence of subsequent systems. We have used this algorithm with excellent results in lattice quantum chromodynamics applications, where hundreds of right-hand sides may be needed. Specifically, about 70 eigenvectors are obtained to full accuracy after solving 24 right-hand sides. Deflating these from the large number of subsequent right-hand sides removes the dreaded critical slowdown, where the conditioning of the matrix increases as the quark mass reaches a critical value. Our experiments show almost a constant number of iterations for our method, regardless of quark mass, and speedups of 8 over original CG for light quark masses. Introduction.The numerical solution of linear systems of equations of large, sparse matrices is central to many scientific and engineering applications. One of the most computationally demanding applications is lattice quantum chromodynamics (QCD) because not only does it involve very large matrix sizes but it also requires the solution of several linear systems with the same matrix but different right-hand sides. Direct methods, although attractive for multiple right-hand sides, cannot be used because of the size of the matrix. Iterative methods provide the only means for solving these problems.QCD is the theory of the fundamental force known as strong interaction, which describes the interactions among quarks, one of the constituents of matter. Lattice QCD is the tool for nonperturbative numerical calculations of these interactions on a Euclidean space-time lattice [58]. The heart of the computations is the solution of the lattice-Dirac equation, which translates to a linear system of equations M x = b, often for a large number of right-hand sides [20]. The Dirac operator M is

K→ππΔI=3/2decay amplitude in the

“…Performing multiple measurements on the same configuration offers two important opportunities for increased efficiency. First if we can use a low-mode deflation method such as eigCG [9] we will be able to amortize the setup costs of such an approach over a large number of inversions. Second we can use the all mode averaging technique [10] and perform most of these many inversions at reduced precision and use a relatively few accurate inversions to determine a correction that guarantees systematic double precision but with an additional (usually small) statistical error that reflects the small number of accurate solves.…”

Practical methods for a direct calculation of \Delta I=1/2 K to \pi\pi Decay

“…The first one is Lüscher's inexact low modes deflation algorithm with the domain-decomposed subspaces that are based on the property called local coherence of the low modes [4]. The second one is the EigCG algorithm by Stathopoulos and Orginos [5]. With the inexact low modes deflation method, we obtained a big factor of improvement with a 16 3 × 32 × 8 lattice on a single node machine.…”

“…We follow very closely the original work of EigCG in [5]. Our goal is to solve Ax = b fast for many right hand side vectors b.…”

Recent development in the tmLQCD software suite

“…The list of available solvers includes CG, BiCG, BiCGstab, FGMRES, CGS, EigCG [17] and GCR. A FGMRES solver applying inexact deflation as discussed in Ref.…”