10 research outputs found
Computing and deflating eigenvalues while solving multiple right hand side linear systems in Quantum Chromodynamics
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, this small window 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 QCD 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.Comment: 22 pages, 26 eps figure
Recycling BiCGSTAB with an Application to Parametric Model Order Reduction
Krylov subspace recycling is a process for accelerating the convergence of
sequences of linear systems. Based on this technique, the recycling BiCG
algorithm has been developed recently. Here, we now generalize and extend this
recycling theory to BiCGSTAB. Recycling BiCG focuses on efficiently solving
sequences of dual linear systems, while the focus here is on efficiently
solving sequences of single linear systems (assuming non-symmetric matrices for
both recycling BiCG and recycling BiCGSTAB).
As compared with other methods for solving sequences of single linear systems
with non-symmetric matrices (e.g., recycling variants of GMRES), BiCG based
recycling algorithms, like recycling BiCGSTAB, have the advantage that they
involve a short-term recurrence, and hence, do not suffer from storage issues
and are also cheaper with respect to the orthogonalizations.
We modify the BiCGSTAB algorithm to use a recycle space, which is built from
left and right approximate invariant subspaces. Using our algorithm for a
parametric model order reduction example gives good results. We show about 40%
savings in the number of matrix-vector products and about 35% savings in
runtime.Comment: 18 pages, 5 figures, Extended version of Max Planck Institute report
(MPIMD/13-21
Estimating the trace of the matrix inverse by interpolating from the diagonal of an approximate inverse
A number of applications require the computation of the trace of a matrix that is implicitly available through a function. A common example of a function is the inverse of a large, sparse matrix, which is the focus of this paper. When the evaluation of the function is expensive, the task is computationally challenging because the standard approach is based on a Monte Carlo method which converges slowly. We present a different approach that exploits the pattern correlation, if present, between the diagonal of the inverse of the matrix and the diagonal of some approximate inverse that can be computed inexpensively. We leverage various sampling and fitting techniques to fit the diagonal of the approximation to the diagonal of the inverse. Depending on the quality of the approximate inverse, our method may serve as a standalone kernel for providing a fast trace estimate with a small number of samples. Furthermore, the method can be used as a variance reduction method for Monte Carlo in some cases. This is decided dynamically by our algorithm. An extensive set of experiments with various technique combinations on several matrices from some real applications demonstrate the potential of our method. (C) 2016 Published by Elsevier Inc
Status and Future Perspectives for Lattice Gauge Theory Calculations to the Exascale and Beyond
In this and a set of companion whitepapers, the USQCD Collaboration lays out
a program of science and computing for lattice gauge theory. These whitepapers
describe how calculation using lattice QCD (and other gauge theories) can aid
the interpretation of ongoing and upcoming experiments in particle and nuclear
physics, as well as inspire new ones.Comment: 44 pages. 1 of USQCD whitepapers