Catálogo de publicaciones - libros

We review recent analysis that has developed finite-range regularised chiral effective field theory as an effecient tool for studying the quark mass variation of QCD observables. Considering a range of regularisation schemes, we study both the expansion of the nucleon mass about the chiral limit and the practical application a of extrapolation for modern lattice QCD.

Iterative solvers based on Krylov subspace techniques form an important collection of tools for numerical linear algebra. The family of generic solvers is made up of a large number of slightly different algorithms with slightly different properties, which leads to redundancy when implementing them. To overcome this, we build the algorithms out of modular parts, which also makes the connections between the algorithms explicit. In addition to the design of this toolkit, we present here a summary of our initial investigations into automatic compiler optimisations of the code for the algorithms built from these parts. These optimisations are intended to mitigate the inefficiency introduced by modularity.

There are classes of linear problems for which a matrix-vector product is a time consuming operation because an expensive approximation method is required to compute it to a given accuracy. One important example is simulations in lattice QCD with Neuberger fermions where a matrix multiply requires the product of the matrix sign function of a large sparse matrix times a vector. The recent interest in this and similar type of applications has resulted in research efforts to study the effect of errors in the matrix-vector products on iterative linear system solvers. In this paper we give a very general and abstract discussion on this issue and try to provide insight into why some iterative system solvers are more sensitive than others.

The Lattice QCD (LQCD) community has occasionally gone through periods of self-examination of its data analysis methods and compared them with methods used in other disciplines [22, 16, 18]. This process has shown that the techniques widely used elsewhere may also be useful in analyzing LQCD data. It seems that we are in such a period now with many groups trying what are generally called Bayesian methods such as Maximal Entropy (MEM) or constrained fitting [19, 15, 1, 7, 5, and many others]. In these proceedings we will attempt to apply this process to a comparison of data modeling techniques used in LQCD and NMR Spectroscopy to see if there are methods which may also be useful when applied to LQCD data.

We investigate optimal choices for the (outer) iteration method to use when solving linear systems with Neuberger’s overlap operator in QCD. Different formulations for this operator give rise to different iterative solvers, which are optimal for the respective formulation. We compare these methods in theory and practice to find the overall optimal one. For the first time, we apply the so-called SUMR method of Jagels and Reichel to the shifted unitary version of Neuberger’s operator, and show that this method is in a sense the optimal choice for propagator computations. When solving the “squared” equations in a dynamical simulation with two degenerate flavours, it turns out that the CG method should be used.

We review the theory of elliptic functions leading to Zolotarev’s formula for the sign function over the range ε≤| x |≤1. We show how Gauss’ arithmetico-geometric mean allows us to evaluate elliptic functions cheaply, and thus to compute Zolotarev coefficients “on the fly” as a function of ε. This in turn allows us to calculate the matrix functions sgn H , $$\sqrt H $$ , and $$1/\sqrt H $$ both quickly and accurately for any Hermitian matrix H whose spectrum lies in the specified range.

We use a continued fraction expansion of the sign-function in order to obtain a five dimensional formulation of the overlap lattice Dirac operator. Within this formulation the inverse of the overlap operator can be calculated by a single Krylov space method and nested conjugate gradient procedures are avoided. We point out that the five dimensional linear system can be made well conditioned using equivalence transformations on the continued fractions.