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Higham, Nicholas (University of Manchester)
MATRIX FUNCTIONS: THEORY AND ALGORITHMS
Wednesday, 02 July, 09:30
Abstract: We describe some recent work on matrix functions, covering both theory and algorithms. We discuss general matrix functions, the special case of $p$th roots, and computation of $f(A)b$ without explicitly forming $f(A)$.
van den Eshof, Jasper (Utrecht University)
ITERATIVE LINEAR SYSTEM SOLVERS WITH INEXACT MATRIX-VECTOR PRODUCTS
Wednesday , 02 July, 11:00
Abstract: Simulations in quantum chromodynamics involving overlap fermions require the frequent solution of a linear system where the operator partly consists of a matrix sign function. To solve this system, a two-level iteration method is often employed where the outer iteration is a standard iterative solver for linear system that invokes, in every iteration step, a vector iteration method to approximate the action of the matrix sign function to a vector (the inner iteration). In this talk we discuss the impact of approximately computed matrix-vector products on a wide class of Krylov subspace methods for linear systems. We will argue that, in general, it is the choice of the basis for the Krylov subspace that determines the sensitivity to inexactness in the matrix-vector products. Subsequently, we give an overview of various choices for the basis that occur in practical methods and we review and discuss known strategies (relaxation strategies) for controlling the accuracy of the matrix-vector products. Unfortunately, a suitable choice for the basis (and therefore method) can depend on the particular problem to be solved. We give a flowchart with suggestions for the selection of an inexact iterative method for general problems. In the second part of the talk, we discuss some practical issues concerning the efficieny of the two-level iteration scheme.
Wenger, Urs (Oxford University)
OPTIMISED CONTINUED FRACTIONS FOR INVERTING THE GINSPARG-WILSON OPERATOR
Wednesday, 02 July, 12:00
Abstract: 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 where nested conjugate gradient procedures are avoided. We show that the five dimensional linear system can be made well conditioned using equivalence transformations on the continued fractions. This is of significant importance when dynamical overlap fermions are simulated.