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Kennedy, Anthony D. (University of Edinburgh)
ZOLOTAREV ON THE FLY
Tuesday, 01 July, 09:30
Abstract: We review the theory of optimal polynomial and rational Chebyshev approximations, and the theory of elliptic functions leading to Zolotarevs formula for the sign function over the range $R = \{z:\varepsilon\leq|z|\leq1\}$. We show how Gauss arithmetic-geometric mean allows us to compute the Zolotarev coefficients on the fly as a function of $\varepsilon$. This allows us to calculate $\sgn(H)$ quickly and accurately for a Hermitian matrix $H$ whose spectrum lies in $R$.
Saad, Yousef (Minnesota University)
INTRODUCTION TO KRYLOV SUBSPACE METHODS
Tuesday, 01 July, 11:15
Abstract: Krylov subspace techniques have been used in many different areas of scientific computing as a means for projecting the original problem into one of smaller dimension. This basic principle is best illustrated when solving matrix problems such as eigenvalue problems or linear systems where they give rise to the Lanczos and the conjugate gradient algorithms. There are however many other uses of what may be termed Krylov projection. This lecture will introduce Krylov subspace methods and give an overview of these applications
Boriçi, Artan (University of Edinburgh)
COMPUTATIONAL METHODS FOR THE FERMION DETERMINANT AND THE LINK BETWEEN OVERLAP AND DOMAIN WALL FERMIONS
Tuesday, 01 July, 14:00
Abstract: This lecture comprises in two parts.
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The first part reviews numerical methods which are used in lattice QCD to compute the determinant of the lattice Dirac operator. These include methods which are based on:
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Gaussian integral representaion
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Krylov subspace evaluation of matrix functions
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Sparse approximate inverses
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The second part of the lecture deals with the formal relationship andalgebraic structure of domain wall and overlap fermions using:
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Truncated overlap fermions as the hybrid of both formulations
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Transfer matrix formalism in 4+1 dimensions
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Dimensional reduction
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Golub, Gene (Stanford University)
VARIANCE REDUCTION BY CONTROL VARIATES IN MONTE CARLO SIMULATIONS OF LARGE SCALE MATRIX FUNCTIONS
Tuesday, 01 July, 15:45
Abstract: In numerical simulation of lattice QCD and physical applications, computational kernels involve the calculation of matrix functions f(A) of a very large matrix A, where f is a smooth function. This includes the determinant and trace of f(A). Based on the theory of moments, we have been able to derive efficient algorithms which lead to highly accurate results. In this talk, we will first briefly review the theory of moments and the corresponding quadrature rules and show how they can be effectively used in these calculations. Then we will focus on a variance reduction technique in the Monte Carlo simulation which uses the first few moments of the matrix A as control variates.