Numerical Analysis and Scientific Computing Seminars

 

25 Oct, 2019

Title : Beyond Chebyshev Technology
Dr Marcus Webb
Lecturer in Department of Mathematics, The University of Manchester.
Abstract : Chebfun is a MATLAB software package for computing numerically with functions, whose inner workings boil down essentially to approximating functions by Chebyshev polynomial expansions. In this talk we’ll discuss problems in which Chebyshev polynomials are not the best basis to use, necessitating the transformation to other bases such as Legendre and other Jacobi polynomials (and back again to Chebyshev). The main part of the talk will be on the state-of-the-art algorithm for transforming between different families of Jacobi polynomials, due to Townsend, myself, and Olver (https://doi.org/10.1090/mcom/3277), which involves Toeplitz matrices, Hankel matrices, low-rank matrix approximation, and the FFT. The analysis involves some rational approximation problems of Zolotarev (https://doi.org/10.1137/19M1244433). We conclude with interesting related miscellanea.
Time : 2:00 PM to 3:00 PM.
Venue : Frank Adams 1, Alan Turing Building.

8 Nov, 2019

Title : Rayleigh quotient optimizations and eigenvalue problems
Prof. Zhaojun Bai
Professor of Computer Science and Mathematics, University of California, Davis.
Abstract :
Many computational science and data analysis techniques lead to optimizing Rayleigh quotient (RQ) and RQ type objective functions, such as computing excitation states (energies) of electronic structures, robust classification to handle uncertainty and constrained data clustering to incorporate a prior information. We will discuss origins of recently emerging RQ optimization problem, variational principles, and reformulations to algebraic linear and nonlinear eigenvalue problems. We will show how to exploit underlying properties of eigenvalue problems for designing eigensolvers, and illustrate the efficacy of these solvers in applications.
Time : 2:00 PM to 3:00 PM.
Venue : Frank Adams 1, Alan Turing Building.

22 Nov, 2019

Title : Compact Finite Differences and Cubic Splines.
Prof. Robert M. Corless
Professor in School of Mathematical and Statistical Sciences, University of Western Ontario.
Abstract :
In this talk I uncover and explain—using contour integrals and residues—a connection between cubic splines and a popular compact finite difference formula. The connection is that on a uniform mesh the simplest Pad\’e scheme for generating fourth-order accurate
compact finite differences gives exactly the derivatives at the interior nodes needed to guarantee twice-continuous differentiability for cubic splines. I also introduce an apparently new spline-like
interpolant that I call a compact cubic interpolant; this is similar to one introduced in 1972 by Swartz and Varga, but has higher order accuracy at the edges. I argue that for mildly nonuniform meshes the compact cubic approach offers some potential advantages, and even for
uniform meshes offers a simple way to treat the edge conditions, relieving the user of the burden of deciding to use one of the three standard options: free (natural), complete (clamped), or “not-a-knot” conditions. Finally, I establish that the matrices defining the compact cubic splines (equivalently, the fourth-order compact finite difference formulas) are positive definite, and in fact
totally nonnegative, if all mesh widths are the same sign.
Time : 2:00 PM to 3:00 PM.
Venue : Frank Adams 1, Alan Turing Building.