Cubature, Approximation, and Isotropy in the Hypercube

ABSTRACT: The hypercube is the standard domain for computation in higher dimensions. We explore two respects in which the anisotropy of this domain has practical consequences. The first is the matter of axis-alignment in low-rank compression of multivariate functions. Rotating a function by a few degrees in two or more dimensions may change its numerical rank completely. The second concerns algorithms based on approximation by multivariate polynomials, an idea introduced by James Clerk Maxwell. Polynomials defined by the usual notion of total degree are isotropic, but in high dimensions, the hypercube is exponentially far from isotropic. Instead one should work with polynomials of a given "Euclidean degree." The talk will include numerical illustrations, a theorem based on several complex variables, and a discussion of "Padua points".

BIO: Professor Trefethen is head of the Numerical Analysis Group in the Mathematical Institute at Oxford University. He was educated at Harvard and Stanford and held professorial positions at NYU, MIT, and Cornell before moving to Oxford in 1997. He is a Fellow of the Royal Society and a member of the US National Academy of Engineering. As an author he is known for his books Numerical Linear Algebra (1997), Spectral Methods in MATLAB (2000), Schwarz- Christoffel Mapping (2002), Spectra and Pseudospectra (2005), Trefethen's Index Cards (2011), and Approximation Theory and Approximation Practice (2013). He is an ISI Highly Cited Researcher, with about 100 journal publications in numerical analysis and applied mathematics, and has served as editor for many of the leading numerical analysis journals. He has lectured in about 20 countries and 30 American states, including invited lectures at both ICM and ICIAM congresses. Some of Trefethen's recent activities include the SIAM 100-Dollar, 100-Digit Challenge, the notion of Ten Digit Algorithms ("ten digits, five seconds, and just one page"), and the Chebfun software system for numerical computation with functions. During 2011-2012 he served as President of the Society for Industrial and Applied Mathema1cs (SIAM).