Mathematics Colloquium: Cubature, Approximation, and Isotropy in the Hypercube
4:10pm Neill Hall 5W
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".