Computational Mathematics and Scientific Computing Seminar
Cubature, approximation, and isotropy in the hypercube
Speaker: Nick Trefethen, University of Oxford and CIMS
Location: Warren Weaver Hall 1302
Date: Oct. 7, 2016, 10 a.m.
The hypercube is the standard domain for computation in higher dimensions. We describe two respects in which the anisotropy of this domain has practical consequences. The first is a matter well known to experts (and to Chebfun users): the importance 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 is new. The standard notion of degree of a multivariate polynomial, total degree, is isotropic -- invariant under rotation. The hypercube, however, is highly anisotropic. We present a theorem showing that as a consequence, the convergence rate of multivariate polynomial approximations in a hypercube is determined not by the total degree but by the Euclidean degree, defined in terms of not the 1-norm but the 2-norm of the exponent vector k of a monomial x1k1 ··· xsks. The consequences, which relate to established ideas of cubature and approximation going back to James Clark Maxwell, are exponentially pronounced as the dimension of the hypercube increases. The talk will include numerical demonstrations.