Nick Trefethen
University of Oxford
16 septembre 2016 de 16 h 00 à 18 h 00 (heure de Montréal/HNE) Sur place
Colloque par Nick Trefethen (University of Oxford)
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 axisalignment in lowrank 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 {\em Euclidean degree,} defined in terms of not the 1norm but the 2norm of the exponent vector $\bf k$ of a monomial $x_1^{k_1}\cdots x_s^{k_s}$. 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.
Adresse
UQAM, Pavillon Président-Kennedy, 201, ave du Président-Kennedy, salle PK5115