Multivariate rational approximation of functions with curves of singularities
CoRR(2023)
摘要
Functions with singularities are notoriously difficult to approximate with
conventional approximation schemes. In computational applications they are
often resolved with low-order piecewise polynomials, multilevel schemes or
other types of grading strategies. Rational functions are an exception to this
rule: for univariate functions with point singularities, such as branch points,
rational approximations exist with root-exponential convergence in the rational
degree. This is typically enabled by the clustering of poles near the
singularity. Both the theory and computational practice of rational functions
for function approximation have focused on the univariate case, with extensions
to two dimensions via identification with the complex plane. Multivariate
rational functions, i.e., quotients of polynomials of several variables, are
relatively unexplored in comparison. Yet, apart from a steep increase in
theoretical complexity, they also offer a wealth of opportunities. A first
observation is that singularities of multivariate rational functions may be
continuous curves of poles, rather than isolated ones. By generalizing the
clustering of poles from points to curves, we explore constructions of
multivariate rational approximations to functions with curves of singularities.
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