Multivariate rational approximation of functions with curves of singularities

Abstract

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.

Figures (12)

• Figure 1: (a) The product grid of sample points used for evaluating a function defined on a square domain. The sample grid is a superposition of product Chebyshev points in the $x$ and $y$ coordinates shown as black dots, and a product of points clustering exponentially fast to $x=0$ with Chebyshev points in the $y$ coordinates, highlighted in red. (b) Sample grid for a function defined on the unit disk with singularities located at $r=3/4$ and $r=1$. The grid is constructed using Chebyshev points in the radial direction and trigonometric points in the angular direction, along with points clustering towards $r=3/4$ (in blue) and $r=1$ (in red) in the radial direction.
• Figure 1: Diagram illustrating the domain decomposition technique for approximating functions with singularities along straight lines. A decomposition of the domain into triangles is performed by a meshing algorithm. Then, the triangles are refined into three quadrilaterals using a barycentric refinement. Finally, each quadrilateral is mapped into a unit square reference element by an affine transformation.
• Figure 1: Sample points used for the numerical examples in \ref{['sec_elliptic', 'sec_green']} (a)-(b), which consist of a Chebyshev grid (black dots) and points clustering exponentially fast towards the singularity curve in the normal direction (red dots).
• Figure 1: The approximation problem of \ref{['lem:gopal2019solving_lemma2']}. The contour $\Gamma = \Gamma_0\cup\Gamma_1$ lies in the closure of the region of analyticity of $f$.
• Figure 2: (a) The function $f_1(x,y) = (x(1-x))^{\frac{1}{4} + y} \sqrt{y(1-y)}$ along with (b) the approximation error $|f_1(x,y)-r(x,y)|$ by a tensor-product rational function $r$ with lines of poles clustering towards the boundary of the square. (c)-(d) Same as (a)-(b) with the function $f_2(x,y)=\sqrt{x+y}$ and its rational approximant with poles located near the lines $x=0$ and $y=0$.

Theorems & Definitions (12)

• Theorem 2.1
• Proof 1
• Remark 2.2
• Theorem 2.3: Multivariate rational convergence analysis
• Proof 2
• Theorem A.1
• Proof 3
• Remark A.2
• Lemma B.1
• Proof 4