Session A1 - Approximation Theory
July 12, 18:20 ~ 18:55 - Room B3
Polynomial approximation of smooth, multivariate functions on irregular domains
Simon Fraser University, Canada - firstname.lastname@example.org
Smooth, multivariate functions defined on tensor domains can be approximated using simple orthonormal bases formed as tensor products of one-dimensional orthogonal polynomials. On the other hand, constructing orthogonal polynomials on irregular domains is a difficult and computationally intensive task. Yet irregular domains arise in many practical problems, including uncertainty quantification, model-order reduction, optimal control and numerical PDEs. In this talk I will introduce a method for approximating smooth, multivariate functions on irregular domains, known as polynomial frame approximation. Importantly, this method corresponds to approximation in a frame, rather than a basis; a fact which leads to several key differences, both theoretical and numerical in nature. However, this method requires no orthogonalization or parametrization of the boundary, thus making it suitable for very general domains. I will discuss theoretical results for the approximation error, stability and sample complexity of this algorithm, and show its suitability for high-dimensional approximation through independence (or weak dependence) of the guarantees on the ambient dimension $d$. I will also present several numerical results, and highlight some open problems and challenges.
Joint work with Daan Huybrechs (K.U. Leuven), Juan Manuel Cardenas (Universidad de Concepcion) and Sebastian Scheuermann (Universidad de Concepcion).