Abstract

Co-author: Daan Huybrechs (KU Leuven)

Smooth, multivariate functions defined on tensor domains can be approximated using orthonormal bases formed as tensor products of one-dimensional orthogonal polynomials. On the other hand, constructing orthogonal polynomials in irregular domains is difficult and computationally intensive. Yet irregular domains arise in many applications, including uncertainty quantification, model-order reduction, optimal control and numerical PDEs. In this talk I will introduce a framework for approximating smooth, multivariate functions on irregular domains, known as polynomial frame approximation. Importantly, this approach 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 approach requires no orthogonalization or parametrization of the domain boundary, thus making it suitable for very general domains, including a priori unknown domains. I will discuss theoretical result s for the approximation error, stability and sample complexity of this approach, 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.