Multilevel weighted least squares polynomial approximation

ESAIM Mathematical Modelling and Numerical Analysis, vol. 54, pp. 649–677

Abstract

Weighted least squares polynomial approximation uses random samples to determine projections of functions onto spaces of polynomials. It has been shown that, using an optimal distribution of sample locations, the number of samples required to achieve quasi-optimal approximation in a given polynomial subspace scales, up to a logarithmic factor, linearly in the dimension of this space. However, in many applications, the computation of samples includes a numerical discretization error. Thus, obtaining polynomial approximations with a single level method can become prohibitively expensive, as it requires a sufficiently large number of samples, each computed with a sufficiently small discretization error. As a solution to this problem, we propose a multilevel method that utilizes samples computed with different accuracies and is able to match the accuracy of single-level approximations with reduced computational cost. We derive complexity bounds under certain assumptions about polynomial approximability and sample work. Furthermore, we propose an adaptive algorithm for situations where such assumptions cannot be verified a priori. Finally, we provide an efficient algorithm for the sampling from optimal distributions and an analysis of computationally favorable alternative distributions. Numerical experiments underscore the practical applicability of our method.

Authors 3

1. Abdul-Lateef Haji-Ali

   Heriot-Watt University

   Affiliation as printed

   Heriot-Watt University, Edinburgh, UK

   Heriot‐Watt University Edinburgh UK

2. Fabio Nobile

   École Polytechnique Fédérale de Lausanne

   Affiliation as printed

   École Polytechnique Fédérale de Lausanne (EPFL), CSQI-MATH, Lausanne, Switzerland

3. Raul F. Tempone Aachen Department of Mathematics

   RWTH Aachen University · King Abdullah University of Science and Technology

   Affiliation as printed

   King Abdullah University of Science and Technology (KAUST), CEMSE, Thuwal, Saudi Arabia

   RWTH Aachen University, Department of Mathematics, Aachen, Germany