Data-driven polynomial chaos expansions: a weighted least-square approximation

TL;DR

This paper introduces a novel data-driven polynomial chaos expansion method combined with a weighted least-squares approach for efficient uncertainty quantification, featuring an input-independent sampling strategy and stable solution procedures.

Contribution

It proposes a new framework integrating data-driven basis construction with a weighted least-squares solver, improving stability and efficiency in uncertainty quantification tasks.

Findings

Sampling strategy is independent of input data.

The approach is quasi-linearly stable.

Numerical tests demonstrate effectiveness.

Abstract

In this work, we combine the idea of data-driven polynomial chaos expansions with the weighted least-square approach to solve uncertainty quantification (UQ) problems. The idea of data-driven polynomial chaos is to use statistical moments of the input random variables to develop an arbitrary polynomial chaos expansion, and then use such data-driven bases to perform UQ computations. Here we adopt the bases construction procedure by following \cite{Ahlfeld_2016SAMBA}, where the bases are computed by using matrix operations on the Hankel matrix of moments. Different from previous works, in the postprocessing part, we propose a weighted least-squares approach to solve UQ problems. This approach includes a sampling strategy and a least-squares solver. The main features of our approach are two folds: On one hand, our sampling strategy is independent of the random input. More precisely, we propose to sampling with the equilibrium measure, and this measure is also independent of the data-driven bases. Thus, this procedure can be done in prior (or in a off-line manner). On the other hand, we propose to solve a Christoffel function weighted least-square problem, and this strategy is quasi-linearly stable -- the required number of PDE solvers depends linearly (up to a logarithmic factor) on the number of (data-driven) bases. This new approach is thus promising in dealing with a class of problems with epistemic uncertainties. Several numerical tests are presented to show the effectiveness of our approach.