Non-intrusive Low-Rank Separated Approximation of High-Dimensional Stochastic Models

TL;DR

This paper introduces a non-intrusive, sampling-based low-rank separated approximation method for high-dimensional stochastic models, effectively addressing the curse of dimensionality with linear growth in required samples.

Contribution

It presents a novel regularized alternating least-squares regression approach with an error indicator for efficient, non-intrusive approximation of high-dimensional stochastic models.

Findings

Number of samples grows linearly with input dimensions

Method achieves quadratic complexity in the number of random inputs

Validated on three numerical examples including high-dimensional ODEs

Abstract

This work proposes a sampling-based (non-intrusive) approach within the context of low-rank separated representations to tackle the issue of curse-of-dimensionality associated with the solution of models, e.g., PDEs/ODEs, with high-dimensional random inputs. Under some conditions discussed in details, the number of random realizations of the solution, required for a successful approximation, grows linearly with respect to the number of random inputs. The construction of the separated representation is achieved via a regularized alternating least-squares regression, together with an error indicator to estimate model parameters. The computational complexity of such a construction is quadratic in the number of random inputs. The performance of the method is investigated through its application to three numerical examples including two ODE problems with high-dimensional random inputs.