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A function approximation algorithm using multilevel active subspaces

Fabio Nobile, Matteo Raviola, Raul Tempone

TL;DR

The paper tackles the computational burden of high-dimensional parametric maps by introducing Multilevel Active Subspaces (MLAS) that exploit hierarchies of discretization levels $f_l$ to reduce sample costs. It then defines level-specific active subspaces $\hat{U}_{L-l,l}$ and aggregates level-wise surrogates via $\mathcal{S}_L^{\mathrm{ML},\star}(f_\infty)$ to approximate $f$ more efficiently. A rigorous complexity analysis under plausible smoothness and sampling assumptions yields bounds on the work and demonstrates potential savings over single-level AS, complemented by a practical, fully discrete framework using optimally weighted discrete least-squares with Hermite bases and adaptive AMLASPA. Numerical experiments on a linear elliptic PDE with lognormal diffusion coefficients confirm substantial cost reductions while preserving accuracy, highlighting MLAS as a viable tool for high-dimensional parametric PDEs.

Abstract

The Active Subspace (AS) method is a widely used technique for identifying the most influential directions in high-dimensional input spaces that affect the output of a computational model. The standard AS algorithm requires a sufficient number of gradient evaluations (samples) of the input output map to achieve quasi-optimal reconstruction of the active subspace, which can lead to a significant computational cost if the samples include numerical discretization errors which have to be kept sufficiently small. To address this issue, we propose a multilevel version of the Active Subspace method (MLAS) that utilizes samples computed with different accuracies and yields different active subspaces across accuracy levels, which can match the accuracy of single-level AS with reduced computational cost, making it suitable for downstream tasks such as function approximation. In particular, we propose to perform the latter via optimally-weighted least-squares polynomial approximation in the different active subspaces, and we present an adaptive algorithm to choose dynamically the dimensions of the active subspaces and polynomial spaces. We demonstrate the practical viability of the MLAS method with polynomial approximation through numerical experiments based on random partial differential equations (PDEs).

A function approximation algorithm using multilevel active subspaces

TL;DR

The paper tackles the computational burden of high-dimensional parametric maps by introducing Multilevel Active Subspaces (MLAS) that exploit hierarchies of discretization levels to reduce sample costs. It then defines level-specific active subspaces and aggregates level-wise surrogates via to approximate more efficiently. A rigorous complexity analysis under plausible smoothness and sampling assumptions yields bounds on the work and demonstrates potential savings over single-level AS, complemented by a practical, fully discrete framework using optimally weighted discrete least-squares with Hermite bases and adaptive AMLASPA. Numerical experiments on a linear elliptic PDE with lognormal diffusion coefficients confirm substantial cost reductions while preserving accuracy, highlighting MLAS as a viable tool for high-dimensional parametric PDEs.

Abstract

The Active Subspace (AS) method is a widely used technique for identifying the most influential directions in high-dimensional input spaces that affect the output of a computational model. The standard AS algorithm requires a sufficient number of gradient evaluations (samples) of the input output map to achieve quasi-optimal reconstruction of the active subspace, which can lead to a significant computational cost if the samples include numerical discretization errors which have to be kept sufficiently small. To address this issue, we propose a multilevel version of the Active Subspace method (MLAS) that utilizes samples computed with different accuracies and yields different active subspaces across accuracy levels, which can match the accuracy of single-level AS with reduced computational cost, making it suitable for downstream tasks such as function approximation. In particular, we propose to perform the latter via optimally-weighted least-squares polynomial approximation in the different active subspaces, and we present an adaptive algorithm to choose dynamically the dimensions of the active subspaces and polynomial spaces. We demonstrate the practical viability of the MLAS method with polynomial approximation through numerical experiments based on random partial differential equations (PDEs).
Paper Structure (6 sections, 3 theorems, 36 equations, 2 algorithms)

This paper contains 6 sections, 3 theorems, 36 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. The Active Subspace method
  3. Multilevel Active Subspaces
  4. Complexity analysis
  5. Polynomial approximation
  6. Optimally weighted least squares

Key Result

theorem thmcountertheorem

constantine2015active Let asmp:poincare hold, then, for a fixed $V \in \mathrm{St}(d,r)$, the $L^2_\mu$ error of the approximation $g^\star \circ V^\ast$, where $g^\star$ is defined in nrt-eq:best-g, satisfies

Theorems & Definitions (5)

  • remark thmcounterremark
  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem