Enhancing $\ell_1$-minimization estimates of polynomial chaos expansions using basis selection (1407.8093v1)

Abstract: In this paper we present a basis selection method that can be used with $\ell_1$-minimization to adaptively determine the large coefficients of polynomial chaos expansions (PCE). The adaptive construction produces anisotropic basis sets that have more terms in important dimensions and limits the number of unimportant terms that increase mutual coherence and thus degrade the performance of $\ell_1$-minimization. The important features and the accuracy of basis selection are demonstrated with a number of numerical examples. Specifically, we show that for a given computational budget, basis selection produces a more accurate PCE than would be obtained if the basis is fixed a priori. We also demonstrate that basis selection can be applied with non-uniform random variables and can leverage gradient information.

• The paper presents an adaptive basis selection method to enhance $\ell_1$-minimization estimates for Polynomial Chaos Expansions (PCE) by tailoring the basis to the specific problem.
• Numerical examples show that this adaptive approach improves the accuracy of PCE estimates compared to fixed bases, particularly for systems with non-uniform variables or when gradient information is used.
• The method offers practical benefits for uncertainty quantification frameworks needing high fidelity and efficiency, while theoretically reducing mutual coherence for better $\ell_1$-minimization.

Enhancing $\ell_1$-minimization Estimates of Polynomial Chaos Expansions Using Basis Selection

The paper under review presents a methodological advancement in the domain of polynomial chaos expansions (PCE) for uncertainty quantification. The authors, Jakeman et al., introduce a basis selection strategy designed to improve $\ell_1$-minimization estimates of PCE, with a focus on adaptive determination of the polynomial basis to accentuate significant coefficients and suppress those contributing to mutual coherence. This technique is particularly important in the context of sparsity-promoting compressed sensing strategies where mutual coherence typically undermines the efficacy of $\ell_1$-minimization.

The primary contribution of this paper is the development of an adaptive basis selection mechanism which constructs anisotropic basis sets tailored to the problem at hand. By allocating more polynomial terms within dimensions of higher significance, this strategy effectively limits the surplus of negligible terms that can degrade computational performance. Notably, the algorithm surpasses the accuracy of conventionally fixed bases, as evidenced through multiple numerical examples where this adaptive construction is applied.

The numerical results cited in the paper demonstrate that within a predetermined computational budget, basis selection enhances the precision of PCE estimates. This enhancement is particularly pronounced when applied to systems described by non-uniform random variables and when leveraging gradient information. Such improvements stem not only from the adaptive nature of the basis but also from the careful orchestration of term selection, emphasizing the dimensions of greater importance.

Implications of this research extend to both practical and theoretical dimensions. Practically, the adaptive basis selection method proposed can be incorporated into computational frameworks dealing with stochastic collocation and uncertainty quantification, particularly in models requiring high fidelity and computational efficiency. From a theoretical perspective, the strategy illustrates a promising avenue for reducing mutual coherence, thus optimizing $\ell_1$-minimization applications within PCE frameworks.

Future work may explore enhancements and extensions to this method, potentially integrating machine learning techniques to refine the basis selection process further. Moreover, expanding its applicability to diverse engineering and scientific domains could substantiate the algorithm's robustness and generalizability. As the field of AI and computational physics progresses, such developments hold promise for significantly advancing the quality of uncertainty quantification analytics.

In conclusion, this paper provides a substantial contribution to the field, offering a viable solution to improve PCE estimates significantly through adaptive basis selection. The work of Jakeman et al. sets a foundational precedent for high-accuracy uncertainty quantification and paves the way for further research and refinement in the stochastic modeling domain.