DIAS: A Data-Informed Active Subspace Regularization Framework for Inverse Problems

Abstract

This paper presents a regularization framework that aims to improve the fidelity of Tikhonov inverse solutions. At the heart of the framework is the data-informed regularization idea that only data-uninformed parameters need to be regularized, while the data-informed parameters, on which data and forward model are integrated, should remain untouched. We propose to employ the active subspace method to determine the data-informativeness of a parameter. The resulting framework is thus called a data-informed (DI) active subspace (DIAS) regularization. Four proposed DIAS variants are rigorously analyzed, shown to be robust with the regularization parameter and capable of avoiding polluting solution features informed by the data. They are thus well suited for problems with small or reasonably small noise corruptions in the data. Furthermore, the DIAS approaches can effectively reuse any Tikhonov regularization codes/libraries. Though they are readily applicable for nonlinear inverse problems, we focus on linear problems in this paper in order to gain insights into the framework. Various numerical results for linear inverse problems are presented to verify theoretical findings and to demonstrate advantages of the DIAS framework over the Tikhonov, truncated SVD, and the TSVD-based DI approaches.

An autoencoder compression approach for accelerating large-scale inverse problems

“…As a capstone result, we show how our proposed autoencoder compression approach can be combined with the DIAS regularization (Nguyen et al 2022) approach on the earth-scale seismic inverse problem. While the DIAS algorithm has the benefit of only applying regularization in the inactive subspace, speed is not one of its strengths.…”

“…Through numerical experiments, we show that this approach is faster than the checkpointing approach and as fast as the state-of-the-art off-the-shelf compression approach, ZFP (Lindstrom 2014), while achieving much higher compression ratios. We then extend the data-informed active subspace (DIAS) regularization method (Nguyen et al 2022) to nonlinear problems and show that autoencoder compression can enable the affordable application of the DIAS prior to large-scale inverse problems.…”

Multi‐fidelity data fusion through parameter space reduction with applications to automotive engineering

“…Linear parameter space reduction with Active Subspaces (AS) 17 can fight such curse using input-output couples obtained by high-fidelity simulations. Successful applications of parameter space reduction with active subspaces can be found in many engineering fields: naval and nautical problems, 18 shape optimization, [19][20][21][22] car aerodynamics studies, 23 inverse problems, 24,25 cardiovascular studies coupled with intrusive model order reduction, 26 for the study of high-dimensional parametric PDEs, 27 and in CFD problems in a data-driven setting, 28,29 among others. New extensions of AS have also been developed in the recent years such as AS for multivariate vector-valued functions, 30 a kernel approach for AS for scalar and vectorial functions, 31 a localization extension for both regression and classification tasks, 32 and sequential learning of active subspaces.…”