Efficient Bayesian inversion for simultaneous estimation of geometry and spatial field using the Karhunen-Lo\`eve expansion

TL;DR

This paper presents a computationally efficient Bayesian inversion method that simultaneously estimates geometry and spatial fields using the Karhunen-Loève expansion, significantly reducing computation time and improving accuracy in geophysical applications.

Contribution

It introduces a novel inversion approach leveraging the domain independence of the K-L expansion, avoiding repeated eigenvalue problem solutions and costly shape derivatives, thus enhancing efficiency.

Findings

Achieves two orders of magnitude faster gradient computation.

Effectively captures abrupt changes in hydraulic conductivity.

Provides accurate boundary and spatial-field uncertainty estimates.

Abstract

Detection of abrupt spatial changes in physical properties representing unique geometric features such as buried objects, cavities, and fractures is an important problem in geophysics and many engineering disciplines. In this context, simultaneous spatial field and geometry estimation methods that explicitly parameterize the background spatial field and the geometry of the embedded anomalies are of great interest. This paper introduces an advanced inversion procedure for simultaneous estimation using the domain independence property of the Karhunen-Lo\`eve (K-L) expansion. Previous methods pursuing this strategy face significant computational challenges. The associated integral eigenvalue problem (IEVP) needs to be solved repeatedly on evolving domains, and the shape derivatives in gradient-based algorithms require costly computations of the Moore-Penrose inverse. Leveraging the domain independence property of the K-L expansion, the proposed method avoids both of these bottlenecks, and the IEVP is solved only once on a fixed bounding domain. Comparative studies demonstrate that our approach yields two orders of magnitude improvement in K-L expansion gradient computation time. Inversion studies on one-dimensional and two-dimensional seepage flow problems highlight the benefits of incorporating geometry parameters along with spatial field parameters. The proposed method captures abrupt changes in hydraulic conductivity with a lower number of parameters and provides accurate estimates of boundary and spatial-field uncertainties, outperforming spatial-field-only estimation methods.