Sensitivity-preserving of Fisher Information Matrix through random data down-sampling for experimental design

TL;DR

This paper introduces a randomized down-sampling method using matrix sketching to select experimental data that preserves Fisher Information Matrix sensitivity, improving inverse problem reconstructions.

Contribution

It presents a novel, efficient framework combining matrix sketching and ensemble sampling to maintain information content in experimental design.

Findings

Effective sensor placement for Schroedinger potential reconstruction.

Preserves Fisher Information Matrix properties with fewer data points.

Applicable to non-smooth or discrete design spaces.

Abstract

The quality of numerical reconstructions for unknown parameters in inverse problems depends fundamentally on the selection of experimental data. To ensure a robust reconstruction, it is crucial to select data that are sensitive to the parameters, a property typically characterized by the conditioning of the Fisher Information Matrix (FIM). In this work, we propose a general framework for an efficient down-sampling strategy that selects experimental setups that preserves the information content of the full-data FIM. Our approach leverages matrix sketching techniques from randomized numerical linear algebra to achieve a sensitivity-preserving approximation. The method involves drawing samples from a sensitivity-informed distribution, which we execute using gradient-free ensemble sampling methods to handle potentially non-smooth or discrete design spaces. Numerical experiments demonstrate the effectiveness of this framework in selecting optimal sensor locations for a Schroedinger potential reconstruction problem.