Sensitivity-preserving of Fisher Information Matrix through random data down-sampling for experimental design
For practitioners in inverse problems, this provides an efficient method to select informative experimental setups without full-data computation.
The paper proposes a framework for down-sampling experimental data while preserving the Fisher Information Matrix's conditioning, using matrix sketching and gradient-free ensemble sampling. Numerical experiments show effectiveness in sensor selection for a Schrödinger potential reconstruction problem.
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.