[M-GI29] Data-driven geosciences

[MGI29-P07] Low-cost and interpretable method for estimating a regularization parameter using resolution-matrix probing

*Tatsu Kuwatani¹ (1.Japan Agency for Marine-Earth Science and Technology)

Keywords:inversion, regularization, hyperparameter, Bayesian estimation

Regularization techniques are often used to solve inverse problems in geosciences, because they are often under determinant. In order to obtain an appropriate solution, it is necessary to properly select the regularization parameter, the weighting factor of the regularization term in the evaluation function. Recently, Kuwatani et al. (2022, Inverse Problems) (https://doi.org/10.1088/1361-6420/ac93ad) clarified the physical meaning and mechanism of hyperparameter estimation in linear inverse problems by reformulating the free-energy minimization in the Bayesian framework using a resolution matrix, which is defined as a matrix, which linearly maps the true model parameter to the estimated one under the ideal noise-less situation. On the other hand, the direct computation of the resolution matrix includes the inverse-matrix calculation of a (the number of parameters) × (the number of parameters) square matrix so that the use of the proposed method is unrealistic for a large-scale inverse problem in geoscience, which generally includes more than several hundreds of thousands. In this study, we developed an high-speed algorithm to estimate the regularization parameter by using a matrix probing technique, which efficiently analyzes matrix properties using random numbers. The proposed algorithm can be applied to large-scale problems with hundreds of thousands or more model parameters with low additional computational cost compared to normal model-parameter estimation without regularization-parameter estimation. Furthermore, the advantages of this algorithm include ease of implementation, interpretability, ability to determine not only regularization parameters but also observation accuracy, and resolution evaluation. In this presentation, we explain the methods and algorithms, and also introduce examples of their application to actual datasets.