One fundamental problem when solving inverse problems is how to find regularization parameters. This article considers solving this problem using data-driven bilevel optimization, i.e. we consider the adaptive learning of the regularization parameter from data by means of optimization. This approach can be interpreted as solving an empirical risk minimization problem, and we analyze its performance in the large data sample size limit for general nonlinear problems. We demonstrate how to implement our framework on linear inverse problems, where we can further show that the inverse accuracy does not depend on the ambient space dimension. To reduce the associated computational cost, online numerical schemes are derived using the stochastic gradient descent method. We prove convergence of these numerical schemes under suitable assumptions on the forward problem. Numerical experiments are presented illustrating the theoretical results and demonstrating the applicability and efficiency of the proposed approaches for various linear and nonlinear inverse problems, including Darcy flow, the eikonal equation, and an image denoising example.
|Original language||English (US)|
|Number of pages||42|
|Journal||Communications in Mathematical Sciences|
|State||Published - Dec 10 2021|
Bibliographical noteKAUST Repository Item: Exported on 2021-12-13
Acknowledgements: NKC acknowledges a Singapore Ministry of Education Academic Research Funds Tier 2 grant [MOE2016-T2-2-135] and KAUST baseline funding. SW is grateful to the DFG RTG1953 “Statistical Modeling of Complex Systems and Processes” for funding of this research. The research of XTT is supported by the National University of Singapore grant R-146-000-292-114.
ASJC Scopus subject areas
- Applied Mathematics