Regularization for inverse problems
Given a forward map \(G:X\rightarrow Y\) between two separable Hilbert spaces $X$ and $Y$, the corresponding inverse problem can usually be written as $y = G(u) + \delta$ where $\delta\in Y$ is a measurement error in the deterministic case, or $y = G(u)+\eta$ where $\eta$ is a $Y$-valued random noise in the stochastic case. Such problems arise frequently in image processing, computed tomography, geophysics, data assimilation, etc. Most inverse problems are ill-posed, posing big challenges for analysis and computation.
This project focuses on the regularization and computation of large-scale linear inverse problems. We aim to develop efficient iterative regularization methods for different types of regularizers. Here are several representative papers:
- Li, H. (2024). Projected Newton method for large-scale Bayesian linear inverse problems. SIAM Journal on Optimization, 35(3),1439–1468.
- Li, H., Feng, J., & Lu, F. (2024). Scalable iterative data-adaptive RKHS regularization. arXiv:2401.00656.
- Li, H. (2024). A preconditioned Krylov subspace method for linear inverse problems with general-form Tikhonov regularization. SIAM Journal on Scientific Computing, 46(4), A2607–A2633.
Matrix-pair problems
Matrix-pair problems frequently emerge from the regularization and computation of inverse problems. Typical examples include the generalized singular value decomposition (GSVD), generalized least squares problems, and least squares problems with linear constrints.
This project focuses on the analysis and computation of such matrix-pair problems from novel perspectives, with particular emphasis on developing efficient iterative algorithms for large-scale instances. Here are several representative papers:
- Li, H. (2025). A new interpretation of the weighted pseudoinverse and its applications. SIAM Journal on Matrix Analysis and Applications, 46(2), 934–956.
- Li, H. (2025). Krylov iterative methods for linear least squares problems with linear equality constraints. Numerical Algorithms, 1-31
- Li, H. (2025). Characterizing GSVD by singular value expansion of linear operators and its computation. SIAM Journal on Matrix Analysis and Applications, 46(1), 439–465.