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 an error in the deterministic case, or $ y = G(u)+\eta $ where $\eta$ is a $Y$-valued random variable 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 mainly focuses on the regularization and its iterative computation for ill-posed inverse problems. We aim to develop efficient iterative regularization methods for different types of regularizers. Here are several related 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.

GSVD and matrix-pair problems

The Generalized Singular Value Decomposition (GSVD) of a matrix-pair is a generalization of the SVD. It is a powerful tool for analysis and computation of many matrix-pair problems.

This project focuses on analyzing GSVD from new perspectives, which leads to new numerical methods for its computation, and also new understanding and computation of the matrix-pair problems. Here are several related 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.

Scientific computing applications

Ab-initio molecular dynamcis simulations, using classical computational methods based on density functional theory (which involves a nonlinear eigenvalue problem as a core component), or machine learning approaches such as the “neural network force field” model. Here are several related papers:

• Liu, R., Guo, Z., Sha, Q., Zhao, T., Li, H., Hu, W., Liu, L., Tan, G., & Jia, W. (2025). Large Scale Finite-Temperature Real-time Time Dependent Density Functional Theory Calculation with Hybrid Functional on ARM and GPU Systems. 2025 IEEE International Parallel and Distributed Processing Symposium (IPDPS).
• Li, H., Wu, X., Liu, L., Wang, L., Wang, L.-W., Tan, G., & Jia, W. (2024). ALKPU: An Active Learning Method for the DeePMD Model with Kalman Filtering. arXiv:2411.13850.
• Yan, Y.-J., Li, H.-B., Zhao, T., Wang, L.-W., Shi, L., Liu, T., Tan, G.-M., Jia, W.-L., & Sun, N.-H. (2024). 10-Million Atoms Simulation of First-Principle Package LS3DF. Journal of Computer Science and Technology, 39(1), 45–62.