题目：Projected Newton method for large-scale Bayesian linear inverse problems.
报告人：李海波（墨尔本大学）
时间：2024年11月5日10：00-11：00
地点：数学院425报告厅
摘要：Computing the regularized solution of Bayesian linear inverse problems as well as the corresponding regularization parameter is highly desirable in many applications. In this talk, I introduce a novel iterative method, termed the Projected Newton method (PNT), that can simultaneously update the regularization parameter and solution step by step without requiring any high-cost matrix inversions or decompositions. By reformulating the Tikhonov regularization as a constrained minimization problem and writing its Lagrangian function, a Newton-type method coupled with a Krylov subspace method is employed for the unconstrained Lagrangian function. The resulting PNT algorithm only needs solving a small-scale linear system to get a descent direction of a merit function at each iteration, thus significantly reducing computational overhead. Rigorous convergence results are proved, showing that PNT always converges to the unique regularized solution and the corresponding Lagrangian multiplier. Experimental results on both small and large-scale Bayesian inverse problems demonstrate its excellent convergence property, robustness and efficiency. Given that the most demanding computational tasks in PNT are primarily matrix-vector products, it is particularly well-suited for large-scale problems.
个人简介：李海波，2021年博士毕业于清华大学数学科学系。2021-2023年在华为公司从事AI for Science的研究工作; 2023年9月至今在澳大利亚墨尔本大学数学与统计学院担任Research Fellow (博士后）; 2024年8-10月在美国约翰斯霍普金斯大学数学系做访问学者。研究方向主要为反问题的计算、数值线性代数、机器学习。在SIAM Journal on Scientific Computing, SIAM Journal on Matrix Analysis and Appliations等计算数学期刊已发表论文7篇。
