报告时间：2025年11月6日 上午10:10开始
报 告 人：张文龙（南方科技大学）
报告地点：9-113
报告题目：Uncertainty quantification for some simple inverse problems and fast algorithms
报告摘要：In this talk, we investigate uncertainty quantification for regular solutions of simple inverse problems governed by partial differential equations. Under randomly noisy pointwise measurement data, we demonstrate the stochastic convergence and optimal finite element probabilistic convergence of these solutions. Regularization error estimates and finite element error estimates depend on noise magnitude, regularization parameter, mesh size, and time step, among other factors. Based on these error estimates, an iterative algorithm for determining the optimal regularization parameter is also proposed. For the inverse source problem of parabolic equations, a data-driven model reduction method is additionally presented. We identify a low-dimensional structure in solutions to parabolic equations within a class of forward problems and construct appropriate POD basis functions to achieve significant computational dimensionality reduction. Under the assumption of weak regularity for parabolic partial differential equations, we prove the convergence of the POD algorithm for solving parabolic inverse problems.
报告人简介：张文龙，本科毕业于南京大学，先后在中国科学院，巴黎高等师范学校获得硕士博士学位，现任南方科技大学助理教授。研究方向包括反问题理论数值计算、不确定性量化、数值分析等。主持国家自然科学基金青年基金和面上基金。在Siam系列，Inverse Problems等杂志发表论文。
