学术报告（李海波 2026.1.23）

摘要：Learning convolution kernels in operators from data arises in numerous applications and represents an ill-posed inverse problem of broad interest. With scant prior information, kernel methods offer a natural nonparametric approach with regularization. However, a major challenge is to select a proper reproducing kernel, especially as operators and data vary. In this talk, I will present a Data-Adaptive RKHS Regularization method for solving ill-posed linear inverse problems arising from both Fredholm integral equation and learning convolution kernels from data.  We show that the input data and forward operator themselves induce an automatic, data-adaptive RKHS, obviating manual kernel selection. We develop both Tikhonov and iterative regularization algorithms using either the automatic basis functions or preselected basis functions. Numerical experiments on integral, nonlocal, and aggregation operators confirm that our automatic RKHS regularization outperforms standard ridge regression and Gaussian process methods with preselected kernels.

主讲人简介：李海波，2021 年在清华大学数学系获得博士学位，2021-2023 年在华为公司从事AI for Science 工作，2023-2025年在澳大利亚墨尔本大学从事博士后研究， 2025年12月入职华中科技大学数学与统计学院担任副教授。主要研究领域包括：反问题的正则化和计算，数值线性代数，科学机器学习等。