Developments of multiscale and probabilistic methods for PDEs and inverse problems
报告人：Yifan Chen (Caltech)
时间：2023-02-26 10:00
地点：腾讯会议 548-666-940
Abstract：
Computation and inference problems are ubiquitous in science and engineering. This talk focuses on using multiscale methods to compute efficiently and probabilistic frameworks to infer consistently and robustly in solving PDEs and inverse problems.
In the first part, I will cover a line of work on multiscale methods for solving a class of heterogeneous and high-frequency PDEs, such as Helmholtz's equation. By a coarse-fine scale decomposition of the solution space and by analyzing the coarse-scale part's low complexity and exploiting the fine-scale part's locality, we obtain an exponentially convergent function approximation of the solution, even when it is highly non-smooth and oscillatory.
In the second part, I will describe using probabilistic Gaussian processes to solve general nonlinear PDEs and inverse problems. The method has the flavor of scientific machine learning automation. GPs are simpler and more interpretable than neural networks; consequently, it is more straightforward to prove the convergence of the learned solutions under regularity assumptions. Algorithmically, we develop a state-of-the-art near-linear complexity solver for dense kernel matrices based on novel statistical screening effects with PDE-type measurements. Thus, the method is scalable to massive collocation and data points.
