摘要：
We consider Bayesian inference for large-scale inverse problems, where computational challenges arise from the need for repeated evaluations of an expensive forward model, which is often given as a black box or is impractical to differentiate. We propose a framework, which is built on Kalman methodology and Fisher-Rao Gradient flow, to efficiently calibrate and provide uncertainty estimations of such models with noisy observation data.
In this talk, I will first explain some basics of variational inference under general metric tensor. In particular, under the Fisher-Rao metric, the gradient flow of the KL divergence has the form of a birth-death process, which has both exponential convergence O(e^-t) and the affine invariant property. The Gaussian approximation of it leads to the natural gradient descent.
Next, I will discuss two different derivative-free approximations of the Fisher-Rao gradient flow. The Gaussian approximation leads to unscented/ensemble Kalman Inversion algorithms. They can also be obtained from a Gaussian approximation of the filtering distribution of a novel mean-field dynamical system. Theoretical guarantees for linear inverse problems are provided.
The Gaussian mixture approximation leads to an efficient derivative-free Bayesian inference approach capable of capturing multiple modes. Finally, I will demonstrate the effectiveness of these approaches in several numerical experiments: learning permeability parameters in subsurface flow; and learning subgrid-scale parameters in a global climate model.
报告人简介：黄政宇是北京大学北京国际数学研究中心助理教授，之前在加州理工学院从事博士后研究、在斯坦福大学获得博士学位、在北京大学获得本科学位。
讨论班简介：北京大学应用数学青年讨论班 (Applied Mathematics Seminar for Youth) 是一个由北京大学卓越研究生计划组织的学术交流平台。该讨论班定期举办一系列读书会、学术报告，涵盖广泛的应用数学领域，旨在为应用数学领域的学生提供一个互相学习、交流和探讨的机会，促进学生们在该领域的学术成长和思维能力的培养。
报名问卷：我们提供午餐（从11:45开始），需要预定午餐的老师同学请填报名问卷 https://wj.qq.com/s2/13373597/9511/
