## Event Date:

## Event Location:

- HSSB 1174

- Department Seminar

### Derivative-Free Bayesian Inference for Large-Scale Inverse Problems

#### Dr. Daniel Zhengyu Huang

California Institute of Technology (Caltech)

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. Next, I will introduce an exploration-exploitation approach to approximating the Fisher-Rao gradient flow in various parametric density spaces, which leads to different derivative-free Bayesian inference approaches. Restricting in the Gaussian density space, we obtain the Kalman inversion algorithm. Theoretical guarantees for linear inverse problems are provided. Restricting in the Gaussian mixture density space, we obtain the Gaussian mixture Kalman inversion algorithm, which is capable of capturing multiple modes in the posterior. 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.

This is based on joint works (https://github.com/Zhengyu-

**Speaker Biography:**

Daniel Zhengyu Huang is a postdoctoral researcher at the Computing + Mathematical Sciences Department at Caltech. His work lies at the interface of computational mathematics and data science, with a focus on solving practical scientific and engineering problems. He received his Ph.D. from Institute for Computational & Mathematical Engineering at Stanford University in 2020.