Variational Bayesian Approximation of Inverse Problems Using Sparse Precision Matrices

Explore a comprehensive lecture on Variational Bayesian Approximation for inverse problems involving partial differential equations (PDEs). Delve into the challenges of ill-posed problems in science and engineering, and discover how Bayesian formulation with prior probability measures can address these issues. Learn about the limitations of traditional Markov Chain Monte Carlo (MCMC) methods for large-scale problems, and understand why Variational Bayes (VB) has emerged as a more computationally tractable alternative. Examine the proposed approach of using Gaussian trial distributions parametrized by precision matrices, leveraging the inherent sparsity of inverse problems in finite element discretization. Gain insights into the use of stochastic optimization for estimating the variational objective and assessing both solution mean errors and uncertainty quantification. Follow along as the speaker demonstrates the application of this method to PDEs based on the Poisson equation in 1D and 2D, and learn about the publicly available Tensorflow implementation on GitHub.