Gaussian processes for Bayesian inverse problems associated with linear partial differential equations

Combining complex mathematical models with observational data is an extremely challenging yet ubiquitous problem in the field of modern applied mathematics and data science. Inverse problems, where one is interested in learning inputs to a mathematical model such as physical parameters and initial conditions given partial and noisy observation of model outputs, are hence of frequent interest. Adopting a Bayesian approach[15, 32], we incorporate our prior knowledge on the inputs into a probability distribution, the prior distribution, and obtain a more accurate representation of the model inputs in the posterior distribution, which results from conditioning the prior distribution on the observed data. The posterior distribution contains all the necessary information about the characteristics of our inputs.