PIMS Mini-course on High-Dimensional Data in Uncertainty Quantification of PDEs

Uncertainty Quantification (UQ) is an increasingly popular subject in computational science and engineering, that deals with assessing the impact on the PDE solution of the uncertainty on the coefficients of the same PDE. Coefficients here is a term that must be understood in a broad sense as "specifics of the PDE", i.e. boundary shape, initial/boundary conditions, forcing terms, diffusion/advection/reaction coefficients etc. Uncertainty on the coefficients can be caused by measurement errors, lack of knowledge or intrinsic aleatoric behavior of such coefficients, and is typically modelled by means of random variables or random fields. Two major computational challenges arise in this setting, namely: the fact that the solution of the PDE is only known to us through an expensive PDE solver (so we would like to minimize the times we query such solver); and the fact that the number of random variables needed to appropriately describe the randomness can be very high (from tens to hundreds, or even countable sequences), depending on the covariance structure of the randomness.

In this mini-course we tackle two relevant problems in this scenario, i.e. 1) how do we compute the expected value of the solution subject to the uncertainty of the coefficients (forward problem)? 2) how do we improve the statistical description of the random coefficients given measurements of the solution (inverse problem)? 

The forward problem can be recast as a high-dimensional quadrature/interpolation problem and will be solved by introducing Multi-Level Monte Carlo/Stochastic Collocation methods (and their advanced "Multi-Index" counterparts), which are state-of-the-art methodology to deal with the discretization of random PDE (i.e., they prescribe both an adequate sampling strategy of the random variables as well as an appropriate sequence of meshes to solve the PDE).

The inverse problem instead is a classic high-dimensional sampling/inference problem and will be tackled by employing standard methods such as Maximum Likelihood or Bayesian Inversion, accelerated with Stochastic Collocation surrogate models (which have been already derived in the context of the forward problem).

Classes will be complemented by hands-on Matlab sessions using the UQ library "Sparse Grids Matlab Kit".