Aretha L. Teckentrup

Tianming Bai, Aretha L. Teckentrup, Konstantinos C. Zygalakis

This work is concerned with the use of Gaussian surrogate models for Bayesian inverse problems associated with linear partial differential equations. A particular focus is on the regime where only a small amount of training data is available. In this regime the type of Gaussian prior used is of critical importance with respect to how well the surrogate model will perform in terms of Bayesian inversion. We extend the framework of Raissi et. al. (2017) to construct PDE-informed Gaussian priors that we then use to construct different approximate posteriors. A number of different numerical experiments illustrate the superiority of the PDE-informed Gaussian priors over more traditional priors.

Elliot J. Addy, Aretha L. Teckentrup

Sparse grids are popular tools for high-dimensional function approximation. In this work, we study the use of sparse grids for interpolation using separable Matérn kernels $Φ_{\boldsymbolν,\boldsymbolλ}(\mathbf{x},\mathbf{x}')=\prod_{j=1}^dϕ_{ν_j,λ_j}(x_j,x_j')$, with a particular focus on the anisotropic setting where the regularity $ν_j$ and the lengthscale $λ_j$ vary with dimension $j$. We combine the construction of anisotropic sparse grids, which exploit anisotropic $ν_j$ to improve convergence rates in smooth dimensions, with the construction of lengthscale-informed sparse grids, which diminish the error contribution of less varying dimensions using anisotropic $λ_j$. We provide theory and numerical experiments to showcase the benefits on asymptotic and pre-asymptotic error behaviour of sparse grid kernel interpolation.