MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Adaptive weighting of Bayesian physics informed neural networks for multitask and multiscale forward and inverse problems

In this paper, we present a novel methodology for automatic adaptive weighting of Bayesian Physics-Informed Neural Networks (BPINNs), and we demonstrate that this makes it possible to robustly address multi-objective and multi-scale problems. BPINNs are a popular framework for data assimilation, combining the constraints of Uncertainty Quantification (UQ) and Partial Differential Equation (PDE). The relative weights of the BPINN target distribution terms are directly related to the inherent uncertainty in the respective learning tasks. Yet, they are usually manually set a-priori, that can lead to pathological behavior, stability concerns, and to conflicts between tasks which are obstacles that have deterred the use of BPINNs for inverse problems with multi-scale dynamics. The present weighting strategy automatically tunes the weights by considering the multi-task nature of target posterior distribution. We show that this remedies the failure modes of BPINNs and provides efficient exploration of the optimal Pareto front. This leads to better convergence and stability of BPINN training while reducing sampling bias. The determined weights moreover carry information about task uncertainties, reflecting noise levels in the data and adequacy of the PDE model. We demonstrate this in numerical experiments in Sobolev training, and compare them to analytically $\epsilon$-optimal baseline, and in a multi-scale Lokta-Volterra inverse problem. We eventually apply this framework to an inpainting task and an inverse problem, involving latent field recovery for incompressible flow in complex geometries

Statistical computation with kernels

Modern statistical inference has seen a tremendous increase in the size and complexity of models and datasets. As such, it has become reliant on advanced computational tools for implementation. A first canonical problem in this area is the numerical approximation of integrals of complex and expensive functions. Numerical integration is required for a variety of tasks, including prediction, model comparison and model choice. A second canonical problem is that of statistical inference for models with intractable likelihoods. These include models with intractable normalisation constants, or models which are so complex that their likelihood cannot be evaluated, but from which data can be generated. Examples include large graphical models, as well as many models in imaging or spatial statistics. This thesis proposes to tackle these two problems using tools from the kernel methods and Bayesian non-parametrics literature. First, we analyse a well-known algorithm for numerical integration called Bayesian quadrature, and provide consistency and contraction rates. The algorithm is then assessed on a variety of statistical inference problems, and extended in several directions in order to reduce its computational requirements. We then demonstrate how the combination of reproducing kernels with Stein's method can lead to computational tools which can be used with unnormalised densities, including numerical integration and approximation of probability measures. We conclude by studying two minimum distance estimators derived from kernel-based statistical divergences which can be used for unnormalised and generative models. In each instance, the tractability provided by reproducing kernels and their properties allows us to provide easily-implementable algorithms whose theoretical foundations can be studied in depth