Plenary Speakers

The Relative Scattering Operator and Imaging with Waves

Sunday, October 3

9:00 - 9:45 am

From Rayleigh's explanation of why the sky is blue, to Rutherford's discovery of the atomic nucleus, through modern applications of computerized tomography and imaging, scattering theory has played a central role in 20th century mathematical physics and continues to be an important area of mathematics.  Although the basic mathematical model of direct scattering theory is deceptively simple, scattering phenomena continues to attract, perplex and challenge mathematicians from diverse disciplines. On the other hand problems in inverse scattering are non-linear and unstable.

This presentation is a journey through some new mathematical developments related to direct and inverse scattering theory for inhomogeneous media. Particular emphasis will be given to  three spectral sets that intrinsically appear in the study of the relative scattering operator, namely scattering poles, non-scattering frequencies and transmission eigenvalues. We show how the  understanding  of these sets guides us to extract nonlinear information about the inhomogeneity from the linear relative scattering operator. Related examples of reconstructions will be presented.

Fully Nonlinear Bayesian Inference for the Geosciences: the Particle Flow Filter

Saturday, October 2

9:00 - 9:45 am

Bayesian Inference in the geosciences, also called data assimilation, is hampered by the high dimension of the state space. For example, for weather forecasting this dimension is of the order of 10^11, and tens of millions of observations are assimilated every 6 hours. Present-day data-assimilation methods are either based on the Ensemble Kalman Filter (older and more efficient than the Unscented or sigma-point Kalman Filter), or variational methods (similar to inverse modelling, but with more sophisticated priors). These methods assume Gaussian priors and allow for weakly nonlinear observation operators that map the state to observation space. Fully nonlinear Bayesian Inference methods are difficult to apply on these problems, while with ever increasing model resolution and more complex observations the problem is becoming highly nonlinear. Particle Filters have been employed via a technique called localization, in which observation are only allowed to influence nearby grid points, but even with localization these filters tend to be degenerate, and ad-hoc tricks such as limiting the minimal weights have to be employed to make them work. However, by limiting the particle weights we through away much of the information in the observations. Markov-Chain Monte-Carlo methods tend to be too expensive because of the complex numerical models involved. Recently, Particle Flow Filters have been introduced in the geosciences, and they do have potential to solve the nonlinearity issues in high-dimensional problems. 

In this talk I will introduce Particle Flow Filters and discuss their properties, including an analytical solution to the fully nonlinear data assimilation problem. Unfortunately, that analytical solution is not practical, for reasons that will be discussed, but via kernel embedding of the flow we can generate algorithms that do work in practise. The resulting technique will be applied to several examples, including high-dimensional and highly nonlinear geophysical systems. The talk will conclude with an outlook into the future.

Data Driven Modeling of Unknown Systems with Deep Neural Networks

Sunday, October 3

9:45 - 10:30 am

We present a framework of predictive modeling of unknown system from measurement data. The method is designed to discover/approximate the unknown evolution operator behind the data. Deep neural network (DNN) is employed to construct such an approximation. Once an accurate DNN model for evolution operator is constructed, it serves as a predictive model for the unknown system and enables us to conduct system analysis. We demonstrate that residual network (ResNet) is particularly suitable for modeling autonomous dynamical systems. Extensions to other types of systems will be discussed, including non-autonomous systems, systems with uncertain parameters, and more importantly, systems with missing variables, as well as partial differential equations (PDEs).

Finite Element Methods with Discontinuous Approximation

Saturday, October 2

1:35 - 2:20 pm

Finite element methods with discontinuous approximation have been active research areas in the past two decades. Discontinuous finite element methods include interior penalty dis- continuous Galerkin (IPDG) method, local discontinuous Galerkin (LDG) method, hybridizable discontinuous Galerkin (HDG) method, weak Galerkin (WG) method, Hybrid high-order (HHO) method and conforming discontinuous Galerkin (CDG) method. Finite element methods with discontinuous approximations use discontinuous Pk elements (Pk denotes a set of polynomials with degree k or less), while their continuous counterpart conforming finite element methods use continuous Pk element. Since discontinuous Pk element introduces many more degrees of freedom, one would expect higher order of convergence for the finite element methods with discontinuous approximation. However, discontinuous finite element methods have the same convergence rate as the classic finite element method using continuous Pk element in general. Can we develop finite element methods using discontinuous Pk element, that can fully utilize all the unknowns of discontinuous Pk element to achieve higher order of convergence than their continuous counterpart? This is the main the question that we like to answer in this talk.