The understanding and incorporation of data within models has become a vital component of applied mathematics. A fundamental one can ask is given noisy measurements of data, how to recover some unknown quantity of interest. Some examples of these fields include in- verse problems which is primarily concerned with parameter estimation and data assimilation for state estimation. Both fields have seen a considerable amount of attention due to recent advance- ments in terms of both classical and statistical approaches. In particular, this mini-symposium will consider particle methods for solving inverse problems with the help optimization tools as well as particle methods aiming to represent the posterior distribution in a bayesian point of view for inverse problems.
The motivation behind this mini-symposium is to bring together experts from both schools. This would provide a complimentary field to the mini-symposium where connections between both areas are currently being developed.
08:30
Discrete Gradient methods for Computational Bayesian inverse problems
Sahani Pathiraja | University of Potsdam | Germany
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Sahani Pathiraja | University of Potsdam | Germany
This talk will consist of 2 parts; starting with a review of various particle-based approaches to solving inverse problems, with the aim of providing an introduction to the Mini symposium. The second part will focus on recent work by the authors concerning numerical schemes for implementing certain continuous-time particle-based formulations of Bayesian inference that have a gradient flow structure. In particular, we focus on the ensemble Kalman-Bucy filter and particle flow Fokker Planck associated to Brownian dynamics, both of which require special numerical methods since they can lead to stiff differential equations. We demonstrate how discrete gradient time stepping methods can be utilised in such situations, by making use of the gradient flow structure of the algorithms. Comparison to semi-implicit and iterative implementations of these algorithms are presented through numerical experiments in an inverse problem setting.
09:00
Bayesian approach to elliptic inverse problems
Svetlana Dubinkina | Centrum Wiskunde & Informatica (CWI) | Netherlands
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Svetlana Dubinkina | Centrum Wiskunde & Informatica (CWI) | Netherlands
Predicting the amount of gas or oil extracted from a subsurface
reservoir depends on the soil properties such as porosity and
permeability. These properties, however, are highly uncertain due to the
lack of measurements. Therefore decreasing these uncertainties is of a
great importance.
Mathematically speaking, permeability can be represented by a random
process, which in turn leads to a random partial differential equation.
The solution of such a partial differential equation, for example
pressure, is only partially observed and, moreover, contaminated with
measurement errors. Therefore, instead of a well-posed forward problem
of finding pressure from certain permeability, we are faced with an
ill-posed inverse problem of finding uncertain random process from a few
pressure measurements. We develop a Bayesian method for inverse problems,
that is both general and computationally affordable.
09:30
A Stein variational Newton method
Gianluca Detommaso | University of Bath | United Kingdom
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Gianluca Detommaso | University of Bath | United Kingdom
Stein variational gradient descent (SVGD) was recently proposed as a general purpose nonpara- metric variational inference algorithm [Liu & Wang, NIPS 2016]: it minimizes the Kullback- Leibler divergence between the target distribution and its approximation by implementing a form of functional gradient descent on a reproducing kernel Hilbert space. In this paper, we accelerate and generalize the SVGD algorithm by including second-order information, thereby approximating a Newton-like iteration in function space. We also show how second-order information can lead to more effective choices of kernel. We observe significant computational gains over the original SVGD algorithm in multiple test cases.
10:00
Kinetic Methods for Inverse Problems
Giuseppe Visconti | RWTH Aachen University | Germany
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Giuseppe Visconti | RWTH Aachen University | Germany
The Ensemble Kalman Filter method can be used as an
iterative numerical scheme for parameter identification or nonlinear
filtering problems. We study the limit of infinitely large ensemble size
and derive the corresponding mean-field limit of the ensemble method,
also in presence of nonlinear constraints. The resulting kinetic
equations allow in simple cases to analyze stability of the solution to
inverse problems as mean of the distribution of the ensembles. Further,
we present a slight but stable modification of the method which leads to
a Fokker-Planck-type kinetic equation. The kinetic methods proposed here
are able to solve the problem with a reduced computational complexity in
the limit of a large ensemble size. We illustrate the properties and the
ability of the kinetic model to provide solution to inverse problems by
using examples from the literature.