Approaches to Multiscale Inverse Problems

Abstract

Many scientific problems involve parameters such as conductivity, permeability or density which vary on multiple spatial and/or temporal scales. When such a parameter is to be estimated from noisy indirect measurements we face a challenging dichotomy: In practical situations the computational cost required to accurately take into account the small scale behaviour can be prohibitive. On the other hand, estimation of the conductivity is an ill-posed inverse problem meaning that any modelling errors, such as neglecting the small scale, which are not accounted for can render estimate useless. A typical way of reducing the computational cost is to simply neglect any behaviour at the smaller scales, enabling a coarse discretisation of the problem. However, if the effects of disregarding smaller scale characteristics along with the coarse discretisation are not taken into account the estimates attained can be misleading. In this thesis, we develop computationally feasible models to tackle the problem of estimating the multiscale distributed parameter of the Poisson equation, which is closely related to the multiscale electrical impedance tomography (EIT) problem. In the case of estimating a low dimensional representation of the conductivity (devoid of the small scale features) we show that by applying the Bayesian approximation error (BAE) approach to marginalise over small scale effects and discretisation errors we are able to substantially reduce the dimension of the problem while maintaining accurate estimates. Moreover, we show that by applying a feature extraction type modification to the BAE approach we can in some cases estimate the amplitude and correlation length of the small scale component of the conductivity. In the case of estimating the full multiscale conductivity we show that multiscale finite element methods (MsFEM) can be implemented at both the forward modelling stage and at the inversion stage, which along with a somewhat coarsened discretisation can reduce overall computational cost. By deriving a closed form expression for the Jacobian matrix which represents the rate at which the electric potential calculated using MsFEM changes with respect to the conductivity we are able to apply gradient based optimisation techniques to estimate the conductivity. The BAE approach was also implemented in this procedure to take into account any modelling discrepancy caused by the use of MsFEM and discretisation errors. Use of such a procedure leads to estimates attained by using 7 multiscale basis functions in line with those calculated using finite elements on 85 linear basis functions.