Parameterizations for Ensemble Kalman Inversion

TL;DR

This paper explores innovative parameterization strategies, including geometric and hierarchical methods, to enhance ensemble Kalman inversion for solving inverse problems like electrical impedance tomography and groundwater flow.

Contribution

It introduces combined geometric and hierarchical parameterizations, such as the level set method, to improve reconstruction quality and adaptivity in ensemble Kalman inversion.

Findings

Geometric ideas enable reconstruction of piecewise continuous fields.

Hierarchical methods improve learning of key parameters like length-scales.

Combined approaches allow for piecewise constant reconstructions with unknown interfaces.

Abstract

The use of ensemble methods to solve inverse problems is attractive because it is a derivative-free methodology which is also well-adapted to parallelization. In its basic iterative form the method produces an ensemble of solutions which lie in the linear span of the initial ensemble. Choice of the parameterization of the unknown field is thus a key component of the success of the method. We demonstrate how both geometric ideas and hierarchical ideas can be used to design effective parameterizations for a number of applied inverse problems arising in electrical impedance tomography, groundwater flow and source inversion. In particular we show how geometric ideas, including the level set method, can be used to reconstruct piecewise continuous fields, and we show how hierarchical methods can be used to learn key parameters in continuous fields, such as length-scales, resulting in improved reconstructions. Geometric and hierarchical ideas are combined in the level set method to find piecewise constant reconstructions with interfaces of unknown topology.