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Basis mapping methods for forward and inverse problems

Schweiger, M; Arridge, S; (2017) Basis mapping methods for forward and inverse problems. International Journal for Numerical Methods in Engineering , 109 (1) pp. 3-28. 10.1002/nme.5271. Green open access

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Abstract

This paper describes a novel method for mapping between basis representation of a field variable over a domain in the context of numerical modelling and inverse problems. In the numerical solution of inverse problems, a continuous scalar or vector field over a domain may be represented in different finite-dimensional basis approximations, such as an unstructured mesh basis for the numerical solution of the forward problem, and a regular grid basis for the representation of the solution of the inverse problem. Mapping between the basis representations is generally lossy, and the objective of the mapping procedure is to minimise the errors incurred. We present in this paper a novel mapping mechanism that is based on a minimisation of the L2 or H1 norm of the difference between the two basis representations. We provide examples of mapping in 2D and 3D problems, between an unstructured mesh basis representative of an FEM approximation, and different types of structured basis including piecewise constant and linear pixel basis, and blob basis as a representation of the inverse basis. A comparison with results from a simple sampling-based mapping algorithm shows the superior performance of the method proposed here.

Type: Article
Title: Basis mapping methods for forward and inverse problems
Open access status: An open access version is available from UCL Discovery
DOI: 10.1002/nme.5271
Publisher version: http://dx.doi.org/10.1002/nme.5271
Language: English
Additional information: © 2016 The Authors. International Journal for Numerical Methods in Engineering Published by John Wiley & Sons Ltd. This is an open access article under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
Keywords: finite element methods; optimisation; inverse problems
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science > Dept of Computer Science
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science > Dept of Med Phys and Biomedical Eng
URI: https://discovery.ucl.ac.uk/id/eprint/1503455
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