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Basis mapping methods for forward and inverse problems
Author(s) -
Schweiger Martin,
Arridge Simon
Publication year - 2016
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.5271
Subject(s) - basis (linear algebra) , basis function , mathematics , inverse problem , scalar (mathematics) , algorithm , piecewise , finite element method , inverse , computer science , mathematical optimization , mathematical analysis , geometry , physics , thermodynamics
Summary 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 L 2 or H 1 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. © 2016 The Authors. International Journal for Numerical Methods in Engineering Published by John Wiley & Sons Ltd.