American Institute of Mathematical Sciences

Advanced
May  2007, 1(2): 371-389. doi: 10.3934/ipi.2007.1.371

Approximation errors and truncation of computational domains with application to geophysical tomography

A. Lehikoinen 1, , S. Finsterle 2, , A Voutilainen 1, , L. M. Heikkinen 1, , M. Vauhkonen 1, and  J. P. Kaipio 1,

1. 

Department of Physics, University of Kuopio, P.O. Box 1627, FIN-70211 Kuopio, Finland, Finland, Finland, Finland
2. 

Lawrence Berkeley National Laboratory (LBNL), University of California, Berkeley, CA 94720, United States

Received  September 2006 Published  April 2007

Numerical realization of mathematical models always induces errors to the computational models, thus affecting both predictive simulations and related inversion results. Especially, inverse problems are typically sensitive to modeling and measurement errors, and therefore the accuracy of the numerical model is a crucial issue in inverse computations. For instance, in problems related to partial differential equation models, the implementation of a numerical model with high accuracy necessitates the use of fine discretization and realistic boundary conditions. However, in some cases realistic boundary conditions can be posed only for very large or even unbounded computational domains. Fine discretization and large domains lead to very high-dimensional models that may be of prohibitive computational cost. Therefore, it is often necessary in practice to use coarser discretization and smaller computational domains with more or less incorrect boundary conditions in order to decrease the dimensionality of the model. In this paper we apply the recently proposed approximation error approach to the problem of incorrectly posed boundary conditions. As a specific computational example we consider the imaging of conductivity distribution of soil using electrical resistance tomography. We show that the approximation error approach can also be applied to domain truncation problems and that it allows one to use significantly smaller scale forward models in the inversion.
Keywords: approximation errors, inverse problems, domain truncation, electrical resistance tomography..
Mathematics Subject Classification: Primary: 65N21, 65N30, 33F05; Secondary: 86A2.
Citation: A. Lehikoinen, S. Finsterle, A Voutilainen, L. M. Heikkinen, M. Vauhkonen, J. P. Kaipio. Approximation errors and truncation of computational domains with application to geophysical tomography. Inverse Problems & Imaging, 2007, 1 (2) : 371-389. doi: 10.3934/ipi.2007.1.371
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