Approximation errors in nonstationary inverse problems
Janne M.J. Huttunen J. P. Kaipio
Inverse Problems & Imaging 2007, 1(1): 77-93 doi: 10.3934/ipi.2007.1.77
Inverse problems are known to be very intolerant to both data errors and errors in the forward model. With several inverse problems the adequately accurate forward model can turn out to be computationally excessively complex. The Bayesian framework for inverse problems has recently been shown to enable the adoption of highly approximate forward models. This approach is based on the modelling of the associated approximation errors that are incorporated in the construction of the computational model. In this paper we investigate the extension of the approximation error theory to nonstationary inverse problems. We develop the basic framework for linear nonstationary inverse problems that allows one to use both highly reduced states and extended time steps. As an example we study the one dimensional heat equation.
keywords: Model reduction State-space models Kalman filtering.
Approximation errors and truncation of computational domains with application to geophysical tomography
A. Lehikoinen S. Finsterle A Voutilainen L. M. Heikkinen M. Vauhkonen J. P. Kaipio
Inverse Problems & Imaging 2007, 1(2): 371-389 doi: 10.3934/ipi.2007.1.371
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.

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