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Abstract
For many inverse parameter problems for partial differential equations in which the domain contains only wellseparated objects, an asymptotic solution to the forward problem involving `polarization tensors' exists. These are functions of the size and material contrast of inclusions, thereby describing the saturation component of the nonlinearity. As such, these asymptotic expansions can allow fast and stable reconstruction of small isolated objects. In this paper, we show how such an asymptotic series can be applied to nonlinear leastsquares reconstruction problems, by deriving an approximate diagonal Hessian matrix for the data mist term. Often, the Hessian matrix can play a vital role in dealing with the nonlinearity, generating good update directions which accelerate the solution towards a global minimum which may lie in a long curved valley, but computational cost can make direct calculation infeasible. Since the polarization tensor approximation assumes sufficient separation between inclusions, our approximate Hessian does not account for nonlinearity in the form of lack of superposition in the inverse problem. It does however account for the nonlinear saturation of the change in the data with increasing material contrast. We therefore propose to use it as an initial Hessian for quasiNewton schemes.
This is demonstrated for the case of electrical impedance tomography in numerical experimentation, but could be applied to any other problem which has an equivalent asymptotic expansion. We present numerical experimentation into the accuracy and reconstruction performance of the approximate Hessian, providing a proof of principle of the reconstruction scheme.
This is demonstrated for the case of electrical impedance tomography in numerical experimentation, but could be applied to any other problem which has an equivalent asymptotic expansion. We present numerical experimentation into the accuracy and reconstruction performance of the approximate Hessian, providing a proof of principle of the reconstruction scheme.
Original language  English 

Journal  Inverse Problems in Science and Engineering 
DOIs  
Publication status  Published  23 Dec 2021 
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 3 Finished

Reducing the Threat to Public Safety: Improved metallic object characterisation, location and detection
Peyton, A. (PI), Lionheart, W. (CoI) & Yin, W. (CoI)
1/01/18 → 31/12/20
Project: Research

Robust Repeatable Respiratory Monitoring with EIT
Lionheart, W. (PI), Parker, G. (CoI) & Wright, P. (CoI)
2/06/14 → 31/12/18
Project: Research

GLOBAL Manchester Image Reconstruction and ANalysis (MIRAN): Step jumps in imaging by Global Exchange of user pull and method push
Lionheart, W. (PI), Cootes, T. (CoI), Dorn, O. (CoI), Gray, N. (CoI), Grieve, B. (CoI), Haigh, S. (CoI), Harris, D. (CoI), Hollis, C. (CoI), Matthews, J. (CoI), Mccann, H. (CoI), Parker, G. (CoI), Villegas Velasquez, R. (CoI) & Withers, P. (CoI)
1/04/12 → 31/03/13
Project: Research