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Domain decomposition method for electromagnetic modelling and quantitative microwave imaging in large-scale three-dimensional configurations

Ivan Voznyuk 1
1 HIPE - HIPE
FRESNEL - Institut FRESNEL
Abstract : The PhD work presented here is devoted to the electromagnetic modeling. As a very important representative of this domain, the inverse problems become more and more popular because of its applications, such as the biomedical or geophysical engineering. The quantitative inverse scattering algorithms attempt to estimate from scattering experiments the physical parameters and features (position, form, size and complex permittivity) of a target. From the mathematical point of view, we can point out two main challenges which are associated with quantitative microwave imagining. These are ill-posedness of the electromagnetic inverse scattering problem and its non-linearity. In order to solve these problems, we considering a Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi- Newton optimization algorithm with line search, which searches for a new estimation of the relative permittivity in the investigation test domain, i.e. a domain where we are sure to have the objects. Taking advantage of the Lagrangian formalism and the definition of an ad-hoc adjoint field, we can calculate the gradients with respect to the relative permittivity. This leads the the necessity of the resolution two direct problems at every iteration step of the inversion algorithm. Various methods have been developed for this resolution. In particular, the Finite-Element Method (FEM) is a numerical technique used to obtain approximate solutions to boundary-value problems treating most complex geometry and physics. However, solving large-scale electromagnetic problems with FEM is very challenging in terms of memory and time of computations. That is why, the part of this PhD work is dedicated to the implementation of the method FETI which is based on the Domain Decomposition technique applied to the Finite-element method. The principal idea of the FETI method consists in dividing the entire domain into non-overlapping subdomains in order to construct a system of linear equations, called the Interface problem, which solution can be used as an unknown boundary condition for every local subdomain. With the aide of these boundary conditions we can then calculate the electromagnetic problem inside each subdomain in independent way. In the framework of this PhD thesis we have proposed a modification of the classical FETI-DPEM2 method in order to improve the transmission conditions between subdomains. The new approach has been called the FETI-DPEM2-full method.
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Ivan Voznyuk. Domain decomposition method for electromagnetic modelling and quantitative microwave imaging in large-scale three-dimensional configurations. Electromagnetism. Aix-Marseille Université, 2014. English. ⟨tel-01279590⟩

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