Amir Moradifam

Amir Moradifam, Adrian Nachman, Alexandre Timonov

We consider the hybrid problem of reconstructing the isotropic electric conductivity of a body $Ω$ from interior Current Density Imaging data obtainable using MRI measurements. We only require knowledge of the magnitude $|J|$ of one current generated by a given voltage $f$ on the boundary $\partialΩ$. As previously shown, the corresponding voltage potential u in $Ω$ is a minimizer of the weighted least gradient problem \[u=\hbox{argmin} \{\int_Ωa(x)|\nabla u|: u \in H^{1}(Ω), \ \ u|_{\partial Ω}=f\},\] with $a(x)= |J(x)|$. In this paper we present an alternating split Bregman algorithm for treating such least gradient problems, for $a\in L^2(Ω)$ non-negative and $f\in H^{1/2}(\partial Ω)$. We give a detailed convergence proof by focusing to a large extent on the dual problem. This leads naturally to the alternating split Bregman algorithm. The dual problem also turns out to yield a novel method to recover the full vector field $J$ from knowledge of its magnitude, and of the voltage $f$ on the boundary. We then present several numerical experiments that illustrate the convergence behavior of the proposed algorithm.

Amir Moradifam, Adrian Nachman

We prove results on weak convergence for the alternating split Bregman algorithm in infinite dimensional Hilbert spaces. We also show convergence of an approximate split Bregman algorithm, where errors are allowed at each step of the computation. To be able to treat the infinite dimensional case, our proofs focus mostly on the dual problem. We rely on Svaiter's theorem on weak convergence of the Douglas-Rachford splitting algorithm and on the relation between the alternating split Bregman and Douglas-Rachford splitting algorithms discovered by Setzer. Our motivation for this study is to provide a convergent algorithm for weighted least gradient problems arising in the hybrid method of imaging electric conductivity from interior knowledge (obtainable by MRI) of the magnitude of one current.

Nassif Ghoussoub, Amir Moradifam

Motivated by the theory of self-duality which provides a variational formulation and resolution for non self-adjoint partial differential equations \cite{G1, G2}, we propose new templates for solving large non-symmetric linear systems. The method consists of combining a new scheme that simultaneously preconditions and symmetrizes the problem, with various well known iterative methods for solving linear and symmetric problems. The approach seems to be efficient when dealing with certain ill-conditioned, and highly non-symmetric systems.