Magnetic resonance electric properties tomography is a non-destructive imaging modality that maps the spatial distribution of the electrical conductivity and permittivity of the human body using standard clinical magnetic resonance imaging systems. From the B + 1 magnetic field maps and the local form of the Maxwell equations, several schemes have been derived to provide direct approximated formulas but they suffer from instabilities. In this paper, we propose to address it as an inverse problem solved by a constrained optimization algorithm where we exploit the weak formulation of the electric Helmholtz equation and a Lagrangian approach. We derive the associated adjoint field equation and employ a Quasi-Newton minimization scheme. We also take advantage of a regularisation strategy based on geometrical a priori information for defining large zones into which the electric parameters are known to be piece-wise constant.
We study the numerical solution to inverse problems in which one reconstructs the coefficients of a parabolic equation depending only on one (space or time) variable. In particular, these classes of problems arise in the study of boundary value problems with nonlocal (integral) boundary conditions. We suggest an approach to the numerical solution to these problems with the use of the line method and the reduction of the original problem to the solution to auxiliary Cauchy problems for systems of ordinary differential equations. We present the results of numerical experiments with test problems.
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