For interpretation of electroencephalography (EEG) and magnetoencephalography (MEG) data, multiple solutions of the respective forward problems are needed. In this paper, we assess performance of the mixed-hybrid finite element method (MHFEM) applied to EEG and MEG modeling. The method provides an approximate potential and induced currents and results in a system with a positive semi-definite matrix. The system thus can be solved with a variety of standard methods (e.g. the preconditioned conjugate gradient method). The induced currents satisfy discrete charge conservation law making the method conservative. We studied its performance on unstructured tetrahedral grids for a layered spherical head model as well as a realistic head model. We also compared its accuracy versus the conventional nodal finite element method (${P}_{1}$FEM). To avoid modeling singular sources, we completed our computations with a subtraction approach; the derived expression for the MEG response different from earlier published and involves integration of finite quantities only. We conclude that although the MHFEM is more computationally demanding than the${P}_{1}$FEM, its use is justified for EEG and MEG modeling on low-resolution head models where${P}_{1}$FEM loses accuracy.

The goal of this paper is to develop a globally convergent numerical method for the inverse problem which would work with the optical experimental data collected by this research group. Only the intensity (modulus square) of the total complex valued wave field is measured at an array of light detectors located on a measurement plane. The phase is not measured. We solve numerically the phaseless coefficient inverse problem to reconstruct locations and refractive indices of unknown scatterers from those measurements. Our method consists of two stages. In the first stage, we obtain an upper estimate for the modulus of the scattered wave field. This estimate allows us to approximately reconstruct the wave field at the measurement plane using a resulting inversion formula. This reduces the phaseless inverse scattering problem to the phased one. In the second stage, we apply a recently developed globally convergent numerical method to reconstruct the desired refractive index from the total wave obtained at the first stage. Unlike the optimization approach, our two-stage numerical method is global in the sense that it does not require a good initial guess of the true solution. A significant part of this work is devoted to testing our numerical method on both computationally simulated and experimental data.

Fast image reconstruction techniques are becoming important with the increasing number of scientific cases in high resolution micro and nano tomography. The processing of the large scale 3D data demands new mathematical tools for the tomographic reconstruction. Due to the high computational complexity of most current algorithms, big data sizes demands powerful hardware and more sophisticated numerical techniques. Several reconstruction algorithms are dependent on a mathematical tool called backprojection (a transposition process). A conventional implementation of the backprojection operator has cubic computational complexity. In the present manuscript we propose a new fast backprojection operator for the processing of tomographic data, providing a low-cost algorithm for this task. We compare our formula against other fast transposition techniques, using real and simulated large data sets.