A hybrid boundary element-finite element method to solve the EEG forward problem

This article presents a hybrid boundary element-finite element (BE–FE) method to solve the EEG forward problem and take advantages of both the boundary element method (BEM) and finite element method (FEM). Although realistic EEG forward problems with heterogeneous and anisotropic regions can be solved by FEM accurately, the FEM modeling of the brain with dipolar sources may lead to singularity. In contrast, the BEM can solve EEG forward problems with isotropic tissue regions and dipolar sources using a suitable integral formulation. This work utilizes both FEM and BEM strengths attained by dividing the regions into some homogeneous BE regions with sources and some heterogeneous and anisotropic FE regions. Furthermore, the BEM is applied for modeling the brain, including dipole sources and the FEM for other head layers. To validate the proposed method, inhomogeneous isotropic/anisotropic three– and four–layer spherical head models are studied. Moreover, a four&-layer realistic head model is investigated. Results for six different dipole eccentricities and two different dipole orientations are computed using the BEM, FEM, and hybrid BE–FE method together with statistical analysis and the related error criteria are compared. The proposed method is a promising new approach for solving realistic EEG forward problems.

Forward Problems Solving of Groundwater Flow using Stochastic Groundwater Vistas Method

In the forward problems, the hydraulic head value can be found by knowing the value of the groundwater parameter. Parameters of groundwater such as hydraulic conductivity, vary over space due to the variation of aquifer properties. Consequently, it is difficult or almost impossible to treat these kinds of variability by a deterministic approach because there is no exact value to be used as input for a parameter. The objective of this research was to obtain a mathematical model of groundwater flow made with the Groundwater Vistas Program that is in accordance with the physical model. Mathematical modeling of groundwater flow using the Groundwater Vistas Program with a stochastic approach and Monte Carlo simulation method where the input data (hydraulic conductivity, hydraulic head) is obtained from the physical model. Results showed that the sum of squares value from the scatter plot diagram of all realization points had a very small value (close to or even zero). The residual mean diagram showed the error value of all realizations had a very low value close to zero. The calculated head value (computed) compared with the results of the observation had a fairly small difference value (ranging from 0.0006−0.009 m). These results were considered quite good because in modeling it is impossible to get modeling results that are exactly the same as those being modeled. The results show that Groundwater Vistas can be used for modeling with very small errors and it can estimate values of hydraulic heads quite well.

Advances on multiobservable thermochemical tomography for the physical state of the upper mantle

<p>Multiobservable thermochemical tomography (MTT) is a recent computational approach to obtain estimates of the physical state (e.g. temperature distribution, compositional structure and rock properties) for the upper mantle [1]. It allows to jointly invert multiple independent datasets (e.g. gravity, seismic, magnetotelluric) within a thermodynamically-constrained and fully probabilistic framework. Evaluation of the plausibility of different physical states of the mantle with Markov Chain Monte Carlo (MCMC) simulations requires the solution of complex forward problems (e.g. Stokes flow, Maxwell’s equations, etc.) millions of times, making MTT computationally demanding for large-scale inverse problems. Furthermore, the number of parameters in a global study can easily reach several millions, making it increasingly difficult to 1) locate the regions of high probability and 2) sample these regions appropriately.</p><p>In order to overcome these limitations, we have combined and implemented a number of techniques, such as reduced-order modelling and efficient parallelization of both the forward problems and the MCMC algorithms, which dramatically accelerate the solution of the forward problems. Our software, LitMod1D_4INV and LitMod3D_4INV, allow to compute a proposal in less than 1 second, even when solving multiple complex forward problems together. We develop a multi-level parallel MPI driver for a collection of advanced MCMC sampling strategies to locate and sample high-probability regions efficiently. The massive amounts of data generated by large-scale MTT inversions need to be managed efficiently. We output results to open-source freeware formats, such as HDF5, TileDB, designed for big data problems. We emphasize that our methods and approaches are not only useful for MTT, but for any demanding inverse problem.</p><p>In this contribution, we will present applications of our software to complex, large-scale MTT problems and discuss its benefits, limitations and future improvements.</p><p>[1] J.C. Afonso, N. Rawlinson, Y. Yang, D. L. Schutt, A. G. Jones, J. Fullea, W. L. Griffin, 3‐D multiobservable probabilistic inversion for the compositional and thermal structure of the lithosphere and upper mantle: III. Thermochemical tomography in the Western‐Central U.S., Journal of Geophysical Research, 121, doi:10.1002/2016JB013049, 2016</p>