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.

Determination of optimal shot peen forming patterns using the theory of non-Euclidean plates

We show how a theoretical framework developed for modelling nonuniform growth can model the shot peen forming process. Shot peen forming consists in bombarding a metal panel with multiple millimeter-sized shot, that induce local bending of the panel. When applied to different areas of the panel, peen forming generates compound curvature profiles starting from a flat state. We present a theoretical approach and its practical realization for simulating peen forming numerically. To achieve this, we represent the panel undergoing peen forming as a bilayer plate, and we apply a geometry-based theory of non-Euclidean plates to describe its reconfiguration. Our programming code based on this approach solves two types of problems: it simulates the effect of a predefined treatment (the forward problem) and it finds the optimal treatment to achieve a predefined target shape (the inverse problem). Both problems admit using multiple peening regimes simultaneously. The algorithm was tested numerically on 200 randomly generated test cases.

Adjoint Numerical Method for a Multiphysical Inverse Problem of Two-Phase Well Testing in Petroleum Reservoirs

Inverse problem solution is an integral part of data interpretation for well testing in petroleum reservoirs. In case of two-phase well tests with water injection, forward problem is based on the multiphase flow model in porous media and solved numerically. The inverse problem is based on a misfit or likelihood objective function. Adjoint methods have proved robust and efficient for gradient calculation of the objective function in this type of problems. However, if time-lapse electrical resistivity measurements during the well test are included in the objective function, both the forward and inverse problems become multiphysical, and straightforward application of the adjoint method is problematic. In this paper we present a novel adjoint algorithm for the inverse problems considered. It takes into account the structure of cross dependencies between flow and electrical equations and variables, as well as specifics of the equations (mixed parabolic-hyperbolic for flow and elliptic for electricity), numerical discretizations and grids, and measurements in the inverse problem. Decomposition is proposed for the adjoint problem which makes possible step-wise solution of the electric adjoint equations, like in the forward problem, after which a cross-term is computed and added to the right-hand side of the flow adjoint equations at this timestep. The overall procedure provides accurate gradient calculation for the multiphysical objective function while preserving robustness and efficiency of the adjoint methods. Example cases of the adjoint gradient calculation are presented and compared to straightforward difference-based gradient calculation in terms of accuracy and efficiency.

Multi-point Sources Continuous Release Inversion Based on Improved Four-Dimensional Variation Method

the continuous release of multi-point sources is one of the most common cases in the field of air pollution. In order to solve the problem of multi-point sources continuous release inversion, a source intensity and location estimation method based on improved Four-Dimensional Variation is proposed. Firstly, by constructing the diffusion equation of multi-point sources continuous release and the monitoring concentration matrix, the source term inversion model of multi-point sources continuous release is formed. Secondly, the joint transformation method and Taylor series upwind difference method are used to solve the convection-diffusion equation of multi-point sources continuous release, and the numerical solution format of the forward problem is formed through simulation analysis. Furthermore, based on the numerical solution scheme of the forward problem, a four-dimensional variation inversion algorithm for multi-point sources continuous release is constructed, and the flower pollination algorithm is used to improve the inversion accuracy and computational efficiency. Finally, the applicability, feasibility, advantages and disadvantages of the improved four-dimensional variation algorithm are verified by numerical simulation analysis. It is found that the improved four-dimensional variation algorithm can realize the source term inversion under three conditions: the location is known and the intensity is unknown, the intensity is known and the location is unknown, the intensity and the location are all unknown, which the accuracy and computational efficiency can basically meet the actual needs.

Multifrequency Topological Derivative Approach to Inverse Scattering Problems in Attenuating Media

Detecting objects hidden in a medium is an inverse problem. Given data recorded at detectors when sources emit waves that interact with the medium, we aim to find objects that would generate similar data in the presence of the same waves. In opposition, the associated forward problem describes the evolution of the waves in the presence of known objects. This gives a symmetry relation: if the true objects (the desired solution of the inverse problem) were considered for solving the forward problem, then the recorded data should be returned. In this paper, we develop a topological derivative-based multifrequency iterative algorithm to reconstruct objects buried in attenuating media with limited aperture data. We demonstrate the method on half-space configurations, which can be related to problems set in the whole space by symmetry. One-step implementations of the algorithm provide initial approximations, which are improved in a few iterations. We can locate object components of sizes smaller than the main components, or buried deeper inside. However, attenuation effects hinder object detection depending on the size and depth for fixed ranges of frequencies.

Solving the inverse problem of time independent Fokker–Planck equation with a self supervised neural network method

The Fokker–Planck equation (FPE) has been used in many important applications to study stochastic processes with the evolution of the probability density function (pdf). Previous studies on FPE mainly focus on solving the forward problem which is to predict the time-evolution of the pdf from the underlying FPE terms. However, in many applications the FPE terms are usually unknown and roughly estimated, and solving the forward problem becomes more challenging. In this work, we take a different approach of starting with the observed pdfs to recover the FPE terms using a self-supervised machine learning method. This approach, known as the inverse problem, has the advantage of requiring minimal assumptions on the FPE terms and allows data-driven scientific discovery of unknown FPE mechanisms. Specifically, we propose an FPE-based neural network (FPE-NN) which directly incorporates the FPE terms as neural network weights. By training the network on observed pdfs, we recover the FPE terms. Additionally, to account for noise in real-world observations, FPE-NN is able to denoise the observed pdfs by training the pdfs alongside the network weights. Our experimental results on various forms of FPE show that FPE-NN can accurately recover FPE terms and denoising the pdf plays an essential role.

A Novel GUI-Based Image Reconstruction Algorithm of EIT Imaging Technique

Electrical impedance tomography (EIT) is a non-invasive technique that is used to estimate the electrical properties of a medical or non-medical object through the boundary data of the object. It used to achieve functional imaging of different objects by measuring electrical conductivity and impedance parameters. In this paper, a novel image reconstruction algorithm is presented, which is based on graphical user interface (GUI) developed on MATLAB software platform. EIT imaging algorithm consists of a forward problem and an inverse problem. The forward problem is formulated with the conductance matrix, and a non-iterative inverse method is used to estimate the conductivity distribution. Image display and data analysis are implemented and controlled directly in the GUI. The numerical simulations and phantom experiments have been carried out to evaluate the performance of the proposed algorithm and other previous research data through quantitative parameters. The obtained result shows satisfactory and comparable results to other EIT imaging algorithm.

Approaches to improve the robustness of the Comparison Model Method for the inverse problem of groundwater hydrology

<p>Indirect inversion approaches are widely used in Geosciences, and in particular also for the identification of the hydraulic properties of aquifers. Nevertheless, their application requires a substantial number of model evaluation (forward problem) runs, a task that for complex problems can be computationally intensive. Reducing this computational burden is an active research topic, and many solutions, including the use of hybrid optimization methods, the use of physical proxies or again machine-learning tools <span>allow to avoid</span> considering the full physics of the problem when running a numerical implementation of the forward problem.</p><p>Direct inversion approaches represent computationally frugal alternatives to indirect approaches, because in general they require a smaller number of runs of the forward problem. The classical drawbacks of these methods can be alleviated by some implementation approaches and in particular by using multiple sets of data, when available.</p><p>This work is an effort to improve the robustness of the Comparison Model Method (CMM), a direct inversion approach aimed at the identification of the hydraulic transmissivity of a confined aquifer. The robustness of the CMM is here ameliorated by (i) improving the parameterization required to handle small hydraulic gradients; (ii) investigating the role of different criteria aimed at merging multiple data-sets corresponding to different flow conditions.</p><p>On a synthetic case study, it is demonstrated that correcting a small percentage of the small hydraulic gradients (about 10%) allows to obtain reliable results, and that a criteria based on the geometric mean is adequate to merge the results coming from multiple data-sets. In addition, the use of multiple-data sets allows to noticeably improve the robustness of the CMM when the input data are affected by noise.</p><p>All the tests are performed by using open source and widely <span>used</span> tools like the USGS Modflow6 and its Python interface flopy to foster the application of the <span>CMM. The scripts and corresponding package</span>, named <em>cmmpy</em>, is available on the Python Package Index (PyPI) and on bitbucket at the following address: https://bitbucket.org/alecomunian/cmmpy.</p>