The Electrical Impedance Tomography (EIT) and electroencephalography (EEG) forward problems in anisotropic inhomogeneous media like the human head belongs to the class of the three-dimensional boundary value problems for elliptic equations with mixed derivatives. We introduce and explore the performance of several new promising numerical techniques, which seem to be more suitable for solving these problems. The proposed numerical schemes combine the fictitious domain approach together with the finite-difference method and the optimally preconditioned Conjugate Gradient- (CG-) type iterative method for treatment of the discrete model. The numerical scheme includes the standard operations of summation and multiplication of sparse matrices and vector, as well as FFT, making it easy to implement and eligible for the effective parallel implementation. Some typical use cases for the EIT/EEG problems are considered demonstrating high efficiency of the proposed numerical technique. 1. Introduction The progress in the forward and inverse modeling in Electrical Encephalography (EEG) and Magnetoencephalography (MEG) source localization as well as in Electrical Impedance Tomography (EIT) depends on efficiency and accuracy of the employed forward solvers for the governing partial differential equations (PDE), in particular, the Poisson equations, describing the electrical potential distribution in highly heterogeneous and anisotropic human head tissues. The modern forward solvers use the variety of computational approaches based on the finite difference (FD), boundary element (BE), and finite element (FE) methods [1–9], multigrid [10] and preconditioned Conjugate Gradient- (CG-) type iterative methods [11–15], and also high performance parallel computing techniques [16–22]. To describe the electrical conductivity in heterogeneous biological media with arbitrary geometry, the method of embedded boundaries or a fictitious domain can be used [23, 24]. In this method, an arbitrarily shaped object of interest is embedded into a rectangular computational domain with extremely low conductivity values in the external complimentary regions modeling the surrounding air. This effectively guarantees that there are no current flows out of the physical area and implicitly sets up the zero flux Neumann boundary condition on the surface of the object. This setup retains the advantages of the finite-difference method (FDM), which is most prominent in the rectangular domain. Previously, we built an iterative finite-difference forward problem solver for an isotropic version of the
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