секция: Дифференциальные уравнения и их приложени

Malony, A.

Prakonina, A.

Turovets, S.

Volkov, V.

Digital Library of Polotsk State University (Репозиторий Полоцкого государственного университета)

PolotskSUДифференциальные уравнения и их приложения 85and the boundary condition. The coercive case is considered and the external perturbationis given in a concrete form. We consider the model of the external perturbation in thespecial analytical form that has not been studied for the Lavrent’ev problem. The existenceof a semiregular solution is proved by the variational method. In addition, if variationalfunctional has no more countable number of points of a global minimum, then there isthe regular solution of the problem, i.e. the semiregular solution with the property ofcorrectness. The regular solutions of the Lavrent’ev problem have not been investigated.The problem on conductor heating is considered under constant voltage and constanttemperature on the conductor surface when electroconductivity of material changes witha jump upon transition through certain temperatures. The results obtained earlier forproblems with a spectral parameter for equations of elliptic type with discontinuous non-linearities are applied to this problem of electrophysics. Restrictions on the gap pointsof nonlinearity are weaken (nonlinearity is electroductivity of the conductor). The theo-rem on both the existence of the nonzero semiregular solution and the estimates for thedifferential operator of the problem under consideration is established.We summarize the results of [1–3] for applications to equations of elliptic type with aspectral parameter and nonlinearity discontinuous with respect to the phase variable.Refrences1. Potapov D.K. On solutions to the Goldshtik problem // Num. Anal. and Appl. 2012. Vol. 5, no. 4.P. 342–347.2. Potapov D.K., Yevstafyeva V.V. Lavrent’ev problem for separated flows with an external perturba-tion // Electron. J. Diﬀer. Eq. 2013. No. 255. P. 1–6.3. Potapov D.K. On one problem of electrophysics with discontinuous nonlinearity // Diﬀer. Equ. 2014.Vol. 50, no. 3.FINITE-DIFFERENCE ITERATIVE SOLVER WITH SPECTRALLYEQUIVALENT PRECONDITIONER FOR ANISOTROPIC ELECTRICALIMPEDANCE TOMOGRAPHY PROBLEMSV.M.Volkov 1, A.V.Prakonina 1, S.I. Turovets 2, A.D. Malony 31 Belarusian State University, Minsk, Belarusv.volkov@tut.by2 Electrical Geodesics, Inc., Eugene, USAsergei@cs.uoregon.edu3 University of Oregon, Eugene, USAThe electrical impedance tomortaphy (EIT) problems in anisotropic inhomogeneousmedia like head tissues belongs to the class of the three-dimensional boundary valueproblems for elliptic equations with mixed derivatives. The eﬃciency of the most discussedand usable in practice numerical methods in context of modeling EIT problems is reviewedin [1,2], where it is shown that the best performance is demonstrated by the algebraicmulti-grid (AMG) methods.We present a novel type of the anisotropic bi-conjugate gradient iterative solver for3D elliptic equations with mixed derivatives. The proposed numerical scheme is basedon the ﬁnite-diﬀerence approximation of the problem in an arbitrary three-dimensionalcomputational domain, augmented to a cuboid with non-conducting claddings and theDirichlet type boundary conditions deﬁned at the facets of the cuboid. The rectangularuniform ﬁnite diﬀerence grid is assumed to have high enough resolution across the cha-racteristic layers of inhomogeneity. The nonconductive claddings imitate the NeumannPolotskSU86 «ЕРУГИНСКИЕ ЧТЕНИЯ–2014»boundary condition on the arbitrarily shaped surface of the embedded object. The discreteproblem is reduced to solving a system of linear algebraic equations with a 19-diagonalsparse matrix.Iterative methods from the BiCG family is most effective with the adequate choice ofpreconditioners. As a preconditioner, we employed combined spectrally-optimal precondi-tioning Fourier-Jacobi (FJ). The key advantages of the proposed technique is to eliminatethe dependence of the convergence rate on the spatial grid resolution and the heterogeneityratio of the discrete model. As a result, the number of iterations to achieve the desiredaccuracy is almost independent of the grid resolution, which puts the proposed techniquein line with the popular multigrid iterative methods. The proposed numerical algorithm in-cludes the standard arithmetic operations with sparse matrices and vectors, as well as FFT,making it easy to implement and readily eligible for the effective parallel implementation.The proposed algorithm has been validated against analytics in a spherical model andtested on the anatomically accurate MRI based human head geometry [3]. Simulationresults show high efficiency of the developed approach. It is capable to solve 128x128x128voxels anisotropic problems with the extreme conductivity tensor eigenvalues ratio of 10:1and the heterogeneity ratio of the piecewise constant coefficients up to 1016 (includingexplicitly titanium clips and air pockets modeling) within a minute runtime in the Matlabimplementation.Refrences1. Mohr M., Vanrumste B. Comparing iterative solvers for linear systems associated with the finitedifference discretization of the forward problem in electro-encephalographic source analysis // Medical andBiological Engineering and Computing. 2003. Vol. 41, no. 1. P. 75–84.2. Barnes D.N., George J. S., Ng K.T. Finite difference iterative solvers for electroencephalography:serial and parallel performance analysis // Medical and biological engineering and computing. 2008. Vol. 46,no. 9. P. 901–910.3. Turovets S., Volkov V., Zherdetsky A., Prakonina A., Malony A.D. A 3D Finite-Difference BiCGIterative Solver with the Fourier-Jacobi Preconditioner for the Anisotropic EIT/EEG Forward Problem //Computational and Mathematical Methods in Medicine. 2014. P. 1–12.

Finite-difference iterative solver with spectrally equivalent preconditioner for anisotropic electrical impedance tomography problems

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Finite-difference iterative solver with spectrally equivalent preconditioner for anisotropic electrical impedance tomography problems

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