PDE-constrained Optimization for Electroencephalographic Source Reconstruction

This paper introduces a novel numerical method for the inverse problem of electroencephalography (EEG). We pose the inverse EEG problem as an optimal control (OC) problem for Poisson’s equation. The optimality conditions lead to a variational system of differential equations. It is discretized direc...

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Bibliographic Details
Published in:Lobachevskii journal of mathematics 2024, Vol.45 (6), p.2875-2894
Main Authors: Malovichko, M. S., Yavich, N. B., Razorenova, A. M., Golubev, V. I., Koshev, N. A.
Format: Article
Language:English
Subjects:
Algebra
Algorithms
Analysis
Differential equations
Discretization
Electroencephalography
Geometry
Inverse problems
Linear equations
Mathematical Logic and Foundations
Mathematics
Mathematics and Statistics
Numerical methods
Optimal control
Optimization
Poisson equation
Probability Theory and Stochastic Processes
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Summary:This paper introduces a novel numerical method for the inverse problem of electroencephalography (EEG). We pose the inverse EEG problem as an optimal control (OC) problem for Poisson’s equation. The optimality conditions lead to a variational system of differential equations. It is discretized directly in finite-element spaces leading to a system of linear equations with a sparse Karush–Kuhn–Tucker matrix. The method uses finite-element discretization and thus can handle MRI-based meshes of almost arbitrary complexity. It extends the well-known mixed quasi-reversibility method (mQRM) in that pointwise noisy data explicitly appear in the formulation making unnecessary tedious interpolation of the noisy data from the electrodes to the scalp surface. The resulting algebraic problem differs considerably from that obtained in the mixed quasi-reversibility, but only slightly larger. The algorithm does not require the formation of the lead-field matrix, which can be beneficial for large matrices. Our tests, both with spherical and MRI-based meshes, demonstrates that the method accurately reconstructs cortical activity.
ISSN:1995-0802
1818-9962
DOI:10.1134/S1995080224603266