V. M. Volkov

The Theory of Spiral Laser Beams in Nonlinear Media

Abstract The analytical theory governing the propagation of spiral laser beams in nonlinear media is developed. It is shown that, taking into account the saturation effect in a Kerr medium, spiral beams have a tube-like structure with a periodicity along the axis of a nonlinear autoguide. In the case of the absence of saturation mechanism the collapse of spiral beams is described. It is found that critical autoguide power depends on the topological charge of the spiral beam. The stability of a circular symmetry structure of the beam is investigated. The analytical results are in good agreement with the numerical experiment.

Auto-waveguide propagation and the collapse of spiral light beams in non-linear media

The equation governing the propagation of spiral beams in a non-linear medium is analysed. It is shown that, taking into account the saturation effect, spiral beams have a tube-like structure with a periodicity along the axis of a non-linear autoguide. A critical power is found which it is necessary to exceed in order to give rise to the indicated autoguided regime of the beam propagation. Furthermore, in this work a strict self-similar solution is obtained of a non-linear Schrodinger-type equation that describes the collapse of spiral beams, and a new class of self-similar solutions is found for the case of spherical symmetry.

Concurrency in electrical neuroinformatics: parallel computation for studying the volume conduction of brain electrical fields in human head tissues

Advances in human brain neuroimaging for high‐temporal and high‐spatial resolutions will depend on localization of electroencephalography (EEG) signals to their cortex sources. The source localization inverse problem is inherently ill‐posed and depends critically on the modeling of human head electromagnetics. We present a systematic methodology to analyze the main factors and parameters that affect the EEG source‐mapping accuracy. These factors are not independent, and their effect must be evaluated in a unified way. To do so requires significant computational capabilities to explore the problem landscape, quantify uncertainty effects, and evaluate alternative algorithms. Bringing high‐performance computing to this domain is necessary to open new avenues for neuroinformatics research. The head electromagnetics forward problem is the heart of the source localization inverse. We present two parallel algorithms to address tissue inhomogeneity and impedance anisotropy. Highly accurate head modeling environments will enable new research and clinical neuroimaging applications. Cortex‐localized dense‐array EEG analysis is the next‐step in neuroimaging domains such as early childhood reading, understanding of resting‐state brain networks, and models of full brain function. Therapeutic treatments based on neurostimulation will also depend significantly on high‐performance computing integration. Copyright

Self-action of Counterpropagating Axially Symmetric Light Beams in a Transparent Cubic-nonlinearity Medium

Abstract The stationary effect of self-action of counterpropagating axially symmetric light beams in a medium with cubic nonlinearity is discussed. Self-similar solutions for initial radiation parameters and nonlinear medium characteristics in the case of slowly varying amplitudes of beams are found. The analysis of approximate solutions obtained on the basis of motion constants is carried out and results are compared with numerical experimental data.

A 3D Finite-Difference BiCG Iterative Solver with the Fourier-Jacobi Preconditioner for the Anisotropic EIT/EEG Forward Problem

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.

Multi-cluster, mixed-mode computational modeling of human head conductivity

A multi-cluster computational environment with mixed-mode (MPI + OpenMP) parallelism for estimation of unknown regional electrical conductivities of the human head, based on realistic geometry from segmented MRI up to 2563 voxels resolution, is described. A finite difference multi-component alternating direction implicit (ADI) algorithm, parallelized using OpenMP, is used to solve the forward problem calculation describing the electrical field distribution throughout the head given known electrical sources. A simplex search in the multi-dimensional parameter space of tissue conductivities is conducted in parallel across a distributed system of heterogeneous computational resources. The theoretical and computational formulation of the problem is presented. Results from test studies based on the synthetic data are provided, comparing retrieved conductivities to known solutions from simulation. Performance statistics are also given showing both the scaling of the forward problem and the performance dynamics of the distributed search.

A 3D Vector-Additive Iterative Solver for the Anisotropic Inhomogeneous Poisson Equation in the Forward EEG problem

We describe a novel 3D finite difference method for solving the anisotropic inhomogeneous Poisson equation based on a multi-component additive implicit method with a 13-point stencil. The serial performance is found to be comparable to the most efficient solvers from the family of preconditioned conjugate gradient (PCG) algorithms. The proposed multi-component additive algorithm is unconditionally stable in 3D and amenable for transparent domain decomposition parallelization up to one eighth of the total grid points in the initial computational domain. Some validation and numerical examples are given.

Next-generation human brain neuroimaging and the role of high-performance computing

Advances in human brain neuroimaging to achieve high-temporal and high-spatial resolution will depend on computational approaches to localize EEG signals to their sources in the cortex. The source localization inverse problem is inherently ill-posed and depends critically on the modeling of human head electromagnetics. In this paper we present a systematic methodology to analyze the main factors and parameters that affect the accuracy of the EEG source-mapping solutions. We argue that these factors are not independent and their effect must be evaluated in a unified way. To do so requires significant computational capabilities to explore the landscape of the problem, to quantify uncertainty effects, and to evaluate alternative algorithms. We demonstrate that bringing HPC to this domain will enable such investigation and will allow new avenues for neuroinformatics research. Two algorithms to the electromagnetics forward problem (the heart of the source localization inverse), incorporating tissue inhomogeneity and impedance anisotropy, are presented and their parallel implementations described. The head model forward solvers are evaluated and their performance analyzed.

A fast BiCG solver for the isotropic Poisson equation in the forward EIT problem in cylinder phantoms

We describe an efficient numerical method for solving the isotropic inhomogeneous 3D Poisson equation in cylindrical coordinates as applied to analysis of EIT phantom experimental systems. The proposed approach is based on the second order accuracy finite-difference scheme and the BiCG iterative method with the FFT preconditioner. Extensive validation, numerical examples and comparison with experimental results are given. The performance and accuracy of the proposed numerical method are investigated and compared with the similar BiCG method with the Jacoby preconditioner.

Evolution of femtosecond solitons in a cubic medium with a two-component relaxing nonlinearity

Soliton propagation in a nonlinear cubic medium with two types of relaxing nonlinearities is investigated. Provided that the pulse duration exceeds the fast relaxation time and be shorter than the slow one, partial compensation of the concerted influence of these relaxations on the soliton dynamics is feasible. A condition for such an optimal regime of soliton pulse propagation is analytically obtained. Computer simulations are in good agreement with analytical predictions.