arXiv subjects

Yury Gryazin

Publications and source records attributed to Yury Gryazin.

3 recordsLinked to original sources

arXiv2026

High-Resolution Solvers for 3D Helmholtz Scattering Problems Using PFFT and Eigenvector-Based Preconditioning

This paper presents an efficient Krylov subspace iterative solver for the three-dimensional (3D) Helmholtz equation with non-constant coefficients and absorbing boundary conditions, combining high-resolution compact schemes with low-order preconditioners. To mitigate numerical dispersion and reduce pollution error, we employ fourth- and sixth-order compact finite-difference schemes, thereby significantly softening the strict points-per-wavelength requirement. The resulting large, ill-conditioned linear systems are solved using a preconditioned GMRES method. The key innovation lies in the construction of the preconditioner: we introduce two highly efficient direct solvers - one based on a low-dimensional eigenvector transformation (EigT) and another on a partial Fast Fourier Transform (PFFT) algorithm - both derived from a lower-order approximation of the original problem that incorporates the absorbing boundary conditions. The motivation and efficacy of this lower-order preconditioning strategy for high-resolution schemes are analyzed through model problems, providing insight into the convergence rate. The theoretical analysis is validated by a comprehensive set of numerical experiments, demonstrating the method's performance for realistic problem sizes and parameters.

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arXiv2020

Scalable Algorithms for High-Order Approximations on Three-Dimensional Compact Stencils

This paper presents an efficient parallel direct algorithm with near-optimal complexity for the compact fourth and sixth-order approximation of the three-dimensional Helmholtz equations [1] with the problem coefficient depending on only one of the coordinate directions. The developed method is based on a combination of the separation of variables technique and a Fast Fourier Transform (FFT) type method. Similar direct solvers for the lower-order approximations of the two and three-dimensional Helmholtz equation were considered in several previous publications by the authors and other researchers (see e.g. [2,3,4,5]). The authors also consider a generalization of the presented algorithm to the solution of a wide class of linear systems obtained from approximation on the compact 27-point three-dimensional stencils on the rectangular grids with similar requirements on the stencil coefficients. The general restrictions on the coefficients in the considered class of compact schemes are developed and presented. This class includes the second, fourth and sixth-order compact approximation schemes for the three-dimensional Helmholtz equation considered in this paper and our previous publications [2,3,6]. As an example of the diversity of applications of the developed general method, the direct parallel implementation of a compact fourth-order approximation scheme for a convection-diffusion equation is considered. The results of the implementation of this method in OpenMP, MPI and hybrid programming environments on the multicore computers and multiple node clusters are presented and discussed. The results demonstrate the high efficiency of the proposed direct solvers for many important applications on the structured grid with the corresponding 27-diagonal matrices.

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arXiv2012

Preconditioned Krylov subspace methods for sixth order compact approximations of the Helmholtz equation

In this paper, we consider an efficient iterative approach to the solution of the discrete Helmholtz equation with Dirichlet, Neumann and Sommerfeld-like boundary conditions based on a compact sixth order approximation scheme and preconditioned Krylov subspace methodology. A sixth order compact scheme for the 3D Helmholtz equation with different boundary conditions is developed to reduce approximation and pollution errors, thereby softening the point-per-wavelength constraint. The resulting systems of finite-difference equations are solved by different preconditioned Krylov subspace-based methods. In the majority of test problems, the preconditioned Generalized Minimal Residual (GMRES) method is the superior choice, but in the case of sufficiently fine grids a simple stationary two-level algorithm proposed in this paper in combination with a lower order approximation preconditioner presents an efficient alternative to the GMRES method. In the analysis of the lower order preconditioning developed here, we introduce the term "$k$-th order preconditioned matrix" in addition to the commonly used "an optimal preconditioner". The necessity of the new criterion is justified by the fact that the condition number of the preconditioned matrix $ AA^{-1}_p $ in some of our test problems improves with the decrease of the grid step size. In a simple 1D case, we are able to prove this analytically. This new parameter could serve as a guide in the construction of new preconditioners. The lower order direct preconditioner used in our algorithms is based on a combination of the separation of variables technique and Fast Fourier Transform (FFT) type methods. The resulting numerical methods allow efficient implementation on parallel computers. Numerical results confirm the high efficiency of the proposed iterative approach.

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