Matrix-Free Parallel Preconditioned Iterative Solvers for the 2D Helmholtz Equation Discretized with Finite Differences

We present a matrix-free parallel iterative solver for the Helmholtz equation related to applications in seismic problems and study its parallel performance. We apply Krylov subspace methods, GMRES, Bi-CGSTAB and IDR(s), to solve the linear system obtained from a second-order finite difference discretization. The Complex Shifted Laplace...

A matrix-free parallel solution method for the three-dimensional heterogeneous Helmholtz equation

The Helmholtz equation is related to seismic exploration, sonar, antennas, and medical imaging applications. It is one of the most challenging problems to solve in terms of accuracy and convergence due to the scalability issues of the numerical solvers. For 3D large-scale applications, high-performance parallel solvers are also needed. In this...

Scalable multi-level deflation preconditioning for highly indefinite time-harmonic waves

Recent research efforts aimed at iteratively solving time-harmonic waves have focused on a broad range of techniques to accelerate convergence. In particular, for the famous Helmholtz equation, deflation techniques have been studied to accelerate the convergence of Krylov subspace methods. In this work, we extend the two-level deflation method...

Inexact Subdomain Solves Using Deflated GMRES for Helmholtz Problems

In recent years, domain decomposition based preconditioners have become popular tools to solve the Helmholtz equation. Notorious for causing a variety of convergence issues, the Helmholtz equation remains a challenging PDE to solve numerically. Even for simple model problems, the resulting linear system after discretisation becomes indefinite...

Pollution and accuracy of solutions of the Helmholtz equation: A novel perspective from the eigenvalues

In researching the Helmholtz equation, the focus has either been on the accuracy of the numerical solution (pollution) or the acceleration of the convergence of a preconditioned Krylov-based solver (scalability). While it is widely recognized that the convergence properties can be investigated by studying the eigenvalues, information from the...

Towards accuracy and scalability: Combining Isogeometric Analysis with deflation to obtain scalable convergence for the Helmholtz equation

Finding fast yet accurate numerical solutions to the Helmholtz equation remains a challenging task. The pollution error (i.e. the discrepancy between the numerical and analytical wave number k) requires the mesh resolution to be kept fine enough to obtain accurate solutions. A recent study showed that the use of Isogeometric Analysis (IgA)...

Scalable multi-level deflation preconditioning for the highly indefinite Helmholtz equation

</p><p class="MsoNormal">Recent research efforts aimed at iteratively solving the Helmholtz equation have focused on incorporating deflation techniques for accelerating the convergence of Krylov subspace methods. In this work, we extend the two-level deflation method in [6] to a multilevel deflation method. By using higher-order deflation...

Scalable Convergence Using Two-Level Deflation Preconditioning for the Helmholtz Equation

Recent research efforts aimed at iteratively solving the Helmholtz equation have focused on incorporating deation techniques for accelerating the convergence of Krylov subpsace methods. The requisite for these efforts lies in the fact that the widely used and well-acknowledged complex shifted Laplacian preconditioner (CSLP) shifts the...