arXiv subjects

Adem Kaya

Publications and source records attributed to Adem Kaya.

3 recordsLinked to original sources

arXiv2023

Low-rank solutions to the stochastic Helmholtz equation

In this paper, we consider low-rank approximations for the solutions to the stochastic Helmholtz equation with random coefficients. A Stochastic Galerkin finite element method is used for the discretization of the Helmholtz problem. Existence theory for the low-rank approximation is established when the system matrix is indefinite. The low-rank algorithm does not require the construction of a large system matrix which results in an advantage in terms of CPU time and storage. Numerical results show that, when the operations in a low-rank method are performed efficiently, it is possible to obtain an advantage in terms of storage and CPU time compared to computations in full rank. We also propose a general approach to implement a preconditioner using the low-rank format efficiently.

math.NA↗

arXiv2021

Application of adapted-bubbles to the Helmholtz equation with large wave numbers in 2d

An adapted bubble approach which is a modifiation of the residual-free bubbles (RFB) method, is proposed for the Helmhotz problem in 2D. A new two-level finite element method is introduced for the approximations of the bubble functions. Unlike the other equations such as the advection-diffusion equation, RFB method when applied to the Helmholtz equation, does not depend on another stabilized method to obtain approximations to the solutions of the sub-problems. Adapted bubbles (AB) are obtained by a simple modification of the sub-problems. This modification increases the accuracy of the numerical solution impressively. The AB method is able to solve the Helmholtz equation efficiently in 2D up to ch = 3.5 where c is the wave number and h is the mesh size. We provide analysis to show how the AB method mitigates the pollution error.

math.NA↗

arXiv2021

Multigrid as an exact solver

We provide an alternative Fourier analysis for multigrid applied to the Poisson problem in 1D, based on explicit derivation of spectra of the iteration matrix. The new Fourier analysis has advantages over the existing one. It is easy to understand and enables us to write the error equation in terms of the eigenvector of the stiffness matrix. When weighted-Jacobi is used as a smoother with two different weights, multigrid is an exact solver.

math.GM↗