Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA Two-Level Preconditioned Helmholtz-Jacobi-Davidson Method for the Maxwell Eigenvalue Problem
85
0
0.0
(
0
)
Added by
Qigang Liang
Publication date
2021
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
In this paper, based on a domain decomposition (DD) method, we shall propose an efficient two-level preconditioned Helmholtz-Jacobi-Davidson (PHJD) method for solving the algebraic eigenvalue problem resulting from the edge element approximation of the Maxwell eigenvalue problem. In order to eliminate the components in orthogonal complement space of the eigenvalue, we shall solve a parallel preconditioned system and a Helmholtz projection system together in fine space. After one coarse space correction in each iteration and minimizing the Rayleigh quotient in a small dimensional Davidson space, we finally get the error reduction of this two-level PHJD method as $gamma=c(H)(1-Cfrac{delta^{2}}{H^{2}})$, where $C$ is a constant independent of the mesh size $h$ and the diameter of subdomains $H$, $delta$ is the overlapping size among the subdomains, and $c(H)$ decreasing as $Hto 0$, which means the greater the number of subdomains, the better the convergence rate. Numerical results supporting our theory shall be given.
rate research
Read More
Pole-swapping algorithms, which are generalizations of the QZ algorithm for the generalized eigenvalue problem, are studied. A new modular (and therefore more flexible) convergence theory that applies to all pole-swapping algorithms is developed. A key component of all such algorithms is a procedure that swaps two adjacent eigenvalues in a triangular pencil. An improved swapping routine is developed, and its superiority over existing methods is demonstrated by a backward error analysis and numerical tests. The modularity of the new convergence theory and the generality of the pole-swapping approach shed new light on bi-directional chasing algorithms, optimally packed shifts, and bulge pencils, and allow the design of novel algorithms.
In this work, we are interested in the determination of the shape of the scatterer for the two dimensional time harmonic inverse medium scattering problems in acoustics. The scatterer is assumed to be a piecewise constant function with a known value inside inhomogeneities, and its shape is represented by the level set functions for which we investigate the information using the Bayesian method. In the Bayesian framework, the solution of the geometric inverse problem is defined as a posterior probability distribution. The well-posedness of the posterior distribution would be discussed, and the Markov chain Monte Carlo (MCMC) methods will be applied to generate samples from the arising posterior distribution. Numerical experiments will be presented to demonstrate the effectiveness of the proposed method.
This paper analyses the following question: let $mathbf{A}_j$, $j=1,2,$ be the Galerkin matrices corresponding to finite-element discretisations of the exterior Dirichlet problem for the heterogeneous Helmholtz equations $ ablacdot (A_j abla u_j) + k^2 n_j u_j= -f$. How small must $|A_1 -A_2|_{L^q}$ and $|{n_1} - {n_2}|_{L^q}$ be (in terms of $k$-dependence) for GMRES applied to either $(mathbf{A}_1)^{-1}mathbf{A}_2$ or $mathbf{A}_2(mathbf{A}_1)^{-1}$ to converge in a $k$-independent number of iterations for arbitrarily large $k$? (In other words, for $mathbf{A}_1$ to be a good left- or right-preconditioner for $mathbf{A}_2$?). We prove results answering this question, give theoretical evidence for their sharpness, and give numerical experiments supporting the estimates. Our motivation for tackling this question comes from calculating quantities of interest for the Helmholtz equation with random coefficients $A$ and $n$. Such a calculation may require the solution of many deterministic Helmholtz problems, each with different $A$ and $n$, and the answer to the question above dictates to what extent a previously-calculated inverse of one of the Galerkin matrices can be used as a preconditioner for other Galerkin matrices.
A thorough backward stability analysis of Hotellings deflation, an explicit external deflation procedure through low-rank updates for computing many eigenpairs of a symmetric matrix, is presented. Computable upper bounds of the loss of the orthogonality of the computed eigenvectors and the symmetric backward error norm of the computed eigenpairs are derived. Sufficient conditions for the backward stability of the explicit external deflation procedure are revealed. Based on these theoretical results, the strategy for achieving numerical backward stability by dynamically selecting the shifts is proposed. Numerical results are presented to corroborate the theoretical analysis and to demonstrate the stability of the procedure for computing many eigenpairs of large symmetric matrices arising from applications.
In this paper we consider the Virtual Element discretization of a minimal surface problem, a quasi-linear elliptic partial differential equation modeling the problem of minimizing the area of a surface subject to a prescribed boundary condition. We derive optimal error estimate and present several numerical tests assessing the validity of the theoretical results.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
