Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userEigenvalues of the truncated Helmholtz solution operator under strong trapping
69
0
0.0
(
0
)
Added by
Euan Spence
Publication date
2021
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
For the Helmholtz equation posed in the exterior of a Dirichlet obstacle, we prove that if there exists a family of quasimodes (as is the case when the exterior of the obstacle has stable trapped rays), then there exist near-zero eigenvalues of the standard variational formulation of the exterior Dirichlet problem (recall that this formulation involves truncating the exterior domain and applying the exterior Dirichlet-to-Neumann map on the truncation boundary). Our motivation for proving this result is that a) the finite-element method for computing approximations to solutions of the Helmholtz equation is based on the standard variational formulation, and b) the location of eigenvalues, and especially near-zero ones, plays a key role in understanding how iterative solvers such as the generalised minimum residual method (GMRES) behave when used to solve linear systems, in particular those arising from the finite-element method. The result proved in this paper is thus the first step towards rigorously understanding how GMRES behaves when applied to discretisations of high-frequency Helmholtz problems under strong trapping (the subject of the companion paper [Marchand, Galkowski, Spence, Spence, 2021]).
rate research
Read More
We study a commonly-used second-kind boundary-integral equation for solving the Helmholtz exterior Neumann problem at high frequency, where, writing $Gamma$ for the boundary of the obstacle, the relevant integral operators map $L^2(Gamma)$ to itself. We prove new frequency-explicit bounds on the norms of both the integral operator and its inverse. The bounds on the norm are valid for piecewise-smooth $Gamma$ and are sharp, and the bounds on the norm of the inverse are valid for smooth $Gamma$ and are observed to be sharp at least when $Gamma$ is curved. Together, these results give bounds on the condition number of the operator on $L^2(Gamma)$; this is the first time $L^2(Gamma)$ condition-number bounds have been proved for this operator for obstacles other than balls.
Over the last ten years, results from [Melenk-Sauter, 2010], [Melenk-Sauter, 2011], [Esterhazy-Melenk, 2012], and [Melenk-Parsania-Sauter, 2013] decomposing high-frequency Helmholtz solutions into low- and high-frequency components have had a large impact in the numerical analysis of the Helmholtz equation. These results have been proved for the constant-coefficient Helmholtz equation in either the exterior of a Dirichlet obstacle or an interior domain with an impedance boundary condition. Using the Helffer-Sjostrand functional calculus, this paper proves analogous decompositions for scattering problems fitting into the black-box scattering framework of Sjostrand-Zworski, thus covering Helmholtz problems with variable coefficients, impenetrable obstacles, and penetrable obstacles all at once. In particular, these results allow us to prove new frequency-explicit convergence results for (i) the $hp$-finite-element method applied to the variable coefficient Helmholtz equation in the exterior of a Dirichlet obstacle, when the obstacle and coefficients are analytic, and (ii) the $h$-finite-element method applied to the Helmholtz penetrable-obstacle transmission problem.
We consider GMRES applied to discretisations of the high-frequency Helmholtz equation with strong trapping; recall that in this situation the problem is exponentially ill-conditioned through an increasing sequence of frequencies. Under certain assumptions about the distribution of the eigenvalues, we prove upper bounds on how the number of GMRES iterations grows with the frequency. Our main focus is on boundary-integral-equation formulations of the exterior Dirichlet and Neumann obstacle problems in 2- and 3-d; for these problems, we investigate numerically the sharpness (in terms of dependence on frequency) of both our bounds and various quantities entering our bounds. This paper is therefore the first comprehensive study of the frequency-dependence of the number of GMRES iterations for Helmholtz boundary-integral equations under trapping.
We study the wave equation in the exterior of a bounded domain $K$ with dissipative boundary condition $partial_{ u} u - gamma(x) u = 0$ on the boundary $Gamma$ and $gamma(x) > 0.$ The solutions are described by a contraction semigroup $V(t) = e^{tG}, : t geq 0.$ The eigenvalues $lambda_k$ of $G$ with ${rm Re}: lambda_k < 0$ yield asymptotically disappearing solutions $u(t, x) = e^{lambda_k t} f(x)$ having exponentially decreasing global energy. We establish a Weyl formula for these eigenvalues in the case $min_{xin Gamma} gamma(x) > 1.$ For strictly convex obstacles $K$ this formula concerns all eigenvalues of $G.$
We describe the non-backtracking spectrum of a stochastic block model with connection probabilities $p_{mathrm{in}}, p_{mathrm{out}} = omega(log n)/n$. In this regime we answer a question posed in DallAmico and al. (2019) regarding the existence of a real eigenvalue `inside the bulk, close to the location $frac{p_{mathrm{in}}+ p_{mathrm{out}}}{p_{mathrm{in}}- p_{mathrm{out}}}$. We also introduce a variant of the Bauer-Fike theorem well suited for perturbations of quadratic eigenvalue problems, and which could be of independent interest.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
