We discuss the problems arising when computing eigenvalues of self-adjoint operators which lie in a gap between two parts of the essential spectrum. Spectral pollution, i.e. the apparent existence of eigenvalues in numerical computations, when no such eigenvalues actually exist, is commonplace in problems arising in applied mathematics. We describe a geometrically inspired method which avoids this difficulty, and show that it yields the same results as an algorithm of Zimmermann and Mertins.
Spectral inclusion and spectral pollution results are proved for sequences of linear operators of the form $T_0 + i γs_n$ on a Hilbert space, where $s_n$ is strongly convergent to the identity operator and $γ> 0$. We work in both an abstract setting and a more concrete Sturm-Liouville framework. The results provide rigorous justification for a method of computing eigenvalues in spectral gaps.
We consider a general framework for investigating spectral pollution in the Galerkin method. We show how this phenomenon is characterised via the existence of particular Weyl sequences which are singular in a suitable sense. For a semi-bounded selfadjoint operator A we identify relative compactness conditions on a selfadjoint perturbation B ensuring that the limiting set of spectral pollution of A and B coincide. Our results show that, under perturbation, this limiting set behaves in a similar fashion as the essential spectrum.
We introduce concepts of essential numerical range for the linear operator pencil $λ\mapsto A-λB$. In contrast to the operator essential numerical range, the pencil essential numerical ranges are, in general, neither convex nor even connected. The new concepts allow us to describe the set of spectral pollution when approximating the operator pencil by projection and truncation methods. Moreover, by transforming the operator eigenvalue problem $Tx=λx$ into the pencil problem $BTx=λBx$ for suitable choices of $B$, we can obtain non-convex spectral enclosures for $T$ and, in the study of truncati…
We establish spectral enclosures and spectral approximation results for the inhomogeneous lossy Drude-Lorentz system with purely imaginary poles, in a possibly unbounded Lipschitz domain of $\mathbb{R}^3$. Under the assumption that the coefficients $θ_e$, $θ_m$ of the material are asymptotically constant at infinity, we prove that: 1) the essential spectrum can be decomposed as the union of the spectrum of a bounded operator pencil in the form $- \operatorname{div} p(ω) \nabla$ and of a second order $\operatorname{curl} \operatorname{curl}_0 - V_{e,\infty}(ω)$ pencil with constant coefficients…
Spectral Pollution and How to Avoid It (With Applications to Dirac and Periodic Schrödinger Operators)
CNRS and Laboratoire de Mathématiques (CNRS UMR 8088), Université de Cergy-Pontoise（法国国家科学研究中心与数学实验室（CNRS UMR 8088），塞吉蓬图瓦兹大学）；Ceremade (CNRS UMR 7534), Université Paris-Dauphine（巴黎道夫菲纳大学与决策数学实验室（CNRS UMR 7534））
A new technique for approximating eigenvalues and eigenvectors of a self-adjoint operator is presented. The method does not incur spectral pollution, uses trial spaces from the form domain, has a self-adjoint algorithm, and exhibits superconvergence.
We study spectral problems for integro-differential equations arising in the theory of Gaussian processes similar to the fractional Brownian motion. We generalize the method of Chigansky--Kleptsyna and obtain the two-term eigenvalue asymptotics for such equations. Application to the small ball probabilities in $L_2$-norm is given.
Using the resolvent operator, we develop an algorithm for computing smoothed approximations of spectral measures associated with self-adjoint operators. The algorithm can achieve arbitrarily high-orders of convergence in terms of a smoothing parameter for computing spectral measures of general differential, integral, and lattice operators. Explicit pointwise and $L^p$-error bounds are derived in terms of the local regularity of the measure. We provide numerical examples, including a partial differential operator, a magnetic tight-binding model of graphene, and compute one thousand eigenvalues …
We study a lattice point counting problem for a class of families of domains in a Euclidean space. This class consists of anisotropically expanding bounded domains, which remain unchanged along some fixed linear subspace and expand in directions, orthogonal to this subspace. We find the leading term in the asymptotics of the number of lattice points in such family of domains and prove remainder estimates in this asymptotics under various conditions on the lattice and the family of domains. As a consequence, we prove an asymptotic formula for the eigenvalue distribution function of the Laplace …
Graph Laplacians computed from weighted adjacency matrices are widely used to identify geometric structure in data, and clusters in particular; their spectral properties play a central role in a number of unsupervised and semi-supervised learning algorithms. When suitably scaled, graph Laplacians approach limiting continuum operators in the large data limit. Studying these limiting operators, therefore, sheds light on learning algorithms. This paper is devoted to the study of a parameterized family of divergence form elliptic operators that arise as the large data limit of graph Laplacians. Th…
We consider the Galerkin method for approximating the spectrum of an operator $T+A$ where $T$ is semi-bounded self-adjoint and $A$ satisfies a relative compactness condition. We show that the method is reliable in all regions where it is reliable for the unperturbed problem - which always contains $\mathbb{C}\backslash\mathbb{R}$. The results lead to a new technique for identifying eigenvalues of $T$, and for identifying spectral pollution which arises from applying the Galerkin method directly to $T$. The new technique benefits from being applicable on the form domain.
In this paper we present how spectral properties of certain linear operators vary when operators are considered in different Hilbert spaces having common dense domain as the space of polynomials in one real variable with complex coefficients. This is done taking differential operator and matrix operator representation of dilation operator $E_{p,d}$ dilating $p$, a polynomial sequence, by $d$, a non-constant sequence of non-zero complex scalars. For the purpose of identification of $E_{p,d}$ with formal differential operator, which includes finite order differential operators as special cases, …
The problem of computing spectra of operators is arguably one of the most investigated areas of computational mathematics. However, the problem of computing spectra of general bounded infinite matrices has only recently been solved. We establish some of the foundations of computational spectral theory through the Solvability Complexity Index (SCI) hierarchy, an approach closely related to Smale's program on the foundations of computational mathematics and McMullen's results on polynomial root finding with rational maps. Infinite-dimensional problems yield an intricate infinite classification t…
We prove bounds for a class of unital homomorphisms arising in the study of spectral sets, by involving extremal functions and vectors. These are used to recover three celebrated results on spectral constants by Crouzeix--Palencia, Okubo--Ando and von Neumann in a unified way and to refine a recent result by Crouzeix--Greenbaum.
We consider operator matrices {\bf H}= (A_0 B_{01} \\ B_{10} A_{1}) with self-adjoint entries A_i, i=0,1, and bounded B_{01}=B_{10}^*, acting in the orthogonal sum {\cal H}={\cal H}_0\oplus{\cal H}_1 of Hilbert spaces {\cal H}_0 and {\cal H}_1. We are especially interested in the case where the spectrum of, say, A_1 is partly or totally embedded into the continuous spectrum of A_0 and the transfer function M_1(z)=A_1-z+V_1(z), where V_1(z)=B_{10}(z-A_0)^{-1}B_{01}, admits analytic continuation (as an operator-valued function) through the cuts along branches of the continuous spectrum of the en…
