FEAST for Differential Eigenvalue Problems

A mechanism is described to symmetrize the ultraspherical spectral method for self-adjoint problems. The resulting discretizations are symmetric and banded. An algorithm is presented for an adaptive spectral decomposition of self-adjoint operators. Several applications are explored to demonstrate the properties of the symmetrizer and the adaptive spectral decomposition.Polynomial approximation theory suggests that we should seek to numerically represent the variable coefficients in Eq. (1) as Chebyshev polynomial expansions [18]. An adaptation of the fast multipole method (FMM) [19] accelerates the Chebyshev-Legendre [20], ultraspherical-ultraspherical [21], and Jacobi-Jacobi [22, §3.3] connection problems to linear complexity in the degree. Provided the Jacobi parameters are not too large, the acceleration of the connection problem enables Jacobi polynomial expansions with nearly the same rapidity as Chebyshev polynomial expansions. Thus in the problems in this paper, the variable coefficients in Eq. (1) are in fact polynomials, but in practice they could very well be good numerical approximations to functions in a particular space. How such approximations, in particular of the leading coefficient p N , affect the spectrum is beyond the scope of this report. A well-known model problemIt is instructive to begin with a well-known model problem for illustrative purposes. 1 In exceptional circumstances, such as when B is zero apart from the first few columns and the nonzero part prepended to L renders the block operator discretization self-adjoint, symmetry may still be within reach. Many common boundary conditions, however, are genuinely infinite-dimensional. This destruction of symmetry has real consequences for the spectra of finite truncations. Whereas Chebyshev and Legendre second-order differentiation matrices by collocation have real spectra [9], it is easy to show by counter-example that finite truncations of ultraspherical discretizations of self-adjoint operators may produce complex spectra.

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“…The symmetrizer may also be useful in the design and analysis of contours for spectral projectors in, e.g. the contFEAST algorithm [64].…”

Section: Discussionmentioning

confidence: 99%

On symmetrizing the ultraspherical spectral method for self-adjoint problems

Aurentz

Slevinsky

2020

Journal of Computational Physics

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“…[7,27,28,40,41,52]), FEAST methods (cf. [17,18,24,31,39,45,49,51]), Beyn methods (cf. [8-10, 32, 47]), and Spectral Indicator Methods (RIM, SIM) (cf.…”

Section: An Algorithm Template For (T)mentioning

confidence: 99%

An algorithm for identifying eigenvectors exhibiting strong spatial localization

Ovall¹,

Reid²

2021

Preprint

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We introduce an approach for exploring eigenvector localization phenomena for a class of (unbounded) selfadjoint operators. More specifically, given a target region and a tolerance, the algorithm identifies candidate eigenpairs for which the eigenvector is expected to be localized in the target region to within that tolerance. Theoretical results, together with detailed numerical illustrations of them, are provided that support our algorithm. A partial realization of the algorithm is described and tested, providing a proof of concept for the approach.

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“…In contrast, we adopt a "solve-then-discretise" approach. The "solve-then-discretise" paradigm has also recently been applied to other spectral problems [64,101,107], extensions of classical methods such as the QL and QR algorithms [30,106], Krylov methods [52], and the computation of resonances [8,9]. We are concerned with the following four types of spectral computations for infinite-dimensional operators:…”

Section: Introductionmentioning

confidence: 99%

Computing spectral properties of topological insulators without artificial truncation or supercell approximation

Colbrook¹,

Horning²,

Thicke³

et al. 2021

Preprint

Self Cite

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Topological insulators (TIs) are renowned for their remarkable electronic properties: quantised bulk Hall and edge conductivities, and robust edge wave-packet propagation, even in the presence of material defects and disorder. Computations of these physical properties generally rely on artificial periodicity (the supercell approximation), or unphysical boundary conditions (artificial truncation). In this work, we build on recently developed methods for computing spectral properties of infinite-dimensional operators. We apply these techniques to develop efficient and accurate computational tools for computing the physical properties of TIs. These tools completely avoid such artificial restrictions and allow one to probe the spectral properties of the infinitedimensional operator directly, even in the presence of material defects and disorder. Our methods permit computation of spectra, approximate eigenstates, spectral measures, spectral projections, transport properties, and conductances. Numerical examples are given for the Haldane model, and the techniques can be extended similarly to other TIs in two and three dimensions.

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FEAST for Differential Eigenvalue Problems

Cited by 20 publications

References 50 publications

On symmetrizing the ultraspherical spectral method for self-adjoint problems

On symmetrizing the ultraspherical spectral method for self-adjoint problems

An algorithm for identifying eigenvectors exhibiting strong spatial localization

Computing spectral properties of topological insulators without artificial truncation or supercell approximation

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