The Generalized Singular Value Decomposition and the Method of Particular Solutions

Betcke, Timo

doi:10.1137/060651057

Cited by 38 publications

(43 citation statements)

References 21 publications

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“…Similar curves appear in [5,7,11]. Equation (1.4) states that the eigenvalue accuracy which may be claimed is controlled by the size of the minimum found.…”

Section: Introduction and Main Resultssupporting

confidence: 51%

“…This problem, and a very similar but independently found cure, is presented in [5]. Recently, Betcke [7] has united the various approaches within a single framework.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…We also find (by extrapolating in ε reg ) that the approximateλ 1 overestimates the true value. Note that Betcke's recent GSVD method [7] includes a regularization similar to the above and can give higher accuracy; we discuss this in comment 2 of section 5. Finally, since only the lowest (or lowest few) eigenvalues are needed, one might expect that an iterative method could improve upon the above O(N 3 ) effort.…”

Section: Computation Of Minimum Boundary Error T M (E)mentioning

confidence: 99%

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Perturbative Analysis of the Method of Particular Solutions for Improved Inclusion of High-Lying Dirichlet Eigenvalues

Barnett¹

2009

SIAM J. Numer. Anal.

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Abstract. The Dirichlet eigenvalue or "drum" problem in a domain Ω ⊂ R 2 becomes numerically challenging at high eigenvalue (frequency) E. In this regime the method of particular solutions (MPS) gives spectral accuracy for many domain shapes. It requires a number of degrees of freedom scaling as √ E, the number of wavelengths on the boundary, in contrast to direct discretization for which this scaling is E. Our main result is an inclusion bound on eigenvalues that is a factor O( √ E) tighter than the classical bound of Moler-Payne and that is optimal in that it reflects the true slopes of curves appearing in the MPS. We also present an MPS variant that cures a normalization problem in the original method, while evaluating basis functions only on the boundary. This method is efficient at high frequencies, where we show that, in practice, our inclusion bound can give three extra digits of eigenvalue accuracy with no extra effort.

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“…Similar curves appear in [5,7,11]. Equation (1.4) states that the eigenvalue accuracy which may be claimed is controlled by the size of the minimum found.…”

Section: Introduction and Main Resultssupporting

confidence: 51%

“…This problem, and a very similar but independently found cure, is presented in [5]. Recently, Betcke [7] has united the various approaches within a single framework.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Computation Of Minimum Boundary Error T M (E)mentioning

confidence: 99%

See 1 more Smart Citation

Perturbative Analysis of the Method of Particular Solutions for Improved Inclusion of High-Lying Dirichlet Eigenvalues

Barnett¹

2009

SIAM J. Numer. Anal.

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“…13,17,35 These different approaches are discussed in Ref. 36. Here, we use the generalized singular value implementation from Ref.…”

Section: A the Methods Of Particular Solutionsmentioning

confidence: 99%

“…Here, we use the generalized singular value implementation from Ref. 36, which is highly accurate and numerically stable. We note that these methods are related to, but improve upon, the plane wave method of Heller.…”

Section: A the Methods Of Particular Solutionsmentioning

confidence: 99%

Quantum mushroom billiards

Barnett

Betcke

2007

Chaos

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We report the first large-scale statistical study of very high-lying eigenmodes ͑quantum states͒ of the mushroom billiard proposed by L. A. Bunimovich ͓Chaos 11, 802 ͑2001͔͒. The phase space of this mixed system is unusual in that it has a single regular region and a single chaotic region, and no KAM hierarchy. We verify Percival's conjecture to high accuracy ͑1.7%͒. We propose a model for dynamical tunneling and show that it predicts well the chaotic components of predominantly regular modes. Our model explains our observed density of such superpositions dying as E −1/3 ͑E is the eigenvalue͒. We compare eigenvalue spacing distributions against Random Matrix Theory expectations, using 16 000 odd modes ͑an order of magnitude more than any existing study͒. We outline new variants of mesh-free boundary collocation methods which enable us to achieve high accuracy and high mode numbers ͑ϳ10 5 ͒ orders of magnitude faster than with competing methods. © 2007 American Institute of Physics. ͓DOI: 10.1063/1.2816946͔ Quantum chaos is the study of the quantum (wave) properties of Hamiltonian systems whose classical (ray) dynamics is chaotic. Billiards are some of the simplest and most studied examples; physically their wave analogs are vibrating membranes, quantum, electromagnetic, or acoustic cavities. They continue to provide a wealth of theoretical challenges. In particular "mixed" systems, where ray phase space has both regular and chaotic regions (the generic case), are difficult to analyze. Six years ago Bunimovich described 1 a mushroom billiard with simple mixed dynamics free of the usual island hierarchies of Kolmogorov-Arnold-Moser (KAM). He concluded by anticipating the growth of "quantum mushrooms;" it is this gardening task that we achieve here, by developing advanced numerical methods to collect an unprecedented large number n of eigenmodes (much higher than competing numerics 2 or microwave studies 3 ). Since uncertainties scale as n −1Õ2 , a large n is vital for accurate spectral statistics and for studying the semiclassical (high eigenvalue) limit. We address three main issues: (i) The conjecture of Percival 4 that semiclassically modes live exclusively in invariant (regular or chaotic) regions, and occur in proportion to the phase space volumes. (ii) The mechanism for dynamical tunneling, or quantum coupling between classically isolated phase space regions. (iii) The distribution of spacings of nearest-neighbor eigenvalues, about which recent questions have been raised. 3 We show many pictures of modes, including the boundary phase space (the so-called Husimi function).

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A Novel Image Fusion Approach Combined Singular Value Decomposition with Averaging Operation

Luo

2012

Lecture Notes in Electrical Engineering

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The Generalized Singular Value Decomposition and the Method of Particular Solutions

Cited by 38 publications

References 21 publications

Perturbative Analysis of the Method of Particular Solutions for Improved Inclusion of High-Lying Dirichlet Eigenvalues

Perturbative Analysis of the Method of Particular Solutions for Improved Inclusion of High-Lying Dirichlet Eigenvalues

Quantum mushroom billiards

A Novel Image Fusion Approach Combined Singular Value Decomposition with Averaging Operation

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