Weyl's Law: Spectral Properties of the Laplacian in Mathematics and Physics

Arendt, Wolfgang; Nittka, Robin; Peter, Wolfgang; Steiner, Frank

doi:10.1002/9783527628025.ch1

Cited by 87 publications

(99 citation statements)

References 124 publications

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“…3. A nice discussion of these issues can be found, for example, in [12]. The consequence of these facts for the effective action S Λ is even more drastic.…”

Section: Sharp Cutoffmentioning

confidence: 99%

Multi-loop zeta function regularization and spectral cutoff in curved spacetime

Bilal

Ferrari

2013

Nuclear Physics B

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We emphasize the close relationship between zeta function methods and arbitrary spectral cutoff regularizations in curved spacetime. This yields, on the one hand, a physically sound and mathematically rigorous justification of the standard zeta function regularization at one loop and, on the other hand, a natural generalization of this method to higher loops. In particular, to any Feynman diagram is associated a generalized meromorphic zeta function. For the one-loop vacuum diagram, it is directly related to the usual spectral zeta function. To any loop order, the renormalized amplitudes can be read off from the pole structure of the generalized zeta functions. We focus on scalar field theories and illustrate the general formalism by explicit calculations at one-loop and two-loop orders, including a two-loop evaluation of the conformal anomaly.

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“…3. A nice discussion of these issues can be found, for example, in [12]. The consequence of these facts for the effective action S Λ is even more drastic.…”

Section: Sharp Cutoffmentioning

confidence: 99%

Multi-loop zeta function regularization and spectral cutoff in curved spacetime

Bilal

Ferrari

2013

Nuclear Physics B

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“…[66]) where he described the asymptotic behaviour 1 of the eigenvalues of the Laplace eigenvalue problem on bounded Lipschitz domains. From then on this field has been investigated by many (see, for example, [1], [4], [25, Chapter VI, Section 4], [58]). The main result reads as follows.…”

Section: Remark B20mentioning

confidence: 99%

“…Finally, the 4 We define the relative residual error of an eigenpair approximation (λ, x) of problem (K, M ) by Kx − λM x 2/ Kx 2. 5 K is a block diagonal matrix and M has a block-sparsity structure similar to the structure of M H (cf.…”

Section: Task (T8+t9+tsi): Transformation Of the Eigensolutionsmentioning

confidence: 99%

“…Furthermore, let (G H ) −1 ∈ H(T I×I , ε) denote the H-matrix approximation of G −1 which is computed by the recursive approach outlined above using an adaptive rank with accuracy ε. Then this approximated inverse is computed in O(N log α N ), see [33], and the error 4 …”

Section: Algorithm 2 Construction Of the Block Cluster Treementioning

confidence: 99%

“…For small-sized interface problems the dense LAPACK solver has been used as well, for large-sized interface problems the iterative eigensolver H-SIL introduced in Section 8.4 has been used. 4. The condensation process, which is performed after task (T4) in Algorithm 5, is applied in such a way that subdomain eigenvalue problems are condensed every 3 levels and only if condition (8.15) has been fulfilled (cf.…”

mentioning

confidence: 99%

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Solving an elliptic PDE eigenvalue problem via automated multi-level substructuring and hierarchical matrices

Gerds

Grasedyck

2013

Comput. Visual Sci.

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To solve an elliptic PDE eigenvalue problem in practice, typically the finite element discretisation is used. From approximation theory it is known that only the smaller eigenvalues and their corresponding eigenfunctions can be well approximated by the finite element discretisation because the approximation error increases with increasing size of the eigenvalue. The number of well approximable eigenvalues or eigenfunctions, however, is unknown. In this work asymptotic estimates of these quantities are derived. For example, it is shown that for three-dimensional problems under certain smoothness assumptions on the data only the smallest Θ(N 2/5 ) eigenvalues and only the eigenfunctions associated to the smallest Θ(N 1/4 ) eigenvalues can be well approximated by the finite element discretisation, when the N -dimensional finite element spaces of piecewise affine functions with uniform mesh refinement are used.To solve the discretised elliptic PDE eigenvalue problem and to compute all well approximable eigenvalues and eigenfunctions, a new method is introduced which combines a recursive version of the automated multi-level substructuring (short AMLS) method with the concept of hierarchical matrices (short H-matrices). AMLS is a domain decomposition technique for the solution of elliptic PDE eigenvalue problems where, after some transformation, a reduced eigenvalue problem is derived whose eigensolutions deliver approximations of the sought eigensolutions of the original problem.Whereas the classical AMLS method is very efficient for elliptic PDE eigenvalue problems posed in two dimensions, it is getting very expensive for three-dimensional problems, due to the fact that it computes the reduced eigenvalue problem via dense matrix operations. This problem is resolved by the use of hierarchical matrices. H-matrices are a data-sparse approximation of dense matrices which, e.g., result from the inversion of the stiffness matrix that is associated to the finite element discretisation of an elliptic PDE operator. The big advantage of H-matrices is that they provide matrix arithmetic with almost linear complexity. This fast H-matrix arithmetic is used for the computation of the reduced eigenvalue problem. Beside this, the size of the reduced eigenvalue problem is bounded by a new recursive version of AMLS which further reduces the costs for the computation and the solution of this problem. Altogether this leads to a new method which is well-suited for three-dimensional problems and which allows us to compute a large amount of eigenpair approximations in optimal complexity.

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Extension of Planck's law of thermal radiation to systems with a steady heat flux

Budaev

Bogy

2011

Ann. Phys.

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Planck's law of thermal radiation is limited to equilibrium systems that have a definite temperature and do not carry any heat flux. Here we extend it to steady‐state systems with a constant heat flux. The obtained formulas explicitly describe the spectrum of thermal radiation in every direction and provide a sound basis for the self‐consistent analysis of radiative heat transport across interfaces, gaps, layered and other important structures.

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Weyl's Law: Spectral Properties of the Laplacian in Mathematics and Physics

Cited by 87 publications

References 124 publications

Multi-loop zeta function regularization and spectral cutoff in curved spacetime

Multi-loop zeta function regularization and spectral cutoff in curved spacetime

Solving an elliptic PDE eigenvalue problem via automated multi-level substructuring and hierarchical matrices

Extension of Planck's law of thermal radiation to systems with a steady heat flux

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