• Corpus ID: 245704345

SpecSolve: Spectral methods for spectral measures

@article{Colbrook2022SpecSolveSM,
  title={SpecSolve: Spectral methods for spectral measures},
  author={Matthew J. Colbrook and Andrew D Horning},
  journal={ArXiv},
  year={2022},
  volume={abs/2201.01314}
}
Self-adjoint operators on infinite-dimensional spaces with continuous spectra are abundant but do not possess a basis of eigenfunctions. Rather, diagonalization is achieved through spectral measures. The SpecSolve package [SIAM Rev., 63(3) (2021), pp. 489–524] computes spectral measures of general (self-adjoint) differential and integral operators by combining state-of-the-art adaptive spectral methods with an efficient resolvent-based strategy. The algorithm achieves arbitrarily high orders of… 

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References

SHOWING 1-10 OF 41 REFERENCES

Computing spectral measures of self-adjoint operators

Using the resolvent operator, an algorithm for computing smoothed approximations of spectral measures associated with self-adjoint operators can achieve arbitrarily high-orders of convergence in terms of a smoothing parameter.

A fast and well-conditioned spectral method for singular integral equations

Numerical approximation of the spectrum of self-adjoint operators in operator preconditioning

We consider operator preconditioning B−1A, which is employed in the numerical solution of boundary value problems. Here, the self-adjoint operators A,B : H 0 (Ω) → H−1(Ω) are the standard

A Spectral Transform Method for Singular Sturm-Liouville Problems with Applications to Energy Diffusion in Plasma Physics

We develop a spectrally accurate numerical method to compute solutions of a model PDE used in plasma physics to describe diffusion in velocity space due to Fokker--Planck collisions. The solution is

The ultraspherical spectral element method

Spectral Theory and Differential Operators

This book gives an account of those parts of the analysis of closed linear operators acting in Banach or Hilbert spaces that are relevant to spectral problems involving differential operators, and

Generalized Spectrum of Second Order Differential Operators

The results presented in this paper extend previous analyses which have addressed elliptic differential operators with scalar coefficient functions by showing that this spectrum can be derived from the spectral decomposition of the operator K=Q \Lambda Q^T.

Laplacian Preconditioning of Elliptic PDEs: Localization of the Eigenvalues of the Discretized Operator

The existence of a one-to-one pairing between the eigenvalues of $\bf{L}^{-1}\bf{A}$ and the intervals determined by the images under $k(x)$ of the supports of the FE nodal basis functions is proved.

Approximating Spectral Densities of Large Matrices

The problem of estimating the spectral density carefully is defined and how to measure the accuracy of an approximate spectral density is discussed, which is generally costly and wasteful, especially for matrices of large dimension.

Finite element approximation of eigenvalue problems

  • D. Boffi
  • Mathematics, Computer Science
    Acta Numerica
  • 2010
The final part tries to introduce the reader to the fascinating setting of differential forms and homological techniques with the description of the Hodge–Laplace eigenvalue problem and its mixed equivalent formulations.