# 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…

## One Citation

### On the Computation of Geometric Features of Spectra of Linear Operators on Hilbert Spaces

- Mathematics, Computer ScienceFoundations of Computational Mathematics
- 2022

The computation of geometric features of spectra in the SCI hierarchy is classified, allowing us to precisely determine the boundaries of what computers can achieve (in any model of computation) and prove that the authors' algorithms are optimal.

## References

SHOWING 1-10 OF 41 REFERENCES

### Computing spectral measures of self-adjoint operators

- Mathematics, Computer ScienceSIAM Rev.
- 2021

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

- MathematicsJ. Comput. Phys.
- 2017

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

- MathematicsNumerical Algorithms
- 2022

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

- MathematicsSIAM J. Appl. Math.
- 2015

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…

### Spectral Theory and Differential Operators

- MathematicsOxford Scholarship Online
- 2018

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

- MathematicsSIAM J. Numer. Anal.
- 2020

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

- MathematicsSIAM J. Numer. Anal.
- 2019

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

- Mathematics, PhysicsSIAM Rev.
- 2016

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

- Mathematics, Computer ScienceActa 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.