2022/01/04 by Matthew J. Colbrook, Colbrook, Matthew J., Andrew Horning +1
Computer Science · Mathematics · #Differential Equations and Numerical Methods #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.2201.01314
openalex publication_date 2022/01/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 convergence in terms of a smoothing parameter, allowing computation of both discrete and continuous spectral components. This article extends SpecSolve to two important classes of operators: singular integro-differential operators and general operator pencils. Essential computational steps are performed with off-the-shelf spectral methods, including spectral methods on the real line, the ultraspherical spectral method, Chebyshev and Fourier spectral methods, and the (hp-adaptive and sparse) ultraspherical spectral element method. This collection illustrates the power and flexibility of SpecSolve's "discretization-oblivious" paradigm.