Foundations Of Computational Mathematics Research Articles

The foundations of spectral computations via the Solvability Complexity Index hierarchy

The problem of computing spectra of operators is arguably one of the most investigated areas of computational mathematics. However, the problem of computing spectra of general bounded infinite matrices has only recently been solved. We establish some of the foundations of computational spectral theory through the Solvability Complexity Index (SCI) hierarchy, an approach closely related to Smale’s program on the foundations of computational mathematics and McMullen’s results on polynomial root finding with rational maps. Infinite-dimensional problems yield an intricate infinite classification theory, determining which spectral problems can be solved and with what types of algorithms. We provide answers to many longstanding open questions on the existence of algorithms. For example, we show that spectra can be computed, with error control, from point sampling operator coefficients for large classes of partial differential operators on unbounded domains. Further results include: computing spectra of (possibly unbounded) operators on graphs and separable Hilbert spaces with error control; determining if the spectrum intersects a compact set; the computational spectral gap problem and computing spectral classifications at the bottom of the spectrum; and computing discrete spectra, multiplicities, eigenspaces and determining if the discrete spectrum is non-empty. Moreover, the positive results with error control can be used in computer-assisted proofs. In contrast, the negative results preclude computer-assisted proofs for classes of operators as a whole. Our proofs are constructive, yielding a library of new algorithms and techniques that handle problems that before were out of reach. We demonstrate these algorithms on challenging problems, giving concrete examples of the failure of traditional approaches (e.g., “spectral pollution”) compared to the introduced techniques.

Analysis of a weak Galerkin method for second-order elliptic equations with minimal regularity on polytopal meshes

We propose an error analysis of a weak Galerkin method for second-order elliptic equations with minimal regularity requirements, i.e., the exact solution u belongs to H1+s(Ω) with s>0 possibly close to zero. In this case, the conormal derivative of u is not well-defined on the mesh faces, which results in a legitimacy problem of the error equation originally derived under smooth assumption. We employ the weak meaning of the conormal derivative introduced in [Ern, A., & Guermond, J. L. (2021). Quasi-optimal Nonconforming Approximation of Elliptic PDEs with Contrasted Coefficients andH1+s,s>0, Regularity. Foundations of Computational Mathematics, 1–36.] to establish new error equation and optimal a priori error estimates for low regular solutions on polytopal meshes. We also investigate the corresponding elliptic eigenvalue problems. The numerical experiments verify the theoretical results and agree with the benchmark eigenvalue problem.

Particle relabelling symmetries and Noether's theorem for vertical slice models

We consider the variational formulation for vertical slice models introduced in Cotter and Holm (Proc Roy Soc, 2013). These models have a Kelvin circulation theorem that holds on all materially-transported closed loops, not just those loops on isosurfaces of potential temperature. Potential vorticity conservation can be derived directly from this circulation theorem. In this paper, we show that this property is due to these models having a relabelling symmetry for every single diffeomorphism of the vertical slice that preserves the density, not just those diffeomorphisms that preserve the potential temperature. This is developed using the methodology of Cotter and Holm (Foundations of Computational Mathematics, 2012).

On a Nonlinear Cell-Average Multiresolution Scheme for Image Compression

The aim of this paper is to present a new nonlinear cell-average multiresolution scheme and its application to the compression of color images. The objective is to obtain an algorithm competing with linear multiresolution transforms in smooth regions but exhibiting a better behaviour (compression, Gibbs phenomenon reduction, . . .) in nonsmooth regions. Several desirable features can be associated to this new algorithm: the reconstruction operator is third-order accurate in smooth regions, the stencil used is always centred with optimal support and it is adapted to the presence of discontinuities. Monotony preservation, order of approximation, convergence of the associated subdivision scheme, elimination of Gibbs effects and stability are analysed. This paper can be considered as the second part of the paper by Amat, Donat, Liandrat and Trillo [ Foundations of Computational Mathematics, 6 (2), 193-225, (2006)] where the point-value framework was considered.