arXiv subjects

Claudio Canuto

Publications and source records attributed to Claudio Canuto.

24 records · Page 2Linked to original sources

arXiv2013

On the decay of the inverse of matrices that are sum of Kronecker products

Decay patterns of matrix inverses have recently attracted considerable interest, due to their relevance in numerical analysis, and in applications requiring matrix function approximations. In this paper we analyze the decay pattern of the inverse of banded matrices in the form $S=M \otimes I_n + I_n \otimes M$ where $M$ is tridiagonal, symmetric and positive definite, $I_n$ is the identity matrix, and $\otimes$ stands for the Kronecker product. It is well known that the inverses of banded matrices exhibit an exponential decay pattern away from the main diagonal. However, the entries in $S^{-1}$ show a non-monotonic decay, which is not caught by classical bounds. By using an alternative expression for $S^{-1}$, we derive computable upper bounds that closely capture the actual behavior of its entries. We also show that similar estimates can be obtained when $M$ has a larger bandwidth, or when the sum of Kronecker products involves two different matrices. Numerical experiments illustrating the new bounds are also reported.

math.NA↗

arXiv2013

Legendre-Gauss-Lobatto grids and associated nested dyadic grids

Legendre-Gauss-Lobatto (LGL) grids play a pivotal role in nodal spectral methods for the numerical solution of partial differential equations. They not only provide efficient high-order quadrature rules, but give also rise to norm equivalences that could eventually lead to efficient preconditioning techniques in high-order methods. Unfortunately, a serious obstruction to fully exploiting the potential of such concepts is the fact that LGL grids of different degree are not nested. This affects, on the one hand, the choice and analysis of suitable auxiliary spaces, when applying the auxiliary space method as a principal preconditioning paradigm, and, on the other hand, the efficient solution of the auxiliary problems. As a central remedy, we consider certain nested hierarchies of dyadic grids of locally comparable mesh size, that are in a certain sense properly associated with the LGL grids. Their actual suitability requires a subtle analysis of such grids which, in turn, relies on a number of refined properties of LGL grids. The central objective of this paper is to derive just these properties. This requires first revisiting properties of close relatives to LGL grids which are subsequently used to develop a refined analysis of LGL grids. These results allow us then to derive the relevant properties of the associated dyadic grids.

math.NA↗

arXiv2013

The derivation of continuum limits of neuronal networks with gap-junction couplings

We consider an idealized network, formed by N neurons individually described by the FitzHugh-Nagumo equations and connected by electrical synapses. The limit for N to infinity of the resulting discrete model is thoroughly investigated, with the aim of identifying a model for a continuum of neurons having an equivalent behaviour. Two strategies for passing to the limit are analysed: i) a more conventional approach, based on a fixed nearest-neighbour connection topology accompanied by a suitable scaling of the diffusion coefficients; ii) a new approach, in which the number of connections to any given neuron varies with N according to a precise law, which simultaneously guarantees the non-triviality of the limit and the locality of neuronal interactions. Both approaches yield in the limit a pde-based model, in which the distribution of action potential obeys a nonlinear reaction-convection-diffusion equation; convection accounts for the possible lack of symmetry in the connection topology. Several convergence issues are discussed, both theoretically and numerically.

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arXiv2012

Robust Preconditioners for DG-Discretizations with Arbitrary Polynomial Degrees

Discontinuous Galerkin (DG) methods offer an enormous flexibility regarding local grid refinement and variation of polynomial degrees for a variety of different problem classes. With a focus on diffusion problems, we consider DG discretizations for elliptic boundary value problems, in particular the efficient solution of the linear systems of equations that arise from the Symmetric Interior Penalty DG method. We announce a multi-stage preconditioner which produces uniformly bounded condition numbers and aims at supporting the full flexibility of DG methods under mild grading conditions. The constructions and proofs are detailed in an upcoming series of papers by the authors. Our preconditioner is based on the concept of the auxiliary space method and techniques from spectral element methods such as Legendre-Gauß-Lobatto grids. The presentation for the case of geometrically conforming meshes is complemented by numerical studies that shed some light on constants arising in four basic estimates used in the second stage.

math.NA↗

arXiv2012

Contraction and optimality properties of adaptive Legendre-Galerkin methods: the 1-dimensional case

As a first step towards a mathematically rigorous understanding of adaptive spectral/$hp$ discretizations of elliptic boundary-value problems, we study the performance of adaptive Legendre-Galerkin methods in one space dimension. These methods offer unlimited approximation power only restricted by solution and data regularity. Our investigation is inspired by a similar study that we recently carried out for Fourier-Galerkin methods in a periodic box. We first consider an "ideal" algorithm, which we prove to be convergent at a fixed rate. Next we enhance its performance, consistently with the expected fast error decay of high-order methods, by activating a larger set of degrees of freedom at each iteration. We guarantee optimality (in the non-linear approximation sense) by incorporating a coarsening step. Optimality is measured in terms of certain sparsity classes of the Gevrey type, which describe a (sub-)exponential decay of the best approximation error.

math.NA↗

arXiv2012

Adaptive Fourier-Galerkin Methods

We study the performance of adaptive Fourier-Galerkin methods in a periodic box in $\mathbb{R}^d$ with dimension $d\ge 1$. These methods offer unlimited approximation power only restricted by solution and data regularity. They are of intrinsic interest but are also a first step towards understanding adaptivity for the $hp$-FEM. We examine two nonlinear approximation classes, one classical corresponding to algebraic decay of Fourier coefficients and another associated with exponential decay. We study the sparsity classes of the residual and show that they are the same as the solution for the algebraic class but not for the exponential one. This possible sparsity degradation for the exponential class can be compensated with coarsening, which we discuss in detail. We present several adaptive Fourier algorithms, and prove their contraction and optimal cardinality properties.

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