Singular-value (and eigenvalue) distribution and Krylov preconditioning of sequences of sampling matrices approximating integral operators

Abstract

Let k({dot operator},{dot operator}) be a continuous kernel defined on Ω×Ω, Ω compact subset of Rd, d ≥1, and let us consider the integral operator K from C(Ω) into C(Ω) (C(Ω) set of continuous functions on Ω) defined as the map f(x) → l(x) =∫Ωk(x,y)f(y)dy,x ∈Ω. K is a compact operator and therefore its spectrum forms a bounded sequence having zero as unique accumulation point. Here, we first consider in detail the approximation of K by using rectangle formula in the case where Ω=[0,1], and the step is h=1/n. The related linear application can be represented as a matrix An of size n. In accordance with the compact character of the continuous operator, we prove that {An}~σ0 and {An}~λ0, that is, the considered sequence has singular values and eigenvalues clustered at zero. Moreover, the cluster is strong in perfect analogy with the compactness of K. Several generalizations are sketched, with special attention to the general case of pure sampling sequences, and few examples and numerical experiments are critically discussed, including the use of GMRES and preconditioned GMRES for large linear systems coming from the numerical approximation of integral equations of the form ((I -K)f(t))(x) = g(x),x ∈Ω, with (Kf(t))(x) =∫Ωk(x,y)f(y)dy and datum g(x)

Block generalized locally Toeplitz sequences: theory and applications in the unidimensional case

“…Block diagonal sampling matrices. If n ∈ N and a : [0, 1] → C s×s , then we define the s-block diagonal sampling matrix D n (a) as the following block diagonal matrix of size sn × sn: If (C s×s ) [0,1] denotes the space of all functions a : [0, 1] → C s×s , then the map…”

“…Local methods are, for example, finite difference methods, finite element methods with "locally supported" basis functions, and collocation methods; in short, all standard numerical methods for the approximation of DEs. Depending on the considered DE and numerical method, the sequence {A n } n might be a scalar GLT sequence (that is, a GLT sequence whose symbol κ is a scalar function) 1 or a block/reduced GLT sequence. In particular, block GLT sequences are encountered in the discretization of vectorial DEs (systems of scalar DEs) as well as in the higher-order finite element or discontinuous Galerkin approximation of scalar DEs.…”

“…We refer the reader to [50,Section 10.5], [51,Section 7.3], and [20,49,75,76] for applications of the theory of GLT sequences in the context of finite difference (FD) discretizations of DEs; to [50,Section 10.6], [51,Section 7.4], and [10,20,42,49,57,67,76] for the finite element (FE) case; to [12] for the finite volume (FV) case; to [50,Section 10.7], [51,, and [36,45,46,47,48,52,57,68] for the case of isogeometric analysis (IgA) discretizations, both in the collocation and Galerkin frameworks; and to [40] for a further application to fractional DEs. We also refer the reader to [50,Section 10.4] and [1,72] for a look at the GLT approach for sequences of matrices arising from IE discretizations.…”

“…Consider one of the above matrices A n , for a large value of n (large, say, with respect to the derivative of a(x)). If, from any entry of A n , we shift downwards by one position along the same diagonal, then the new entry differs from the old one by a quantity which tends to zero as n tends to infinity (the difference is O(1/n) if, for example, a(x) is Lipschitz continuous over [0,1]). Now consider any given diagonal of A n (the main diagonal, for instance).…”

“…Equations (1.9)-(1.10) are usually referred to as the Szegő formulas for block Toeplitz matrices; see [80] for their proof. Now, consider the classical Lagrangian p-degree finite element discretization of the translation-invariant problem (1.3) on a uniform mesh in [0, 1] with stepsize 1 n . If p = 1, then the resulting discretization matrix is again a scalar Toeplitz matrix, namely T n−1 (2 − 2 cos θ).…”