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

Abstract

SUMMARYLet k(·,·) be a continuous kernel defined on Ω × Ω, Ω bounded subset of R d , d≥ 1, and let us consider theintegral operator K˜ from C(Ω) into C(Ω) (C(Ω) set of continuous functions on Ω) defined as the mapf(x) → l(x) =Z Ω k(x,y)f(y)dy, x∈ Ω.K˜ is a compact operator and therefore its spectrum forms a bounded sequence having zero as uniqueaccumulation point. Here we first consider in detail the appr oximation of K˜ by using rectangle formulain the case where Ω = [0,1] and the step is h= 1/n. The related linear application can be represented asa matrix A n of size n. In accordance with the compact character of the continuous operator, we prove that{A n } ∼ σ 0 and {A n } ∼ λ 0, i.e., the considered sequence has singular values and eigenvalues clustered atzero. Moreover the cluster is strong in perfect analogy with the compactness of K˜. Several generalizationsare sketched, with special attention to the general case of pure sampling sequences, and few examples andnumerical experiments are critically discussed, including the use of GMRES and preconditioned GMRESfor large linear systems coming from the numerical approximation of integral equations of the form((I− K˜)f(t))(x) = g(x), x∈ Ω, (1)with (Kf˜ (t))(x) =R