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

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).