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Singular‐value (and eigenvalue) distribution and Krylov preconditioning of sequences of sampling matrices approximating integral operators
Author(s) -
AlFhaid A.S.,
SerraCapizzano S.,
Sesana D.,
Zaka Ullah M.
Publication year - 2014
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.1922
Subject(s) - mathematics , generalized minimal residual method , eigenvalues and eigenvectors , sequence (biology) , operator (biology) , bounded function , singular integral , compact space , matrix (chemical analysis) , integral equation , singular value , kernel (algebra) , rectangle , mathematical analysis , linear system , discrete mathematics , biochemistry , chemistry , physics , materials science , repressor , quantum mechanics , biology , gene , transcription factor , composite material , genetics , geometry
SUMMARY Let k ( ⋅ , ⋅ ) be a continuous kernel defined on Ω × Ω, Ω compact subset ofR d , d ⩾ 1 , and let us consider the integral operatorK ̃ from C ( Ω ) into C ( Ω ) ( C ( Ω ) set of continuous functions on Ω) defined as the mapf ( x ) → l ( x ) = ∫ Ω k ( x , y ) f ( y ) d y , 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 ofK ̃ 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 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, that is, the considered sequence has singular values and eigenvalues clustered at zero. Moreover, the cluster is strong in perfect analogy with the compactness ofK ̃ . 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 (K ̃ f ( t ) ) ( x ) = ∫ Ω k ( x , y ) f ( y ) d y and datum g ( x ). Copyright © 2014 John Wiley & Sons, Ltd.