The rate of convergence of Toeplitz based PCG methods for second order nonlinear boundary value problems

Abstract

In previous works [21][22][23] we proposed the use of τ [5] and band Toeplitz based preconditioners for the solution of 1D and 2D boundary value problems (BVP) by means of the preconditioned conjugate gradient (PCG) methods. As τ and band Toeplitz linear systems can be solved [4] by using fast sine transforms [8], these methods become especially attractive in a parallel environment of computation. In this paper we extend this technique to the nonlinear, nonsymmetric case and, in addition, we prove some clustering properties for the spectra of the preconditioned matrices showing why these methods exhibit a convergence speed which results to be more than linear. Therefore these methods work much finer than those based on separable preconditioners [18,45], on incomplete LU factorizations [36,13,27], and on circulant preconditioners [9,30,35] since the latter two techniques do not assure a linear rate of convergence. On the other hand, the proposed technique has a wider range of application since it can be naturally used for nonlinear, nonsymmetric problems and for BVP in which the coefficients of the differential operator are not strictly positive and only piecewise smooth. Finally the several numerical experiments performed here and in [22,23] confirm the effectiveness of the theoretical analysis.

A Korovkin-type theory for finite Toeplitz operators via matrix algebras

“…In the current literature, the tools of function theory or approximation theory have been used in order to analyze properties in structured linear algebra: some references are the following [15,17,18,28,16,45,46,49,50]. In this paper we have partially overturned the point of view: in Sect.…”

“…Now we are ready to prove the matrix versions of the Korovkin result. In these theorems the test functions are given by 1, sin x and cos x in the 2π periodic case and by 1, cos x and cos 2x in the case where the functions are also even (this situation typically occurs in inherently symmetric problems [29,11,50] in which we use inherently symmetric algebras as the τ class, the Hartley class or some cosine algebras). Moreover, the results are stated for f being real-valued because we are interested in the Hermitian case.…”

A unifying approach to abstract matrix algebra preconditioning

“…By construction it is true that the minimum among the values f j is asymptotical to n −p [16] and therefore the preconditioner τ (H f ) has the desired property in the sense that its condition number is asymptotical to the one of A n (f ). It is interesting to point out that a similar idea has been firstly developed in the context of the preconditioning for elliptic differential problems (see [33] and [37,Theorem 5.3]). The second preconditioner is nothing other than an approximation of the first one where the integrals f j = n 2π I j f (t)dt have been approximated by the trapezoidal rule.…”

“…Finally we want to stress that the case of symmetric Toeplitz matrices generated by trigonometric polynomials with only one zero at x = 0 and with order 2m, m ≥ 1, is not academic because it comes from and has applications in the discretization of elliptic (and semielliptic) differential boundary value problems of order 2m [33,37].…”

Spectral Analysis of Nonsymmetric Quasi-Toeplitz matrices with Applications to Preconditioned Multistep Formulas

“…As stated in [24,22], if a(x) has a zero at the origin of order α the smallest eigenvalue shows an asymptotic behaviour like (∆x) 2 /m max(2,α) .…”