Efficient implementation of Radau collocation methods

In this paper we define an efficient implementation of Runge-Kutta methods of Radau IIA type, which are commonly used when solving stiff ODE-IVPs problems. The proposed implementation relies on an alternative low-rank formulation of the methods, for which a splitting procedure is easily defined. The linear convergence analysis of this splitting procedure exhibits excellent properties, which are confirmed by its performance on a few numerical tests.Comment: 19 pages, 3 figures, 9 table

Spectral solution of ODE-IVPs by using SHBVMs

Recently, Hamiltonian Boundary Value Methods (HBVMs), have been used as spectral methods in time for effectively solving multi-frequency, highly-oscillatory and/or stiffly-oscillatory problems. A complete analysis of their use in such a fashion has been also carried out, providing a theoretical framework explaining their effectiveness. We report here a few numerical examples showing their potentialities to provide a fully accurate solver for general ODE problems

PTPN22 R620W polymorphism in the ANCA-associated vasculitides

Objectives. PTPN22 is involved in T-cell activation and its R620W single-nucleotide polymorphism ( SNP) has been shown to predispose to different autoimmune diseases. The aims of this study were to investigate the role of the PTPN22 R620W SNP in conferring susceptibility to the ANCA-associated vasculitides (AAVs), and to explore potential associations between the PTPN22 genotype and the disease manifestations. Methods. PTPN22 R620W SNP was genotyped in a cohort of 344 AAV patients [143 with granulomatosis with polyangiitis (Wegener's) (GPA), 102 with microscopic polyangiitis (MPA) and 99 with Churg-Strauss syndrome (CSS)] and in 945 healthy controls. Results. The frequency of the minor allele (620W) was significantly higher in GPA patients than in controls [P = 0.005, chi(2) = 7.858, odds ratio (OR) = 1.91], while no statistically significant association was found with MPA or CSS. Among GPA patients, the 620W allele was particularly enriched in ANCA-positive patients as compared with controls (P = 0.00012, chi(2) = 14.73, OR = 2.31); a particularly marked association was also found with ENT involvement (P = 0.0071, chi(2) = 7.258, OR = 1.98), lung involvement (P = 0.0060, chi(2) = 7.541, OR = 2.07) and skin manifestations of all kinds (P = 0.000047, chi(2) = 16.567, OR = 3.73). Conclusion. The PTPN22 620W allele confers susceptibility to the development of GPA (but not of MPA or CSS), and particularly of its ANCA-positive subset

Blended General Linear Methods based on Generalized BDF

General Linear Methods were introduced in order to encompass a large family of numerical methods for the solution of ODE-IVPs, ranging from LMF to RK formulae. In so doing, it is possible to obtain methods able to overcome typical drawbacks of the previous classes of methods. For example, stability limitations of LMF and order reduction for RK methods. Nevertheless, these goals are usually achieved at the price of a higher computational cost. Consequently, many efforts have been done in order to derive GLMs with particular features, to be exploited for their efficient implementation. In recent years, the derivation of GLMs from particular Boundary Value Methods (BVMs), namely the family of Generalized BDF (GBDF), has been proposed for the numerical solution of stiff ODE-IVPs. Here, this approach is further developed in order to derive GLMs combining good stability and accuracy properties with the possibility of efficiently solving the generated discrete problems via the blended implementation of the methods