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  • 1
    Electronic Resource
    Electronic Resource
    Adams-Type Methods for the Numerical Solution of Stochastic Ordinary Differential Equations (2000)
    Springer
    BIT 40 (2000), S. 451-470 
    Details
    ISSN: 1572-9125
    Keywords: Stochastic ODEs ; strong convergence ; linear multistep formulae ; Adams methods ; predictor-corrector methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The modelling of many real life phenomena for which either the parameter estimation is difficult, or which are subject to random noisy perturbations, is often carried out by using stochastic ordinary differential equations (SODEs). For this reason, in recent years much attention has been devoted to deriving numerical methods for approximating their solution. In particular, in this paper we consider the use of linear multistep formulae (LMF). Strong order convergence conditions up to order 1 are stated, for both commutative and non-commutative problems. The case of additive noise is further investigated, in order to obtain order improvements. The implementation of the methods is also considered, leading to a predictor-corrector approach. Some numerical tests on problems taken from the literature are also included.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1022363612387
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  • 2
    Electronic Resource
    Electronic Resource
    Stochastic Methods for Ill-Posed Problems (2000)
    Springer
    BIT 40 (2000), S. 226-240 
    Details
    ISSN: 1572-9125
    Keywords: Stochastic differential equations ; regularisation ; stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper is devoted to the numerical analysis of ill-posed problems of evolution equations in Banach spaces using certain classes of stochastic one-step methods. The linear stability properties of these methods are studied. Regularisation is given by the choice of the regularisation parameter as α = $$\sqrt {\tau _n }$$ , where τ n is the stepsize and provides the convergence on smooth initial data. The case of the approximation of well-posed problems is also considered.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1022386822865
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  • 3
    Electronic Resource
    Electronic Resource
    Parallel iterated methods based on variable step‐size multistep Runge—Kutta methods of Radau type for stiff problems (2000)
    Springer
    Advances in computational mathematics 13 (2000), S. 257-270 
    Details
    ISSN: 1572-9044
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In our previous paper [3], the performance of a variable step‐size implementation of Parallel Iterated Methods based on Multistep Runge–Kutta methods (PIMRK) is far from satisfactory. This is due to the fact that the underlying parameters of the Multistep Runge–Kutta (MRK) method, and the splitting matrices W that are needed to solve the nonlinear system are designed on a fixed step‐size basis. Similar unsatisfactory results based on this method were also noted by Schneider [12], who showed that the method is only suitable when the step‐size does not vary too often. In this paper, we design the Variable step‐size Multistep Runge–Kutta (VMRK) method as the underlying formula for Parallel Iterated methods. The numerical results show that Parallel Iterated Variable step‐size MRK (PIVMRK) methods improve substantially on the PIMRK methods and are usually competitive with Parallel Iterated Runge–Kutta methods (PIRKs).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1018906311204
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