A quasi-randomized Runge-Kutta method

Coulibaly, Ibrahim; Lécot, Christian

doi:10.1090/s0025-5718-99-01056-x

Cited by 15 publications

(11 citation statements)

References 10 publications

(8 reference statements)

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“…Let us stress, that the mapping f is not necessarily continuous with respect to the temporal variable t. In addition, the mappings L and K are not assumed to be bounded, in contrast to other results found in the literature [6,20,30,31]. Moreover, from (17) and (18) we directly deduce the linear growth condition |f (t, x)| ≤ K(t)(1 + |x|) (19) for almost all t ∈ [0, T ] and In the following proposition we collect a few properties of the solution u to (1).…”

Section: Numerical Approximation Of Carathéodory Odesmentioning

confidence: 97%

Error Analysis of Randomized Runge–Kutta Methods for Differential Equations with Time-Irregular Coefficients

Kruse

2017

Computational Methods in Applied Mathematics

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Abstract. This paper contains an error analysis of two randomized explicit Runge-Kutta schemes for ordinary differential equations (ODEs) with timeirregular coefficient functions. In particular, the methods are applicable to ODEs of Carathéodory type, whose coefficient functions are only integrable with respect to the time variable but are not assumed to be continuous. A further field of application are ODEs with coefficient functions that contain weak singularities with respect to the time variable.The main result consists of precise bounds for the discretization error with respect to the L p (Ω; R d )-norm. In addition, convergence rates are also derived in the almost sure sense. An important ingredient in the analysis are corresponding error bounds for the randomized Riemann sum quadrature rule. The theoretical results are illustrated through a few numerical experiments.

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Section: Numerical Approximation Of Carathéodory Odesmentioning

confidence: 97%

Error Analysis of Randomized Runge–Kutta Methods for Differential Equations with Time-Irregular Coefficients

Kruse

2017

Computational Methods in Applied Mathematics

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“…However, the statistical implications of the reliance on a numerical approximation to the actual solution of the differential equation have not been addressed in the statistics literature to date and this is the open problem comprehensively addressed in this paper. Earlier work in the literature including randomisation in the approximate integration of ordinary differential equations (ODEs) includes (Coulibaly and Lécot 1999;Stengle 1995). Our strategy fits within the emerging field known as Probabilistic Numerics (Hennig et al 2015), a perspective on computational methods pioneered by Diaconis (1988), and subsequently (Skilling 1992).…”

Section: Review Of Existing Workmentioning

confidence: 99%

Statistical analysis of differential equations: introducing probability measures on numerical solutions

et al. 2016

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In this paper, we present a formal quantification of uncertainty induced by numerical solutions of ordinary and partial differential equation models. Numerical solutions of differential equations contain inherent uncertainties due to the finite-dimensional approximation of an unknown and implicitly defined function. When statistically analysing models based on differential equations describing physical, or other naturally occurring, phenomena, it can be important to explicitly account for the uncertainty introduced by the numerical method. Doing so enables objective determination of this source of uncertainty, relative to other uncertainties, such as those caused by data contaminated with noise or model error induced by missing physical or inadequate descriptors. As ever larger scale mathematical models are being used in the sciences, often sacrificing complete resolution of the differential equation on the grids used, formally accounting for the uncertainty in the numerical method is becoming increasingly more important. This paper provides the formal means to incorporate this uncer- School of Mathematics, University of Edinburgh, Edinburgh, Scotland tainty in a statistical model and its subsequent analysis. We show that a wide variety of existing solvers can be randomised, inducing a probability measure over the solutions of such differential equations. These measures exhibit contraction to a Dirac measure around the true unknown solution, where the rates of convergence are consistent with the underlying deterministic numerical method. Furthermore, we employ the method of modified equations to demonstrate enhanced rates of convergence to stochastic perturbations of the original deterministic problem. Ordinary differential equations and elliptic partial differential equations are used to illustrate the approach to quantify uncertainty in both the statistical analysis of the forward and inverse problems.

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“…Here N denotes again the computational cost of the scheme. Such approximation schemes for CDEs were studied in [4,11]. See also [7] for an analysis in the case of delay differential equations.…”

Section: Other Approximation Schemes For Cdesmentioning

confidence: 99%