Abstract
The local error of single step methods is modelled as a function of the state derivative multiplied by bias and zero-mean white noise terms. The deterministic Taylor series expansion of the local error depends on the state derivative meaning that the local error magnitude is zero in steady state and grows with the rate of change of the state vector. The stochastic model of the local error may include a constant, “catch-all” noise term. A continuous time extension of the local error model is developed and this allows the original continuous time state differential equation to be represented by a combination of the simulation method and a stochastic term. This continuous time stochastic differential equation model can be used to study the propagation of the simulation error in Monte Carlo experiments, for step size control, or for propagating the mean and variance. This simulation error model can be embedded into continuous-discrete state estimation algorithms. Two illustrative examples are included to highlight the application of the approach.
Highlights
In the solution of ordinary differential equations using numerical methods the problem of local and global errors arises because of finite order methods and finite step size
The ideas in [1] are developed in [3] to establish stronger convergence results for probabilistic integrators, and as their results apply to general additive noise models, they apply to the the state dependent noise with bias model used to model the local error in this paper
This section develops the continuous time extension of the results above by representing the continuous time system via a stochastic differential equation. [1] modelled the local error as a stochastic term added to the differential equation and calibrated this as a zero mean, iid Gaussian sequence, with a covariance scaled by h2p+1
Summary
In the solution of ordinary differential equations using numerical methods the problem of local and global errors arises because of finite order methods and finite step size. (Equation (4) can be represented using a single, equivalent term, ξr,i ∼ N(mr, Qr).) For LTI systems with pth order method and a fixed step size, the local error multiplier, mr(h) is constant (depending on h), as the local error is, ei = hp+1 M(h) Axi with M(h) =. Use a higher order method or sub-steps (possibly with Richardson extrapolation) to get an accurate estimate of the state at the time, xi+1,[k] ≈ x[k](ti + h), and estimate the local error, ei,[k] = xi+1,[k] − xψi+1,[k] This step is only performed with high precision during calibration but typically there would be some form of (cheap) step size control to limit the worst-case local error during normal operation. The choice of metric, s (e.g. one- or two-norm) for the fit between γr2,i and θ2r,i will affect the result of the optimisation
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