Randomised numerical schemes

This talk is given by Dr Yue Wu (University of Strathclyde)

In this talk, I will introduce randomised numerical schemes for a wide range of differential equations with time-irregular coefficients. The initial scheme considered is a randomised Runge-Kunta for Caratheodory ODEs.  Our motivation for studying Caratheodory type initial value problems stems from the fact that certain rough differential equations that are driven by an additive noise can be transformed into a problem of this form. it is well-known that there is lack of convergence for deterministic algorithms in this case, and our proposed method in [1], by incorporating stratified Monte-Carlo simulation into the corresponding deterministic ones, showed that even with very mild conditions, the order of convergence can be at least half with respect to the Lp norm.

The idea was later extended to stochastic settings. We proposed a drift-randomized Milstein method to achieve a higher order approximation to non-autonomous SDEs where the standard smoothness and growth requirements of standard Milstein-type methods are not fulfilled. We also investigated the numerical solution of non-autonomous semilinear stochastic evolution equations (SEEs) driven by an additive Wiener noise. Usually quite restrictive smoothness requirements are imposed in order to achieve a high order of convergence rate.  To relax such conditions, we proposed a novel numerical method in [3] for the approximation of the solution to the semilinear SEE that combines the drift-randomization technique from [2] with a Galerkin finite element method. It turns out that the resulting method converges with a higher rate with respect to the temporal discretization parameter without requiring any differentiability of the nonlinearity. Our approach also relaxes the smoothness requirements of the coefficients with respect to the time variable considerably.

Recently, we pushed this idea to Caratheodory delay ODEs (CDDEs), where a randomised Euler scheme is proposed to approximate the exact solution [4]. It is worth mentioning that, mainly due to CDDEs being considered interval-by-interval, we developed a suitable proof technique that is based on mathematical induction. The randomised technique from [2] is only applicable for the initial inductive step; as the systems iterate over time, a different strategy is required to handle the effect of the delay variable.

[1] Kruse, R. and Wu, Y., 2017. Error analysis of randomized Runge–Kutta methods for differential equations with time-irregular coefficients. Computational Methods in Applied Mathematics, 17(3), pp.479-498.

[2] Kruse, R. and Wu, Y., 2019. A randomized Milstein method for stochastic differential equations with non-differentiable drift coefficients. Discrete and Continuous Dynamical Systems - Series B, 24(8), pp.3475-3502.

[3] Kruse, R. and Wu, Y., 2019. A randomized and fully discrete Galerkin finite element method for semilinear stochastic evolution equations. Mathematics of Computation, 88(320), pp.2793-2825.

[4] Difonzo, F.V., PrzybyƂowicz, P. and Wu, Y., 2022. Existence, uniqueness and approximation of solutions to Carath\'eodory delay differential equations. arXiv preprint arXiv:2204.02016.