Programme 2023

All seminars will take place online via Zoom and all are welcome

25 October 2023, Wednesday, 15:00-16:00 BST Seminar

Speaker: Prof Anne Kvarno, Norwegian University of Science and Tech., Norway

Title: B-series for non-autonomous semi-linear SDEs

Abstract: In this talk, we will describe how well known results for B-series applied SDEs can be used to derive order conditions for exponential integrators for non-autonomous semilinear problems. This approach seems to make it possible both to simplify and generalize existing results.

11 October 2023, Wednesday, 15:00-16:00 BST Seminar

Speaker: Prof Gabriel J. Lord, Radboud University, Netherlands

Title: A numerical method for an SDE with a WIS integral

Abstract: We consider a one-dimensional SDE with linear multiplicative noise from fractional Brownian motion fBM. The fBM is parametrized by $H\in(0,1)$ the Hurst parameter and we interpret the integral as a WIS integral which allows us to understand the SDE for all values of $H$. This is in contrast to, for example, interpretation through rough paths where $H>1/3$. We will introduce fBM and some of the theory needed to understand the WIS integral such as the Wick product [Mishura]. We will consider the numerical approximation proposed in [Mishura], propose some modifications and indicate how the method may be implemented. We present numerical results and observe for $H<1/2$ faster rates of convergence than predicted from the theory. This is joint work with Roy Schieven and Utku Erdogan. [Mishura] Y. S. Mishura Stochastic calculus for fractional Brownian motion and related processess. Springer. [2008]

7June 2023, Wednesday, 15:00-16:00 BST Seminar

Speaker: Xuerong Mao, University of Strathclyde, UK

Title: Positivity and Boundedness Preserving Numerical Scheme for the Stochastic Epidemic Model

Abstract: This work concerns about the numerical solution to the stochastic epidemic model proposed by Cai et al. in 2019. The typical features of the model including the positivity and boundedness of the solution and the presence of the square-root diffusion term make this an interesting and challenging work. By modifying the classical Euler-Maruyama (EM) scheme, we generate a positivity and boundedness preserving numerical scheme, which is proved to have a strong convergence to the true solution over finite time intervals. We also demonstrate that the principle of this method is applicable to a bunch of popular stochastic differential equation (SDE) models, e.g. the mean-reverting square-root process, an important financial model, and the multi-dimensional SDE SIR epidemic model. This is a joint work with Y. Cai and J. Hu.

10 May 2023, Wednesday, 15:00-16:00 BST Seminar

Speaker: Dr Conall Kelly, University College Cork, Ireland

Title: Two new variations on adaptivity for the numerical solution of nonlinear stochastic systems

Abstract: An RSE Saltire online seminar held on May 11, 2022 explored the use of path-dependent adaptive timestepping with standard explicit and semi-implicit Euler methods to overcome the instability and convergence issues that arise for these methods when applied to some nonlinear systems of SDEs over uniform grids. Specifically we (i) investigated strategies for maintaining dynamical consistency with solutions of nonlinear SDEs where the diffusion coefficient determines local/global existence and uniqueness, (ii) ensured order-1/2 convergence for a semi-explicit Euler-Maruyama method applied to systems of nonlinear SDEs where the drift ensures global existence and uniqueness of solutions, (iii) described an adaptive explicit scheme for the Cox-Ingersoll-Ross (CIR) model, with a square-root diffusion. In this follow-up we look at how our framework for adaptivity can be extended to stochastic systems perturbed by a Poisson jump process with finite intensity. We also investigate a novel splitting method for the CIR model that relies upon adaptive timestepping to ensure well-posedness of the scheme when the intensity of the stochastic perturbation is large. This is joint work with Gabriel Lord (Radboud University, The Netherlands) and Fandi Sun (Heriot-Watt University, Edinburgh, UK).

12 April 2023, Wednesday, 15:00-16:00 BST Seminar

Speaker: Prof Tomas Caraballo, Universidad de Sevilla, Spain

Title: Dynamics of stochastic PDEs: a survey

Abstract: The aim of this talk is to provide a survey on different aspects related to the dynamics of systems modeled by stochastic partial differential equations. We first report on the dynamics of random dynamical systems generated by stochastic PDE with linear multiplicative or additive noise. The main technique is a conjugation transformation which allows us to obtain a random dynamical system generated by the original stochastic problem. However, when the noise is more general than additive or multiplicative, this transformation does not work, but we still have two other alternatives to handle our problem. The first one consists in approximating the noisy term by a Wong-Zakai approximation (also called colored noise). In this way we can consider a random partial differential equation which generates a random dynamical system and the theory of random dynamical systems can be applied. A reason justifying this approach is that the random dynamical system generated by the Wong-Zakai approximation converges to the random dynamical system generated by the stochastic problem when the noise is additive or multiplicative. However, the convergence for general multiplicative noise is still unsolved. The second approach to analyze the case of nonlinear multiplicative noise is to apply the theory of weak mean random attractor which provides interesting information about the dynamics of the problem in an appropriate phase space. Additional results on the existence of invariant measures will complete our talk.

8 March 2023, Wednesday, 15:00-16:00 GMT, Seminar

Speaker: Prof Evelyn Buckwar, Johannes Kepler University, Austria

Title: A stochastic hierarchical model for low grade glioma evolution

Abstract: A stochastic hierarchical model for the evolution of low grade gliomas is proposed. Starting with the description of cell motion using piecewise diffusion Markov processes (PDifMPs) at the cellular level, we derive an equation for the density of the transition probability of this Markov process using the generalised Fokker-Planck equation. Then a macroscopic model is derived via parabolic limit and Hilbert expansions in the moment equations. After setting up the model, we perform several numerical tests to study the role of the local characteristics and the extended generator of the PDifMP in the process of tumour progression. The main aim focuses on understanding how the variations of the jump rate function of this process at the microscopic scale and the diffusion coefficient at the macroscopic scale are related to the diffusive behaviour of the glioma cells. This is joint work with Amira Meddah, JKU, and Martina Conte, Politecnico di Torino

8 February 2023, Wednesday, 15:00-16:00 GMT, Seminar

Speaker: Prof Mireille Bossy, Universite Cote d'Azur, France

Title: Stochastic SDEs for turbulent transport: some modelling and numerical issues

Abstract: In this talk, I would like to introduce some recent work related to the stochastic modelling of particle transport by turbulent flow. Pollutant dispersion, pollens, microplastics, ash, examples of particles transported in a fluid with turbulent fluctuations are numerous, both in the environment and in industrial processes. When the particles are small and light in relation to the turbulence, the numerical engineering solvers all make the simplification that the particles are point-like and spherical. Taking into account a non-spherical shape implies describing a model for the rotational dynamics. I will introduce the approach we have proposed leading to SDEs on angles and orientation, which must then be robustly integrated. I will also show, how the model can be confronted with the observation of turbulence. In particular, I will show some limits of the Brownian diffusion hypothesis in this framework and introduce some models of anomalous diffusions aiming to better capture the richness of the phenomena to be described. These anomalous diffusions lead in turn to the development of suitable numerical integration schemes.

11 January 2023, Wednesday, 15:00-16:00 GMT, Seminar

Speaker: Prof John Appleby, City University of Dublin, Ireland

Title: Mean Square Asymptotic Behaviour of Perturbed Linear Stochastic Functional Differential Equations

Abstract: In recent work, it has proved possible to obtain necessary and sufficient conditions under which linear autonomous stochastic functional differential equations converge in the mean square to a zero equilibrium. These conditions can be expressed directly in terms of the problem data, and are not significantly more difficult to check than the corresponding conditions for deterministic functional differential equations. In the deterministic theory of functional differential equations, it is well-known that the solution of perturbed equations inherit the properties of the state-independent perturbation when the unperturbed system is uniformly asymptotically stable (which is equivalent to the fundamental solution being integrable). However, we show, in both the deterministic and stochastic case, that the situation is better than some of the classical deterministic literature suggests: the stability of the underlying equation can allow the perturbation to lie in a less-well behaved space, and yet the solution of the perturbed equation may be better behaved than the perturbation. Very roughly, our new results for the stochastic equations have the following character. Suppose that the unperturbed linear SFDE is uniformly mean square asymptotically stable. Then the mean square of the perturbed equation is in a nice space of functions V for any choice of initial function if and only if the state-independent perturbations lie in a space W, of which V is a proper subspace. From the perspective of applications, it is interesting that perturbing functions in W can have very bad pointwise behaviour, but in a certain average sense, will be as well behaved as functions in V. The choice of spaces V covers many important spaces for applications: it includes spaces in which functions which tend to zero, are p-integrable, exponentially convergent, bounded, or tend to a non-trivial limit. We can also give a characterisation of mean square stability with respect to perturbations, as well as when equations with persistent perturbations will have weakly stationary solutions. The results appear to hold for a large class of autonomous or asymptotically autonomous equations, including Volterra equations, but the talk will concentrate largely on scalar, finite memory equations. Some details for the finite dimensional case will be sketched. The work is joint with Emmet Lawless (DCU).