## One World Virtual Seminar Series - Stochastic Numerics and Inverse Problems

This is the webpage for the One World Stochastic Numerics and Inverse Problems Seminar Series

The seminars occur weekly on a Wednesday (1pm British Summer Time (BST), 2pm Central European Summer Time (CEST))

### Next Seminar

The next seminar is planned for 8 July 2020 - 1pm (BST)

08 July:Georg Gottwald (The University of Sydney)

Title: Simulation of non-Lipschitz stochastic differential equations driven by α-stable noise: a method based on deterministic homogenisation

Abstract.     The talk introduces an explicit method to integrate $\alpha$-stable stochastic differential equations (SDEs) with non-Lipschitz coefficients. To mitigate against numerical instabilities caused by unbounded increments of the L\'evy noise, we use a deterministic map which has the desired SDE as its homogenised limit. Moreover, our method naturally overcomes difficulties in expressing the Marcus integral explicitly. We present an example of an SDE with a natural boundary showing that our method respects the boundary whereas Euler-Maruyama discretisation fails to do so. As a by-product we devise an entirely deterministic method to construct $\alpha$-stable laws.(Joint work with Ian Melbourne)

JulyTBC (TBC)

Title: TBC

### To Be Rescheduled

Date:TBC-Time TBC

Evelyn Buckwar (Johannes Kepler University)

Title: Structure preserving methods for a finite dimensional stochastic Landau-Lifshitz-Gilbert equation

Abstract: In this talk we discuss a finite dimensional stochastic Landau-Lifshitz-Gilbert equation. The equation describes the precessional motion of the magnetisation of magnetic nanoparticles subject to thermal noise and comes with a number of features. In particular, the modulus if its solution is conserved over time and the expectation of the energy is bounded, these are features that need to be preserved in a numerical approximation. We discuss approaches of designing structure-preserving numerical methods in this setting.

### Previous Seminars

Speaker: Kostas Zygalakis (Edinburgh)

Title: Explicit stabilised Runge-Kutta methods and their application to Bayesian inverse problems

A recording of this seminar is avialable here.

Speaker: Xuerong Mao (Strathclyde)

Title:The Truncated Euler-Maruyama Method for Stochastic Differential Delay Equations

A recording of this seminar is avialable here.

Speaker: Charles-Edouard Bréhier (Claude Bernard Lyon)

Title: Analysis of splitting schemes for the stochastic Allen-Cahn equation

A recording of this seminar is available here.

Speaker: Conall Kelly University College Cork

Title: A hybrid, adaptive numerical method for the Cox-Ingersoll-Ross model

A recording of this seminar is available here

Speaker: Abdul Lateef Haji-Ali(Heriot Watt University)

Title: Sub-sampling and other considerations for efficient risk estimation in large portfolios

A recording of this seminar is available here

Speaker:David Cohen (Umeå University)

Title:Drift-preserving schemes for stochastic Hamiltonian and Poisson systems

This seminar was not recorded.

Title: Numerics and SDE a model for the stochastically forced vorticity equation

A recording of this seminar is avialable here.

Speaker: Marco Iglesias (University of Nottingham)

Title: Ensemble Kalman Inversion: from subsurface environments to composite materials

A recording of this seminar is avialable here.

Speaker: Ray Kawai (University of Tokyo)

Title: Stochastic approximation in adaptive Monte Carlo variance reduction

This talk was NOT recorded.

Kody Law (University of Manchester)

Bayesian Static Parameter Estimation using Multilevel and multi-index Monte Carlo

Akash Sharma & Michael Tretyakov (University of Nottingham)

Title: Computing ergodic limits of reflected diffusions and sampling from distributions with compact support

Abstract.     A simple-to-implement weak-sense numerical method to approximate reflected stochastic differential equations (RSDEs) is proposed and analysed. It is proved that the method has the first order of weak convergence. Together with the Monte Carlo technique, it can be used to numerically solve linear parabolic and elliptic PDEs with Robin boundary condition. One of the key results of this work  is the use of the proposed method for computing ergodic limits, i.e. expectations with respect to the invariant law of RSDEs, both inside a domain and on its boundary. This allows to efficiently sample from distributions with compact support. Both time-averaging and ensemble-averaging estimators are considered and analysed. A number of extensions are considered including a second-order weak approximation, the case of arbitrary oblique direction of reflection, and a new adaptive weak scheme to solve a Poisson PDE with Neumann boundary condition. The presented theoretical results re supported by several numerical experiments. The talk is based on a joint work of Ben Leimkuhler (Edinburgh), Akash Sharma (Nottingham) and M

This is a One World Seminar.

Zoom is the online platform being used to deliver this seminar series. Any questions relating to the seminar, please contact liam.holligan@icms.org.uk

This seminar series is supported as part of the ICMS/INI Online Mathematical Sciences Seminars.