# LMS-Birmingham Workshop on Stochastics/Partial Differential Equations, June 2022

- Location
- Watson Building - Lecture Theatre B

- Dates
- Monday 27 June (13:00) - Tuesday 28 June 2022 (17:00)

- Contact

The aim of this workshop is to discuss recent developments in the fields of stochastic and partial differential equations from both analytical and computational aspects.

### Monday, 27th June

13:00-13:40, Ben Leimkuhler (Edinburgh): Numerical Algorithms for SDEs on Bounded Domains

13:40-14:20, Yue Wu (Strathclyde): The backward Euler-Maruyama method for invariant measures of stochastic differential equations with super-linear coefficients

14:20-14:40, Coffee/Tea Break

14:40-15:20, Tony Shardlow (Bath): Dean Kawasaki models with inertia

15:20-16:00, Chunrong Feng (Durham): Random periodic processes and periodic measures

16:00-16:20: Coffee/Tea Break

16:20-17:00, Xiaocheng Shang (Birmingham): Accurate and Efficient Splitting Methods for Dissipative Particle Dynamics

17:00-17:30, Early Career Researchers Short Talks

17:30-18:00, Wine Reception

### Tuesday, 28th June

09:00-09:40, Michela Ottobre (Heriot-Watt): Non mean-field Vicsek type models for collective behaviour

09:40-10:20, Johannes Zimmer (TU Munich): From particles to macroscopic models: the role of fluctuations

10:20-10:50, Coffee/Tea Break

10:50-11:30, Urbain Vaes (Inria Paris): Mobility for Langevin-like dynamics in a periodic potential: Scaling limits and efficient numerical estimation

11:30-12:10, Hong Duong (Birmingham): Coarse-graining of SDEs and PDEs

12:10-12:40, Early Career Researcher Short Talks

12:40-13:50: Lunch Break

13:50-14:30, Jiahua Jiang (Birmingham):Offline-Enhanced Reduced Basis Method through adaptive construction of the Surrogate Parameter Domain

14:30-15:10, Eyal Neuman (Imperial College): Scaling Properties of a Moving Polymer

15:10-15:40, Coffee/Tea Break

15:40-16:20, Mohammud Foondun (Strathclyde): Small Ball Probabilities for solutions to stochastic partial differential equations

16:20-17:00, Jingyu Huang (Birmingham): Stochastic heat equation with super-linear drift and multiplicative noise on ℝ^{d}.

**The lunches and breaks will be in Maths Learning Centre close to the Lecture Theatre B.**

### For Early Career Researchers (ECRs)

We have a section for talks from ECRs scheduled on Monday 27 June, 17:00-17:30. We are very keen to support the participation of as many ECRs as possible. Please could you let us know, by sending an email to Dr X. Shang stating whether you would be willing to give a short talk in that session and also if you need financial support to attend the workshop.

### Titles and abstracts

### Ben Leimkuhler (Edinburgh)

Title: Numerical Algorithms for SDEs on Bounded Domains

Abstract: I will describe recent progress in devising weakly accurate numerical schemes for SDEs, including algorithms for SDEs on compact subdomains of Euclidean space which are subject to (hard) reflections at domain boundaries [1]. Applications of these types of methods include models for physical phenomena as well as Bayesian sampling (e.g. in econometrics).

[1] Simplest random walk for approximating Robin boundary value problems and ergodic limits of reflected diffusions, B. Leimkuhler, A. Sharma, M.V. Tretyakov. Preprint, 2022. https://arxiv.org/abs/2006.15670

### Yue Wu (Strathclyde)

Title: The backward Euler-Maruyama method for invariant measures of stochastic differential equations with super-linear coefficients

Abstract: The backward Euler-Maruyama (BEM) method is employed to approximate the invariant measure of stochastic differential equations, where both the drift and the diffusion coefficient are allowed to grow super-linearly. I will discuss the existence and uniqueness of the invariant measure of the numerical solution generated by the BEM method and show the convergence of the numerical invariant measure to the underlying one. Simulations are provided to illustrate the theoretical results and demonstrate the application of our results in the area of system control. This is joint work with Wei Liu (Shanghai Normal University) and Xuerong Mao (Strathclyde).

### Tony Shardlow (Bath)

Title: Dean Kawasaki models with inertia

Abstract: The Dean Kawasaki model is a well known stochastic PDE describing the empirical density of a large number of interacting diffusions. We consider an inertial and regularised version of the Dean Kawasaki model and discuss its well posedness and numerical approximation, considering especially the non-negativity of solutions.

### Jiahua Jiang (Birmingham)

Title: Offline-Enhanced Reduced Basis Method through adaptive construction of the Surrogate Parameter Domain

Abstract: The Reduced Basis Method (RBM) is a popular certified model reduction approach for solving parametrized partial differential equations. One critical stage of the offline portion of the algorithm is a greedy algorithm, requiring maximization of an error estimate over parameter space. In practice this maximization is usually performed by replacing the parameter domain continuum with a discrete 'training' set. When the dimension of parameter space is large, it is necessary to significantly increase the size of this training set in order to effectively search parameter space. Large training sets diminish the attractiveness of RBM algorithms since this proportionally increases the cost of the offline phase.

In this work we propose novel strategies for offline RBM algorithms that mitigate the computational difficulty of maximizing error estimates over a training set. The main idea is to identify a subset of the training set, a 'surrogate parameter domain' (SPD), on which to perform greedy algorithms. The SPD’s we construct are much smaller in size than the full training set, yet our examples suggest that they are accurate enough to represent the solution manifold of interest at the current offline RBM iteration. We propose two algorithms to construct the SPD: Our first algorithm, the Successive Maximization Method (SMM) method, is inspired by inverse transform sampling for non-standard univariate probability distributions. The second constructs an SPD by identifying pivots in the Cholesky Decomposition of an approximate error correlation matrix. We demonstrate the algorithm through numerical experiments, showing that the algorithm is capable of accelerating offline RBM procedures without degrading accuracy, assuming that the solution manifold has low Kolmogorov width.

### Xiaocheng Shang (Birmingham)

Title: Accurate and Efficient Splitting Methods for Dissipative Particle Dynamics

Abstract: We study numerical methods for dissipative particle dynamics (DPD), which is a system of stochastic differential equations and a popular stochastic momentum-conserving thermostat for simulating complex hydrodynamic behavior at mesoscales. We propose a new splitting method that is able to substantially improve the accuracy and efficiency of DPD simulations in a wide range of friction coefficients, particularly in the extremely large friction limit that corresponds to a fluid-like Schmidt number, a key issue in DPD. Various numerical experiments on both equilibrium and transport properties are performed to demonstrate the superiority of the newly proposed method over popular alternative schemes in the literature.

### Michela Ottobre (Heriot-Watt)

Title: Non mean-field Vicsek type models for collective behaviour

Abstract: We consider Interacting Particle dynamics with Vicsek type interactions, and their macroscopic PDE limit, in the non-mean-field regime; that is, we consider the case in which each particle/agent in the system interacts only with a prescribed subset of the particles in the system (for example, those within a certain distance). It was observed by Motsch and Tadmore that in this non-mean-field regime the influence between agents (i.e. the interaction term) can be scaled either by the total number of agents in the system (global scaling) or by the number of agents with which the particle is effectively interacting at time t (local scaling). We compare the behaviour of the globally scaled and the locally scaled system in many respects; in particular we observe that, while both models exhibit multiple stationary states, such equilibria are unstable (for certain parameter regimes) for the globally scaled model, with the instability leading to travelling wave solutions, while they are always stable for the locally scaled one. This observation is based on a careful numerical study of the model, supported by formal analysis.

Based on a soon-to-be preprint in collaboration with P. Buttà, B. Goddard, T. Hodgson, and K. Painter.

### Johannes Zimmer (TU Münich)

Title: From particles to macroscopic models: the role of fluctuations

Abstract: Processes can often be described on a microscopic level, through particles, and on a macroscopic level, through partial differential equations (PDEs). This talk will describe, at the hand of examples, how the macroscopic structure of the PDE is linked to the underlying particle dynamics. Fluctuations will be seen to play a central role, as they encode the macroscopic dissipation. Specifically, suitable infinite-dimensional fluctuation-dissipation relations are, under suitable assumptions, capable of describing the full macroscopic evolution operator in a numerically tractable way.

### Urbain Vaes (Inria Paris)

Title: Mobility for Langevin-like dynamics in a periodic potential: Scaling limits and efficient numerical estimation

Abstract: In the first part of this presentation, we study the diffusive limit of solutions to the generalized Langevin equation (GLE) in a periodic potential. We first present sharp estimates on the rate of convergence to equilibrium for a simple Markovian approximation of the GLE, and then we obtain asymptotic results for the effective diffusion coefficient in the small correlation time, overdamped and underdamped regimes. In the second part of the talk, we present a computational variance reduction approach based on control variates for estimating numerically the mobility of Langevin-like dynamics. The efficiency of the proposed approach is demonstrated through theoretical results and numerical experiments.

### Hong Duong (Birmingham)

Title: Coarse-graining of SDEs and PDEs

Abstract: Coarse-graining is the procedure of approximating a complex system by a simpler or lower-dimensional one. It thus often offers cheaper and more efficient computational methods and is of interest in a variety of fields.

In this talk, I will present methods for qualitative and quantitative coarse-graining of several SDEs and PDEs, in the presence or absence of a scale-separation.

### Chunrong Feng (Durham) Online

Title: Random periodic processes and periodic measures

Abstract: In my talk, first I will introduce the concepts of random periodic processes and periodic measures. Secondly, I will discuss the analytical and numerical approach of random periodic solutions and periodic measures for some SDEs. For the numerical part, we use Euler-Maruyama scheme and the error will be given. This is a joint work with Huaizhong Zhao.

### Eyal Neuman (Imperial College)

Title: Scaling Properties of a Moving Polymer

Abstract: We set up an SPDE model for a moving, weakly self-avoiding polymer with intrinsic length *J* taking values in (0,∞). Our main result states that the effective radius of the polymer is approximately *J*^{5/3}; evidently for large *J* the polymer undergoes stretching. This contrasts with the equilibrium situation without the time variable, where many earlier results show that the effective radius is approximately *J*.

For such a moving polymer taking values in ℝ^{2}, we offer a conjecture that the effective radius is approximately *J*^{5/4}.

This is a joint work with Carl Mueller.

### Mohammud Foondun (Strathclyde)

Title: Small Ball Probabilities for solutions to stochastic partial differential equations

Abstract: Finding bounds on small ball probability has been a major problem in various branches of probability theory. Surprisingly, there has not been much work about this sort of problems for solutions to stochastic partial differential equations (SPDEs). In this talk, we present some new results about small ball probabilities for SPDEs in the Holder semi norms. This is based on joint work with M. Joseph and K. Kim.

### Jingyu Huang (Birmingham)

Title: Stochastic heat equation with super-linear drift and multiplicative noise on ℝ^{d}

Abstract: Consider the stochastic heat equation on ℝ^{d}, ∂*u*/∂*t* = 1/2 Δ*u* + *b*(*u*) + σ(*u*)*Ẇ*, where *Ẇ* is a centred Gaussian noise which is white in time and coloured in space, the initial condition is assumed to be a positive measure. The functions *b*(z) and σ(*z*) are locally Lipschitz and as |*z*| → ∞, |*b*(*z*)| = O(|*z*|log |*z*|) and |σ(*z*)| = o(|*z*|(log |*z*|)^{α}) for some 0 < α ≤ 1/2. We show that under improved Dalang's condition, there is a unique global solution. This is based on joint work with Le Chen.