Maria Bruna, University of Oxford

**Model reduction for slow-fast stochastic systems with bistability**

*The quasi-steady-state approximation (or stochastic averaging principle) is a useful tool in the study of multiscale stochastic systems, giving a practical method by which to reduce the number of degrees of freedom in a model. We extend this method to slow–fast systems in which the fast variables exhibit metastable behaviour. The key parameter that determines the form of the reduced model is the ratio of the timescale for the switching of the fast variables between metastable states to the timescale for the evolution of the slow variables. Our approach uses a perturbation analysis at the level of the Fokker–Planck equation for the joint probability density. The method is illustrated with two simple examples (a chemical switch and a prey-predator system), and tested with several numerical simulations.*

Yves Capdeboscq, University of Oxford

**Homogenisation and Localisation Phenomena in bounded domains**

*In this talk, we will discuss some homogenisation results for models with vanishing diffusion - or, equivalently, increasing domains. These phenomena appear in a variety of domain in the physical sciences: in neutron diffusion models for the determination of production regimes of nuclear reactors, but also in bio-motor models in mathematical biology. Such models usually involve cascades of equations, but (some of) the underlying issues can already be observed in diffusion equations with drift, with periodic or piecewise periodic coefficients. Classical homogenization methods may fail to predict the effective limit; viscosity solutions approaches to homogenization have been more successful, and generalised to stochastic settings. We will focus on issues related to boundary conditions and interfaces, where both approaches have complementary advantages.*

Jon Chapman, University of Oxford

**Diffusion in porous media**

*We consider several different approaches to the problem of calculating the effective diffusion coefficient in a material with microstructure. The particular microstructure we consider here comprises a collection of impenetrable spheres. We first extend the method of multiple scales to account for non-uniform porosity distributions. We then compare and contrast this result to averaging over a stochastic distribution of spheres.*

Ulrich Dobramysl, University of Oxford

**Spatial Multi-scale Modelling: Applications to Biological Systems**

*Many spatially-extended modelling problems in biology require high accuracy only in a small region of the computational domain, such as cell receptor binding sites. Hence they can be efficiently simulated via a coupling of a high-accuracy algorithm in the region of interest with a more efficient algorithm in the remainder of the domain. In particular, we couple a Brownian dynamics in the high-accuracy regime with an on-lattice Gillespie algorithm. I will give an introduction of two different ways of implementing the interface between the two algorithms, namely the Two-Regime Method and the Ghost-Cell method. I will then present applications of these methods to the growth and collapse of filopodia, which are cytoplasmic projections that are important to environmental sensing and chemotaxis, cell motility and many other cell functions, as well as the dynamics of intra-cellular calcium ions.*

*This is joint work with Radek Erban.*

Andrew Duncan, Imperial College London

**A Hybrid Discrete-Continuum Approach to the Chemical Master Equation**

*It is well known that stochasticity can play a fundamental role in various biochemical processes, such as cell regulatory networks and enzyme cascades. Such systems can be adequately modelled by Markov processes, the dynamics of which are characterised by the Chemical Master Equation (CME), a difference-differential equation for the probability of a given state at a certain time. If the state space is truncated, the CME reduces to a finite dimensional system of ODEs which can be solved numerically. However, the size of the problem grows exponentially with the number of chemical species and solving the CME quickly becomes intractable. Various methods have been proposed to mitigate this curse of dimensionality. One such approach is via the Chemical Fokker-Planck equation (CFPE), a PDE approximation to the CME, valid in the limit of large volume size. Using the CFPE is computationally advantageous as it permits the use of far sparser computational grids. However, significant errors can arise for systems exhibiting low copy number fluctuations in one or more chemical species.*

*In this talk, I will present a hybrid scheme for approximating the CME, using the CFPE in regions with large volume along with the CME in localised regions where the Fokker-Planck approximation fails. The two schemes are coupled via an intermediate region in which a "blended'' jump-diffusion model is introduced. After formulating the model, I will describe a finite element discretisation of the hybrid system, and provide numerical examples to demonstrate the efficacy of this scheme.*

*This is joint work with Radek Erban.*

Mike Giles, University of Oxford

**Application of multilevel Monte Carlo to the simulation of dilute polymers**

*Polymers immersed in a fluid can be modelled as chains connected by finitely-extensible bonds with nonlinear elastic potentials, subject to random forcing. In this work we discuss the simulation of the system of coupled SDEs which results from the modelling, and the use of multilevel Monte Carlo (MLMC) to efficiently estimate expectations arising from the associated equilibrium distribution. One important element is the use of adaptive timestepping which can be incorporated fairly easily into MLMC. Another is the use of a new multilevel coupling idea due to Glynn and Rhee (2014) for the simulation of equilibrium expectations in the context of contracting Markov Chains.*

*This is joint work with Endre Suli, James Whittle and Shenghan Ye.*

Ben MacArthur, University of Southampton

**Stochastic models of stem cell dynamics**

*There is a long-running debate amongst stem cell biologists over stochastic versus instructive models of stem cell dynamics. In this talk I will outline some simple models and discuss recent data that favours the stochastic perspective. I will also discuss some of the difficulties of distinguishing biologically functional cell-cell variability from experimental noise, and suggest areas for further work that may benefit from increased collaboration between experimentalists and mathematicians.*

Grigorios Pavliotis, Imperial College London

**Accelerating convergence to equilibrium for nonreversible diffusions**

*A standard technique for sampling from a probability distribution that is known up to a constant is to run stochastic dynamics that is ergodic with respect to the distribution from which we want to sample. There are (infinitely) many diffusion processes that are ergodic with respect to the same distribution and it is desirable to choose the one that converges as quickly as possible to the target distribution and, at the same time, minimizes the asymptotic variance. In this talk we show that the addition of a nonreversible perturbation to reversible stochastic dynamics can accelerate convergence to equilibrium. We also construct the optimal nonreversible perturbation for diffusions with linear drift. Finally, we address the issue of minimizing the asymptotic variance.*

Christoph Reisinger, University of Oxford

**Multiscale problems in computational finance**

*We present three different examples of (stochastic) differential equations arising in financial engineering where some sort of multiple scales are involved. The first example is that of a so-called penalty method for a free boundary problem for an American option, where we use matched asymptotic expansions to describe the boundary layer of the penalty solution around the exercise boundary. In the second example, we use the strong anisotropy of the high-dimensional diffusion process describing equity baskets to derive accurate expansion approximations to option prices with respect to the small diffusion coefficients. Finally, we develop a multi-level simulation method for a large credit pool, where the levels correspond to nested pools of increasing size, which allows us to construct efficient estimators for expected tranche losses.*

Dimitry Savostyanov

**Numerical solution of chemical master equations using new tensor product algorithm**

*Chemical master equation (CME) describes reactions between chemical or biological species on mesoscopic scale - i.e. when the number of molecules is too small for a proper statistical ensemble, and the macroscopic description via the concentrations and deterministic reaction rates fails. CME describes system dynamics by the probability distribution function (p.d.f.), a notably high-dimensional function. The number of unknowns in this description (hence the complexity of the problem) grows exponentially with dimension (number of components), which makes the computations difficult. *

* When high-dimensional problems are concerned, not much algorithms can break the curse of dimensionality, and solve them efficiently and reliably. Among those, tensor product algorithms, which implement the idea of separation of variables for multi-index arrays (tensors), seem to be the most general and also very promising. They originated in quantum physics and chemistry and descent broadly from the density matrix renormalization group (DMRG) [7] and matrix product states (MPS) [4] formalisms. The same tensor formats were recently re-discovered in the numerical linear algebra (NLA) community as the tensor train (TT) format [5].*

* We have recently proposed the alternating minimal energy (AMEn) algorithm for linear systems [1,2], that bridges the classical iterative algorithms and DMRG/MPS methods from quantum physics, and has proven global convergence. The AMEn algorithm has already proved to be extremely useful in quantum chemistry to simulate the NMR spectra of large molecules (such as ubiquitin) [6], and in quantum physics to find the ground state of a periodic spin chain [3]. In this talk we show the application of the AMEn method to the chemical master equation describing the mesoscopic model of gene regulatory networks [2].*

*This is a joint work with Sergey Dolgov the from Max-Planck Institute for Mathematics in the Sciences, Leipzig, Germany.*

*References:*

*1. S. V. Dolgov and D. V. Savostyanov. Alternating minimal energy methods for linear systems in higher dimensions. Part I: SPD systems. arXiv preprint 1301.6068, 2013. URL: http://arxiv.org/abs/1301.6068.*

*2. S. V. Dolgov and D. V. Savostyanov. Alternating minimal energy methods for linear systems in higher dimensions. Part II: Faster algorithm and application to non- symmetric systems. arXiv preprint 1304.1222, 2013. URL: http://arxiv.org/abs/ 1304.1222.1*

*3. S. V. Dolgov and D. V. Savostyanov. Corrected one-site density matrix renormalization group and alternating minimal energy algorithm. In Proc. ENUMATH 2013, accepted, 2014. URL: http://arxiv.org/abs/1312.6542.*

*4. M. Fannes, B. Nachtergaele, and R.F. Werner. Finitely correlated states on quantum spin chains. Comm. Math. Phys., 144(3):443–490, 1992. doi:10.1007/BF02099178.*

*5. I. V. Oseledets. Tensor-train decomposition. SIAM J. Sci. Comput., 33(5):2295–2317, 2011. doi:10.1137/090752286.*

*6. D. V. Savostyanov, S. V. Dolgov, J. M. Werner, and Ilya Kuprov. Exact NMR simulation of protein-size spin systems using tensor train formalism. Phys. Rev. B, 90:085139, 2014. doi:10.1103/PhysRevB.90.085139.*

*7. Steven R. White. Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett., 69(19):2863–2866, 1992. doi:10.1103/PhysRevLett.69.2863.*

Markus Schmuck, Heriot Watt University

**Renormalization and Entropy Methods for a New Stochastic Mode Reduction**

*Abstract. We consider dissipative systems such as the generalized Kuramoto-Sivashinsky equation [1,2] which serves as a prototype modelling nonlinear media with energy sup- ply, energy dissipation, and dispersion. Our main result represents a new stochastic mode reduction methodology that systematically accounts for the neglected degrees of freedom thanks to a strategy relying on a dynamic renormalization group argument combined with an adapted maximum entropy principle. The resulting equations show a stochastic driving force and are characterized by rigorous error estimates [2]. Herewith, our methodology systematically introduces randomness into an originally deterministic problem by accounting for the unresolved degrees of freedom by an information theoretic argument. *

*This is joint work with Marc Pradas, Grigorios A. Pavliotis, and Serafim Kalliadasis.*

*References:*

*1. M. Schmuck, M. Pradas, S. Kalliadasis, and G.A. Pavliotis, Phys. Rev. Lett. 110(24):244101 **2013.*

*2. M. Schmuck, M. Pradas, G.A. Pavliotis, S. Kalliadasis, IMA J. Appl. Math. doi:10.1093/imamat/hxt041 2013. *

Kostantinos Zygalakis, University of Southampton

**On long time approximation of ergodic stochastic differential equations**

*In this talk we will discuss a variety of different problems related to questions related to the long time behaviour of solutions of stochastic differential equations. This is an important question with relevance in many field of applied mathematics, such as modelling turbulent diffusion and molecular dynamics and can be addressed with a variety of different techniques such as for example homogenization. Once the limiting behaviour has been established, one needs to design appropriate numerical algorithms that are able to capture it correctly. This is a very delicate problem and in addressing this, we will discuss how the recently developed theory of modified equations/backward error analysis for SDEs can explain the behaviour of existing stochastic numerical methods and guide the construction of more efficient ones.*