On the pathwise approximation of stochastic differential equations

We consider one-step methods for integrating stochastic differential equations and prove pathwise convergence using ideas from rough path theory. In contrast to alternative theories of pathwise convergence, no knowledge is required of convergence in pth mean and the analysis starts from a pathwise bound on the sum of the truncation errors. We show how the theory is applied to the Euler-Maruyama method with fixed and adaptive time-stepping strategies. The assumption on the truncation errors suggests an error-control strategy and we implement this as an adaptive time-stepping Euler-Maruyama method using bounded diffusions. We prove the adaptive method converges and show some computational experiments.Comment: 21 page

Well-posedness for a regularised inertial Dean-Kawasaki model for slender particles in several space dimensions

A stochastic PDE, describing mesoscopic fluctuations in systems of weakly interacting inertial particles of finite volume, is proposed and analysed in any finite dimension $d\in\mathbb{N}$. It is a regularised and inertial version of the Dean-Kawasaki model. A high-probability well-posedness theory for this model is developed. This theory improves significantly on the spatial scaling restrictions imposed in an earlier work of the same authors, which applied only to significantly larger particles in one dimension. The well-posedness theory now applies in $d$-dimensions when the particle-width $\epsilon$ is proportional to $N^{-1/\theta}$ for $\theta>2d$ and $N$ is the number of particles. This scaling is optimal in a certain Sobolev norm. Key tools of the analysis are fractional Sobolev spaces, sharp bounds on Bessel functions, separability of the regularisation in the $d$-spatial dimensions, and use of the Fa\`a di Bruno's formula.Comment: 28 pages, no figure

The Regularised Inertial Dean-Kawasaki equation:discontinuous Galerkin approximation and modelling for low-density regime

The Regularised Inertial Dean-Kawasaki model (RIDK) -- introduced by the authors and J. Zimmer in earlier works -- is a nonlinear stochastic PDE capturing fluctuations around the mean-field limit for large-scale particle systems in both particle density and momentum density. We focus on the following two aspects. Firstly, we set up a Discontinuous Galerkin (DG) discretisation scheme for the RIDK model: we provide suitable definitions of numerical fluxes at the interface of the mesh elements which are consistent with the wave-type nature of the RIDK model and grant stability of the simulations, and we quantify the rate of convergence in mean square to the continuous RIDK model. Secondly, we introduce modifications of the RIDK model in order to preserve positivity of the density (such a feature only holds in a ''high-probability sense'' for the original RIDK model). By means of numerical simulations, we show that the modifications lead to physically realistic and positive density profiles. In one case, subject to additional regularity constraints, we also prove positivity. Finally, we present an application of our methodology to a system of diffusing and reacting particles. Our Python code is available in open-source format.Comment: 35 pages, 13 figure

A walk outside spheres for the fractional Laplacian:fields and first eigenvalue

The Feynman-Kac formula for the exterior-value problem for the fractional Laplacian leads to a walk-outside-spheres algorithm via sampling alpha-stable Levy processes on their exit from maximally inscribed balls and sampling their occupation distribution. Kyprianou, Osojnik, and Shardlow (2017) developed this algorithm, providing a complexity analysis and an implementation, for approximating the solution at a single point in the domain. This paper shows how to efficiently sample the whole field by generating an approximation in L_2(D), for a domain D . The method takes advantage of a hierarchy of triangular meshes and uses the multilevel Monte Carlo method for Hilbert space-valued quantities of interest. We derive complexity bounds in terms of the fractional parameter alpha and demonstrate that the method gives accurate results for two problems with exact solutions. Finally, we show how to couple the method with the variable-accuracy Arnoldi iteration to compute the smallest eigenvalue of the fractional Laplacian. A criteria is derived for the variable accuracy and a comparison is given with analytical results of Dyda (2012)

A coupled Cahn–Hilliard particle system

A Cahn-Hilliard equation is coupled to a system of stochastic differential equations to model a random growth process. We show the model is well posed and analyze the model asymptotically (in the limit of the interfacial distance becoming small), to recover a free boundary problem. A numerical method together with example solutions is presented

Deep surrogate accelerated delayed-acceptance HMC for Bayesian inference of spatio-temporal heat fluxes in rotating disc systems

We study the Bayesian inverse problem of inferring the Biot number, a spatio-temporal heat-flux parameter in a PDE model. This is an ill-posed problem where standard optimisation yields unphysical inferences. We introduce a training scheme that uses temperature data to adaptively train a neural-network surrogate to simulate the parametric forward model. This approach approximates forward and inverse solution together, by simultaneously identifying an approximate posterior distribution over the Biot number, and weighting the forward training loss according to this approximation. Utilising random Chebyshev series, we outline how to approximate an arbitrary Gaussian process prior, and using the surrogate we apply Hamiltonian Monte Carlo (HMC) to efficiently sample from the corresponding posterior distribution. We derive convergence of the surrogate posterior to the true posterior distribution in the Hellinger metric as our adaptive loss function approaches zero. Furthermore, we describe how this surrogate-accelerated HMC approach can be combined with a traditional PDE solver in a delayed-acceptance scheme to a-priori control the posterior accuracy, thus overcoming a major limitation of deep learning-based surrogate approaches, which do not achieve guaranteed accuracy a-priori due to their non-convex training. Biot number calculations are involved turbo-machinery design, which is safety critical and highly regulated, therefore it is important that our results have such mathematical guarantees. Our approach achieves fast mixing in high-dimensional parameter spaces, whilst retaining the convergence guarantees of a traditional PDE solver, and without the burden of evaluating this solver for proposals that are likely to be rejected. Numerical results compare the accuracy and efficiency of the adaptive and general training regimes, as well as various Markov chain Monte Carlo proposals strategies