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In this paper, we investigate stochastic heat equation with sublinear diffusion coefficients. By assuming certain concavity of the diffusion coefficient, we establish non-trivial moment upper bounds and almost sure spatial asymptotic…
We present a Bayesian non-parametric way of inferring stochastic differential equations for both regression tasks and continuous-time dynamical modelling. The work has high emphasis on the stochastic part of the differential equation, also…
In this paper, a semi-discrete spatial finite volume (FV) method is proposed and analyzed for approximating solutions of anomalous subdiffusion equations involving a temporal fractional derivative of order $\alpha \in (0,1)$ in a…
In this article, we consider the solution to elliptic diffusion problems on a class of random domains obtained by log-Gaussian random homothety of the unit disk respectively an annulus. We model the problem under consideration and verify…
In this article, we analyse the domain mapping method approach to approximate statistical moments of solutions to linear elliptic partial differential equations posed over random geometries including smooth surfaces and bulk-surface…
This work introduces structure preserving hierarchical decompositions for sampling Gaussian random fields (GRF) within the context of multilevel Bayesian inference in high-dimensional space. Existing scalable hierarchical sampling methods,…
This paper studies formulations of second-order elliptic partial differential equations in nondivergence form on convex domains as equivalent variational problems. The first formulation is that of Smears \& S\"uli [SIAM J.\ Numer.\ Anal.\…
Diffusion processes are a class of stochastic differential equations (SDEs) providing a rich family of expressive models that arise naturally in dynamic modelling tasks. Probabilistic inference and learning under generative models with…
We overview a series of recent works addressing numerical simulations of partial differential equations in the presence of some elements of randomness. The specific equations manipulated are linear elliptic, and arise in the context of…
Spatially distributed problems are often approximately modelled in terms of partial differential equations (PDEs) for appropriate coarse-grained quantities (e.g. concentrations). The derivation of accurate such PDEs starting from finer…
In this study a stabilized finite element method for solving advection-diffusion-reaction equation with spatially variable coefficients has been carried out. Here subgrid scale approach along with algebraic approximation to the sub-scales…
We consider an implicit finite difference scheme on uniform grids in time and space for the Cauchy problem for a second order parabolic stochastic partial differential equation where the parabolicity condition is allowed to degenerate. Such…
We construct a class of discontinuous superprocesses with dependent spatial motion and general branching mechanism. The process arises as the weak limit of critical interacting-branching particle systems where the spatial motions of the…
L\'evy processes, known for their ability to model complex dynamics with skewness, heavy tails and discontinuities, play a critical role in stochastic modeling across various domains. However, inference for most L\'evy processes, whether in…
Process convolutions yield random fields with flexible marginal distributions and dependence beyond Gaussianity, but statistical inference is often hampered by a lack of closed-form marginal distributions, and simulation-based inference may…
Linear systems with large differences between coefficients ("discontinuous coefficients") arise in many cases in which partial differential equations(PDEs) model physical phenomena involving heterogeneous media. The standard approach to…
In "I. Smears, E. S\"{u}li, \emph{Discontinuous Galerkin finite element approximation of nondivergence form elliptic equations with Cord\'{e}s coefficients. SIAM J. Numer Anal., 51(4):2088-2106, 2013}" the authors designed and analysed a…
In this paper we study the existence of a unique solution for linear stochastic differential equations driven by a L\'evy process, where the initial condition and the coefficients are random and not necessarily adapted to the underlying…
This paper presents new results allowing an unknown non-Gaussian positive matrix-valued random field to be identified through a stochastic elliptic boundary value problem, solving a statistical inverse problem. A new general class of…
A new method is proposed to numerically extract the diffusivity of a (typically nonlinear) diffusion equation from underlying stochastic particle systems. The proposed strategy requires the system to be in local equilibrium and have…