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1. A blob method for inhomogeneous diffusion with applications to multi-agent control and sampling

   https://doi.org/10.1090/mcom/3841  

   Craig, Katy; Elamvazhuthi, Karthik; Haberland, Matt; Turanova, Olga (November 2023, Mathematics of Computation)

   As a counterpoint to classical stochastic particle methods for linear diffusion equations, such as Langevin dynamics for the Fokker-Planck equation, we develop a deterministic particle method for the weighted porous medium equation and prove its convergence on bounded time intervals. This generalizes related work on blob methods for unweighted porous medium equations. From a numerical analysis perspective, our method has several advantages: it is meshfree, preserves the gradient flow structure of the underlying PDE, converges in arbitrary dimension, and captures the correct asymptotic behavior in simulations. 

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2. A martingale formulation for stochastic compartmental susceptible-infected-recovered (SIR) models to analyze finite size effects in COVID-19 case studies

   https://doi.org/10.3934/nhm.2022009  

   Li, Xia; Wang, Chuntian; Li, Hao; Bertozzi, Andrea L. (January 2022, Networks & Heterogeneous Media)

   Deterministic compartmental models for infectious diseases give the mean behaviour of stochastic agent-based models. These models work well for counterfactual studies in which a fully mixed large-scale population is relevant. However, with finite size populations, chance variations may lead to significant departures from the mean. In real-life applications, finite size effects arise from the variance of individual realizations of an epidemic course about its fluid limit. In this article, we consider the classical stochastic Susceptible-Infected-Recovered (SIR) model, and derive a martingale formulation consisting of a deterministic and a stochastic component. The deterministic part coincides with the classical deterministic SIR model and we provide an upper bound for the stochastic part. Through analysis of the stochastic component depending on varying population size, we provide a theoretical explanation of finite size effects. Our theory is supported by quantitative and direct numerical simulations of theoretical infinitesimal variance. Case studies of coronavirus disease 2019 (COVID-19) transmission in smaller populations illustrate that the theory provides an envelope of possible outcomes that includes the field data. 

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3. Computationally Efficient Simulations of Stochastically Perturbed Nonlinear Dynamical Systems

   https://doi.org/10.1115/1.4054932  

   Breunung, Thomas; Balachandran, Balakumar (July 2022, Journal of Computational and Nonlinear Dynamics)

   Abstract A probabilistic approach is needed to address systems with uncertainties arising in natural processes and engineering applications. For computational convenience, however, the stochastic effects are often ignored. Thus, numerical integration routines for stochastic dynamical systems are rudimentary compared to those for the deterministic case. In this work, the authors present a method to carry out stochastic simulations by using methods developed for the deterministic case. Thereby, the well-developed numerical integration routines developed for deterministic systems become available for studies of stochastic systems. The convergence of the developed method is shown and the method's performance is demonstrated through illustrative examples. 

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4. From microfacets to participating media: A unified theory of light transport with stochastic geometry

   Seyb, Dario; d'Eon, Eugene; Bitterli, Benedikt; Jarosz, Wojciech (July 2024, ACM transactions on graphics)

   Stochastic geometry models have enjoyed immense success in graphics for modeling interactions of light with complex phenomena such as participating media, rough surfaces, fibers, and more. Although each of these models operates on the same principle of replacing intricate geometry by a random process and deriving the average light transport across all instances thereof, they are each tailored to one specific application and are fundamentally distinct. Each type of stochastic geometry present in the scene is firmly encapsulated in its own appearance model, with its own statistics and light transport average, and no cross-talk between different models or deterministic and stochastic geometry is possible. In this paper, we derive a theory of light transport on stochastic implicit surfaces, a geometry model capable of expressing deterministic geometry, microfacet surfaces, participating media, and an exciting new continuum in between containing aggregate appearance, non-classical media, and more. Our model naturally supports spatial correlations, missing from most existing stochastic models. Our theory paves the way for tractable rendering of scenes in which all geometry is described by the same stochastic model, while leaving ample future work for developing efficient sampling and rendering algorithms. 

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5. Mean field limits of particle-based stochastic reaction-drift-diffusion models ^*

   https://doi.org/10.1088/1361-6544/ad8fea  

   Heldman, M; Isaacson, S A; Liu, Q; Spiliopoulos, K (January 2025, Nonlinearity)

   Abstract We consider particle-based stochastic reaction-drift-diffusion models where particles move via diffusion and drift induced by one- and two-body potential interactions. The dynamics of the particles are formulated as measure-valued stochastic processes (MVSPs), which describe the evolution of the singular, stochastic concentration fields of each chemical species. The mean field large population limit of such models is derived and proven, giving coarse-grained deterministic partial integro-differential equations (PIDEs) for the limiting deterministic concentration fields’ dynamics. We generalize previous studies on the mean field limit of models involving only diffusive motion, with care to formulating the MVSP representation to ensure detailed balance of reversible reactions in the presence of potentials. Our work illustrates the more general set of PIDEs that arise in the mean field limit, demonstrating that the limiting macroscopic reactive interaction terms for reversible reactions obtain additional nonlinear concentration-dependent coefficients compared to the purely diffusive case. Numerical studies are presented which illustrate that two-body repulsive potential interactions can have a significant impact on the reaction dynamics, and also demonstrate the empirical numerical convergence of solutions to the PBSRDD model to the derived mean field PIDEs as the population size increases. 

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