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We develop $H$(div)-conforming mixed finite element methods for the unsteady Stokes equations modeling single-phase incompressible fluid flow. A projection method in the framework of the incremental pressure correction methodology is…
Multiscale spatial structure complicates temporal prediction because small-scale spatial fluctuations influence large-scale evolution, yet resolving all scales is often intractable. Standard diffusion models do not address this problem…
In this paper, we present a high order finite difference solver for anisotropic diffusion problems based on the first-order hyperbolic system method. In particular, we demonstrate that the construction of a uniformly accurate fifth-order…
Generative models such as denoising diffusion models are quickly advancing their ability to approximate highly complex data distributions. They are also increasingly leveraged in scientific machine learning, where samples from the implied…
In this paper, an alternative approximation to the innovation method is introduced for the parameter estimation of diffusion processes from partial and noisy observations. This is based on a convergent approximation to the first two…
A hydrogeological model for the spread of pollution in an aquifer is considered. The model consists in a convection-diffusion-reaction equation involving the dispersion tensor which depends nonlinearly of the fluid velocity. We introduce an…
We propose a novel algorithm for the approximation of surface-quasi geostrophic (SQG) flows modeled by a nonlinear partial differential equation coupling transport and fractional diffusion phenomena. The time discretization consists of an…
Diffusion approximation provides weak approximation for stochastic gradient descent algorithms in a finite time horizon. In this paper, we introduce new tools motivated by the backward error analysis of numerical stochastic differential…
In this article we propose two novel 3D finite element models, denoted method A and B, for electron and hole Drift-Diffusion (DD) current densities. Method A is based on a primal-mixed formulation of the DD model as a function of the…
Diffusion models emerged as a leading approach in text-to-image generation, producing high-quality images from textual descriptions. However, attempting to achieve detailed control to get a desired image solely through text remains a…
Diffusion models offer stable training and state-of-the-art performance for deep generative modeling tasks. Here, we consider their use in the context of multivariate subsurface modeling and probabilistic inversion. We first demonstrate…
We proposed a structure-preserving stabilized parametric finite element method (SPFEM) for the evolution of closed curves under anisotropic surface diffusion with an arbitrary surface energy $\hat{\gamma}(\theta)$. By introducing a…
Parametric finite elements lead to very efficient numerical methods for surface evolution equations. We introduce several computational techniques for curvature driven evolution equations based on a weak formulation for the mean curvature.…
We consider an arbitrary-Lagrangian-Eulerian evolving surface finite element method for the numerical approximation of advection and diffusion of a conserved scalar quantity on a moving surface. We describe the method, prove optimal order…
A singularly perturbed parabolic problem of convection-diffusion type with a discontinuous initial condition is examined. A particular complimentary error function is identified which matches the discontinuity in the initial condition. The…
We present and analyse an implicit-explicit timestepping procedure with finite element spatial approximation for a semilinear reaction-diffusion systems on evolving domains arising from biological models, such as Schnakenberg's (1979). We…
A singularly perturbed linear system of second order ordinary differential equations of reaction-diffusion type with given boundary conditions is considered. The leading term of each equation is multiplied by a small positive parameter.…
A singularly perturbed linear system of second order ordinary differential equations of reaction-diffusion type with given boundary conditions is considered. The leading term of each equation is multiplied by a small positive parameter.…
We extend to multi-dimensions the work of [1], where new fully explicit kinetic methods were built for the approximation of linear and non-linear convection-diffusion problems. The fundamental principles from the earlier work are retained:…
We present a finite element approach for diffusion problems with thermal fluctuations based on a fluctuating hydrodynamics model. The governing transport equations are stochastic partial differential equations with a fluctuating forcing…