Related papers: Invariant meshless discretization schemes
We introduce a meshless method for solving both continuous and discrete variational formulations of a volume constrained, nonlocal diffusion problem. We use the discrete solution to approximate the continuous solution. Our method is…
The gradient discretisation method (GDM) is a generic framework for designing and analysing numerical schemes for diffusion models. In this paper, we study the GDM for the porous medium equation, including fast diffusion and slow diffusion…
Motivated by the mathematics literature on the algebraic properties of so-called polynomial vector flows, we propose a technique for approximating nonlinear differential equations by linear differential equations. Although the idea of…
We are interested in numerical schemes for the simulation of large scale gas networks. Typical models are based on the isentropic Euler equations with realistic gas constant. The numerical scheme is based on transformation of conservative…
We describe a framework in which is possible to develop and implement algorithms for the approximation of invariant measures of dynamical systems with a given bound on the error of the approximation. Our approach is based on a general…
The gradient scheme framework is based on a small number of properties and encompasses a large number of numerical methods for diffusion models. We recall these properties and develop some new generic tools associated with the gradient…
In the process of constructing invariant difference schemes which approximate partial differential equations we write down a procedure for discretizing an arbitrary partial differential equation on an arbitrary lattice. An open problem is…
The transport and continuum equations exhibit a number of conservation laws. For example, scalar multiplication is conserved by the transport equation, while positivity of probabilities is conserved by the continuum equation. Certain…
Numerical solutions of differential equations are usually not smooth functions. However, they should resemble the smoothness of the corresponding real solutions in one way or another. In two of our recent papers, a kind of spacial…
In this paper we propose a (non-linear) smoothing algorithm for group-affine observation systems, a recently introduced class of estimation problems on Lie groups that bear a particular structure. As most non-linear smoothing methods, the…
In this paper, we consider a class of stochastic midpoint and trapezoidal Lawson schemes for the numerical discretization of highly oscillatory stochastic differential equations. These Lawson schemes incorporate both the linear drift and…
This paper addresses the challenging numerical simulation of nonlinear hybrid stochastic functional differential equations with infinite delays. We first propose an explicit scheme using space and time truncation, requiring only finite…
We present a new meshless method for scalar diffusion equations which is motivated by their compatible discretizations on primal-dual grids. Unlike the latter though, our approach is truly meshless because it only requires the graph of…
Mathematical descriptions of flow phenomena usually come in the form of partial differential equations. The differential operators used in these equations may have properties such as symmetry, skew-symmetry, positive or negative…
Diffusion models have achieved remarkable success in imaging inverse problems owing to their powerful generative capabilities. However, existing approaches typically rely on models trained for specific degradation types, limiting their…
We study a Lagrangian numerical scheme for solution of a nonlinear drift diffusion equation on an interval. The discretization is based on the equation's gradient flow structure with respect to the Wasserstein distance. The scheme inherits…
In this paper we present a general, axiomatical framework for the rigorous approximation of invariant densities and other important statistical features of dynamics. We approximate the system trough a finite element reduction, by composing…
We consider overlap splines that are defined by connecting the patches of piecewise functions via common values at given finite sets of nodes, without using any partitions of the computational domain. It is shown that some classical finite…
Equivariant diffusion policies (EDPs) combine the generative expressivity of diffusion models with the strong generalization and sample efficiency afforded by geometric symmetries. While steering these policies with reinforcement learning…
Diffusion models have emerged as powerful generative tools with applications in computer vision and scientific machine learning (SciML), where they have been used to solve large-scale probabilistic inverse problems. Traditionally, these…