Related papers: Domain Decomposition for the Closest Point Method
We study the homogeneous Cauchy-Dirichlet Problem (CDP) for a nonlinear and nonlocal diffusion equation of singular type of the form $\partial_t u =-\mathcal{L} u^m$ posed on a bounded Euclidean domain $\Omega\subset\mathbb{R}^N$ with…
High-dimensional partial-differential equations (PDEs) arise in a number of fields of science and engineering, where they are used to describe the evolution of joint probability functions. Their examples include the Boltzmann and…
In this article we consider a Bayesian inverse problem associated to elliptic partial differential equations (PDEs) in two and three dimensions. This class of inverse problems is important in applications such as hydrology, but the…
Developments in dynamical systems theory provides new support for the discretisation of \pde{}s and other microscale systems. By systematically resolving subgrid microscale dynamics the new approach constructs asymptotically accurate,…
Although synthetic data can alleviate acquisition challenges in image dehazing tasks, it also introduces the problem of domain bias when dealing with small-scale data. This paper proposes a novel dual-branch collaborative unpaired dehazing…
In this paper we present a high-order kernel method for numerically solving diffusion and reaction-diffusion partial differential equations (PDEs) on smooth, closed surfaces embedded in $\mathbb{R}^d$. For two-dimensional surfaces embedded…
This paper presents a numerical method for variable coefficient elliptic PDEs with mostly smooth solutions on two dimensional domains. The PDE is discretized via a multi-domain spectral collocation method of high local order (order 30 and…
Gas transport and other complex real-world challenges often require solving and controlling partial differential equations (PDEs) defined on graph structures, which typically demand substantial memory and computational resources. The Random…
The reduction of computational costs in the numerical solution of nonstationary problems is achieved through splitting schemes. In this case, solving a set of less computationally complex problems provides the transition to a new level in…
A numerical method for solving elliptic PDEs with variable coefficients on two-dimensional domains is presented. The method is based on high-order composite spectral approximations and is designed for problems with smooth solutions. The…
We propose a double obstacle phase field approach to the recovery of piece-wise constant diffusion coefficients for elliptic partial differential equations. The approach to this inverse problem is that of optimal control in which we have a…
Medical image segmentation is a challenging task, made more difficult by many datasets' limited size and annotations. Denoising diffusion probabilistic models (DDPM) have recently shown promise in modelling the distribution of natural…
Mixed dimensional partial differential equations (PDEs) are equations coupling unknown fields defined over domains of differing topological dimension. Such equations naturally arise in a wide range of scientific fields including geology,…
In this paper, we extend the Generalized Finite Difference Method (GFDM) on unknown compact submanifolds of the Euclidean domain, identified by randomly sampled data that (almost surely) lie on the interior of the manifolds. Theoretically,…
Reduced-order models have long been used to understand the behavior of nonlinear partial differential equations (PDEs). Naturally, reduced-order modeling techniques come at the price of computational accuracy for a decrease in computation…
We consider solving the Laplace-Beltrami problem on a smooth two dimensional surface embedded into a three dimensional space meshed with tetrahedra. The mesh does not respect the surface and thus the surface cuts through the elements. We…
We study the convergence of the new family of mimetic finite difference schemes for linear diffusion problems recently proposed in [38]. In contrast to the conventional approach, the diffusion coefficient enters both the primary mimetic…
In this paper, we propose a domain decomposition dynamical low-rank method to solve high-dimensional radiative transfer problems and similar kinetic equations. The algorithm uses a separate low-rank approximation on each spatial subdomain,…
Chemically-active colloids modify the concentration of chemical solutes surrounding them in order to self-propel. In doing so, they generate long-ranged hydrodynamic flows and chemical gradients that modify the trajectories of other…
A discretization scheme for variable coefficient Helmholtz problems on two-dimensional domains is presented. The scheme is based on high-order spectral approximations and is designed for problems with smooth solutions. The resulting system…