Related papers: Kernel-based active subspaces with application to …
In this paper, we develop a local multiscale model reduction strategy for the elastic wave equation in strongly heterogeneous media, which is achieved by solving the problem in a coarse mesh with multiscale basis functions. We use the…
We present a finite element implementation for the steady-state nonlocal Dirichlet problem with homogeneous volume constraints. Here, the nonlocal diffusion operator is defined as integral operator characterized by a certain kernel…
A discontinuous Galerkin (dG) method for the numerical solution of initial/boundary value multi-compartment partial differential equation (PDE) models, interconnected with interface conditions, is presented and analysed. The study of…
In this paper, a class of high order numerical schemes is proposed to solve the nonlinear parabolic equations with variable coefficients. This method is based on our previous work [10] for convection-diffusion equations, which relies on a…
Modern Bayesian optimization and adaptive sampling methods increasingly rely on nonlinear parametric models, yet theoretical guarantees for such models under adaptive data collection remain limited. Existing analyses largely focus on…
Motivated by applications to numerical simulation of flows in highly heterogeneous porous media, we develop multiscale finite element methods for second order elliptic equations. We discuss a multiscale model reduction technique in the…
A discontinuous Galerkin (DG) method suitable for large-scale astrophysical simulations on Cartesian meshes as well as arbitrary static and moving Voronoi meshes is presented. Most major astrophysical fluid dynamics codes use a finite…
We present a new approach to using neural networks to approximate the solutions of variational equations, based on the adaptive construction of a sequence of finite-dimensional subspaces whose basis functions are realizations of a sequence…
We develop the general form of the variational multiscale method in a discontinuous Galerkin framework. Our method is based on the decomposition of the true solution into discontinuous coarse-scale and discontinuous fine-scale parts. The…
We present a new high-order accurate Lagrangian discontinuous Galerkin (DG) hydrodynamic method to simulate material dynamics (for e.g., gasses, fluids, and solids) with up to fourth-order accuracy on cubic meshes. The variables, such as…
We have developed the formalism necessary to employ the discontinuous-Galerkin approach in general-relativistic hydrodynamics. The formalism is firstly presented in a general 4-dimensional setting and then specialized to the case of…
The discontinuous Galerkin (DG) algorithm is a representative high order method in Computational Fluid Dynamics (CFD) area which possesses considerable mathematical advantages such as high resolution, low dissipation, and dispersion.…
We introduce an immersed high-order discontinuous Galerkin method for solving the compressible Navier-Stokes equations on non-boundary-fitted meshes. The flow equations are discretised with a mixed discontinuous Galerkin formulation and are…
We extend the discontinuous Galerkin (DG) framework to the analysis of first-order hyperbolic and advection-dominated problems posed on implicitly defined surfaces. The focus will be on the hyperbolic part, which is discretised using a…
Kernel methods for solving partial differential equations on surfaces have the advantage that those methods work intrinsically on the surface and yield high approximation rates if the solution to the partial differential equation is smooth…
This paper is concerned with developing accurate and efficient numerical methods for fully nonlinear second order elliptic and parabolic partial differential equations (PDEs) in multiple spatial dimensions. It presents a general framework…
Historically, the machine learning community has derived spectral decompositions from graph-based approaches. We break with this approach and prove the statistical and computational superiority of the Galerkin method, which consists in…
This paper presents a fully discrete numerical scheme for one-dimensional nonlocal wave equations and provides a rigorous theoretical analysis. To facilitate the spatial discretization, we introduce an auxiliary variable analogous to the…
As an extension of our previous work in Sun et.al (2018) [41], we develop a discontinuous Galerkin method for solving cross-diffusion systems with a formal gradient flow structure. These systems are associated with non-increasing entropy…
This work focuses on the conservation of quantities such as Hamiltonians, mass, and momentum when solution fields of partial differential equations are approximated with nonlinear parametrizations such as deep networks. The proposed…