Related papers: Kernel-based methods for Solving Time-Dependent Ad…
We develop a kernel-based solver for path-dependent PDEs (PPDEs) along with a convergence theory. Our numerical scheme leverages signature kernels, a recently introduced class of kernels on path-space. Specifically, we solve an optimal…
This work is devoted to the design of interior penalty discontinuous Galerkin (dG) schemes that preserve maximum principles at the discrete level for the steady transport and convection-diffusion problems and the respective transient…
This paper is concerned with the numerical solution of porous-media flow and transport problems , i. e. heterogeneous, advection-diffusion problems. Its aim is to investigate numerical schemes for these problems in which different time…
We present an error analysis for the discontinuous Galerkin method applied to the discrete-ordinate discretization of the steady-state radiative transfer equation. Under some mild assumptions, we show that the DG method converges uniformly…
The discretization of elliptic PDEs leads to large coupled systems of equations. Domain decomposition methods (DDMs) are one approach to the solution of these systems, and can split the problem in a way that allows for parallel computing.…
We present and analyze an a posteriori error estimator for a space-time hybridizable discontinuous Galerkin discretization of the time-dependent advection-diffusion problem. The residual-based error estimator is proven to be reliable and…
Diffusion models show promise for image restoration, but existing methods often struggle with inconsistent fidelity and undesirable artifacts. To address this, we introduce Kernel Density Steering (KDS), a novel inference-time framework…
We consider a model convection-diffusion problem and present our recent numerical and analysis results regarding mixed finite element formulation and discretization in the singular perturbed case when the convection term dominates the…
Gaussian processes (GPs) provide a principled and direct approach for inference and learning on graphs. However, the lack of justified graph kernels for spatio-temporal modelling has held back their use in graph problems. We leverage an…
We introduce the concept of Hypoelliptic Diffusion Maps (HDM), a framework generalizing Diffusion Maps in the context of manifold learning and dimensionality reduction. Standard non-linear dimensionality reduction methods (e.g., LLE,…
In this paper, we use an implicit two-derivative deferred correction time discretization approach and combine it with a spatial discretization of the discontinuous Galerkin spectral element method to solve (non-)linear PDEs. The resulting…
Diffusion models have recently emerged as powerful stochastic frameworks for high-dimensional inference and generation. However, existing applications to partial differential equations (PDEs) predominantly rely on physics-informed training…
In this paper we demonstrate a computational method to solve the inverse scattering problem for a star-shaped, smooth, penetrable obstacle in 2D. Our method is based on classical ideas from computational geometry. First, we approximate the…
Pansharpening seeks to fuse high-resolution panchromatic (PAN) and low-resolution multispectral (LRMS) images into a single image with both fine spatial and rich spectral detail. Despite progress in deep learning-based approaches, existing…
We develop a cut Discontinuous Galerkin method (cutDGM) for a diffusion-reaction equation in a bulk domain which is coupled to a corresponding equation on the boundary of the bulk domain. The bulk domain is embedded into a structured,…
Calculus and geometry are ubiquitous in the theoretical modelling of scientific phenomena, but have historically been very challenging to apply directly to real data as statistics. Diffusion geometry is a new theory that reformulates…
In this paper, we present a sparse grid-based Monte Carlo method for solving high-dimensional semi-linear nonlocal diffusion equations with volume constraints. The nonlocal model is governed by a class of semi-linear partial…
Recently, diffusion models have achieved great success in generative tasks. Sampling from diffusion models is equivalent to solving the reverse diffusion stochastic differential equations (SDEs) or the corresponding probability flow…
In this work, we consider the numerical solution of an initial boundary value problem for the distributed order time fractional diffusion equation. The model arises in the mathematical modeling of ultra-slow diffusion processes observed in…
In theory, diffusion curves promise complex color gradations for infinite-resolution vector graphics. In practice, existing realizations suffer from poor scaling, discretization artifacts, or insufficient support for rich boundary…