Related papers: High-order BDF convolution quadrature for subdiffu…
This work introduces an extension of the high order, single stage Lax-Wendroff Flux Reconstruction (LWFR) of Babbar et al., JCP (2022) to solve second order time-dependent partial differential equations in conservative form on curvilinear…
Thanks to the singularity of the solution of linear subdiffusion problems, most time-stepping methods on uniform meshes can result in $O(\tau)$ accuracy where $\tau$ denotes the time step. The present work aims to discover the reason why…
In this article, we consider the diffusion equation with multi-term time-fractional derivatives. We first derive that the solution is positive when the source term is nonpositive by a subordination principle for the solution. As an…
In this paper, we construct a quadrature scheme to numerically solve the nonlocal diffusion equation $(\mathcal{A}^\alpha+b\mathcal{I})u=f$ with $\mathcal{A}^\alpha$ the $\alpha$-th power of the regularly accretive operator $\mathcal{A}$.…
In this paper, we propose a parallel space-time domain decomposition method for solving an unsteady source identification problem governed by the linear convection-diffusion equation. Traditional approaches require to solve repeatedly a…
Our aim is to study the backward problem, i.e. recover the initial data from the terminal observation, of the subdiffusion with time dependent coefficients. First of all, by using the smoothing property of solution operators and a…
We consider a fully discrete and explicit scheme for the mean curvature flow of boundaries, based on an elementary diffusion step and a precise redistancing operation. We give an elementary convergence proof for the scheme under the…
In this paper we consider a sub-diffusion problem where the fractional time derivative is approximated either by the L1 scheme or by Convolution Quadrature. We propose new interpretations of the numerical schemes which lead to a posteriori…
In this work, we investigate the two-step backward differentiation formula (BDF2) with nonuniform grids for the Allen-Cahn equation. We show that the nonuniform BDF2 scheme is energy stable under the time-step ratio restriction…
Hyperbolic conservation laws with stiff source terms appear in the study of a variety of physical systems. Early work showed that the use of formally second-order accurate semi-implicit methods could lead to a substantial loss of accuracy,…
Minimizing the Mumford-Shah functional is frequently used for smoothing signals or time series with discontinuities. A significant limitation of the standard Mumford-Shah model is that linear trends -- and in general polynomial trends -- in…
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…
Part of the success of diffusion models stems from their ability to perform iterative refinement, i.e., repeatedly correcting outputs during generation. However, modern masked discrete diffusion lacks this capability: when a token is…
Stability and convergence of a time-weighted discrete scheme with nonuniform time steps are established for linear reaction-subdiffusion equations. The Caupto derivative is approximated at an offset point by using linear and quadratic…
A novel efficient and high accuracy numerical method for the time-fractional differential equations (TFDEs) is proposed in this work. We show the equivalence between TFDEs and the integer-order extended parametric differential equations…
The subdiffusion equations with a Caputo fractional derivative of order $\alpha \in (0,1)$ arise in a wide variety of practical problems, which is describing the transport processes, in the force-free limit, slower than Brownian diffusion.…
Denoising diffusion models (DDMs) offer a flexible framework for sampling from high dimensional data distributions. DDMs generate a path of probability distributions interpolating between a reference Gaussian distribution and a data…
It is a promising extension of the quantum mechanical/molecular mechanical (QM/MM) approach to incorporate the solvent molecules surrounding the QM solute into the QM region to ensure the adequate description of the electronic polarization…
This paper studies the original discrete-time denoising diffusion probabilistic model (DDPM) from a probabilistic point of view. We present three main theoretical results. First, we show that the time-dependent score function associated…
On Bakhvalov-type mesh, uniform convergence analysis of finite element method for a 2-D singularly perturbed convection-diffusion problem with exponential layers is still an open problem. Previous attempts have been unsuccessful. The…