Related papers: Computer-assisted proofs for some nonlinear diffus…

Finding correspondences between images is a fundamental problem in computer vision. In this paper, we show that correspondence emerges in image diffusion models without any explicit supervision. We propose a simple strategy to extract this…

Score-based diffusion models are a recently developed framework for posterior sampling in Bayesian inverse problems with a state-of-the-art performance for severely ill-posed problems by leveraging a powerful prior distribution learned from…

In this paper, we propose a new adaptation of the D-iteration algorithm to numerically solve the differential equations. This problem can be reinterpreted in 2D or 3D (or higher dimensions) as a limit of a diffusion process where the…

We consider nonlinear parabolic equations involving fractional diffusion of the form $\partial_t u + (-\Delta)^s \Phi(u)= 0,$ with $0<s<1$, and solve an open problem concerning the existence of solutions for very singular nonlinearities…

We consider the reconstruction of a diffusion coefficient in a quasilinear elliptic problem from a single measurement of overspecified Neumann and Dirichlet data. The uniqueness for this parameter identification problem has been established…

We propose and analyze a one-dimensional multi-species cross-diffusion system with non-zero-flux boundary conditions on a moving domain, motivated by the mod- eling of a Physical Vapor Deposition process. Using the boundedness by entropy…

An equation containing a fractional power of an elliptic operator of second order is studied for Dirichlet boundary conditions. Finite difference approximations in space are employed. The proposed numerical algorithm is based on solving an…

The paper proves Liouville-type results for stable solutions of semilinear elliptic PDEs with convex nonlinearity, posed on the entire Euclidean space. Extensions to solutions which are stable outside a compact set are also presented.

Computational fluid dynamics (CFD) is a powerful tool for modeling turbulent flow and is commonly used for urban microclimate simulations. However, traditional CFD methods are computationally intensive, requiring substantial hardware…

We formulate a numerical method to solve the porous medium type equation with fractional diffusion \[\frac{\partial u}{\partial t}+(-\Delta)^{1/2} (u^m)=0.\] The problem is posed in $x\in \mathbb{R}^N$, $m\geq 1$ and with nonnegative…

We prove global existence and uniqueness of strong solutions to the logarithmic porous medium type equation with fractional diffusion $$ \partial_tu+(-\Delta)^{1/2}\log(1+u)=0, $$ posed for $x\in \mathbb{R}$, with nonnegative initial data…

A finite difference method is constructed to solve singularly perturbed convection-diffusion problems posed on smooth domains. Constraints are imposed on the data so that only regular exponential boundary layers appear in the solution. A…

In this paper we study the following one-dimensional reaction-diffusion problem $$ u_t+(-\Delta)^s u=f(x-c t, u) \;\:\textrm{ in } \mathbb{R}\times (0,+\infty), $$ where $s>\frac{1}{2}$, $c \in \mathbb{R}$ is a prescribed velocity, and $f$…

Continuous diffusion models have demonstrated remarkable performance in data generation across various domains, yet their efficiency remains constrained by two critical limitations: (1) the local adjacency structure of the forward Markov…

Diffusion models are powerful tools for sampling from high-dimensional distributions by progressively transforming pure noise into structured data through a denoising process. When equipped with a guidance mechanism, these models can also…

We seek solutions $u\in\R^n$ to the semilinear elliptic partial difference equation $-Lu + f_s(u) = 0$, where $L$ is the matrix corresponding to the Laplacian operator on a graph $G$ and $f_s$ is a one-parameter family of nonlinear…

Nonintegrable systems thermalize, leading to the emergence of fluctuating hydrodynamics. Typically, this hydrodynamics is diffusive. We use the effective field theory (EFT) of diffusion to compute higher-point functions of conserved…

The theory of degenerate parabolic equations of the forms \[ u_t=(\Phi(u_x))_{x} \quad {\rm and} \quad v_{t}=(\Phi(v))_{xx} \] is used to analyze the process of contour enhancement in image processing, based on the evolution model of…

Denoising diffusion models have emerged as the go-to generative framework for solving inverse problems in imaging. A critical concern regarding these models is their performance on out-of-distribution tasks, which remains an under-explored…

We present a novel artificial diffusion method to circumvent the instabilities associated with the standard finite element approximation of convection-diffusion equations. Motivated by the micromorphic approach, we introduce an auxiliary…