一种基于小波分析的各向异性图象去噪方法

本文利用非线性各向异性扩散方程结合小波变换提出一种图象去噪的方法。首先对图像进行离散小波变换,然后对其各个分量分别用各向异性的方法实现去噪。实验结果表明,该方法能够较好的去除噪声的同时,很好的保留边缘信息。     

Quasi-neutral Limit of a Nonlinear Drift Diffusion Model for Semiconductors

The limit of the vanishing Debye length (the charge neutral limit) in a nonlinear bipolar drift-diffusion model for semiconductors without a pn-junction (i.e., with a unipolar background charge) is studied. The quasi-neutral limit (zero-Debye-length limit) is determined rigorously by using the so-called entropy functional which yields appropriate uniform estimates.     

Optimal Control Approach to Nonlinear Diffusion Equations Driven by Wiener Noise

The stochastic nonlinear infinite-dimensional equations of gradient type and with additive Wiener noise can be reduced to an optimal convex control problem via Brezis–Ekeland duality device. This approach is illustrated here on a few classes of nonlinear stochastic parabolic equations which are relevant as diffusion models under stochastic Gaussian perturbations, and image restoring technique.     

A Nonlinear Diffusion Problem with Localized Large Diffusion

Abstract

This paper is concerned with a nonlinear parabolic problem, with nonlinear boundary conditions, for which the diffusion coefficient becomes very large in a sub-region of the physical domain.     

半导体中非线性漂流扩散模型的拟中性极限:快扩散情形

本文研究无Pn-联结的非线性双极半导体漂流扩散模型的消失Debye长度极限(即粒子中性极限)问题．使用熵方法和弱紧性方法从数学上严格证明了快扩散情形的拟中性极限．     

A Nonlinear Diffusion System with Coupled Nonlinear Boundary Flux

This paper studies a nonlinear diffusion system with coupled nonlinear boundary flux and two kinds of inner sources （positive for the first and negative for the second）, where the four nonlinear mechanisms are described by eight nonlinear parameters. The critical exponent of the system is determined by a complete classification of the eight nonlinear parameters, which is represented via the characteristic algebraic system introduced to the problem.     

A Nonlinear Parabolic Equation with Noise

In this paper we study a class of parabolic equations with a nonlinear gradient term. The system is disturbed by white noise in time. We show that the unique solution of this problem can be represented as the Wick product between a normalized random variable of exponential form and the solution of a nonlinear parabolic equation. We allow random initial data which might be anticipating. A relation between the Wick product with a normalized exponential and translation is proved in order to establish our results.     

具有非线性扩散项的多维趋化趋触模型的整体有界性

本文主要研究一类带有齐次Neumann边界条件且具有非线性扩散项的趋化趋触模型,在较宽的条件下,证明了系统具有整体有界古典解.推广了XU等(2019)和JIA等(2020)得到的整体有界的古典解的结论.     

Nonlinear Filtering of Diffusion Processes in Correlated Noise: Analysis by Separation of Variables

Abstract. An approximation to the solution of a stochastic parabolic equation is constructed using the Galerkin approximation followed by the Wiener chaos decomposition. The result is applied to the nonlinear filtering problem for the time-homogeneous diffusion model with correlated noise. An algorithm is proposed for computing recursive approximations of the unnormalized filtering density and filter, and the errors of the approximations are estimated. Unlike most existing algorithms for nonlinear filtering, the real-time part of the algorithm does not require solving partial differential equations or evaluating integrals. The algorithm can be used for both continuous and discrete time observations. \par     

A Duality Approach to Nonlinear Diffusion Equations

We provide new existence results for a nonlinear diffusion equation with a monotonically increasing multivalued time-dependent nonlinearity, under minimal growth and coercivity conditions. The results given in this paper prove that a generalized solution to the nonlinear equation is provided by a solution to an equivalent minimization problem for a convex functional involving the potential of the nonlinearity and its conjugate, in the case when the potential is time and space depending. If the potential is time depending only and it has a symmetry at infinity, the null minimizer in the minimization problem is found to coincide with a weak solution to the nonlinear equation.     

Nonlinear Filtering of Diffusion Processes in Correlated Noise: Analysis by Separation of Variables

   Abstract. An approximation to the solution of a stochastic parabolic equation is constructed using the Galerkin approximation followed by the Wiener chaos decomposition. The result is applied to the nonlinear filtering problem for the time-homogeneous diffusion model with correlated noise. An algorithm is proposed for computing recursive approximations of the unnormalized filtering density and filter, and the errors of the approximations are estimated. Unlike most existing algorithms for nonlinear filtering, the real-time part of the algorithm does not require solving partial differential equations or evaluating integrals. The algorithm can be used for both continuous and discrete time observations. \par     

A Nonlinear Filter with Fractional Gaussian Noise

Using a measure change, an exact estimate and an approximate recursive estimate are obtained for the conditional density of a hidden signal and a parameter in a state space model, where the hidden signal has deterministic dynamics and it is observed in fractional Gaussian noise.     

特慢扩散的一种分数阶结构导数模型

自然界和工程中存在很多比幂率慢扩散(sub-diffusion)过程更慢的扩散,即特慢扩散(ultra-slow diffusion).特慢扩散难以用传统的反常扩散建模方法来描述.Sinai(西奈)随机模型描述了一种特殊的对数关系特慢扩散.运用Mittag-Leffler(米塔格-累夫勒)函数的反函数,将Sinai扩散拓展为一般的特慢扩散.此外,该文的模型引入初始状态参量,解决了Sinai对数扩散不适用于初始时刻附近的问题.作为分数阶导数的一般情况,该文也引入了分数阶结构导数的概念,并用来建立特慢扩散的控制微分方程.     

Nonclassical Symmetries for Nonlinear Diffusion and Absorption

Nonclassical symmetry methods are used to study the nonlinear diffusion equation with a nonlinear source. In particular, exponential and power law diffusivities are examined and we obtain mathematical forms of the source term which permit nonclassical symmetry reductions. In addition to the known source terms obtained by classical symmetry methods, we find new source terms which admit symmetry reductions. We also deduce a class of nonclassical symmetries which are valid for arbitrary diffusivity and deduce corresponding new solution types. This is an important result since previously only traveling wave solutions were known to exist for arbitrary diffusivity. A number of examples are considered and new exact solutions are constructed.     

非线性扩散方程的一个新的近似解

本文研究了文献[1]中的同样问题.本文作者的解近似地满足全部基本方程(1.1)和(1.2)和全部边界条件(1.3)～(1.5).而刘氏的解却不满足连续性方程(1.2).     

Data-Driven Tight Frame Construction for Impulsive Noise Removal

The method of data-driven tight frame has been shown very useful in image restoration problems.We consider in this paper extending this important technique,by incorporating L₁ data fidelity into the original data-driven model,for removing impulsive noise which is a very common and basic type of noise in image data.The model contains three variables and can be solved through an efficient iterative alternating minimization algorithm in patch implementation,where the tight frame is dynamically updated.It constructs a tight frame system from the input corrupted image adaptively,and then removes impulsive noise by the derived system.We also show that the sequence generated by our algorithm converges globally to a stationary point of the optimization model.Numerical experiments and comparisons demonstrate that our approach performs well for various kinds of images.This benefits from its data-driven nature and the learned tight frames from input images capture richer image structures adaptively.     

Rigorous Derivation of a Nonlinear Diffusion Equation as Fast-Reaction Limit of a Continuous Coagulation-Fragmentation Model with Diffusion

Weak solutions of the spatially inhomogeneous (diffusive) Aizenmann–Bak model of coagulation-breakup within a bounded domain with homogeneous Neumann boundary conditions are shown to converge, in the fast reaction limit, towards local equilibria determined by their mass. Moreover, this mass is the solution of a nonlinear diffusion equation whose nonlinearity depends on the (size-dependent) diffusion coefficient. Initial data are assumed to have integrable zero order moment and square integrable first order moment in size, and finite entropy. In contrast to our previous result [5 Carrillo , J. A. , Desvillettes , L. , Fellner , K. ( 2008 ). Fast-reaction limit for the inhomogeneous aizenman-bak model . Kinetic and Related Models 1 : 127 – 137 . [Google Scholar]], we are able to show the convergence without assuming uniform bounds from above and below on the number density of clusters.     

A Nonlinear Singularly Perturbed Problem for Reaction Diffusion Equations with Boundary Perturbation

A nonlinear singularly perturbed problems for reaction diffusion equation with boundary perturbation is considered. Under suitable conditions, the asymptotic behavior of solution for the initial boundary value problems of reaction diffusion equations is studied using the theory of differential inequalities.     

Higher-Order Relations for Derivatives of Nonlinear Diffusion Semigroups

We show that a special choice of the Cameron–Martin direction in the characterization of the Wiener measure via the formula of integration by parts leads to a set of natural representations for derivatives of nonlinear diffusion semigroups. In particular, we obtain a final solution of the non-Lipschitz singularities in the Malliavin calculus.     

Self-similar Asymptotics for Linear and Nonlinear Diffusion Equations

The long-time asymptotic solutions of initial value problems for the heat equation and the nonlinear porous medium equation are self-similar spreading solutions. The symmetries of the governing equations yield three-parameter families of these solutions given in terms of their mass, center of mass, and variance. Unlike the mass and center of mass, the variance, or “time-shift,” of a solution is not a conserved quantity for the nonlinear problem. We derive an optimal linear estimate of the long-time variance. Newman's Lyapunov functional is used to produce a maximum entropy time-shift estimate. Results are applied to nonlinear merging and time-dependent, inhomogeneously forced diffusion problems.