Complex diffusion filtering of three dimensional turbulent flows

The linear and nonlinear complex diffusion filtering methods are proposed to extract the organized coherent part as well as the random incoherent part from forced and decaying turbulent flows. An attempt to examine the robustness of the two methods in filtering the turbulent flow field without the transformation into the frequency domain is carried out. The velocity fields of the forced and decaying cases are decomposed into coherent and incoherent parts in the spatial domain. The complex diffusion process can be obtained by combining the linear diffusion equation and the free particle Schrodinger equation. The imaginary parts in the two methods serve as a robust edge-detector with increasing confidence in time. The numerical implementations of the 3D linear and nonlinear complex diffusion partial differential equations are done using the finite difference method. The flatness, skewness and spectrum of the extracted fields are also studied for each filtering method. Results show that the two filtering methods can identify the coherent fields and preserve the features of the turbulent fields. Comparisons to the wavelet and Fourier decompositions are also considered.     

EVANS FUNCTIONS AND INSTABILITY OF A STANDING PULSE SOLUTION OF A NONLINEAR SYSTEM OF REACTION DIFFUSION EQUATIONS

In this paper, we consider a nonlinear system of reaction diffusion equations arising from mathematical neuroscience and two nonlinear scalar reaction diffusion equations under some assumptions on their coefficients. The main purpose is to couple together linearized stability criterion (the equivalence of the nonlinear stability, the linear stability and the spectral stability of the standing pulse solutions) and Evans functions to accomplish the existence and instability of standing pulse solutions of the nonlinear system of reaction diffusion equations and the nonlinear scalar reaction diffusion equations. The Evans functions for the standing pulse solutions are constructed explicitly.     

Non-self-similar solutions of multidimensional nonlinear diffusion equations

An original construction of the exact nonnegative solution of multidimensional nonlinear diffusion equations is proposed and studied. On substituting this construction into the original equation, we obtain a system of algebro-differential equations in which the number of equations exceeds the number of unknown functions. It is proved that the resulting system possesses nontrivial solutions. On the basis of this result, we construct exact non-self-similar explicit nonnegative solutions anisotropic with respect to the space variables both of the class of equations of a porous medium (nonstationary filtration) and of the class of equations involving nonlinear thermal conductivity with negative exponent. In particular, this class contains the so-called equations of rapid and limit diffusion. Translated fromMatematicheskie Zametki, Vol. 67, No. 2, pp. 250–256, February, 2000.     

圆形三向网架非线性动力稳定性分析

用拟板法将网架简化为平板,给出表层应变与中面位移的非线性关系．根据薄板的非线性动力学理论,建立了在直角坐标系中三向网架的非线性动力学方程,又将此方程转化为极坐标系轴对称非线性动力学方程．在周边固定条件下,引入异于等厚度板的无量纲量,对基本方程无量纲化．利用Galerkin法得到一个三次非线性振动方程,在无外激励情况下,讨论了稳定性与分岔问题．在外激励情况下,用Melnikov方法研究了圆形三向网架可能发生的混沌运动．通过数字仿真绘出了发生混沌的相平面图．     

Finite element analysis and application for a nonlinear diffusion model in image denoising

The stability analysis and error estimates are presented for a nonlinear diffusion model, which appears in image denoising and solved by a fully discrete time Galerkin method with kth (k ≥ 1) order conforming finite element spaces. Numerical experiments are provided with denoising several grayscale noisy images by our Galerkin method on bilinear finite elements. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 649–662, 2002; DOI 10.1002/num.10017     

A regularization of nonlinear diffusion equations in a multiresolution framework

We develop a regularization technique for Perona–Malik diffusion equations that relies on multiresolution techniques. The main result of this paper is to show that the chosen discretization overcomes the ill posedness of the nonlinear Perona–Malik model. The resulting algorithm is tested and the results are compared with pixel‐based methods. Copyright © 2007 John Wiley & Sons, Ltd.     

一类非线性时滞反应扩散系统奇摄动问题

研究了一类具有非线性时滞反应扩散系统奇摄动问题.在适当的条件下,利用比较定理讨论了问题解的渐近性态.     

Nonlinear source in diffusion filtering methods for image processing

A diffusion filtering algorithm is proposed based on the solution to an initial-boundary value problem for the two-dimensional diffusion equation with a special nonlinear source.     

THE NONLINEAR SINGULARLY PERTURBED PROBLEMS FOR REACTION DIFFUSION EQUATIONS WITH BOUNDARY PERTURBATION

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

Construction of parameterizations of masks for tight wavelet frames with two symmetric/antisymmetric generators and applications in image compression and denoising

In this paper, we present a general construction framework of parameterizations of masks for tight wavelet frames with two symmetric/antisymmetric generators which are of arbitrary lengths and centers. Based on this idea, we establish the explicit formulas of masks of tight wavelet frames. Additionally, we explore the transform applicability of tight wavelet frames in image compression and denoising. We bring forward an optimal model of masks of tight wavelet frames aiming at image compression with more efficiency, which can be obtained through SQP (Sequential Quadratic Programming) and a GA (Genetic Algorithm). Meanwhile, we present a new model called Cross-Local Contextual Hidden Markov Model (CLCHMM), which can effectively characterize the intrascale and cross-orientation correlations of the coefficients in the wavelet frame domain, and do research into the corresponding algorithm. Using the presented CLCHMM, we propose a new image denoising algorithm which has better performance as proved by the experiments.     

变质量非线性非完整系统相对运动动力学方程的积分方法

本文给出积分变质量非线性非完整系统相对于非惯性系动力学方程的梯度法,单分量法和场方法。首先,将这类问题的动力学方程表示为正则形式和场方程形式;然后,分别用梯度法,单分量法和场方法积分相应常质量完整系统相对于惯性系的动力学方程,并加上非完整约束对初始条件的限制而得到变质量非线性非完整系统相对于非惯性系动力学方程的解。     

Nonlinear reaction diffusion problems with ultra parabolic limiting equations

In this paper the nonlinear reaction diffusion problems with ultraparabolic equa- tions are considered.By using comparison theorem,the existence,uniqueness and asymptotic behavior of solution for the problem are studied.     

Nonlinear PDE Model for Image Restoration Using Second-Order Hyperbolic Equations

In this article we consider a novel nonlinear PDE-based image denoising technique. The proposed restoration model uses second-order hyperbolic diffusion equations. It represents an improved nonlinear version of a linear hyperbolic PDE model developed recently by the author, providing more effective noise removal results while preserving the edges and other image features. A rigorous mathematical investigation is performed on this new differential model and its well-posedness is treated. Next, a consistent finite-difference numerical approximation scheme is proposed for this nonlinear diffusion-based approach. Our successful image denoising experiments and method comparisons are also described.     

Error estimates of CFVE method for fully nonlinear convection‐dominated diffusion problems

Finite volume method and characteristics finite element method are two important methods for solving the partial differential equations. These two methods are combined in this paper to establish a fully discrete characteristics finite volume method for fully nonlinear convection‐dominated diffusion problems. Through detailed theoretical analysis, optimal order H¹ norm error estimates are obtained for this fully discrete scheme. Copyright © 2010 John Wiley & Sons, Ltd.     

非线性扩散耦合系统的广义解的最优判据

This paper studies coupled nonlinear diffusion equations with more general nonlinearities, subject to homogeneous Neumann boundary conditions. The necessary and sufficient conditions are obtained for the existence of generalized solutions of the system, which extend the known results for nonlinear diffusion systems with more special nonlinearities.     

A class of nonlinear singularly perturbed initial boundary value problems for reaction diffusion equations with boundary perturbation

A class of nonlinear initial boundary value problems for reaction diffusion equations with boundary perturbation is considered. Under suitable conditions and using the theory of differential inequalities the asymptotic solution of the initial boundary value problems is studied.     

The Role of Frame Force in Quantum Detection

A general method is given to solve tight frame optimization problems, borrowing notions from classical mechanics. In this article, we focus on a quantum detection problem, where the goal is to construct a tight frame that minimizes an error term, which in quantum physics has the interpretation of the probability of a detection error. The method converts the frame problem into a set of ordinary differential equations using concepts from classical mechanics and orthogonal group techniques. The minimum energy solutions of the differential equations are proven to correspond to the tight frames that minimize the error term. Because of this perspective, several numerical methods become available to compute the tight frames. Beyond the applications of quantum detection in quantum mechanics, solutions to this frame optimization problem can be viewed as a generalization of classical matched filtering solutions. As such, the methods we develop are a generalization of fundamental detection techniques in radar.      

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.     

Local convergence of quasi-Newton methods for B-differentiable equations

We study local convergence of quasi-Newton methods for solving systems of nonlinear equations defined by B-differentiable functions. We extend the classical linear and superlinear convergence results for general quasi-Newton methods as well as for Broyden's method. We also show how Broyden's method may be applied to nonlinear complementarity problems and illustrate its computational performance on two small examples.