扩展链Gel'fand-Dikii型方程族及其解

借助广义Cauchy矩阵方法,本文给出扩展链Gel’fand-Dikii(GD)型方程族,包括扩展链GD方程族和扩展修正链GD方程族.这些方程族可用定义在特定点上的标量函数S(i,j)进行表示.通过分析矩阵K和K′的特征值结构,本文得到扩展链GD型方程族的解.这些解,如孤子解和Jordan块解,均含有γ个平面波因子.     

一个组合方程的单孤子解和周期尖波解

构造一个组合方程的单孤子解和周期尖波解.应用格林函数的性质,以及求一个非线性偏微分方程(简称PDE)弱解的方法.求出了这个组合方程的单孤子解和周期尖波解,推广了前人的研究成果.     

改进偏微分方程的脑MRI图像降噪比较研究

对于噪声降低的时候传统的脑MRI医学图像降噪算法会使脑MRI医学图像的纹理、边缘和血管等的重要信息产生丢失.而偏微分方程(PDE)的脑MRI医学图像降噪算法能够在降低噪声的同时可以非常有效的缓解上述的情形确保细节的存留.主要介绍了几种PDE降噪模型.研究发现全变分的模型与四阶PDE模型降噪情况好于其余算法的降噪情况,但是水平线的生成对于成为水平集算法的初始水平集的情形较差,而四阶PDE模型迭代次数较多,运行时间长,在实际应用中有较强的限制.     

PARALLEL STABILIZATION

1.IntroductionLetusconsideranevolutionsystemwhosestateisgivenbythesolutionofaPartialDifferentialEquation(PDE)whichiswritten(formallyforthetimebeing)asIn(1.1),(1.2),whichmaybealinearoranonlinearPDE,ydenotesthestate,andvdenotesthecontrol.TheoperatorA,w...     

A QUASI-NEWTON METHOD IN INFINITE-DIMENSIONAL SPACES AND ITS APPLICATION FOR SOLVING A PARABOLIC INVERSE PROBLEM

1.IntroductionQuasi-Newtonmethodsplayanimportantroleinnumericallysolvingnon--linearsystemsofequationsontheEuclideanspaces.Blltitseemsthatthequasi-Newtonmethodshavenotbeenapplieddirectlytosolvinginverseproblemsinpartialdifferentialequations(PDE)uptonowifwe…     

构造GD设计的组合递推方法

引言 本文给出构造GD设计的一类组合递推方法;当r-λ_1=1时GD设计存在的充要条件(定理9);附表中列出在r≤10范围内新得的设计或与表[3]所列设计不同构的。 以GD[k,λ_1,λ_2,n,m]记GD设计:v=mn个处理分割为大小为n的m个(结合)组;v个处理安排在大小为k的b个区组B_j中(j=1,2,…,b),使同组的两不同处理在λ_1个区组中相遇,不同组的两个处理在λ_2个区组中相遇。这时每个处理恰出现在r个     

Adaptive Image Zo oming Algorithm Combining Total Variation Minimization and a Second-order Functional

An image zooming algorithm by using partial differential equations(PDEs) is proposed here. It combines the second-order PDE with a fourth-order PDE. The combined algorithm is able to preserve edges and at the same time avoid the blurry effect in smooth regions. An adaptive function is used to combine the two PDEs. Numerical experiments illustrate advantages of the proposed model.     

Commutation of Geometry-Grids and Fast Discrete PDE Eigen-Solver GPA

A geometric intrinsic pre-processing algorithm(GPA for short) for solving largescale discrete mathematical-physical PDE in 2-D and 3-D case has been presented by Sun(in 2022–2023). Different from traditional preconditioning, the authors apply the intrinsic geometric invariance, the Grid matrix G and the discrete PDE mass matrix B,stiff matrix A satisfies commutative operator BG = GB and AG = GA, where G satisfies G^m= I, m << dim(G). A large scale system solvers can be replaced t...     

时间周期折现Hamilton-Jacobi方程里1-周期解的存在性

主要运用PDE方法,在时间1-周期的哈密尔顿函数H(x,t,p)关于(x,t,p)连续、关于p强制且关于t,x周期、关于t线性的条件下,证明了比较定理,从而得到了时间周期折现Hamilton-Jacobi方程λu(x,t)+u_t(x,t)+H(x,t,D_xu(x,t))=0里唯一1-周期解的存在性.     

空间齐次和非齐次下活化-抑制模型动力学分析

该文讨论齐次Neumann边界条件下Gierer-Meinhardt活化-抑制扩散模型.对于空间均匀(ODE)系统,分析了内部平衡态的渐近行为及其附近极限环的存在性和稳定性;对于空间异质(PDE)系统,给出了内部平衡态的Turing不稳定性条件,说明了Turing模式和时空周期模式的存在性.最后,通过数值算例验证了相应理论结果.     

An accelerated algebraic multigrid algorithm for total-variation denoising

The variational partial differential equation (PDE) approach for image denoising restoration leads to PDEs with nonlinear and highly non-smooth coefficients. Such PDEs present convergence difficulties for standard multigrid methods. Recent work on algebraic multigrid methods (AMGs) has shown that robustness can be achieved in general but AMGs are well known to be expensive to apply. This paper proposes an accelerated algebraic multigrid algorithm that offers fast speed as well as robustness for image PDEs. Experiments are shown to demonstrate the improvements obtained.     

基于全变分与二阶泛函联合的自适应图像放大算法（英文）

An image zooming algorithm by using partial differential equations(PDEs) is proposed here. It combines the second-order PDE with a fourth-order PDE. The combined algorithm is able to preserve edges and at the same time avoid the blurry effect in smooth regions. An adaptive function is used to combine the two PDEs. Numerical experiments illustrate advantages of the proposed model.     

A PDE approach to image restoration problem with observation on a meager domain

We present here a new nonlinear PDE approach to image restoration (the inpainting problem) using a meager blurred image or a finite number of observation points. To this end, one uses a least square approach with the H⁻¹ distributional metric. Some important theoretical and numerical results are provided in this paper.     

Review of Methods Inspired by Algebraic-Multigrid for Data and Image Analysis Applications

Algebraic Multigrid (AMG) methods were developed originally for numerically solving Partial Differential Equations (PDE), not necessarily on structured grids. In the last two decades solvers inspired by the AMG approach, were developed for non PDE problems, including data and image analysis problems, such as clustering, segmentation, quantization and others. These solvers share a common principle that there is a crosstalk between fine and coarse representations of the problems, with flow of information in both directions, fine-to-coarse and coarse-to-fine. This paper surveys some of these problems and the AMG-inspired algorithms for their solution.     

Fast inversion of matrices arising in image processing

In recent years, new nonlinear partial differential equation (PDE) based approaches have become popular for solving image processing problems. Although the outcome of these methods is often very promising, their actual realization is in general computationally intensive. Therefore, accurate and efficient schemes are needed. The aim of this paper is twofold. First, we will show that the three dimensional alignment problem of a histological data set of the human brain may be phrased in terms of a nonlinear PDE. Second, we will devise a fast direct solution technique for the associated structured large systems of linear equations. In addition, the potential of the derived method is demonstrated on real-life data. This revised version was published online in June 2006 with corrections to the Cover Date.     

Adaptive level set evolution starting with a constant function

In this paper, we propose a novel level set evolution model in a partial differential equation (PDE) formulation. According to the governing PDE, the evolution of level set function is controlled by two forces, an adaptive driving force and a total variation (TV)-based regularizing force that smoothes the level set function. Due to the adaptive driving force, the evolving level set function can adaptively move up or down in accordance with image information as the evolution proceeds forward in time. As a result, the level set function can be simply initialized to a constant function rather than the widely-used signed distance function or piecewise constant function in existing level set evolution models. Our model completely eliminates the needs of initial contours as well as re-initialization, and so avoids the problems resulted from contours initialization and re-initialization. In addition, the evolution PDE can be solved numerically via a simple explicit finite difference scheme with a significantly larger time step. The proposed model is fast enough for near real-time segmentation applications while still retaining enough accuracy; in general, only a few iterations are needed to obtain segmentation results accurately.     

Dynamic PDE parametric curves

Dynamic partial differential equation (PDE) parametric curves which can be expressed as a coupled system of two hyperbolic equations are developed. In curve design, dynamic PDE parametric curves can be modified intuitively and are more flexible than ordinary differential equation (ODE) curves. The calculation of dynamic PDE parametric curves must recur to numerical methods and a three-level finite difference scheme is proposed. Approximation and stability properties for the scheme are proved and convergence property is derived. An example of interpolating PDE curves is presented as an application of dynamic PDE parametric curves.     

Extended crystal PDE’s stability,I: The general theory

This work, divided in two parts, follows some our previous works devoted to the algebraic topological characterization of PDE’s. In this first part, the stability of PDE’s is studied in some details in the framework of the geometric theory of PDE’s, and bordism groups theory of PDE’s. In particular we identify criteria to recognize PDE’s that are stable (in extended Ulam sense) and in their regular smooth solutions do not occur finite time unstabilities, (stable extended crystal PDE’s). Applications to some important PDE’s are carefully considered. (In the second part a stable extended crystal PDE, encoding anisotropic incompressible magnetohydrodynamics is obtained Ref. [A. Prástaro, Extended crystal PDE’s (submitted for publication)].)     

图像反问题中的数学与深度学习方法

我们生活在数字的时代,数据已经成为了我们生活中不可或缺的一部分,而图像无疑是最重要的数据类型之一.图像反问题,包括图像降噪,去模糊,修复,生物医学成像等,是图像科学中的重要领域.计算机技术的飞速发展使得我们可以用精细的数学和机器学习工具来为图像反问题设计有效的解决方案.本文主要回顾图像反问题中的三大类方法,即以小波（框架）为代表的计算调和分析法、偏微分方程（PDE）方法和深度学习方法.我们将回顾这些方法的建模思想和一些具体数学形式,探讨它们之间的联系与区别,优点与缺点,探讨将这些方法有机融合的可行性与优势.