一类偏积分-微分方程中的反问题

对一类偏积分-微分方程中参数校准的反问题进行研究.在弱解的框架下,原问题可转化为含具体IE则化项的最优化问题.文中证明了该最优化问题的解的1竽在性和稳定性,并考察了最优解存在的一阶必要条件.另外,证明了当正则化参数足够人时,该最优化问题关于参数a的凸性性质.基于偏积分.微分方程反问题的研究对于金融市场中的模型校准问题具有重要的意义.     

偏微分方程在图像去噪中的应用

本文介绍用于图像去噪的偏微分模型、方法的发展历程．从理论上分析了线性模型、简单非线性模型、复杂非线性模型、多步处理模型出现的背景和优缺点，并从空域和频域上对偏微分方程模型的去噪原理进行了分析．最后，指出了偏微分方程去噪与小波去噪结合的途径，据此对偏微分方程未来的发展方向进行了展望．     

一类线性积分—偏微分方程Cauchy问题的求解公式及其应用

本文讨论了一类线性积分一偏微分方程Cauchy问题，得出了它们的求解公式，并举例说明公式的应用。     

一类非线性积分偏微分方程的初值问题

讨论初值问题 整体经典解的存在性。该问题来源于粘弹性力学。在关于已知函数的一些正则性假设和p'(s)≥c1>0,|q'(s)|≤const,λ(0)2(0)的条件下,通过能量估计,证明了该问题整体经典解的存在性。     

带跳反射倒向随机微分方程与相应的 积分-偏微分方程障碍问题

证明由 Brown 运动和 Poisson 随机测度共同驱动的终端为停时的反射倒向随机微分方程存在唯一解,并且在 Markov框架下该解为积分-偏微分方程障碍问题黏性解提供了概率解释.     

一维双曲型积分微分方程的一个反问题

本文研究一类一维双曲型微分方程的一个反问题,即确定方程ｕｔｔ（ｘ,ｔ）－ｕｘｘ（ｘ,ｔ）＝∫ｋ（τ）ｕ（ｘ,ｔ－τ）ｄτ＋ｆ（ｘ,ｔ）中的ｕ（ｘ,ｔ）和积分核ｋ（ｔ）,得到了解的存在唯一性。     

二维椭圆型偏微分方程的反势问题

将Radon变换及其反投影变换原理应用于二维椭圆型偏微分方程反势问题的求解，从另一个角度解决了小扰动情况下椭圆型偏微分方程的反势问题．     

一类二变量积分不等式及其在积分-微分方程中的应用

建立了一类二变量的积分不等式,该不等式包含了一个一重积分和两个二重积分.利用分析技巧,给出了积分不等式中未知函数的估计.这一结果可以作为研究积分-微分方程解的定性性质的工具.     

一类非线性积分偏微分方程初边值问题的整体解

讨论初边值问题整体经典解的存在性．在P′（s）≥0,p′（s）─q′（s）|≤const．的条件下,用Galerkin方法证明了该问题整体经典解的存在唯一性．     

一类偏积分微分方程二阶差分全离散格式

本文给出了数值求解一类偏积分微分方程的二阶差分全离散格式．时间方向采用了二阶向后差分格式,积分项的离散利用了Lubich的二阶卷积求积公式,给出了稳定性的证明、误差估计及收敛性的结果,并给出了数值例子．     

Wavelet regularization for an inverse heat conduction problem

In this paper we consider an inverse heat conduction problem which appears in some applied subjects. This problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. The Meyer wavelets are applied to formulate a regularized solution which is convergent to exact one on an acceptable interval when data error tends to zero.     

LOCAL REGULARIZATION METHODS FOR THE STABILIZATION OF LINEAR ILL-POSED EQUATIONS OF VOLTERRA TYPE

Inverse problems based on first-kind Volterra integral equations appear naturally in the study of many applications, from geophysical problems to the inverse heat conduction problem. The ill-posedness of such problems means that a regularization technique is required, but classical regularization schemes like Tikhonov regularization destroy the causal nature of the underlying Volterra problem and, in general, can produce oversmoothed results. In this paper we investigate a class of local regularization methods in which the original (unstable) problem is approximated by a parameterized family of well-posed, second-kind Volterra equations. Being Volterra, these approximating second-kind equations retain the causality of the original problem and allow for quick sequential solution techniques. In addition, the regularizing method we develop is based on the use of a regularization parameter which is a function (rather than a single constant), allowing for more or less smoothing at localized points in the domain. We take this approach even further by adopting the flexibility of an additional penalty term (with variable penalty function) and illustrate the sequential selection of the penalty function in a numerical example.     

Arnoldi–Tikhonov regularization methods

Tikhonov regularization for large-scale linear ill-posed problems is commonly implemented by determining a partial Lanczos bidiagonalization of the matrix of the given system of equations. This paper explores the possibility of instead computing a partial Arnoldi decomposition of the given matrix. Computed examples illustrate that this approach may require fewer matrix–vector product evaluations and, therefore, less arithmetic work. Moreover, the proposed range-restricted Arnoldi–Tikhonov regularization method does not require the adjoint matrix and, hence, is convenient to use for problems for which the adjoint is difficult to evaluate.     

ON STABILITY AND REGULARIZATION FOR BACKWARD HEAT EQUATION

Consider a backward heat equation in a bounded domain Ω (?) R2 with the noisy data in the initial time geometry. The aim is to find the temperature for 0 < ε < t < T. For this ill-posed problem, the authors give a continuous dependence estimate of the solution. Moreover, the convergence rate of the approximate solution is also given.     

A control type method for solving the Cauchy–Stokes problem

In this work, we propose an approximate optimal control formulation of the Cauchy problem for the Stokes system. Here the problem is converted into an optimization one. In order to handle the instability of the solution of this ill-posed problem, a regularization technique is developed. We add a term in the least square function which happens to vanish while the algorithm converges. The efficiency of the proposed method is illustrated by numerical experiments.     

A modified method for a non-standard inverse heat conduction problem

A non-standard inverse heat conduction problem is considered. Data are given along the line x = 1 and the solution at x = 0 is sought. The problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. In order to solve the problem numerically it is necessary to employ some regularization method. In this paper, we study a modification of the equation, where a fourth-order mixed derivative term is added. Error estimates for this equation are given, which show that the solution of the modified equation is an approximation of the heat equation. A numerical implementation is considered and a simple example is given. Some numerical results show the usefulness of the modified method.     

一个抛物型方程不适定问题的小波正则化方法

一维抛物型方程如下定解问题狌狋＋狌狓＝狌狓狓, ０≤狓＜ ∞,０≤狋＜ ∞,狌（１,狋）＝犵（狋）, ０≤狋＜ ∞,狌（狓,０）＝０, 狓≥０烅烄烆．是一个不适定问题．数据犵的微小变化可以引起解的巨大误差．该文通过构造一个在频域具紧支集的小波并在尺度空间上展开数据和解,滤除了高频分量,并结和Ｇａｌｅｒｋｉｎ方法,建立了一种逼近准确解的正则化方法,恢复了解对数据的连续依赖性,并建立了误差估计．     

Two regularization methods for identifying the unknown source of Sobolev equation with fractional Laplacian

In this paper, an inverse source problem for the Sobolev equation with fractional Laplacian is investigated. We prove that this kind of problem is ill-posed and apply the Quasi-boundary regularization method and fractional Landweber iterative regularization method to solve this inverse problem. Based on the result of conditional stability, the error estimates between the exact solution and the regularization solution are given under the priori and posteriori regularization parameter selection rules. Finally, three examples are given to illustrate the effectiveness and feasibility of these methods.