识别Rayleigh-Stokes方程源项的分数阶Landweber迭代正则化方法

该文研究具有Riemann-Liouville时间分数阶导数的Rayleigh-Stokes方程未知源识别问题.首先证明这个问题是不适定的,并应用分数阶Landweber正则化方法求解此反问题.基于条件稳定性结果,在先验和后验正则化参数选取规则下,分别给出精确解与正则解之间的误差估计.最后通过数值例子说明此方法求解此类...     

一种基于积分方程的数值微分方法

本文研究了目前一些求解数值微分的方法无法求出端点导数或是求出的端点附近导数不可用的问题.利用构造一类积分方程的方法,将数值微分问题转化为这类积分方程的求解,并用一种加速的迭代正则化方法来求解积分方程. 数值实验结果表明该算法可以有效求出端点的导数,且具有数值稳定、计算简单等优点.     

双调和方程柯西问题的修正的Tikhonov正则化方法

本文研究了双调和方程柯西问题,这类是不适定的,即问题的解(如果存在)不连续依赖于测量数据.首先在精确解的先验假设下给出问题的条件稳定性结果.接着利用修正的Tikhonov正则化方法求解此不适定问题.在先验和后验正则化参数选取规则下,给出正则解和精确解之间的误差估计式.最后给出几个数值例子验证此正则化方法求解此类反问题的有效性.     

求解非光滑方程组的三次正则化方法

考虑求解非光滑方程组的三次正则化方法及其收敛性分析.利用信赖域方法的技巧,保证该方法是全局收敛的.在子问题非精确求解和BD正则性条件成立的前提下,分析了非光滑三次正则化方法的局部收敛速度.最后,数值实验结果验证了该算法的有效性.     

Poisson方程热源识别反问题

探讨了半带状区域上二维Poisson方程只含有一个空间变量的热源识别反问题.这类问题是不适定的,即问题的解(如果存在的话)不连续依赖于测量数据.利用Carasso-Tikhonov正则化方法,得到了问题的一个正则近似解,并且给出了正则解和精确解之间具有Holder型误差估计.数值实验表明Carasso-Tikhonov正则化方法对于这种热源识别是非常有效的.     

线性不适定问题的渐近正则化方法

对线性不适定问题考虑了一类近似求解方法，即渐近正则化方法，当数据精确给定时，考虑了渐近正则化解的收敛性及收敛速度，并给出了一些逆结果。如果右端数据是近似给定的，证明了所叙方法确实为正则化方法，并考虑了偏差原理对它的应用。为了使的工作更加实际可行，还考虑了算子和右端数据同时近似给定的情形，得到了一系列的结果。     

求解l1极小化问题的Bregman迭代算法

在Tikhonov正则化方法的基础上将其转化为一类l1极小化问题进行求解,并基于Bregman迭代正则化构建了Bregman迭代算法,实现了l1极小化问题的快速求解.数值实验结果表明,Bregman迭代算法在快速求解算子方程的同时,有着比最小二乘法和Tikhonov正则化方法更高的求解精度.     

非线性不适定问题的正则化方法

本文考虑非线性不适定问题Tx=y的近似求解,利用Тихоноь正则化方法来逼近问题的x-极小模解,当算子和右端都近似已知时,给出一种决定正则化参数的方法,并给出正则解的收效性和渐近收敛阶估计。     

对称锥权互补问题的正则化非单调非精确光滑牛顿法

该文提出正则化非单调非精确光滑牛顿法求解对称锥权互补问题(wSCCP).算法将正则化参数视为一个独立变量,因此它与许多现有的算法相比,更简单易实现.在每次迭代中,算法只需求得方程组的近似解.另外,算法中的非单调线搜索包含了两种常用的非单调形式.在单调假设下,证明算法全局收敛且局部二阶收敛.最后,一些数值结果表明了算法的...     

一类球型区域上变系数反向热传导问题

考虑了一类球型区域上变系数反向热传导问题.这个问题是不适定的,即问题的解(若存在)并不连续依赖于测量数据.构造了投影迭代正则化方法,得到了该反问题的正则近似解,同时给出了在先验和后验参数选取规则下精确解与正则近似解之间的收敛性误差估计.最后,通过数值结果验证了该方法的有效性.     

Recovering the local volatility in Black–Scholes model by numerical differentiation

In this article, a numerical method for recovering the local volatility in Black–Scholes model is proposed based on the Dupire formula in which the numerical derivatives are used. By Tikhonov regularization, a new numerical differentiation method in two-dimensional (2-D) case is presented. The convergent analysis and numerical examples are also given. It shows that our method is efficient and stable.     

Numerical differentiation from a viewpoint of regularization theory

In this paper, we discuss the classical ill-posed problem of numerical differentiation, assuming that the smoothness of the function to be differentiated is unknown. Using recent results on adaptive regularization of general ill-posed problems, we propose new rules for the choice of the stepsize in the finite-difference methods, and for the regularization parameter choice in numerical differentiation regularized by the iterated Tikhonov method. These methods are shown to be effective for the differentiation of noisy functions, and the order-optimal convergence results for them are proved.

     

First and second order numerical differentiation with Tikhonov regularization

This work deals with the numerical differentiation for an unknown smooth function whose data on a given set are available. The numerical differentiation is an ill-posed problem. In this work, the first and second derivatives of the smooth function are approximated by using the Tikhonov regularization method. It is proved that the approximate function can be chosen as a minimizer to a cost functional. The existence and uniqueness theory of the minimizer is established. Errors in the derivatives between the smooth unknown function and the approximate function are obtained, which depend on the mesh size of the grid and the noise level in the data. The numerical results are provided to support the theoretical analysis of this work. Selected from Numerical Mathematics (A Journal of Chinese Universities), 2004, 26(1):62–74     

STABILIZED NUMERICAL APPROXIMATIONS OF THE BACKWARD PROBLEM OF A PARABOLIC EQUATION

1　IntroductionLetΩ be a bounded domain in Rn and Ω be its boundary.ThenΣ =Ω× ( 0 ,1 ) is abounded domain in Rn+1 .We consider the following backwad problem of a prabolic equa-tion: u t= ni,j=1 xiaij( x) u xj -c( x) u,　　 ( x,t)∈Σ,( 1 )u| Ω× [0 ,1 ] =0 , ( 2 )u| t=1 =g( x) . ( 3 )　　 Where { aij( x) } are smooth functions given onΩ satisfyingaij( x) =aji( x) ,　　 1≤ i,j≤ n, ( 4)α0 ni=1ζ2i ≤ ni,j=1aij( x)ζiζj≤α1 ni=1ζ2i,　　 ζ∈ Rn,x∈Ω. ( 5)　　Where0 <α…     

正则参数求解的微分进化算法

研究了正则化方法中正则参数的求解问题,提出了利用微分进化算法获取正则参数.微分进化算法属于全局最优化算法,具有鲁棒性强、收敛速度快、计算精度高的优点.把正则参数的求解问题转化为非线性优化问题,通过保持在解空间不同区域中各个点的搜索,以最大的概率找到问题的全局最优解,同时还利用数值模拟将此方法与广义交叉原理、L-曲线准则、逆最优准则等进行了对比,数值模拟结果表明该方法具有一定的可行性和有效性.     

用Tikhonov正则化方法求一阶和两阶的数值微分

Numerical differentiation is an ill-posed problem, which is important in scientific research and practical applications.In this paper, we use the Tikhonov regularization method to discuss the first and secord order derivatives of a smooth function. The error estimate is also given. And the numerical results prove that our method is applicable.     

Numerical differentiation by radial basis functions approximation

Based on radial basis functions approximation, we develop in this paper a new com-putational algorithm for numerical differentiation. Under an a priori and an a posteriori choice rules for the regularization parameter, we also give a proof on the convergence error estimate in reconstructing the unknown partial derivatives from scattered noisy data in multi-dimension. Numerical examples verify that the proposed regularization strategy with the a posteriori choice rule is effective and stable to solve the numerical differential problem. *The work described in this paper was partially supported by a grant from CityU (Project No. 7001646) and partially supported by the National Natural Science Foundation of China (No. 10571079).     

Adaptive Choice of the Regularization Parameter in Numerical Differentiation

We investigate a novel adaptive choice rule of the Tikhonov regularization parameter in numerical differentiation which is a classic ill-posed problem. By assuming a general unknown Hölder type error estimate derived for numerical differentiation, we choose a regularization parameter in a geometric set providing a nearly optimal convergence rate with very limited a-priori information. Numerical simulation in image edge detection verifies reliability and efficiency of the new adaptive approach.     

On the extraction technique in boundary integral equations

In this paper we develop and analyze a bootstrapping algorithm for the extraction of potentials and arbitrary derivatives of the Cauchy data of regular three-dimensional second order elliptic boundary value problems in connection with corresponding boundary integral equations. The method rests on the derivatives of the generalized Green's representation formula, which are expressed in terms of singular boundary integrals as Hadamard's finite parts. Their regularization, together with asymptotic pseudohomogeneous kernel expansions, yields a constructive method for obtaining generalized jump relations. These expansions are obtained via composition of Taylor expansions of the local surface representation, the density functions, differential operators and the fundamental solution of the original problem, together with the use of local polar coordinates in the parameter domain. For boundary integral equations obtained by the direct method, this method allows the recursive numerical extraction of potentials and their derivatives near and up to the boundary surface.

     

关于一类广义Tikhonov 正则化方法的饱和效应分析

Tikhonov正则化方法是研究不适定问题最重要的正则化方法之一,但由于这种方法的饱和效应出现的太早,使得无法随着对解的光滑性假设的提高而提高正则逼近解的收敛率,也即对高的光滑性假设,正则解与准确解的误差估计不可能达到阶数最优.Schrroter T 和Tautenhahn U给出了一类广义Tikhonov正则化方法并重点讨论了它的最优误差估计, 但却未能对该方法的饱和效应进行研究.本文对此进行了仔细分析,并发现此方法可以防止饱和效应,而且数值试验结果表明此方法计算效果良好.