全变差正则化在抛物型方程初始条件反问题的应用

当反问题反演的函数不连续时,一般的正则化算法反演效果不令人满意,用全变差正则化方法对抛物型方程初始条件反问题进行求解,并进行了数值分析和数值模拟,结果显示数值解与真解吻合较好,表明该方法对于不连续函数求解具有高效、稳定等优点.     

求解Symm积分方程的信赖域Lanczos法

Symm积分方程在位势理论中具有重要的作用,它是Hadamard意义下的不适定问题.本文基于信赖域算法并结合Lanczos迭代方法,得到了Symm积分方程的数值解法,与通常的正则化方法相比,在数据出现噪声的情况下,该方法克服了正则化方法中正则参数选取的困难,同时具有较高的精度.     

一维热传导问题时变边界上热通量重构问题

具有Neumann边界条件的抛物方程的初边值问题是偏微分方程研究领域的一类经典问题.正问题是由已知的边界条件和初始条件来求区域温度场的问题.如果边界条件不足,但给出了区域内部的一些额外信息,这样便构成了一类热通量重构的反问题.本文讨论了一维热传导问题时动边界上的热通量重构问题,借助于位势理论方法,引入密度函数,将反问题本质上转化为一类关于密度函数的具有弱奇性核的第一类Volterra积分方程,采用了Tikhonov正则化,在正则化参数的选取上采用了后验的模型函数方法,数值结果验证了反演方法的有效性.     

Fredholm积分方程的正则化GMRES算法

利用数值求积公式,对二维第1类Fredholm积分方程进行离散处理,引入正则化GMRES算法,将离散后的积分方程转化为离散适定问题,通过广义极小残余算法得到其数值解。数值模拟结果表明,正则化GMRES算法求解二维第1类Fredholm积分方程计算速度快、精度高。     

数值求导的离散正则化方法

数值微分作为一个典型的反问题,在Hadamard意义下是不适定的,即在求导中函数的微小扰动就可能导致计算上很大的误差。本文首次利用目前处理不适定问题的、广为采用且相当有效的Tikhonov 正则化方法,讨论了用离散正则化方法处理数值求导的有关理论和技术问题,包括离散正则解的收敛性、稳定性以及在原始数据误差水平已知和未知情况下的正则参数选取问题,给出了稳定和有效的算法,并在Matlab环境下加以实现,而且进行了成功的数值试验和对比试验研究。理论分析和数值试验表明：利用本文所给给的离散正则化算法求导,具有精度高和数值稳定性好、抗干扰性能强等优点。     

电磁逆散射成像的一种混合正则化方法

电磁逆散射成像问题数值求解中,非线性逆散射方程及其对应的离散方程组具有明显的不适定性,针对求解通常所用Tikhonov正则化方法的参数选择在先验选取时缺乏有效的误差信息,而后验选取时需要更多计算量求解有关参数的方程的困难,本文中将小参数Tikhonov正则化方法与共轭梯度法结合,提出了不适定方程组的混合正则化方法。数据仿真表明,该方法既可保证正则化效果,也减少了计算量。     

超声逆散射图象重建问题中截断奇异值分解正则化方法研究

为解决超声逆散射成像问题中的非线性性,人们需要反复地求解前向散射方程和逆散射方程,以达到对全场和未知函数的精确近似,从而根据这一未知函数的精确近似,较好地重建物体内部的断层图象.前向散射方程是一个适定的方程组,可以采用通常的方法进行求解;而逆散射方程则是一个不适定性的方程组,即使数据中存在一个微小的误差,都可能引起解的较大偏离,因此,对这个不适定方程组的求解问题是整个迭代算法成功的关键.而在不适定性问题的求解过程中,正则化参数的选取又是非常重要的.求解不适定性方程的传统方法是Tikhonov正则化方法,这一方法的实质是在传统最小二乘方法上加上一个小于1的滤波因子,对于超声逆散射成像问题来说,效果并不太好.本文将截断奇异值分解正则化方法应用于逆散射方程的求解问题中,并对正则化参数的选取方法进行修正.数值仿真结果表明,这一方法配合适当的正则化参数选取,可以更好地滤除噪声,提高重建图象的质量与可信度,同时还可以减小迭代过程中的计算量.     

用演化算法求解抛物型方程扩散系数的识别问题

基于演化算法给出了一类求解参数识别反问题的一般方法,该方法表明只要找到好的、求解相应的正问题的数值方法,演化算法就可以用于求解此类反问题。设计有效的求解反问题的演化算法的关键是寻找一种适合反问题的解空间的编码表示形式、适当的适应值函数形式以及有效的计算正问题的数值方法。该文结合算法、传统的求解反问题的工方法和正则化技术,设计了一类求解参数识别反问题的方法。为验证此类方法,将其用于求解一维扩散方程的     

同伦支持向量机

支持向量机(SVM)作为一种新颖的人工智能技术,已越来越广泛地运用于各个学科领域。同伦正则化方法也是近几年新兴的一种正则化方法,在数理方程反问题中得到了广泛的应用。将同伦正则化的思想应用到支持向量机中,建立了一种新的支持向量机模型,并对最常用的核函数——高斯核进行了修改。与传统的正则化方法相比,新模型最大的优点就是正则化参数的取值范围由无限区间变成了有限区间(0,1),从而大大缩短了正则化参数的优化时间。     

求解二维Fredholm积分方程的参数化信赖域方法

给出了求解二维第一类Fredholm积分方程信赖域方法。通过引入正则化参数将离散后的Fredholm积分方程转化带参数的最优化问题,借助于KKT条件将二次信赖域子问题参数化,并进行分析求解,最后给出了数值模拟。     

非线性演化方程的新Jacobi椭圆函数解

基于sinh-Gordon方程的椭圆函数解,构造新的试探解来扩展sinh-Gordon方程展开法.利用该方法研究了KdV-mKdV方程,双sine-Gordon方程和BBM方程,获得了这些方程的新Jacobi椭圆函数解.该方法也能用来求解其他数学物理中的非线性演化方程.     

A Class of Sparse Spectral Operators for Inversion of Powers of the Laplacian in N Dimensions

For inversion of the Laplacian subject to Dirichlet boundary conditions and, more generally, for the kth power of the Laplacian subject to boundary conditions on the function and its first k – 1 derivatives in the normal coordinate, there is a sparse symmetric, well-conditioned, projection of the operator that results from an expansion in associated Legendre polynomials.     

Zakai equation of nonlinear filtering with unbounded coefficients. The case of dependent noises

In this paper, we prove that the unnormalized filter associated with nonlinear filtering problems with dependent noises and a one-dimensional observations process the coefficients of which are unbounded solves a parabolic stochastic partial differential equation, the Zakai equation. The robust form of the Zakai equation is also computed.     

Numerical solution of GRLW equation using Sinc-collocation method

In this paper, we present a meshfree technique for the numerical solution of the generalized regularized long wave (GRLW) equation. This approach is based on a global collocation method using Sinc basis functions. The propagation of single solitons and the interaction of two solitary waves are used to validate the method which is found to be accurate and efficient. The three invariants of the motion are evaluated to determine the conservation properties of the method.     

Matrix algorithms for solving (in)homogeneous bound state equations

In the functional approach to quantum chromodynamics, the properties of hadronic bound states are accessible via covariant integral equations, e.g. the Bethe–Salpeter equation for mesons. In particular, one has to deal with linear, homogeneous integral equations which, in sophisticated model setups, use numerical representations of the solutions of other integral equations as part of their input. Analogously, inhomogeneous equations can be constructed to obtain off-shell information in addition to bound-state masses and other properties obtained from the covariant analogue to a wave function of the bound state. These can be solved very efficiently using well-known matrix algorithms for eigenvalues (in the homogeneous case) and the solution of linear systems (in the inhomogeneous case). We demonstrate this by solving the homogeneous and inhomogeneous Bethe–Salpeter equations and find, e.g. that for the calculation of the mass spectrum it is as efficient or even advantageous to use the inhomogeneous equation as compared to the homogeneous. This is valuable insight, in particular for the study of baryons in a three-quark setup and more involved systems.     

On the quantum master equation under feedback control

The nature of the quantum trajectories, described by stochastic master equations, may be jump-like or diffusive, depending upon different measurement processes. There are many different unravelings corresponding to different types of stochastic master equations for a given master equation. In this paper, we study the relationship between the quantum stochastic master equations and the quantum master equations in the Markovian case under feedback control. We show that the corresponding unraveling no longer exists when we further consider feedback control besides measurement. It is due to the fact that the information gained by the measurement plays an important role in the control process. The master equation governing the evolution of ensemble average cannot be restored simply by eliminating the noise term unlike the case without a control term. By establishing a fundamental limit on performance of the master equation with feedback control, we demonstrate the differences between the stochastic master equation and the master equation via theoretical proof and simulation, and show the superiority of the stochastic master equation for feedback control.     

求一类非线性偏微分方程解析解的简洁构造算法

通过引入一个变换,利用齐次平衡原理和选准一个待定函数来构造求解一类非线性偏微分方程解析解的算法.作为实例,我们将该算法应用到了mKdV方程,KdV-Burgers方程和KdV-Burgers-Kuramoto方程.借助符号计算软件Mathematica获得了这些方程的解析解.不难看出,该方法不仅简洁,而且有望进一步扩展.     

Numerical scheme based on similarity reductions for the regularized long wave equation

Numerical solution based on similarity reductions for partial differential equations used to get the numerical scheme for the regularized long wave (RLW) equation. The similarity reductions for RLW equation are obtained locally on subdomains defined by the classical three-point stencil. The ordinary differential equation, which deduced from the similarity reduction can be linearized, integrated analytically and then obtain the solution. This approch eliminates the difficulties associated with boundary conditions for the similarity reduction over the whole solution domain. Numerical results are obtained for test problem. The computed results using our scheme confirm the accuracy of our scheme.     

An analytic algorithm of Lane-Emden type equations arising in astrophysics using modified Homotopy analysis method

Lane-Emden type equation models many phenomena in mathematical physics and astrophysics. It is a nonlinear differential equation which describes the equilibrium density distribution in self-gravitating sphere of polytropic isothermal gas, has a singularity at the origin, and is of fundamental importance in the field of stellar structure, radiative cooling, modeling of clusters of galaxies. An efficient analytic algorithm is provided for Lane-Emden type equations using modified homotopy analysis method, which is different from other analytic techniques as it itself provides us with a convenient way to adjust convergence regions even without Pade technique. Some examples are given to show its validity.     

正倒向随机微分方程与一类线性二次随机最优控制问题

讨论一类正倒向随机微分方程解的存在唯一性及其对应的一类线性二次随机最优控制 问题,利用单调性方法证明了一类特殊的正倒向随机微分方程解的存在唯一性定理,利用该结果 研究一类耦合了一个倒向随机微分方程的线性随机控制系统广义最优指标随机控制问题,得到 由正倒向随机微分方程的解所表示的唯一最优控制的显式表达式,并得到精确的线性反馈及其 对应的Riccati方程.