求解非线性反问题的大范围收敛梯度正则化算法

基于同伦映射的思想，改进了求解非线性反问题的梯度正则化算法。通过路径跟踪有效地拓宽了梯度正则化算法求解的收敛范围。对于正则化参数的修正，通过引入拟Sigmoid函数，提出了一种下降速率可调的连续化参数修正方法，在保证迭代稳定的条件下，得到较好的计算效率，同时保证该算法具有很好的抵抗观测噪声能力。实际算例表明，该方法收敛范围宽，计算效率高，在存在较强观测噪声的条件下也能得到很好的反演结果。     

一种用于重力梯度动态测量的载体环境引力梯度补偿方法

重力梯度仪动态测量时,重力梯度敏感器一直稳定在地理坐标系下,载体姿态变化使载体质量分布相对敏感器的位置发生变化,形成载体环境引力梯度变化。为提高重力梯度仪动态测量精度,提出一种基于Tikhonov正则化的载体环境引力梯度补偿方法。首先,推导了载体环境引力梯度的解析模型,建立了引力梯度变化的回归方程。然后,针对回归算子病态性问题,提出了Tikhonov正则化方法,通过半物理仿真确定最优正则化参数,使补偿量的误差控制在2%以内。最后,利用该参数处理船载试验实测数据,结果表明:所提出的方法对载体环境梯度变化补偿具有明显的效果,可将两路重力梯度测量信号内符合中误差分别降低19 E和21 E,补偿后重力梯度测量精度达到10 E的精度水平。     

基于正则化等效源模型的航空重力梯度测量信息降噪方法

航空重力梯度测量能获取重力梯度的多个分量,在保证各梯度分量内部固有约束条件下降低测量噪声,是一个巨大的挑战。为提高航空重力梯度测量精度,提出一种基于正则化等效源模型的测量降噪方法。根据矩形棱柱体与重力梯度之间的关系构建了等效源线性方程组,针对该方程组可能存在病态性的问题,引入截断奇异值TSVD和Tikhonov正则化的方法,对等效源模型进行正则化改造。模拟试验表明,正则化方法能够有效抑制病态系数矩阵小奇异值放大噪声对未知参数的污染,提高反演的精度和稳定性,其中基于L曲线确定正则化参数的截断奇异值TSVD法精度较高。实测数据表明,利用正则化等效源滤波转换的垂直梯度与利用傅里叶变换转换的结果吻合。可见,正则化等效源是重力场滤波和分量转换的有效工具。     

微分方程反问题的梯度正则化法

本文在微分方程反问题研究当前发展状况的基础上,提出了一种求解微分方程反问题的比较统一的方法——梯度正则化法。该方法将反问题的求解分为两个主要部分,一是展开以求补充条件对未知参数的梯度矩阵,二是正则化以解不适定的线性方程组。梯度正则化法从反问题的共性入手,未附加任何特殊的约束,所以可以适用各种类型的反问题,且在求解时不受空间维数的限制。     

基于1范数正则化的模型修正方法在结构损伤识别中的应用

以基于灵敏度分析的有限元模型修正方法为基础，提出了一种基于1范数正则化过程的结构损伤识别方法。通过与以Tikhonov正则化为代表的二次型正则化过程相比较，本文的理论分析表明1范数正则化方法在迭代计算过程中能根据上一迭代步损伤识别结果自适应地调整正则化项中的损伤参数权系数，从而显著改善了Tikhonov正则化识别结果过度光滑的缺陷，更利于识别结构的局部损伤。为解决引入1范数造成的数值计算困难，文中还对基于1范数正则化的模型修正算法进行了改进。以二维框架模型为例的损伤识别数值模拟表明：1范数正则化方法与模型修正方法相结合可以有效抑制实测模态参数中噪声的影响，体现出较好的鲁棒性；在模态噪声水平达到10%的情况下，仍能有效抑制噪声干扰，凸显结构局部损伤位置，准确识别损伤程度。     

网络RTK参考站间模糊度动态解算中病态方程的一种解算方法

在网络RTK参考站间的模糊度估计中，若误差方程严重病态，将导致模糊度解与其准确值偏差较大或整周模糊度无法固定，因此提出了一种适于网络RTK模糊度动态解算的新方案：1）法方程病态性的判断；2）Tikhonov正则化解算病态方程；3）LAMBDA方法搜索固定整周模糊度。同时，深入研究了Tikhonov正则化矩阵的构造方法和正则化参数的选取准则。最后以实例验证了采用此方案解算病态方程是可行的，通过选取合适的正则化参数可以解得准确的整周模糊度；详细讨论了选择不同的正则化参数对模糊度解算结果的影响。     

跨音速翼型反设计的一种大范围收敛方法

求解跨音速翼型的反设计问题时，传统的梯度型方法一般均为局部收敛. 为增大求解的收敛范围，依据同伦方法的思想，通过构造不动点同伦，将原问题的求解 转化为其同伦函数的求解，并依据拟Sigmoid函数调整同伦参数以提高计算效率，进而构造 出一种具有较高计算效率的大范围收敛反设计方法. 数值算例以RAE2822翼型的表面压力分 布为拟合目标，分别采用B样条方法, PARSEC方法及正交形函数方法等3种不同的 参数化方法，并分别以NACA0012, OAF139及VR15翼型为初始翼型进行迭代计 算. 计算结果证明，该方法适用于多种参数化方法，且具有较好的计算效率，从多 个不同的初始翼型出发，经较少次数迭代后， 均能与目标翼型很好地拟合，是一种高效的大范围收敛方法.     

基于模态参数灵敏度的损伤方程组求解正则化方法研究

基于模态参数的结构损伤识别方法是振动损伤识别领域中应用最为广泛的方法。利用模态参数灵敏度构建结构损伤方程组,对其进行求解可以识别结构损伤位置和程度。由于实际工程中模态参数不完备性和噪声的影响,结构损伤方程易出现病态问题,直接求解可能产生错误的结果。为了解决这一问题,可以引入正则化方法进行求解。然而,各类正则化方法的基本原理、区别和联系及其在结构损伤识别中的应用没有系统的研究和对比。本文梳理了几类常用的正则化方法,对比分析其在基于模态参数灵敏度的损伤方程组求解中的适用性,讨论损伤程度、噪声水平和测点数目对几类方法识别结果的影响,为结构损伤识别中的正则化方法选择提供依据。通过连续梁和框架结构数值算例分析表明,在求解损伤方程的应用中,L1范数正则化方法鲁棒性较强,贝叶斯正则化方法次之,奇异值截断算法和L2范数正则化方法的鲁棒性较差;L1范数正则化方法能够产生更少的假阳性损伤单元,受噪声和测点数目影响较小,更适合损伤识别的应用。     

含水层参数识别的正则化方法

采用正则化最小二乘方法研究了含水层参数识别问题．通过合理选取正则化参数能够解决参数识别中的不稳定问题．同时通过合理布置水头观测点的位置和补充流量条件解决了参数识别中的唯一性问题．数值计算结果表明，在水头观测误差为1cm～2cm的条件下，参数识别结果的误差不超过1％．     

松散堆积体工程边坡变形机理分析及支护对策研究

利用表面温度测量来反演热传导方程中的热源项是一类典型的热传导逆问题，在采用有 限体积法对三维非稳态热传导问题进行数值求解的基础上，将该热传导逆问题转化为优化问 题，建立了伴随方程法和共轭梯度法这两类反演算法. 采用这两类算法对一个典型算例的计 算结果表明：建立的两类反演算法是有效的，具有较好的抗噪性能. 此外，对反演算法 中计算收敛准则的选取进行了较深入的分析，结果表明，由于热传导逆问题的不适定性，优 化过程中目标函数值越小并不意味着反演结果与真值更为接近，可以通过设定合适的收敛准 则来模拟正则化项的作用，克服不适定性的影响.     

New regularization method and iteratively reweighted algorithm for sparse vector recovery

Motivated by the study of regularization for sparse problems,we propose a new regularization method for sparse vector recovery.We derive sufficient conditions on the well-posedness of the new regularization,and design an iterative algorithm,namely the iteratively reweighted algorithm(IR-algorithm),for efficiently computing the sparse solutions to the proposed regularization model.The convergence of the IR-algorithm and the setting of the regularization parameters are analyzed at length.Finally,we present numerical examples to illustrate the features of the new regularization and algorithm.     

具有多个极限环非线性动力系统的解析近似

应用一种新的解析方法------同伦分析法，研究了一种具有多个 极限环的Rayleigh振子问题. 与所有其他传统方法不同，该方法不依赖于小参数， 且提供了一个简便的途径以确保级数解的收敛, 因此，特别适用于强非线性问题. 将同伦分析法与平均法以及四阶的龙格库塔方法(数值解)做了比较. 结果 表明，平均法在强非线性情况失效, 四阶的龙格库塔法不能找到非稳定的极限环，而同伦分析法不仅适用于强非线性情 况，而且给出了非稳定的极限环.     

NEW THEORY FOR EQUATIONS OF NON-FUCHSIAN TYPE——REPRESENTATION THEOREM OF TREE SERIES SOLUTION (Ⅱ)

Our main result consists in proving the representation theorem. Irregular integral is a new type of analytic function, represented by a compound Taylor-Fourier tree series, in which each coefficient of the Fourier series is a Taylor series, while the Taylor coefficients are tree series in terms of equations parameters, higher order correction terms to each coefficient having tree structure with inexhaustible proliferation.The solution obtained is proved to be convergent absolutely and uniformly in the region defined by coefficient functions of the original equation, provided the structure parameter is less than unity. Direct substitution shows that our tree series solution satisfies the equation explicity generation by generation.As compared with classical theory our method not only furnishes explicit expression of irregular integral, leading to the solution of Poincare problem, but also provides possibility of extending the scope of investigation for analytic theory to equations with various kinds of singularities in a unifying way.Exact explicit analytic expression for irregular integrals can be obtained by means of correspondence principle.It is not difficult to prove the convergence of the tree series solution obtained. Direct substitution shows it satisfies the equation.The tree series is automorphic, which agrees completely with Poincaré’s conjecture.     

Some theoretical problems on variational data assimilation

Theoretical aspects of variational data assimilation (VDA) for a simple model with both global and local observational data are discussed. For the VDA problems with global observational data, the initial conditions and parameters for the model are revisited and the model itself is modified. The estimates of both error and convergence rate are theoretically made and the validity of the method is proved. For VDA problem with local observation data, the conventional VDA method are out of use due to the ill-posedness of the problem. In order to overcome the difficulties caused by the ill-posedness, the initial conditions and parameters of the model are modified by using the improved VDA method, and the estimates of both error and convergence rate are also made. Finally, the validity of the improved VDA method is proved through theoretical analysis and illustrated with an example, and a theoretical criterion of the regularization parameters is proposed.     

A hot wire method for measuring turbulence in transonic or supersonic heated flows

This work presents a method for measuring fluctuating quantities such as temperature or velocity using a constant current hot wire anemometer. The scope of this method has been extended to include not only supersonic flows, but also transonic flows with low Reynolds numbers and transonic or supersonic heated flows. After examining the dependence of the different coefficients of sensitivity to aerodynamic and thermal parameters, the result of the study was applied to a turbulent boundary layer using a suitable processing method.     

Stress-based nonlocal damage model

Progressive microcracking in brittle or quasi-brittle materials, as described by damage models, presents a softening behavior that in turn requires the use of regularization methods in order to maintain objective results. Such regularization methods, which describe interactions between points, provide some general properties (including objectivity and the non-alteration of a uniform field) as well as drawbacks (damage initiation, free boundary).A modification of the nonlocal integral regularization method that takes the stress state into account is proposed in this contribution. The orientation and intensity of nonlocal interactions are modified in accordance with the stress state. The fundamental framework of the original nonlocal method has been retained, making it possible to maintain the method’s advantages. The modification is introduced through the weight function, which in this modified version depends not only on the distance between two points (as for the original model) but also on the stress state at the remote point.The efficiency of this novel approach is illustrated using several examples. The proposed modification improves the numerical solution of problems observed in numerical simulations involving regularization techniques. Damage initiation and propagation in mode I as well as shear band formation are analyzed herein.     

ON THE REGULARIZATION METHOD OF THE FIRST KIND OFFREDHOLM INTEGRAL EQUATION WITH A COMPLEX KERNEL AND ITS APPLICATION

The regularized integrodifferential equation for the first kind of Fredholm integral equation with a complex kernel is derived by generalizing the Tikhonov regularization method and the convergence of approximate regularized solutions is discussed. As an application of the method, an inverse problem in the two-demensional wave-making problem of a flat plate is solved numerically, and a practical approach of choosing optimal regularization parameter is given. Project supported by the National Natural Science Foundation, of China     

演化算法求解桁架多目标拓扑优化的收敛速度研究

演化算法能够同时满足结构拓扑优化的前沿领域对全局优化、黑箱函数优化、组合优化和多目标优化的需求,但采用此类算法的可行性与必要性由其收敛性与计算效率决定。本文以应力约束桁架多目标拓扑优化问题为求解对象,致力于揭示在收敛性与计算效率两方面具有竞争力的算法。首先提出评估演化算法求解拓扑优化问题收敛性与计算效率的通用方法,采用穷举法严格推导了典型桁架多目标拓扑优化问题的全局最优解,并采用超体积指标定义了多层次收敛性能准则。最后通过比较研究得到不同收敛性需求下具有最快收敛速度的演化算法,并揭示了具有竞争力的算法机制。本研究为演化算法求解多目标拓扑优化问题的收敛速度奠定了理论基础,同时为高效求解实际工程拓扑优化问题提供算法支持。