【作者】 周广禄；

【导师】 吴勃英；

【作者基本信息】 哈尔滨工业大学 ， 数学， 2016， 博士

【摘要】 反问题和奇异摄动问题广泛存在于科学和工程实践中,反问题大多具有不适定性,由于奇异摄动参数的影响,奇异摄动问题的精确解会在求解区域内产生剧烈的变化,这使得通常的数值方法不能够得到这些问题数值稳定的近似解。所以,越来越多的学者开始关注这些问题稳定的数值算法。本文主要研究反问题和奇异摄动问题的变分方法,该方法以变分理论为基础,其主要优势在于该方法能为求解实际问题提供强大的理论框架,可使病态问题表现为一个适定问题。主要研究内容如下:第二章给出偏微分方程源项反问题和系数反问题的反演方法,提出利用间接法对问题进行反演。基于这种思想,通过给定的附加条件得到与原方程等价的非线性方程,用等价方程的解对精确解进行近似。由于要考虑求解过程中误差的积累对方程数值解的影响,本文引入变分迭代法给出等价的非线性方程的数值解,该方法的特点在于能够通过较少的迭代步数得到很好的数值结果。在数值例子与实验方面,举例说明方法的实现过程,并对附加条件进行扰动实验。第三章研究了奇异摄动问题的数值算法,提出三种改进的变分迭代法用于求解非线性二阶奇异摄动初边值问题。一方面,针对传统变分迭代法的不足,提出了求解长区间上的Lane–Emden方程的分段变分迭代法。该方法是在长区间上,将区间分段,在每个小区间上应用变分迭代法,克服了传统的变分迭代法只在小区间范围收敛的缺陷,并为了保证方法的稳定性,进行了收敛性分析与误差估计;另一方面,对于摄动问题,将变分迭代法与摄动技术巧妙的结合在一起,小参数的摄动技术能够消除摄动参数对数值求解过程的影响,而变分迭代法在很多情况下能够得到快速收敛解。变分迭代-摄动方法结合了这两种方法的优势,能够有效的求解非线性二阶摄动初值问题;此外,对于边值问题,由于变分迭代法给出的迭代公式不能直接推导出迭代结果满足边值条件,已有的处理办法是在迭代初值中加入待定常数,但当非线性项很复杂时,这个待定常数很难确定,所以本文将打靶法与变分迭代法结合,很好的解决了这个问题,得到了收敛快、精度高的打靶-变分迭代法。在第四章中,针对变分原理在反问题方面的实际应用——图像去噪问题,提出一种基于凸与非凸函数耦合的能量泛函的变分模型,基于该变分模型利用变分原理得到对应的偏微分方程用于图像去噪。该方程是正倒向扩散的,非常适用于对分片常值图像进行图像去噪。由于非凸函数的引入,要考虑方程解的存在唯一性,本文提出用Young测度理论论证方程弱解的存在性,在一定的附加条件下,论证Young测度解的唯一性,并利用Young测度解的性质说明模型的合理性。在数值实验方面,本文提出用PM方法的离散格式和AOS离散格式对所得偏微分方程进行数值计算,另外,提出用图像的峰值信噪比和结构相似度来评价去噪效果。实验的数值结果表明,本文提出的模型不但可以消除PM模型的斑点效应,避免TV模型的阶梯效应,还可以很好的保护边界。与这两种经典模型相比,新模型无论从视觉效果上还是从图像去噪评价指标上都优于其它模型。更多还原

【Abstract】 The inverse problems and singular perturbation problems appear widely in the science and physical engineering,inverse problems are almost ill-posed,the exact solution of singular perturbation problems can produce dramatic changes in the solution area caused by the influence of the singular perturbation parameters,which are the reasons that the usual numerical methods can not get the numerically stable approximate solution of these problems.So more and more scholars pay attention to the stable numerical algorithms.In this thesis,we study on the variational methods of inverse problems and singular perturbation problems,the advantage of the variational methods based on the variational principle is that it provides a powerful theoretical framework for solving practical problems in which the ill-posed problems can be expressed as well posed problems.The main research contents are as follows:The second section gives the inversion methods of the unknown source and coefficient inverse problems of partial differential equations,the direct method is proposed to recover the inverse problems.Based on this idea,an equivalent nonlinear equation is obtained by the given additional conditions,then the approximate solution of the equivalent equation is used to approximate the exact solution.Due to the error accumulation of the solution progress,the variation iteration method is employed to giving the numerical solution of the equivalent equation,the characteristic of this method is that it can get very good numerical results with fewer iteration steps.Some examples are given to explain the implementation process and disturbance experiments are carried out by disturbing the additional conditions.The third section studies on singular perturbation problems,three modified variation iteration methods are presented for solving the nonlinear singular perturbation of second order initial and boundary value problems.On the one hand,in view of the deficiency of the traditional variation iteration method,a piecewise variation iteration method is developed for solving the long interval Lane-Emden equation.In the long interval,the long interval is divided into some subintervals in which we apply the variation iteration method to solve the equation,this can overcome the deficiency that the traditional variational iteration method is only convergent in a small region.In order to assure the stability of this method,the convergence and error estimation are analyzed.On the other hand,for the perturbation problems,variation iteration method is combined with the perturbation technique,the perturbation technique of small parameters can eliminate the influence of a small perturbation parameter to the numerical process.Meanwhile the variation iteration method can obtain the convergent solution rapidly in many cases.The variation iterationperturbation method combines the advantages of these two methods so that it can solve the nonlinear second order perturbation problems effectively.In addition,for the boundary value problems,the iteration function from the variation iteration method can not be deduced to satisfy the boundary value conditions directly,the existing method is to join the undetermined constants in the initial iteration function,but when the nonlinear item is very complex,the undetermined constants are difficult to determine.Therefore,the shooting method and variation iteration method are used to solve this problem well,which leads the shooting-variation iteration method of high convergence speed and precision.In the fourth section,the practical application of the variational principle in inverse problems is the image denoising.A variational model based on the coupling energy functional of convex and nonconvex functions is proposed,under this model,the corresponding partial differential equation is deduced to image denoising by the variational principle.The partial differential equation is a forward–backward diffusion equation which is very suitable for removing noises of the piecewise constant image.Owing to the introduction of a nonconvex function,it is needed to consider the existence and uniqueness of solution,so the Young measure theory is introduced to prove the existence of weak solution.Under certain conditions,the uniqueness of solution is proved.At the same time,the properties of the Young measure solution explain the rationality of the model.In the numerical experiments,the discrete formate of PM method and the AOS discrete formate of the partial differential equation to numerical calculations are given.Moreover,the peak signal to noise ratio and structural similarity are proposed to evaluate the effect of denoising.The numerical results show that the presented model not only can eliminate the speckle effect of PM model,avoid the step effect of TV model,but also can protect borders.Compared with the two classical models,the new model is superior to other models in the visual effect and the image denoising evaluation indexes.更多还原