非线性偏微分方程多解计算大范围收敛算法及其应用研究

发布时间：2018-10-05 06:31

【摘要】：本文主要研究非线性偏微分方程多解计算的大范围收敛性算法及其相关应用。非线性偏微分方程解的多重性和不稳定性,给计算方法的设计和相关理论的研究带来了诸多本质的困难,尤其是直接针对非线性偏微分方程本身的具有大范围收敛性的数值算法的研究尚处于起步阶段。如何设计稳定的数值算法去逼近不稳定的解,同时减少非线性偏微分方程多解计算对初值的依赖性,从而实现大范围收敛性,又保证每次所计算出来的解一定为新解,从而使得每次计算都有效。上述内容都是非常重要且富有挑战性的科学问题。文章主要包含两部分内容。首先第一部分内容针对具有山路型变分结构的一类非线性偏微分方程,首先介绍基于标准化非精确搜索准则的局部极小极大方法(LMM)的基本概念和思想,并回答“最优化理论中Goldstein线性搜索策略是否能够推广应用到无限维Hilbert空间非线性偏微分方程多解的计算中”这一问题。文中借助能量泛函J的梯度与局部峰选择p(v)的有界变差的关系给出标准化Goldstein搜索准则,该准则克服了标准化Armijo搜索准则在算法中需要人为设置一个最小迭代步长的缺陷。值得注意的是,在原来的LMM算法的可行性证明中,局部峰选择p(v)满足局部Lipschitz连续是一个非常重要的条件。本文将借助X.D.Yao在文献[114]中定义的局部峰选择p(v)所谓的“超线性”性质将基于标准化Goldstein搜索准则和Armijo搜索准则的LMM算法的可行性证明中,p(v)的局部Lipschitz连续性条件降低为连续即可,并给出了在这种较弱的假定之下,上述两个算法的全局收敛性。第二部分的内容讨论旨在计算新解的增广部分牛顿法(APNM)。通过已找的解的信息构造合适的增广奇异变换(AST),再利用APN-M方法求解相应的增广奇异方程。该方法将迭代限制在一类广义的Nehari流形MG内进行,打破了经典Newton法的奇异线-局部场结构和对称不变性,这是区别于其他Newton型算法的最大亮点。值得指出的是,该算法不受变分结构的限制,并保证了每次计算出来的解必定为新解,但其核心是构造合适的增广奇异变换。在我们已有工作的基础上[115],本文将提出一类新的巧妙地增广奇异变换G,其在形式上虽然只与文[115]中的增广奇异变换G发生了看似细微的改变,但其数学结构却发生了巨大变化。事实上,利用该新的增广奇异变换G在计算新解时所需条件将大为减弱,且条件易于验证。此外上述利用新的增广奇异变换G求新解的思想对于非齐次问题同样适用,从而扩大了APNM方法的应用范围。本部分内容将给出基于这类新的增广奇异变换G的APNM方法的理论分析,并将其直接应用到几类非线性偏微分方程多解计算中,其中包含Henon方程、Gross-Pitaevskii方程以及一类非齐次非线性偏微分方程。
[Abstract]:In this paper, the large range convergence algorithm of nonlinear partial differential equations with multiple solutions and its related applications are studied. The multiplicity and instability of the solutions of nonlinear partial differential equations bring many essential difficulties to the design of calculation methods and the study of related theories. Especially, the study of numerical algorithms with large range convergence for nonlinear partial differential equations is still in its infancy. How to design a stable numerical algorithm to approximate the unstable solution, and at the same time to reduce the dependence of the multiple solutions of nonlinear partial differential equations on the initial value, so as to achieve the convergence of a wide range, and to ensure that the solution calculated every time must be a new solution. So that every calculation is effective. All of these are very important and challenging scientific issues. The article mainly contains two parts. In the first part, for a class of nonlinear partial differential equations with mountain path variational structure, the basic concepts and ideas of the local minimax method (LMM) based on the standard inexact search criterion are introduced. The question whether the Goldstein linear search strategy in optimization theory can be extended to the computation of multiple solutions of nonlinear partial differential equations in infinite dimensional Hilbert spaces is answered. Based on the relationship between the gradient of the energy functional J and the bounded variation of the local peak selection p (v), a normalized Goldstein search criterion is presented, which overcomes the defect that the normalized Armijo search criterion needs to set a minimum iterative step size artificially in the algorithm. It is worth noting that the local peak selection of p (v) to satisfy the local Lipschitz continuity is a very important condition in the feasibility of the original LMM algorithm. In this paper, with the help of the local peaks defined by X.D.Yao in reference [114], the so-called "superlinear" property of p (v) is chosen. The feasibility of the LMM algorithm based on standardized Goldstein search criteria and Armijo search criteria is proved by reducing the local Lipschitz continuity condition of p (v) to continuity. The global convergence of the two algorithms is given under this weak assumption. The second part discusses the augmented partial Newton method (APNM).) for calculating the new solution. The appropriate augmented singular transformation (AST),) is constructed from the information of the solution, and the APN-M method is used to solve the corresponding augmented singular equation. The method limits the iteration to a class of generalized Nehari manifold MG and breaks the singular line-local field structure and symmetry invariance of the classical Newton method, which is the highlight of other Newton algorithms. It is worth pointing out that the algorithm is not restricted by the variational structure and ensures that the solution calculated every time must be a new solution, but its core is to construct an appropriate augmented singular transformation. On the basis of our previous work, this paper will propose a new class of ingeniously augmented singular transformations G, whose mathematical structure has changed greatly although it only appears to change slightly with the augmented singular transformation G in [115]. As a matter of fact, the conditions required for the calculation of the new solution by using the new augmented singular transformation G will be greatly weakened, and the conditions are easy to verify. In addition, the idea of using the new augmented singular transformation G to find the new solution is also applicable to the nonhomogeneous problem, thus extending the application of the APNM method. In this part, the theoretical analysis of the APNM method based on the new augmented singular transformation G is given, and it is directly applied to the multiple solutions of several nonlinear partial differential equations. It contains Henon equation, Gross-Pitaevskii equation and a class of nonhomogeneous nonlinear partial differential equations.
【学位授予单位】：湖南师范大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175.29

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