若干非线性问题特殊解的迭代解法
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摘要
在现代科学工程计算领域中,会遇到一些具有许多解(或方案)的问题.对于这类问题,人们感兴趣的往往是它的一些特殊解.本文研究的问题便属于这类问题.对核子物理中提出的非对称代数Riccati方程,人们关心的是它的最小非负解.对多材料鼓面形状设计问题和非线性磁材料中的裂纹和不均匀性检测问题,人们想得到的是最优形状.如何快速有效地求得人们所关心的解,吸引了众多研究者的兴趣.
关于非线性方程的求解,已经存在很多成熟的迭代方法,如牛顿法,Eu-ler迭代族和Halley迭代族等.对于同一个问题,不同迭代法的优劣性也会因Jacobian矩阵的计算和函数求值的复杂性而不同.King-Werner迭代法自被King(?)Werner分别提出后,因其每步迭代中Jacobian矩阵和函数值的计算次数与牛顿法相同,而其收敛阶在非奇异时为1+(?)的优势而引起了研究者的注意.对于最佳形状设计和形状重构问题,其用数学刻画是个约束优化问题,这可以有许多算法求解,如拉格朗日乘子法,增广拉格朗日法等.
本文将求解三个问题的特殊解.第一个问题是在中子迁移理论中提出的非对称代数Riccati方程的最小正解,第二个问题是鼓面设计中的符合特定设计要求的最佳形状问题.第三个问题是非线性磁材料中的裂纹和材料不均匀性检测问题.
在第二章中,我们首先介绍一些已有的求解非对称代数Riccati方程的最小正解的算法,然后用King-Werner方法求非对称代数Riccati方程的最小正解并给出King-Werner迭代法求解该非线性方程的最小正解的收敛性分析以及误差分析.数值例子表明King-Werner方法求奇异情况下的最小正解具有一定的优势.在求解奇异的非对称代数Riccati方程的最小正解时,许多已有的算法迭代速度会比较慢,借助Guo等提出的移技术,一些算法收敛速度慢的问题得以解决.在移技术中,会涉及到一个正实参数,并且该参数的不同取值对迭代算法的收敛速度的影响会有所不同.我们将算法得到的迭代点列看作是该参数的函数列,通过分析该参数对函数列的影响来分析该参数对简单迭代法,修改的简单迭代法,非线性块雅可比迭代法和非线性块高斯-赛德尔迭代法四种迭代法求最小正解时的收敛速率的影响.随后的数值例子验证了理论分析的正确性.
在第三章中,我们将研究鼓面的最佳材料分布问题.我们首先简单介绍目标函数关于密度函数的Gateaux导数,然后选取密度函数的一个能使目标函数的增量尽量大于零的更新方向,通过这一方式来设计一个算法,最后通过数值实验来验证算法求解该类问题的有效性和合理性.
在第四章里,我们解决的问题是非线性磁材料中裂纹和材料不均性的检测.这个问题属于形状重构问题.我们首先用分片常数水平集方法来表示材料的组成,然后将该问题转化为一个约束优化问题,并通过拉格朗日乘子法将该约束优化问题转化成无约束优化问题,最后通过梯度法来求解该无约束问题.数值实验显示我们的算法不但对水平集函数的初始值依赖不大,而且有效稳定.
In the modern scientific computing and engineering, people often encounter many problems which have several solutions. For problems of this kind, what people are interested in are usually some special solutions. The problems we investigate in this paper belong to this kind of problems. For the nonsymmetric algebraic Riccati equation arising in nuclear physics, what people concern is its minimal positive solution. For the problems of the optimal shape design and the shape reconstruction problems, the goal is to find the optimal shape. How to obtain the solutions in which people are interested in an effective way, has attracted the attentions of many researchers.
For the solutions of some nonlinear equations, there are many mature meth-ods such as Newton method, Euler-family and Halley-family. For the same prob-lem, due to the complexity of the computation of Jacobian matrix and the value of the function, the merits and demerits of different methods are different. Since the proposal of the King-Werner iteration method by King and Wiener, respec-tively, due to the advantage that the computation of the Jacobian matrix and the value of function is almost the same as the Newton iteration but the con-vergent order in the nonsingular case is high up to1+(?), it has attracted lots of attentions of many researchers. For the problems of the optimal shape design and the shape reconstruction, they can be transformed into the unconstrained optimization ones which can be solved by many methods such as the Lagrangian multiplier method, augmented Lagrangian method.
In this paper, we will discuss the special solutions of three problems of this kind. The first problem is to seek the minimal positive solution of the nonsymmetric algebraic Riccati equation. The second one is to find the optimal shape of the head of an drum according to a certain criteria. The last problem is to seek the shape which is the most close to the actual shape of the crack or the inhomogeneity in the nonlinear magnetic material.
In Chapter2, we will present the existing methods for the minimal positive solution of the nonsymmetric algebraic Riccati equation. We use the King-Werner iteration method to seek the minimal positive solution and give the convergence analysis and the error analysis. Numerical tests show King-Werner method has some advantage at the singular case. For the singular case, the existing methods converge slowly. By means of the shift technique proposed by Guo et al., the phenomenon that some existing methods converge slowly has been solved. In the shift technique, it will involve a positive real parameter. The value of this parameter has some effects on the convergence rates of some algorithms. By con-sidering the iterative sequences as the functions sequence of this parameter, we analyze the effects of the parameter on the convergence rates of these four meth-ods including the Simple Iteration method, Modified Simple; iteration method, nonlinear block Jaccobi iteration method and nonlinear block Gauss-Seidel iter-ation method. Numerical tests verified our theoretical analysis.
In Chapter3, we solve a problem of finding the optimal shape of the head of an drum. We first introduce the Gatcaux derivative of the objective with respect to the density function, then design an algorithm by choosing a direction of updating the density function which can make the increment of the objective functional larger than0as soon as possible. We also do some numerical tests to show the efficiency and feasibility of the algorithm.
In Chapter4. we want to find the crack or the inhomogeneity in the nonlinear magnetic material. This problem belongs to the shape reconstruction problem. We first use the piecewise constant level set method to represent the composition of the material, then transform this problem into a constrained optimization one and further transform the constrained optimization one into an unconstrained one by the Lagrangian multiplier method, finally solve the unconstrained opti-mization by the gradient method. Numerical tests show that the algorithm is not dependent on the initial choice of the level set function, and the algorithm is not only efficient, but also robust.
关于非线性方程的求解,已经存在很多成熟的迭代方法,如牛顿法,Eu-ler迭代族和Halley迭代族等.对于同一个问题,不同迭代法的优劣性也会因Jacobian矩阵的计算和函数求值的复杂性而不同.King-Werner迭代法自被King(?)Werner分别提出后,因其每步迭代中Jacobian矩阵和函数值的计算次数与牛顿法相同,而其收敛阶在非奇异时为1+(?)的优势而引起了研究者的注意.对于最佳形状设计和形状重构问题,其用数学刻画是个约束优化问题,这可以有许多算法求解,如拉格朗日乘子法,增广拉格朗日法等.
本文将求解三个问题的特殊解.第一个问题是在中子迁移理论中提出的非对称代数Riccati方程的最小正解,第二个问题是鼓面设计中的符合特定设计要求的最佳形状问题.第三个问题是非线性磁材料中的裂纹和材料不均匀性检测问题.
在第二章中,我们首先介绍一些已有的求解非对称代数Riccati方程的最小正解的算法,然后用King-Werner方法求非对称代数Riccati方程的最小正解并给出King-Werner迭代法求解该非线性方程的最小正解的收敛性分析以及误差分析.数值例子表明King-Werner方法求奇异情况下的最小正解具有一定的优势.在求解奇异的非对称代数Riccati方程的最小正解时,许多已有的算法迭代速度会比较慢,借助Guo等提出的移技术,一些算法收敛速度慢的问题得以解决.在移技术中,会涉及到一个正实参数,并且该参数的不同取值对迭代算法的收敛速度的影响会有所不同.我们将算法得到的迭代点列看作是该参数的函数列,通过分析该参数对函数列的影响来分析该参数对简单迭代法,修改的简单迭代法,非线性块雅可比迭代法和非线性块高斯-赛德尔迭代法四种迭代法求最小正解时的收敛速率的影响.随后的数值例子验证了理论分析的正确性.
在第三章中,我们将研究鼓面的最佳材料分布问题.我们首先简单介绍目标函数关于密度函数的Gateaux导数,然后选取密度函数的一个能使目标函数的增量尽量大于零的更新方向,通过这一方式来设计一个算法,最后通过数值实验来验证算法求解该类问题的有效性和合理性.
在第四章里,我们解决的问题是非线性磁材料中裂纹和材料不均性的检测.这个问题属于形状重构问题.我们首先用分片常数水平集方法来表示材料的组成,然后将该问题转化为一个约束优化问题,并通过拉格朗日乘子法将该约束优化问题转化成无约束优化问题,最后通过梯度法来求解该无约束问题.数值实验显示我们的算法不但对水平集函数的初始值依赖不大,而且有效稳定.
In the modern scientific computing and engineering, people often encounter many problems which have several solutions. For problems of this kind, what people are interested in are usually some special solutions. The problems we investigate in this paper belong to this kind of problems. For the nonsymmetric algebraic Riccati equation arising in nuclear physics, what people concern is its minimal positive solution. For the problems of the optimal shape design and the shape reconstruction problems, the goal is to find the optimal shape. How to obtain the solutions in which people are interested in an effective way, has attracted the attentions of many researchers.
For the solutions of some nonlinear equations, there are many mature meth-ods such as Newton method, Euler-family and Halley-family. For the same prob-lem, due to the complexity of the computation of Jacobian matrix and the value of the function, the merits and demerits of different methods are different. Since the proposal of the King-Werner iteration method by King and Wiener, respec-tively, due to the advantage that the computation of the Jacobian matrix and the value of function is almost the same as the Newton iteration but the con-vergent order in the nonsingular case is high up to1+(?), it has attracted lots of attentions of many researchers. For the problems of the optimal shape design and the shape reconstruction, they can be transformed into the unconstrained optimization ones which can be solved by many methods such as the Lagrangian multiplier method, augmented Lagrangian method.
In this paper, we will discuss the special solutions of three problems of this kind. The first problem is to seek the minimal positive solution of the nonsymmetric algebraic Riccati equation. The second one is to find the optimal shape of the head of an drum according to a certain criteria. The last problem is to seek the shape which is the most close to the actual shape of the crack or the inhomogeneity in the nonlinear magnetic material.
In Chapter2, we will present the existing methods for the minimal positive solution of the nonsymmetric algebraic Riccati equation. We use the King-Werner iteration method to seek the minimal positive solution and give the convergence analysis and the error analysis. Numerical tests show King-Werner method has some advantage at the singular case. For the singular case, the existing methods converge slowly. By means of the shift technique proposed by Guo et al., the phenomenon that some existing methods converge slowly has been solved. In the shift technique, it will involve a positive real parameter. The value of this parameter has some effects on the convergence rates of some algorithms. By con-sidering the iterative sequences as the functions sequence of this parameter, we analyze the effects of the parameter on the convergence rates of these four meth-ods including the Simple Iteration method, Modified Simple; iteration method, nonlinear block Jaccobi iteration method and nonlinear block Gauss-Seidel iter-ation method. Numerical tests verified our theoretical analysis.
In Chapter3, we solve a problem of finding the optimal shape of the head of an drum. We first introduce the Gatcaux derivative of the objective with respect to the density function, then design an algorithm by choosing a direction of updating the density function which can make the increment of the objective functional larger than0as soon as possible. We also do some numerical tests to show the efficiency and feasibility of the algorithm.
In Chapter4. we want to find the crack or the inhomogeneity in the nonlinear magnetic material. This problem belongs to the shape reconstruction problem. We first use the piecewise constant level set method to represent the composition of the material, then transform this problem into a constrained optimization one and further transform the constrained optimization one into an unconstrained one by the Lagrangian multiplier method, finally solve the unconstrained opti-mization by the gradient method. Numerical tests show that the algorithm is not dependent on the initial choice of the level set function, and the algorithm is not only efficient, but also robust.
引文
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