原对偶方法求解偏微分方程优化问题的研究 Alternative Title On primal-dual method for PDE optimization problems 王奎 Thesis Advisor 黄玉梅 2016-05-22 Degree Grantor 兰州大学 Place of Conferral 兰州 Degree Name 硕士 Keyword 偏微分方程约束优化问题 原对偶方法 交替迭代 鞍点问题 全变分模型 高斯白噪声 图像恢复 收敛性 Abstract 原对偶方法是求解优化问题的一种高效方法，该方法通过对原始变量和对偶变量交替迭代而求得原问题的全局最优解。原对偶方法在很多领域的优化问题求解中有着重要的应用。在本文中，我们对基于原对偶方法求解两类优化问题的方法进行了研究。一类是图像恢复问题。我们研究了图像全变分范数正则化恢复问题，考虑了对图像中的高斯白噪声及模糊进行去除的情形。已有的非原对偶方法不能很好的解决全变分项的非可微性问题，通过采用对偶方法实现了对全变分模型的精确求解，很好的恢复了图像。另一类是椭圆分布控制约束优化问题。我们提出了基于原对偶方法求解该问题的算法并进行了理论分析。通过对椭圆分布控制约束优化问题离散，得到了一个具有鞍点结构的优化问题，再基于原对偶方法对该鞍点问题进行求解，进而得到此类偏微分方程约束优化问题的解。理论分析和数值实验表明我们所提方法对椭圆分布控制约束优化问题的求解是有效的。 Other Abstract The primal-dual algorithm is an efficient algorithm which obtains the global solution by iterating the primal and dual variables alternatively. The primal-dual algorithm has wide applications in different areas. In this paper, we consider solving two PDE optimization problems by using the primal-dual method. The one is the image restoration problem. We focus on the total variation image restoration model to remove the Gaussian white noise and blur in the observed image. The existing methods can not handle the nondifferentiability of the total variation term effectively. But we can solve the total variation model exactly by using primal-dual algorithm and the result of the image restoration is well. The other one is the PDE-constrained optimization problem, we propose an algorithm based on primal-dual method for it and we also give theoretical analysis. After discretizing the optimization problem and combining Lagrange multiplier, the original problem is converted into a saddle-point problem. We apply the primal-dual method to the resulting problems and the solution of the original problems thus is obtained. Both theoretical analysis and numerical experiments show that the proposed method is very efficient for solving the PDE-constrained optimization problems. URL 查看原文 Language 中文 Document Type 学位论文 Identifier https://ir.lzu.edu.cn/handle/262010/224372 Collection 数学与统计学院 Recommended CitationGB/T 7714 王奎. 原对偶方法求解偏微分方程优化问题的研究[D]. 兰州. 兰州大学,2016.
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