【作者】 张宇；

【导师】 于波； 宋晓良；

【作者基本信息】 大连理工大学 ， 计算数学， 2024， 博士

【摘要】 最优控制问题是一类重要的数学模型,在石油勘探、化工与能源领域的流体控制、医学成像、结构设计等领域有广泛的应用背景.许多实际问题的解本身或者它在某种变换下会呈现出稀疏性,对于这类问题通过引入稀疏正则项来建立稀疏最优控制模型.关于稀疏最优控制问题的理论和解法研究刚刚起步,主要成果有:以控制装置的最优布放问题等为应用背景的带L1-控制成本项的椭圆最优控制问题的非光滑牛顿方法、序列二次规划法(Sequential quadratic programming,SQP)、(加速、不精确)邻近梯度算法、有限元-不精确异构ADMM(FE-ihADMM)、有限元-基于对称高斯-赛德尔分解技巧的不精确majorized加速块坐标下降法(FE-sGS-imABCD)等.非凸正则化在诱导解的稀疏性方面比凸正则化更具优势.因此,一个很自然的想法是将非凸正则化技术和相应的算法推广到无穷维的最优控制问题中.然而直接的推广会在理论分析和数值计算上带来许多问题.比如,在有限维情形,l1-2正则化项总是非负的,该性质在凸差(Difference of convex algorithm,DCA)算法收敛性证明中至关重要,但是在无穷维空间中,这一结论却并不成立.另外,带L1-2-正则项的最优控制问题的非凸非光滑性和不可分性,给该问题的数值求解带来更多挑战.本博士论文具体关注了带L1-2控制成本的椭圆型最优控制问题,分析了其高效的数值求解方法,并对其收敛性进行了探讨.本文的主要研究成果如下:首先,我们研究了 一类带L1-2-控制成本和椭圆偏微分方程(Partial difference equa-tion,PDE)约束的最优控制(L1-2-EOC)问题,求解L1-2-EOC 问题可以被看做为求解一个无穷维的DC问题.一种常见的策略是利用DCA将非凸问题表示为一系列仅包含l1正则化项的凸子问题进行求解.在本节中我们提出了一种带筛选策略的不精确凸差算法(Inexact difference of convex algorithm with sieving strategy,s-iDCA)用于求解L1-2-EOC问题.需要指出的是,我们将l1-2正则项推广到无穷维PDE约束优化问题中会在理论分析和数值计算上带来许多问题.在有限维问题中,求解带l1-2范数最小二乘问题的DC算法的收敛性证明是利用目标函数的非负性实现的,但是这一点在无穷维空间中无法保证.在本文中我们通过构造了一个新的能量泛函,并证明其下降性;次梯度的有界性以及能量泛函的KL性质进而给出无穷维空间中的DC算法的全局收敛性证明.另外,为了降低计算成本,我们给出了无穷维空间意义下的一种更适用的不精确凸差算法(Inexact difference of convex algorithm,iDCA).这里我们将‖δk‖≤‖uk-uk-1‖作为iDCA子问题的终止条件,并给出了与精确DCA类似的收敛性证明结果.注意到上述终止条件中δk与控制变量uk耦合,这可能会导致iDCA无法终止.因此,我们进一步给出了一种可执行的带筛选策略的不精确DC算法(s-iDCA).数值实验结果验证了在无穷维空间中带L1-2正则项依然比L1正则项具有更强的诱导解的稀疏性的能力,并且也验证了 s-iDCA在求解该稀疏模型上的有效性.其次,我们提出了一种用于求解带L1-EOC问题的不精确自适应多重网格交替方向乘子法(Inexact adaptive multilevel alternating direction method of multipliers,Am-iADMM)算法.受到多重网格技术与牛顿法结合在求解非线性偏微分方程(PDE)问题中展示出的高效性的影响,在本文中,我们将后延误差估计与多重网格方法和ADMM型算法结合用于求解无穷维空间中L1-EOC问题.该问题是一个凸的非光滑问题.相比于传统的逐层加细的多重网格方法,我们进一步考虑了在多重网格方法中用于判断是否进行网格加细的连续误差δk与离散误差δhk校之间的关系.具体的,我们将后验误差分析与多重网格策略进行结合,提出了一种Am-iADMM算法.在Am-iADMM算法中我们将离散解uhkk看做是由函数空间中不精确交替方向乘子法(Inexact alternating direction method of multipliers,iADMM)得到的不精确解.因此离散误差δhkk也需要满足iADMM算法的误差条件.但是需要注意的是,函数空间中iADMM算法的误差‖δk‖L2无法直接计算得到,只能计算问题后验误差估计值εocp.这样,我们就得到连续误差所满足的条件‖δk‖L2≤εocp≤ρk,这里ρk是我们提前给定的每层的容许精度.该条件不仅保证了 Am-iADMM算法中求解子问题的误差精度,同时也作为判断是否进行网格细化的条件.数值实验结果表明,在相同的计算精度下,我们所提出的Am-iADMM算法所需的迭代次数和计算时间明显优于其它ADMM型算法.并且随着离散网格尺寸h越小,Am-iADMM算法在数值性能方面的优势就越明显.这也验证的自适应多重网格算法在求解大规模问题上的高效性.最后,我们将自适应多重网格策略应用到求解非凸非光滑的L1-2-EOC问题,并提出了不精确自适应多重网格凸差算法(Inexact adaptive multilevel difference of convex algorithm,Am-iDCA).首先,我们利用不精确凸差算法(iDCA)求解L1-2-EOC问题.为了保证iDCA算法的收敛性,我们要求iDCA算法中的误差满足‖δk‖L2≤C‖uk-uk-1‖L2。由于L1-2-EOC问题是一个非凸非光滑问题,因此利用传统的数值求解方法求解,相应的计算量会很大.因此在本节中,为了降低计算成本我们考虑将多重网格策略引入到我们所提出的iDCA算法中,基于多重网格算法的思想,我们首先提出了两种基于不精确求解子问题的多重网格凸差算法(Inexact multilevel difference of convex algorithm,imDCA),一种是基于算法迭代次数来判断是否对网格进行加细,另一种是基于iDCA的求解精度来决定.对于以上两种imDCA算法,他们的基本思想是在粗网格上获得一个不精确解,然后将其投影到更细一层的网格上作为初始解进行迭代.最后在本章中提出了一种不精确自适应多重网格DC算法(Am-iDCA).该方法通过比较后验误差估计值与有限元解的近似误差来判断当前层是否能够满足目标函数足够的下降,进而决定当前层网格是否需要细化.最后,利用L1-2-EOC问题的特殊结构,我们在理论上给出了 Am-iDCA的收敛性结果以及有限元误差估计.数值实验结果验证了 Am-iDCA的有效性.更多还原

【Abstract】 Partial differential equation(PDE)constrained optimization problems are an important class of mathematical models and have extensive application backgrounds in areas such as optimal control,oil exploration,medical imaging,and structural design.PDE constrained optimization problems are a type of infinite-dimensional inverse problems,which are usually ill-posed and require regularization methods(such as Tikhonov regularization)for processing.The solutions of many practical problems themselves or under certain transformations will exhibit sparsity.In such cases,sparse regularization is a very effective method.The research on convex regularization methods(such as l1-regularization,nuclear norm regularization,etc.)and non-convex regularization methods(such as lp-regularization,l1-2regularization,exponential concave function regularization,etc.)in finite-dimensional sparse optimization problems has been a research hotspot in the fields of applied mathematics,data science,operations research,information science,and artificial intelligence in the past two decades.There are rich theoretical achievements and many successful applications.There are also some achievements in the theoretical and solution studies of sparse PDE constrained optimization problems.For example,nonsmooth Newton methods,sequential quadratic programming methods(SQP),(accelerated,inexact)proximal gradient algorithms,finite element-inexact heterogeneous ADMM(FE-ihADMM),finite element-inexact majorized accelerated block coordinate descent method based on symmetric Gauss-Seidel decomposition technique(FE-sGS-imABCD),etc.for elliptic optimal control problems with L1-control cost in application backgrounds such as the optimal placement problem of control devices.Non-convex regularization has more advantages than convex regularization in inducing the sparsity of solutions.Therefore,a natural idea is to extend non-convex regularization techniques and corresponding algorithms to infinite-dimensional PDE constrained optimization problems.However,direct extensions will bring many problems in theoretical analysis and numerical computation.For example,in the finite-dimensional case,the l1-2-regularization term is always non-negative,and this property is crucial in the proof of the convergence of the DC algorithm,but in the infinite-dimensional space,this conclusion does not hold.In addition,the non-convexity,non-smoothness and inseparability of PDE constrained optimization problems with L1-2-regularization terms bring more challenges to the numerical solution of this problem.Therefore,this paper focuses on designing efficient numerical solution algorithms that can fully utilize the problem structure and the sparsity of the solutions.This paper studies the efficient numerical solution methods and their convergence analysis of elliptic PDE constrained optimization problems with L1-2-control costs.The main research results are as follows:In the first place,We studied a class of optimal control(L1-2-EOC)problems with L1-2 control costs and elliptic PDE constraints.Solving the L1-2-EOC problem can be regarded as solving an infinite-dimensional DC problem.A common strategy is to use DCA to represent the non-convex problem as a series of convex subproblems containing only L1 regularization terms for solution.In this section,we propose an inexact difference of convex algorithm with a sieving strategy(s-iDCA)for solving the L1-2-EOC problem.It should be pointed out that extending the l1-2 norm to the infinite-dimensional PDE constrained optimization problem will bring many problems in theoretical analysis and numerical computation.In the finite-dimensional problem,the convergence proof of the DC algorithm for solving the least squares problem with the l1-2 norm is achieved by utilizing the non-negativity of the objective function,but this cannot be guaranteed in the infinite-dimensional space.In this paper,by constructing a new energy functional and proving its decreasing property,the boundedness of the subgradient,and the KL property of the energy functional,we then give the proof of the global convergence of the DC algorithm in the infinite-dimensional space.In addition,to reduce the computational cost,we present a more applicable inexact difference of convex algorithm(iDCA)in the sense of the infinite-dimensional space.Here,we take‖δk‖≤‖uk-uk-1‖ as the termination condition of the iDCA subproblem and give the convergence proof result similar to the exact DCA.Notice that δ is coupled with the control variable u in the above termination condition,which may cause iDCA not to terminate.Therefore,we further present an implementable inexact DC algorithm with a sieving strategy(s-iDCA).The numerical experiment results verify that the L1-2 regularization term still has a stronger ability to induce the sparsity of the solution than the L1 regularization term in the infinite-dimensional space,and also verify the effectiveness of s-iDCA in solving this sparse model.In the second place,We studied an inexact adaptive multilevel alternating direction method of multipliers(Am-iADMM)algorithm for solving the L1-EOC problem.Influenced by the high efficiency of combining multilevel technology and Newton’s method in solving nonlinear partial differential equation(PDE)problems,in this paper,we combine a posteriori error estimation with the multilevel method and ADMM-type algorithms to solve the L1-EOC problem in infinite-dimensional space.This problem is a convex non-smooth problem.Compared with the traditional multilevel method with layer-by-layer refinement,we further consider the relationship between the continuous error δk and the discrete error δhk used to determine whether to perform grid refinement in the multilevel method.Specifically,we combine a posteriori error analysis with the multilevel strategy and propose an inexact adaptive multilevel ADMM algorithm(Am-iADMM).In the Am-iADMM algorithm,we regard the discrete solution uhkk as an inexact solution obtained by the iADMM algorithm in function space.Therefore,the discrete error δhkk also needs to satisfy the error condition of the iADMM algorithm.However,it should be noted that the error ‖δk‖L2 of the iADMM algorithm in function space cannot be directly calculated.We replace it by calculating the a posteriori error estimate eocp.Thus,we obtain the condition satisfied by the continuous error‖δk‖L2≤εocp≤ρk,where ρk is the allowable accuracy of each layer given in advance.This condition not only guarantees the error accuracy of solving subproblems in the AmiADMM algorithm but also serves as a condition for determining whether to perform grid refinement.Numerical experiment results show that under the same computational accuracy,the number of iterations and computational time required by the proposed Am-iADMM algorithm are significantly better than other ADMM-type algorithms.And as the discrete grid size h becomes smaller,the advantage of the Am-iADMM algorithm in numerical performance becomes more obvious.This also verifies that the adaptive multilevel algorithm can significantly reduce computational costs.Lastly,we apply the adaptive multilevel strategy to solve the non-convex and nonsmooth L1-2-EOC problem and propose an inexact adaptive multilevel difference of convex algorithm(Am-iDCA).First,we use the inexact difference of convex algorithm(iDCA)to solve the L1-2-EOC problem.To ensure the convergence of the iDCA algorithm,we require the error in the iDCA algorithm to satisfy ‖δk‖L2≤C‖uk-uk-1‖L2.Since the L1-2-EOC problem is a non-convex and non-smooth problem,solving it using traditional numerical methods will result in a large amount of computation.Therefore,in this section,to reduce computational costs,we consider introducing the multilevel strategy into our proposed iDCA algorithm.Based on the idea of the multilevel algorithm,we propose two multilevel DC algorithms based on inexact solution of subproblems(imDCA),one is based on the number of algorithm iterations to determine whether to refine the grid,and the other is based on the solution accuracy of iDCA to decide.For the above two imDCA algorithms,their basic idea is to obtain an inexact solution on a coarse grid and then project it onto a finer grid as the initial solution for iteration.Finally,in this chapter,an inexact adaptive multilevel DC algorithm(Am-iDCA)is proposed.This method determines whether the current layer can achieve a sufficient decrease in the objective function by comparing the a posteriori error estimate with the approximate error of the finite element solution,and then decides whether the grid of the current layer needs to be refined.Finally,by using the special structure of the L1-2-EOC problem,we theoretically give the convergence result of Am-iDCA and the finite element error estimate.Numerical experiment results verify the effectiveness of Am-iDCA.更多还原