PDE和RPDE最优控制问题的交替方向乘子法
发布时间：2018-09-01 13:13
【摘要】：现实生活中,多数物理,医药,金融等问题均可由偏微分方程(PDEs)或者随机偏微分方程(RPDEs)来描述.很多时候,人们不只关心PDEs或者RPDEs解本身的性质,更关心能否通过控制方程中的某些变量,使得另一些变量达到预期的状态,同时保证代价最小,这就是典型的PDE或者RPDE最优控制问题.由于实际需求的驱动,系数确定和系数随机情况下的PDE最优控制问题得到了广泛的研究和关注([44,56,58,131,144]).对于PDE最优控制问题,其求解难点在于数值离散后,离散系统的规模较大,直接求解较困难,且在优化过程中,需要求解大量代数方程组.对于RPDE最优控制问题,它在PDE最优控制问题的基础上引入了随机空间,因而问题更为复杂,一方面需要考虑适合于此问题结构的随机空间离散方法,另一方面原有定问题中的求解难点也因随机空间的引入而增大.现有算法,对随机空间离散后,多采用迭代法求解,在每步迭代中都需要求解大量定的PDEs,且内存占用量很大.本文主要针对以上困难,提出了合理可行的数值算法,大大缩减求解方程的数量并降低了存储代价.下面介绍本文中所用到的两种核心数值方法.交替方向乘子法(Alternating Direction Method of Multipliers,ADMM),是一种求解结构凸优化问题的高效数值方法,主要利用分裂的思想,基于问题变量可分的特点,将大规模优化问题分解成若干个规模较小的子问题.该方法具有全局收敛性,且最差情况下的收敛速度为O(1/k),其中k为迭代步数.多重模式展开方法(Multi-Mode Expansion,MME),是求解随机系数下RPDEs的有效算法.其主要思想是将随机解按扰动量级的幂级数进行展开,展开系数满足一系列迭代方程,并且迭代方程中系数为确定且相同的函数,只有右端含有随机项,从而降低了求刚度矩阵逆带来的计算压力.同时,该方法具有易模拟和高效的特点.本论文主要由两部分构成.首先,我们将ADMM应用到PDE最优控制问题上,求解了几类带椭圆型PDE约束的最优控制问题,给出完整的收敛性分析.其次,结合MME方法和Monte Carlo方法,我们将ADMM推广到RPDE最优控制问题上,在理论上证明了算法的收敛性,在数值上验证了算法的有效性.具体工作如下:第一部分是本文的第二章,以Poisson方程为例,研究带椭圆型PDE约束的最优控制问题.主要考虑了在无约束和盒子约束下,控制变量分别为分布控制,Dirichlet边界控制和Robin边界控制等六种情况.问题描述为其中,针对分布控制问题和边界控制问题,积分区间B分别取做D或(?)D,相应地,dz取做dx或ds.这里,控制约束条件分别取Uad=U(无约束)和Uad={u(a)∈U|ua≤u(x)≤ub}(盒子约束)两种不同情况.状态方程e(y,u)= 0分别取做以下三类控制问题对应的变分形式:本文首先考虑对上述问题进行有限元离散,并将三类状态方程的离散格式写成统一的线性方程组形式.至此,模型问题化成了有限维的优化问题,形式如下:接着,我们分别针对无控制约束条件和盒子控制约束条件,采用ADMM和two-block ADMM求解上述离散问题.与现有算法相比,本文所提算法的优势在于避免了求解大规模离散系统,且在每步迭代时无需求解方程组,只需在迭代循环外,求两次系数矩阵的逆即可.此外,本文还给出了所提算法的收敛性分析,结果如下:定理 设h为有限元方法网格步长,k为ADMM的迭代步数.令u*和uhk分别表示原问题的解和ADMM求出的迭代解.则有误差估计‖u*-Ru_h~k‖_(L~2(D))=O(hp)+O(k~(-1/2)),成立,其中p为常数(对不同的模型取不同值,参见论文第二章).最后,数值实验表明,本文所提算法能够有效地处理PDE最优控制问题.第二部分由本文的三,四章组成,我们将ADMM推广到了如下的RPDE最优控制问题在第三章中,上述模型中的状态方程s(ξ,y(x,ξ),u(x))=0取为如下的随机Poisson方程,基于Monte Carlo方法和有限元离散,容易得到上述模型的离散优化问题为:传统的方法(例如采用Monte Carlo方法离散随机空间,并结合Newton迭代法或SQP等算法求解离散后的优化问题),在计算过程中各个样本之间会耦合在一起形成一个超大规模系统.在本章里,我们首先用Monte Carlo方法结合有限元方法,给出问题的离散格式.随后,根据所得离散问题的全局一致特性,采用ADMM分裂求解.迭代过程中,每个样本都对应一个与定的PDE最优控制问题同规模的子问题,且基于不同样本,我们可以采用并行计算.对比来说,本章所提算法的优势依然在于ADMM,避免求解大规模离散系统.只需在迭代过程中对于每个样本求解相应的低维子问题.特别地,全程不涉及求解PDEs,只需在ADMM迭代循环外,对每个样本点求系数矩阵的逆,保存下来,迭代过程中直接使用.本文还给出了算法的整体误差估计,结果如下:定理 设M为采用的Monte Carlo样本数,h为有限元方法网格步长,k为ADMM迭代步数.令u*和u≤分别为原问题的解和ADMM迭代解,则有如下离散误差估计‖u*-Ruhk‖L2(D)≤O(M1/2)+O(h2)+O(k-1/2),在第四章中,我们继续研究RPDE最优控制问题.主要考虑状态方程s(ξ,y(x,ξ).u(x))= 0取为如下的随机Helmholtz方程.对于该问题,本章采取了三种算法来求解.第一种算法是直接将第三章的方法,平行推广到带随机Helmholtz方程约束的最优控制问题.由于本章模型问题需要在复空间中求解,且对网格精度有一定要求,故算法1存在一定的缺陷:(1).需要求解M+1次系数矩阵的逆,M为样本数;(2).算法内存花费量为O(MN~2),N为物理空间自由度.当M和N很大时,算法需要的计算量和内存量是难以忍受的.值得一提的是,对于求解RPDE最优控制问题的一般方法,如Stochastic Collocation结合CG法,Monte Carlo方法结合SQP方法等,也存在上述两个问题.针对以上两个难点,本文采用MME方法,对最优控制问题进行预处理.在算法2中,我们基于MME,有限元离散和Monte Carlo方法,将原问题转化为一个离散的最优控制问题,然后利用ADMM进行迭代求解.这种方法全程只需要求解两次系数矩阵的逆;在计算量上已经较一般方法有了显著的优势,但此算法的内存花费量依然为O(MN~2).接着,我们依据MME的一些迭代特性,提出了一个重要的等式关系,并利用此等式关系,在算法2的基础上设计了算法3.此方法能使得RPDE最优控制问题中的随机域全部转移到目标泛函带有期望的系数上,使随机更容易处理.经过简单计算后,原随机问题可变为定的PDE最优控制问题,从而避免了求解随机最优控制问题时会遇到的困难.该算法全程只需求两次系数矩阵的逆,无需额外求解方程组,且内存花费量仅为O(N~2),与M无关,从本质上解决了上述两个难点,是三种算法中效果最好的.算法3的收敛阶估计如下:定理设M为利用Monte Carlo方法的样本数,ε为随机折射率中的扰动量级,Q为MME方法的展开项数,h为有限元方法的网格步长,k为迭代步数.则原问题最优解u*和本文所提算法得到的数值解uhQ,k间的误差估计为最后,我们通过数值模拟验证了本文所提算法的高效性.
[Abstract]:In real life, most physical, medical and financial problems can be described by partial differential equations (PDEs) or stochastic partial differential equations (RPDEs). In many cases, people are not only concerned about the properties of PDEs or RPDEs solutions, but also about whether some variables in the governing equations can make other variables reach the desired state, while ensuring that This is a typical PDE or RPDE optimal control problem with minimal cost. Driven by actual demand, the PDE optimal control problem with deterministic coefficients and stochastic coefficients has attracted extensive research and attention ([44,56,58,131,144]). It is difficult to solve directly and a large number of algebraic equations need to be solved in the process of optimization. For RPDE optimal control problem, stochastic space is introduced on the basis of PDE optimal control problem, so the problem is more complicated. On the one hand, the stochastic space discretization method suitable for the structure of the problem should be considered, on the other hand, the original definite problem should be solved. The difficulty of solving the problem is also increased by the introduction of the random space. The existing algorithms, after the discretization of the random space, usually use iterative method to solve the problem. In each iteration, they need to solve a large number of PDEs, and the amount of memory occupied is very large. In this paper, two core numerical methods are introduced. Alternating Direction Method of Multipliers (ADMM) is an efficient numerical method for solving structural convex optimization problems. Based on the idea of splitting, large-scale optimization problems are decomposed into discrete variables. Several small-scale sub-problems. The method has global convergence and the worst-case convergence rate is O(1/k), where k is the iterative step. Multimode Expansion (MME) is an effective algorithm for solving RPDEs with random coefficients. The expansion coefficients satisfy a series of iteration equations, and the coefficients in the iteration equations are deterministic and the same function. Only the right end of the iteration equation contains random terms, which reduces the computational pressure caused by the inverse of stiffness matrix. In the optimal control problem, several kinds of optimal control problems with elliptic PDE constraints are solved and the complete convergence analysis is given. Secondly, combining MME method with Monte Carlo method, we extend ADMM to RPDE optimal control problem. The convergence of the algorithm is proved theoretically and the effectiveness of the algorithm is verified numerically. The first part is the second chapter of this paper. Taking Poisson equation as an example, the optimal control problem with elliptic PDE constraints is studied. The control variables are distributed control, Dirichlet boundary control and Robin boundary control respectively under unconstrained and box constraints. The problem is described as distributed control problem and boundary control problem. For control problems, the integral interval B is taken as D or (?) D respectively, and DZ is taken as DX or ds. Here, the control constraints are taken as two different cases: Uad = U (unconstrained) and Uad = {u (a) - {u | u a U < u (x) < {UB box constraints. The state equation E (y, u) = 0 is taken as the corresponding variational form of the following three types of control problems: Firstly, the above-mentioned question is considered. The problem is discretized by finite element method, and the discrete schemes of three kinds of state equations are written into a unified linear system of equations. Thus, the model problem is transformed into a finite-dimensional optimization problem in the following forms: Then, for uncontrolled constraints and box control constraints, we use ADMM and two-block ADMM to solve the above discrete problems respectively. Compared with other algorithms, the advantage of the proposed algorithm is that it avoids solving large-scale discrete systems and does not need to solve equations at each iteration. It only needs to find the inverse of the coefficient matrix twice outside the iteration cycle. K is the iteration step of ADMM. Let u* and u H K represent the solution of the original problem and the iterative solution of ADMM, respectively. Then the error estimate _*-Ru_h~k_ (L~2(D))= O(h p) + O (k~(-1/2)) holds, where p is a constant (for different models, see Chapter 2 of this paper). Finally, numerical experiments show that the proposed algorithm can deal with PDE most effectively. The second part is composed of three and four chapters of this paper. We generalize ADMM to the following RPDE optimal control problems. In the third chapter, the state equation s (zeta, y (x, zeta), u (x))= 0 in the above model is taken as the following stochastic Poisson equation. Based on Monte Carlo method and finite element discretization, the discrete optimization problem of the above model can be easily obtained. The Title is: Traditional methods (such as Monte Carlo method to discretize random space and Newton iterative method or SQP algorithm to solve discrete optimization problems) will be coupled to form a very large-scale system. In this chapter, we first use Monte Carlo method and finite element method to solve the discrete optimization problems. Then, according to the global uniformity of the discrete problem, the ADMM splitting method is used to solve the problem. Each sample corresponds to a sub-problem of the same size as the fixed PDE optimal control problem in the iterative process, and based on different samples, we can adopt parallel computing. The advantages of the proposed algorithm in this chapter are still in comparison. In ADMM, the solution of large-scale discrete systems is avoided. Only the corresponding low-dimensional sub-problems are solved for each sample in the iterative process. Particularly, the whole process does not involve solving PDEs, only the inverse of the coefficient matrix is obtained for each sample point outside the iterative cycle of ADMM, which is saved and used directly in the iterative process. The results are as follows: Theorem M is the number of Monte Carlo samples used, h is the mesh step of the finite element method, and K is the ADMM iterative step. If U * and U < are the solution of the original problem and the ADDM iterative solution respectively, then the following discrete error estimates -Ruhk L2 (D) O (M1/2) + O (h2) + O (k-1/2) are obtained. In Chapter 4, we continue to study RPDE optimal control. This paper mainly considers the stochastic Helmholtz equation in which the state equation s (zeta, y (x, zeta).U (x) = 0 is taken as the following. Three algorithms are used to solve this problem. The first algorithm is to extend the method in Chapter 3 directly to the optimal control problem with stochastic Helmholtz equation constraints. Because the model problem in this chapter needs to be solved in complex space. It needs to solve the inverse of M+1 coefficient matrix and M is the number of samples; (2) The memory cost of the algorithm is O (MN~2), and N is the degree of freedom of physical space. When M and N are very large, the amount of computation and memory required by the algorithm is unbearable. The general methods of DE optimal control problems, such as Stochastic Collocation combined with CG method, Monte Carlo method combined with SQP method, also have the above two problems. To solve the above two difficulties, MME method is used to preprocess the optimal control problem. In algorithm 2, based on MME, finite element discretization and Monte Carlo method, the original problem is solved. This method only needs to solve the inverse of the coefficient matrix twice in the whole process. It has a remarkable advantage over the general method in computation, but the memory cost of this algorithm is still O (MN~2). Then, according to some iterative characteristics of MME, we propose a new method. Using this equation, an algorithm is designed on the basis of algorithm 2. This method can transfer all the random fields in the RPDE optimal control problem to the target functional with expected coefficients, making the stochastic process easier. After simple calculation, the original stochastic problem can be changed into a PDE optimal control problem. The algorithm only needs the inverse of the coefficient matrix twice in the whole process, and does not need to solve the equations. The memory cost is only O (N~2), and it is independent of M. It essentially solves the above two difficulties and is the best of the three algorithms. In order to use the sample number of Monte Carlo method, e is the order of perturbation in random refractive index, Q is the expansion term of MME method, h is the mesh step of finite element method and K is the iteration step, then the error estimation between the optimal solution u* and the numerical solution u h Q, K obtained by the proposed algorithm is finally verified by numerical simulation. The efficiency of the algorithm.
【学位授予单位】：吉林大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O241.82;O232
本文编号：2217309
[Abstract]:In real life, most physical, medical and financial problems can be described by partial differential equations (PDEs) or stochastic partial differential equations (RPDEs). In many cases, people are not only concerned about the properties of PDEs or RPDEs solutions, but also about whether some variables in the governing equations can make other variables reach the desired state, while ensuring that This is a typical PDE or RPDE optimal control problem with minimal cost. Driven by actual demand, the PDE optimal control problem with deterministic coefficients and stochastic coefficients has attracted extensive research and attention ([44,56,58,131,144]). It is difficult to solve directly and a large number of algebraic equations need to be solved in the process of optimization. For RPDE optimal control problem, stochastic space is introduced on the basis of PDE optimal control problem, so the problem is more complicated. On the one hand, the stochastic space discretization method suitable for the structure of the problem should be considered, on the other hand, the original definite problem should be solved. The difficulty of solving the problem is also increased by the introduction of the random space. The existing algorithms, after the discretization of the random space, usually use iterative method to solve the problem. In each iteration, they need to solve a large number of PDEs, and the amount of memory occupied is very large. In this paper, two core numerical methods are introduced. Alternating Direction Method of Multipliers (ADMM) is an efficient numerical method for solving structural convex optimization problems. Based on the idea of splitting, large-scale optimization problems are decomposed into discrete variables. Several small-scale sub-problems. The method has global convergence and the worst-case convergence rate is O(1/k), where k is the iterative step. Multimode Expansion (MME) is an effective algorithm for solving RPDEs with random coefficients. The expansion coefficients satisfy a series of iteration equations, and the coefficients in the iteration equations are deterministic and the same function. Only the right end of the iteration equation contains random terms, which reduces the computational pressure caused by the inverse of stiffness matrix. In the optimal control problem, several kinds of optimal control problems with elliptic PDE constraints are solved and the complete convergence analysis is given. Secondly, combining MME method with Monte Carlo method, we extend ADMM to RPDE optimal control problem. The convergence of the algorithm is proved theoretically and the effectiveness of the algorithm is verified numerically. The first part is the second chapter of this paper. Taking Poisson equation as an example, the optimal control problem with elliptic PDE constraints is studied. The control variables are distributed control, Dirichlet boundary control and Robin boundary control respectively under unconstrained and box constraints. The problem is described as distributed control problem and boundary control problem. For control problems, the integral interval B is taken as D or (?) D respectively, and DZ is taken as DX or ds. Here, the control constraints are taken as two different cases: Uad = U (unconstrained) and Uad = {u (a) - {u | u a U < u (x) < {UB box constraints. The state equation E (y, u) = 0 is taken as the corresponding variational form of the following three types of control problems: Firstly, the above-mentioned question is considered. The problem is discretized by finite element method, and the discrete schemes of three kinds of state equations are written into a unified linear system of equations. Thus, the model problem is transformed into a finite-dimensional optimization problem in the following forms: Then, for uncontrolled constraints and box control constraints, we use ADMM and two-block ADMM to solve the above discrete problems respectively. Compared with other algorithms, the advantage of the proposed algorithm is that it avoids solving large-scale discrete systems and does not need to solve equations at each iteration. It only needs to find the inverse of the coefficient matrix twice outside the iteration cycle. K is the iteration step of ADMM. Let u* and u H K represent the solution of the original problem and the iterative solution of ADMM, respectively. Then the error estimate _*-Ru_h~k_ (L~2(D))= O(h p) + O (k~(-1/2)) holds, where p is a constant (for different models, see Chapter 2 of this paper). Finally, numerical experiments show that the proposed algorithm can deal with PDE most effectively. The second part is composed of three and four chapters of this paper. We generalize ADMM to the following RPDE optimal control problems. In the third chapter, the state equation s (zeta, y (x, zeta), u (x))= 0 in the above model is taken as the following stochastic Poisson equation. Based on Monte Carlo method and finite element discretization, the discrete optimization problem of the above model can be easily obtained. The Title is: Traditional methods (such as Monte Carlo method to discretize random space and Newton iterative method or SQP algorithm to solve discrete optimization problems) will be coupled to form a very large-scale system. In this chapter, we first use Monte Carlo method and finite element method to solve the discrete optimization problems. Then, according to the global uniformity of the discrete problem, the ADMM splitting method is used to solve the problem. Each sample corresponds to a sub-problem of the same size as the fixed PDE optimal control problem in the iterative process, and based on different samples, we can adopt parallel computing. The advantages of the proposed algorithm in this chapter are still in comparison. In ADMM, the solution of large-scale discrete systems is avoided. Only the corresponding low-dimensional sub-problems are solved for each sample in the iterative process. Particularly, the whole process does not involve solving PDEs, only the inverse of the coefficient matrix is obtained for each sample point outside the iterative cycle of ADMM, which is saved and used directly in the iterative process. The results are as follows: Theorem M is the number of Monte Carlo samples used, h is the mesh step of the finite element method, and K is the ADMM iterative step. If U * and U < are the solution of the original problem and the ADDM iterative solution respectively, then the following discrete error estimates -Ruhk L2 (D) O (M1/2) + O (h2) + O (k-1/2) are obtained. In Chapter 4, we continue to study RPDE optimal control. This paper mainly considers the stochastic Helmholtz equation in which the state equation s (zeta, y (x, zeta).U (x) = 0 is taken as the following. Three algorithms are used to solve this problem. The first algorithm is to extend the method in Chapter 3 directly to the optimal control problem with stochastic Helmholtz equation constraints. Because the model problem in this chapter needs to be solved in complex space. It needs to solve the inverse of M+1 coefficient matrix and M is the number of samples; (2) The memory cost of the algorithm is O (MN~2), and N is the degree of freedom of physical space. When M and N are very large, the amount of computation and memory required by the algorithm is unbearable. The general methods of DE optimal control problems, such as Stochastic Collocation combined with CG method, Monte Carlo method combined with SQP method, also have the above two problems. To solve the above two difficulties, MME method is used to preprocess the optimal control problem. In algorithm 2, based on MME, finite element discretization and Monte Carlo method, the original problem is solved. This method only needs to solve the inverse of the coefficient matrix twice in the whole process. It has a remarkable advantage over the general method in computation, but the memory cost of this algorithm is still O (MN~2). Then, according to some iterative characteristics of MME, we propose a new method. Using this equation, an algorithm is designed on the basis of algorithm 2. This method can transfer all the random fields in the RPDE optimal control problem to the target functional with expected coefficients, making the stochastic process easier. After simple calculation, the original stochastic problem can be changed into a PDE optimal control problem. The algorithm only needs the inverse of the coefficient matrix twice in the whole process, and does not need to solve the equations. The memory cost is only O (N~2), and it is independent of M. It essentially solves the above two difficulties and is the best of the three algorithms. In order to use the sample number of Monte Carlo method, e is the order of perturbation in random refractive index, Q is the expansion term of MME method, h is the mesh step of finite element method and K is the iteration step, then the error estimation between the optimal solution u* and the numerical solution u h Q, K obtained by the proposed algorithm is finally verified by numerical simulation. The efficiency of the algorithm.
【学位授予单位】：吉林大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O241.82;O232
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