PDE和RPDE最優(yōu)控制問題的交替方向乘子法
發(fā)布時(shí)間：2018-09-01 13:13
【摘要】：現(xiàn)實(shí)生活中,多數(shù)物理,醫(yī)藥,金融等問題均可由偏微分方程(PDEs)或者隨機(jī)偏微分方程(RPDEs)來描述.很多時(shí)候,人們不只關(guān)心PDEs或者RPDEs解本身的性質(zhì),更關(guān)心能否通過控制方程中的某些變量,使得另一些變量達(dá)到預(yù)期的狀態(tài),同時(shí)保證代價(jià)最小,這就是典型的PDE或者RPDE最優(yōu)控制問題.由于實(shí)際需求的驅(qū)動(dòng),系數(shù)確定和系數(shù)隨機(jī)情況下的PDE最優(yōu)控制問題得到了廣泛的研究和關(guān)注([44,56,58,131,144]).對(duì)于PDE最優(yōu)控制問題,其求解難點(diǎn)在于數(shù)值離散后,離散系統(tǒng)的規(guī)模較大,直接求解較困難,且在優(yōu)化過程中,需要求解大量代數(shù)方程組.對(duì)于RPDE最優(yōu)控制問題,它在PDE最優(yōu)控制問題的基礎(chǔ)上引入了隨機(jī)空間,因而問題更為復(fù)雜,一方面需要考慮適合于此問題結(jié)構(gòu)的隨機(jī)空間離散方法,另一方面原有定問題中的求解難點(diǎn)也因隨機(jī)空間的引入而增大.現(xiàn)有算法,對(duì)隨機(jī)空間離散后,多采用迭代法求解,在每步迭代中都需要求解大量定的PDEs,且內(nèi)存占用量很大.本文主要針對(duì)以上困難,提出了合理可行的數(shù)值算法,大大縮減求解方程的數(shù)量并降低了存儲(chǔ)代價(jià).下面介紹本文中所用到的兩種核心數(shù)值方法.交替方向乘子法(Alternating Direction Method of Multipliers,ADMM),是一種求解結(jié)構(gòu)凸優(yōu)化問題的高效數(shù)值方法,主要利用分裂的思想,基于問題變量可分的特點(diǎn),將大規(guī)模優(yōu)化問題分解成若干個(gè)規(guī)模較小的子問題.該方法具有全局收斂性,且最差情況下的收斂速度為O(1/k),其中k為迭代步數(shù).多重模式展開方法(Multi-Mode Expansion,MME),是求解隨機(jī)系數(shù)下RPDEs的有效算法.其主要思想是將隨機(jī)解按擾動(dòng)量級(jí)的冪級(jí)數(shù)進(jìn)行展開,展開系數(shù)滿足一系列迭代方程,并且迭代方程中系數(shù)為確定且相同的函數(shù),只有右端含有隨機(jī)項(xiàng),從而降低了求剛度矩陣逆帶來的計(jì)算壓力.同時(shí),該方法具有易模擬和高效的特點(diǎn).本論文主要由兩部分構(gòu)成.首先,我們將ADMM應(yīng)用到PDE最優(yōu)控制問題上,求解了幾類帶橢圓型PDE約束的最優(yōu)控制問題,給出完整的收斂性分析.其次,結(jié)合MME方法和Monte Carlo方法,我們將ADMM推廣到RPDE最優(yōu)控制問題上,在理論上證明了算法的收斂性,在數(shù)值上驗(yàn)證了算法的有效性.具體工作如下:第一部分是本文的第二章,以Poisson方程為例,研究帶橢圓型PDE約束的最優(yōu)控制問題.主要考慮了在無約束和盒子約束下,控制變量分別為分布控制,Dirichlet邊界控制和Robin邊界控制等六種情況.問題描述為其中,針對(duì)分布控制問題和邊界控制問題,積分區(qū)間B分別取做D或(?)D,相應(yīng)地,dz取做dx或ds.這里,控制約束條件分別取Uad=U(無約束)和Uad={u(a)∈U|ua≤u(x)≤ub}(盒子約束)兩種不同情況.狀態(tài)方程e(y,u)= 0分別取做以下三類控制問題對(duì)應(yīng)的變分形式:本文首先考慮對(duì)上述問題進(jìn)行有限元離散,并將三類狀態(tài)方程的離散格式寫成統(tǒng)一的線性方程組形式.至此,模型問題化成了有限維的優(yōu)化問題,形式如下:接著,我們分別針對(duì)無控制約束條件和盒子控制約束條件,采用ADMM和two-block ADMM求解上述離散問題.與現(xiàn)有算法相比,本文所提算法的優(yōu)勢(shì)在于避免了求解大規(guī)模離散系統(tǒng),且在每步迭代時(shí)無需求解方程組,只需在迭代循環(huán)外,求兩次系數(shù)矩陣的逆即可.此外,本文還給出了所提算法的收斂性分析,結(jié)果如下:定理 設(shè)h為有限元方法網(wǎng)格步長(zhǎng),k為ADMM的迭代步數(shù).令u*和uhk分別表示原問題的解和ADMM求出的迭代解.則有誤差估計(jì)‖u*-Ru_h~k‖_(L~2(D))=O(hp)+O(k~(-1/2)),成立,其中p為常數(shù)(對(duì)不同的模型取不同值,參見論文第二章).最后,數(shù)值實(shí)驗(yàn)表明,本文所提算法能夠有效地處理PDE最優(yōu)控制問題.第二部分由本文的三,四章組成,我們將ADMM推廣到了如下的RPDE最優(yōu)控制問題在第三章中,上述模型中的狀態(tài)方程s(ξ,y(x,ξ),u(x))=0取為如下的隨機(jī)Poisson方程,基于Monte Carlo方法和有限元離散,容易得到上述模型的離散優(yōu)化問題為:傳統(tǒng)的方法(例如采用Monte Carlo方法離散隨機(jī)空間,并結(jié)合Newton迭代法或SQP等算法求解離散后的優(yōu)化問題),在計(jì)算過程中各個(gè)樣本之間會(huì)耦合在一起形成一個(gè)超大規(guī)模系統(tǒng).在本章里,我們首先用Monte Carlo方法結(jié)合有限元方法,給出問題的離散格式.隨后,根據(jù)所得離散問題的全局一致特性,采用ADMM分裂求解.迭代過程中,每個(gè)樣本都對(duì)應(yīng)一個(gè)與定的PDE最優(yōu)控制問題同規(guī)模的子問題,且基于不同樣本,我們可以采用并行計(jì)算.對(duì)比來說,本章所提算法的優(yōu)勢(shì)依然在于ADMM,避免求解大規(guī)模離散系統(tǒng).只需在迭代過程中對(duì)于每個(gè)樣本求解相應(yīng)的低維子問題.特別地,全程不涉及求解PDEs,只需在ADMM迭代循環(huán)外,對(duì)每個(gè)樣本點(diǎn)求系數(shù)矩陣的逆,保存下來,迭代過程中直接使用.本文還給出了算法的整體誤差估計(jì),結(jié)果如下:定理 設(shè)M為采用的Monte Carlo樣本數(shù),h為有限元方法網(wǎng)格步長(zhǎng),k為ADMM迭代步數(shù).令u*和u≤分別為原問題的解和ADMM迭代解,則有如下離散誤差估計(jì)‖u*-Ruhk‖L2(D)≤O(M1/2)+O(h2)+O(k-1/2),在第四章中,我們繼續(xù)研究RPDE最優(yōu)控制問題.主要考慮狀態(tài)方程s(ξ,y(x,ξ).u(x))= 0取為如下的隨機(jī)Helmholtz方程.對(duì)于該問題,本章采取了三種算法來求解.第一種算法是直接將第三章的方法,平行推廣到帶隨機(jī)Helmholtz方程約束的最優(yōu)控制問題.由于本章模型問題需要在復(fù)空間中求解,且對(duì)網(wǎng)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次系數(shù)矩陣的逆,M為樣本數(shù);(2).算法內(nèi)存花費(fèi)量為O(MN~2),N為物理空間自由度.當(dāng)M和N很大時(shí),算法需要的計(jì)算量和內(nèi)存量是難以忍受的.值得一提的是,對(duì)于求解RPDE最優(yōu)控制問題的一般方法,如Stochastic Collocation結(jié)合CG法,Monte Carlo方法結(jié)合SQP方法等,也存在上述兩個(gè)問題.針對(duì)以上兩個(gè)難點(diǎn),本文采用MME方法,對(duì)最優(yōu)控制問題進(jìn)行預(yù)處理.在算法2中,我們基于MME,有限元離散和Monte Carlo方法,將原問題轉(zhuǎn)化為一個(gè)離散的最優(yōu)控制問題,然后利用ADMM進(jìn)行迭代求解.這種方法全程只需要求解兩次系數(shù)矩陣的逆;在計(jì)算量上已經(jīng)較一般方法有了顯著的優(yōu)勢(shì),但此算法的內(nèi)存花費(fèi)量依然為O(MN~2).接著,我們依據(jù)MME的一些迭代特性,提出了一個(gè)重要的等式關(guān)系,并利用此等式關(guān)系,在算法2的基礎(chǔ)上設(shè)計(jì)了算法3.此方法能使得RPDE最優(yōu)控制問題中的隨機(jī)域全部轉(zhuǎn)移到目標(biāo)泛函帶有期望的系數(shù)上,使隨機(jī)更容易處理.經(jīng)過簡(jiǎn)單計(jì)算后,原隨機(jī)問題可變?yōu)槎ǖ腜DE最優(yōu)控制問題,從而避免了求解隨機(jī)最優(yōu)控制問題時(shí)會(huì)遇到的困難.該算法全程只需求兩次系數(shù)矩陣的逆,無需額外求解方程組,且內(nèi)存花費(fèi)量?jī)H為O(N~2),與M無關(guān),從本質(zhì)上解決了上述兩個(gè)難點(diǎn),是三種算法中效果最好的.算法3的收斂階估計(jì)如下:定理設(shè)M為利用Monte Carlo方法的樣本數(shù),ε為隨機(jī)折射率中的擾動(dòng)量級(jí),Q為MME方法的展開項(xiàng)數(shù),h為有限元方法的網(wǎng)格步長(zhǎng),k為迭代步數(shù).則原問題最優(yōu)解u*和本文所提算法得到的數(shù)值解uhQ,k間的誤差估計(jì)為最后,我們通過數(shù)值模擬驗(yàn)證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.
【學(xué)位授予單位】：吉林大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2017
【分類號(hào)】：O241.82;O232
本文編號(hào)：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.
【學(xué)位授予單位】：吉林大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2017
【分類號(hào)】：O241.82;O232
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