The ADMM For Optimal Control Problems Constrained By PDE Or RPDE

Many physical,medical,and financial phenomena can be modeled by partial differential equations?PDEs?or random partial differential equations?RPDEs?.Generally speaking,we not only concern on the numerical solution of PDEs/RPDEs,but also focus on the control systems governed by PDEs/RPDEs,which are called PDE/RPDE-optimal control problems.In the last decades,many researchers in engineering and science pay signifi-cant attention to PDE constrained optimal control problem with deterministic or random coefficients for the necessary of practical problems?[44,56,58,131,144]?.For the PDE-constrained optimal control problem,the efficient numerical method for solving the discretized large-scale optimality system is an issue,and many PDE solves required in the optimization procedures.While for the RPDE-optimal control problem,which adds a random space to the PDE-optimal control problems,has the bigger challenges in the design of numerical method.We should consider both the suitable discrete methods for the random space and the efficient solver for the discretized large-scale optimality system.Existent algorithms for this problem always need to solve a large number of deterministic PDE at each optimization iteration,and the memory consumption is huge.To resolve these issues,we shall propose efficient and robust algorithms which can reduce the computation and memory requirements.In the following,we will introduce two cardinal numerical methods used in this paper.The alternating direction method of multipliers?ADMM?is an efficient merical algorithm for convex optimization problem,which combines the benefits of dual decompo-sition and augmented Lagrangian methods.Firstly,the ADMM algorithm possess a global convergence and a convergence rate of O?1/k?with k denoting the number of iterations.Sec-ondly,since the cost functional and the constrain both have variables separable structure,the control variable and the state variable are decoupled when using the ADMM to con-struct the algebraic system.This strategy splits the problem into several sub-problems,which cut down the computational scale.Finally when assembling the algebraic system,the stiff matrix of left hand side is invariant in each iteration,so that we only need to cal-culate the inverse matrices for one time in the whole process which reduces the computing time of solving the algebraic system sharply.The multi-mode expansion?MME?technique is an efficient numerical algorithm for optimal control problems constrained by RPDE.We expand the random state variable with multi-modes representation as a power series of the perturbation parameter,where the system decouple into recursive PDEs with random source terms.This method combines the merits of projection-based methods and sample-based methods,which reduces the computing the number of computing the inverse of matrix sharply.Meanwhile,it is easy to implement and robust.This thesis consists two parts.Firstly,we apply the ADMM to the PDE-constrained optimal control problems.We have solved several standard elliptic equation constrained optimization problems,and present convergence analysis systematically.Secondly,com-bining the multi-modes expansion and Monte Carlo technique,we extend the ADMM to the RPDE-constrained optimal control problems,and we also present the theoretical results and convergence analysis for this methods.In the first part?Chapter 2?,we mainly focus on using an ADMM to solve the following elliptic equation constrained optimization problem where,B is D or???D and dz is dx or ds,corresponding to the distribution control prob-lem or boundary control problem,respectively.Here,the problem is studyed under two different control constraint cases,which are unconstrained case and Box-constrained case.We also concentrate on three model problems with different types of control variables in s?y,u?= 0-a right hand side control?RHSC?,a Dirichlet boundary control?DBC?and a Robin boundary control?RBC?-for which our ADMM is applicable.The state equations s?y,u?= 0 of these above model problems are given by the variational forms of respectively,where f and g are given functions.Therefore,we totally have six cases of the elliptic equation constrained optimization problem,which include three types of elliptic equation constraints with two different control constraints.In this chapter,we first apply the finite element method?FEM?to the above system,and rewrite the three types of elliptic equation constraints as an uniform discritized system,and then the model problem can be reformulated as We apply the ADMM and two-block ADMM to the above discrete optimal systems for unconstrained case and Box-constrained case,respectively.Comparing to the existent methods,our algorithm does not need to solve large scale systems,and we only need to solve and storage two inverse matrices instead of solving equations in each iterative steps.The overall convergence result is given by the following theorem:Theorem Set h be the meshsize of the FEM,and k be the number of iterations.Let u*and uhk be the optimal control of the original problem and the iterative solution of the algorithm respectively.Then we have the follouwing error estimate?u*-Ruhk?L2?D?=O?hp?+O?k-1/2?,where p is a constant.The numerical experiments are performed to verify the efficiency of the method.In the second part?Chapter 3 and 4?,we expand the ADMM to the optimal control problems governed by RPDEs.Consider the following RPDE-constrained optimal control problemIn Chapter 3,the state equation s??,y?x,??,u?x??= 0 is given by the following random Poisson problem.Based on the Monte Carlo method and finite element method,the discrete problem for the above problem isEach realization of optimal control problems constrained by RPDE are always coupled together?e.g.the standard Monte Carlo technique and Newton/SQP method for the random discretization and optimization solving respectively?,which result in large-scale linear systems and require unbearable computational cost in each iteration.In this chapter,we first discrete the problem by the finite element method and Monte Carlo technique,then use the ADMM for the resulting discrete form due to its global consensus properties.Meanwhile,during the ADMM iteration,the scale of sub-problem resulting from each realization is same as the PDE-constrained control problem.Finally,the MC method is very easy to implement and suitable for MPI parallel programming,although it suffers from a relative low probabilistic convergence rate.Comparing with existent method for solving the optimal control problems governed by RPDEs,our algorithm decouples the large system and solve a low dimensional sub-problems for each realization.In particular,this algorithm only need to compute and store the inverse matrices for all the realizations outside the iterations,which avoids solving the linear systems for every realizations in each iterative step.We also obtain the following error estimates:Theorem Set M be the number of samples,h be the meshsize of the FEM,and k be denotes the number of iterations.Let u*and uhk be the solution of the original optimal control problem and ADMM respectively.Then,we have?u*-Ru_h^k?_L²?D??O(M^1/2)+O?h²?+O(k^-1/2),In Chapter 4,we study another RPDE-constrained optimal control problem,where the state equation s??,y?x,??,u?x??=0 is given by the following random Helmholtz equation.In this chapter,we develop three numerical algorithms incrementally for solving the above control problems.First,we apply the standard Monte Carlo technique and finite clement method for the random and spatial discretization respectively,and then ADMM is adopted for the resulting system.Since the solution of the model problem is complex and high resolution is needed,the algorithm 1 is not efficient.Generally speaking,there are two challenges in the design of numerical methods for this type RPDE-constrained optimization problem:?1?.Calculate the inverse of matrixes M + 1 times,where M is the sample number in random space;?2?.The cost of CPU memory is O?MN²?,where N is the number of points in one spatial direction.It is still unbearable for the computational resources and the memory consumption as M and N being large.General methods,such as the Stochastic Collocation with CG,and the Monte Carlo method with SQP,also have above two problems.To address these two challenges,we adopt the the multi-modes expansion to re-duce the computational resources and memory consumption.Combining the multi-modes expansion,Monte Carlo technique,finite element method,and ADMM,we propose the second algorithm.In this algorithm,the stiff matrix of the left hand side is invariant for all realizations,which means we only need to compute the inverse for two stiff matrices.However,the memory cost is still O?MN²?.Take advantage of the properties of the MME,we propose an important equation,which can change the RPDE-constrained optimization problem to a deterministic one after some simple processing.Base on this,we preprocess certain quantities before the ADMM iteration,so that nearly no random variable is in the inner iteration,which is called algorithm 3.The computational cost of this algorithm is the same as algorithm 2,but the memory cost is reduced to O?N²?.This algorithm is the most efficient and simple to implement.The error estimate of algorithm 3 is given by the following theorem:Theorem Set M be the number of samples,? be the magnitude of the random perturbation,Q be the truncated number of modes in MME,h be the meshsize of the FEM,and k be the number of iterations.Then the error estimate between the optimal control u*of the original problem and the iterative solution uQ,kh of the algorithm 3 isThe numerical experiments verify the efficiency of our algorithms.