A quasi-sequential algorithm for PDE-constrained optimization based on space–time orthogonal collocation on finite elements

Highlights

• Both space and time domain are discretized by OCFE.

• A quasi-sequential optimization algorithm based on space–time OCFE.

• PDEs with different types and state constraints are taken into account.

• The proposed algorithm shows a high numerical accuracy.

Abstract

Orthogonal collocation on finite elements (OCFE) has been used universally to approximate ODEs to date. For PDEs, this contribution presents a novel discretization scheme applying the methodology of OCFE to discretize both space and time domain simultaneously, named as space–time orthogonal collocation on finite elements (ST-OCFE). Due to the existence of boundary conditions, the selection of discrete points and the constitution of discretized algebraic equations are different in space and time domain. Furthermore, for solving optimal control problems constrained by PDEs, a discretize-then-optimize algorithm based on ST-OCFE and quasi-sequential approach is proposed. The formulation of discretized optimization problems and the procedure of sensitivity computation are deduced. In the algorithm, diverse types of PDEs, state constraints, and general control parameterization are considered. The proposed method has the advantages of generality, higher numerical accuracy, and easy handling of state constraints, as demonstrated by three examples.

Introduction

Optimal control problems (OCPs) constrained by partial differential equations (PDEs), which is known as PDE-constrained optimization or distributed optimal control, arise naturally in most physical and industrial fields. Dynamic optimization, referring to OCPs constrained by differential algebraic equations (DAEs), have been widely studied in the past few decades where ordinary differential equations (ODEs) are in majority. Promoted by the development of numerical computation, PDE-constrained optimization has gained increasing significance in engineering during recent years.

The well-known classification of solution strategies for dynamic optimization is indirect methods and direct methods [1], [2]. Based on Pontryagin’s Maximum Principle that originated from the variational approach, indirect methods concentrate on obtaining a solution of the first-order necessary optimality conditions leading to a multiple-point boundary value problem. Recently, saturation functions and interior penalty methods have attracted the attention, where state and input constraints are incorporated in a new unconstrained OCP that can be solved by indirect methods [3], [4], [5], [6]. In direct methods, the optimal solution is found by transforming an infinite-dimensional optimization problem into a finite-dimensional nonlinear programming (NLP) problem [1]. According to the level of discretization, direct methods are separated into two categories: sequential strategies and simultaneous strategies. In the sequential approach, also known as control vector parameterization (CVP), only the control variables are discretized; while in the simultaneous approach, control variables and state variables are all discretized, especially using orthogonal collocation on finite elements (OCFE) [7], [8], [9]. The quasi-sequential approach proposed by Hong et al. [10] is a hybrid approach of the sequential and simultaneous strategies, trading off the computational advantages of them. Moreover, further research about the quasi-sequential approach has been extended in both theory and application [11], [12], [13], [14], [15], [16].

In a similar way, numerical algorithms for solving PDE-constrained OCPs can be classified into optimize-then-discretize (OD) and discretize-then-optimize (DO)approaches, corresponding to indirect and direct methods, respectively [17], [18]. In the OD approach, the first-order necessary continuous optimality conditions are derived analytically, and then the continuous optimality system is discretized with appropriate discretization schemes [19]. For instance, Rezazadeh et al. [20] solved the optimality system of a parabolic constrained OCP with space–time spectral collocation method, where spectral collocation is an alternative name for the pseudospectral method and can be recognized as global orthogonal collocation [21], [22].

In the DO approach, a fully discretized finite-dimensional optimization problem is obtained based on diverse discretization schemes, and then solved by numerical optimization algorithms. Analogous to direct methods, sequential and simultaneous strategies are inherited in the DO approach. Chen et al. [23] presented the optimal boundary control for 2D colloid transport in a dead-end microchannel by virtue of the CVP technique. In order to handle inequality state constraints in PDE-constrained optimization, Schultz et al. [24] performed CVP first and obtained a new semi-infinite program. Holmqvist and Magnusson [25] dealt with a PDE-constrained OCP in process engineering based on the simultaneous method, where the space domain was discretized with an adaptive finite volume weighted essentially non-oscillatory scheme and the time domain by OCFE. Aiming at the predictive control of transport-reaction processes, Galerkin’s method was used to derive ODE system from PDEs and an economic model predictive control framework was developed in [26], [27]. Due to the benefits of treating complex state constraints and bounds flexibly, the DO approach (more specifically, the quasi-sequential strategy) is applied in this study. As mentioned before, the quasi-sequential approach is a balanced and efficient method for large-scale dynamic optimization problems. Furthermore, the finite difference method, finite volume method, and finite element method are used more frequently in most research about the DO approach. They have less accuracy than OCFE and more difficulty in dealing with the stiffness of differential equations. Also, in some studies, OCFE is utilized in the time domain but other low-order methods are used in the space domain. Therefore, applying OCFE to both space and time domain is a promising strategy for improving the performance of numerical computation.

Based on the above perceptions, we propose a novel discretization scheme called space–time orthogonal collocation on finite elements (ST-OCFE) in order to discretize the space and time domains simultaneously. Considering the general control parameterization rather than piecewise-constant control signal, a DO algorithm for solving PDE-constrained OCP is presented, which combines the ST-OCFE discretization and quasi-sequential approach. The new method incorporates the superiority of OCFE and quasi-sequential approach, such as higher approximation accuracy and easy-to-handle path constraints. In addition, the intermediate results in our proposed approach are useful, which means that it is a feasible path method and benefit to engineering applications. Limited by complicated geometry boundaries of high-dimensional space domain, this study just focuses on 1D space.

The paper is organized as follows. Section 2 introduces the formulation of PDE-constrained OCP that involves 1D PDEs. In Section 3, the ST-OCFE discretization scheme and the discretized optimization problem are proposed. Section 4 describes the procedure of computing gradients based on the quasi-sequential approach, and the algorithm is summarized in Section 5. Section 6 presents three numerical cases to illustrate the viability of the proposed method. Section 7 concludes the work and gives future prospectives.