Automatic differentiation of explicit Runge-Kutta methods for optimal control

This paper considers the numerical solution of optimal control problems based on ODEs. We assume that an explicit Runge-Kutta method is applied to integrate the state equation in the context of a recursive discretization approach. To compute the gradient of the cost function, one may employ Automatic Differentiation (AD). This paper presents the integration schemes that are automatically generated when differentiating the discretization of the state equation using AD. We show that they can be seen as discretization methods for the sensitivity and adjoint differential equation of the underlying control problem. Furthermore, we prove that the convergence rate of the scheme automatically derived for the sensitivity equation coincides with the convergence rate of the integration scheme for the state equation. Under mild additional assumptions on the coefficients of the integration scheme for the state equation, we show a similar result for the scheme automatically derived for the adjoint equation. Numerical results illustrate the presented theoretical results.     

Direct trajectory optimization and costate estimation of finite-horizon and infinite-horizon optimal control problems using a Radau pseudospectral method

A method is presented for direct trajectory optimization and costate estimation of finite-horizon and infinite-horizon optimal control problems using global collocation at Legendre-Gauss-Radau (LGR) points. A key feature of the method is that it provides an accurate way to map the KKT multipliers of the nonlinear programming problem to the costates of the optimal control problem. More precisely, it is shown that the dual multipliers for the discrete scheme correspond to a pseudospectral approximation of the adjoint equation using polynomials one degree smaller than that used for the state equation. The relationship between the coefficients of the pseudospectral scheme for the state equation and for the adjoint equation is established. Also, it is shown that the inverse of the pseudospectral LGR differentiation matrix is precisely the matrix associated with an implicit LGR integration scheme. Hence, the method presented in this paper can be thought of as either a global implicit integration method or a pseudospectral method. Numerical results show that the use of LGR collocation as described in this paper leads to the ability to determine accurate primal and dual solutions for both finite and infinite-horizon optimal control problems.     

An adaptive finite element method for the sparse optimal control of fractional diffusion

We propose and analyze an a posteriori error estimator for a partial differential equation (PDE)-constrained optimization problem involving a nondifferentiable cost functional, fractional diffusion, and control-constraints. We realize fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly PDE and propose an equivalent optimal control problem with a local state equation. For such an equivalent problem, we design an a posteriori error estimator which can be defined as the sum of four contributions: two contributions related to the approximation of the state and adjoint equations and two contributions that account for the discretization of the control variable and its associated subgradient. The contributions related to the discretization of the state and adjoint equations rely on anisotropic error estimators in weighted Sobolev spaces. We prove that the proposed a posteriori error estimator is locally efficient and, under suitable assumptions, reliable. We design an adaptive scheme that yields, for the examples that we perform, optimal experimental rates of convergence.     

A Priori Error Estimates for Least-Squares Mixed Finite Element Approximation of Elliptic Optimal Control Problems

In this paper, a constrained distributed optimal control problem governed by a first-order elliptic system is considered. Least-squares mixed finite element methods, which are not subject to the Ladyzhenkaya-Babuska-Brezzi consistency condition, are used for solving the elliptic system with two unknown state variables. By adopting the Lagrange multiplier approach, continuous and discrete optimality systems including a primal state equation, an adjoint state equation, and a variational inequality for the optimal control are derived, respectively. Both the discrete state equation and discrete adjoint state equation yield a symmetric and positive definite linear algebraic system. Thus, the popular solvers such as preconditioned conjugate gradient (PCG) and algebraic multi-grid (AMG) can be used for rapid solution. Optimal a priori error estimates are obtained, respectively, for the control function in $L^2(Ω)$-norm, for the original state and adjoint state in $H^1(Ω)$-norm, and for the flux state and adjoint flux state in $H$(div; $Ω$)-norm. Finally, we use one numerical example to validate the theoretical findings.     

An Eulerian–Lagrangian method for optimization problems governed by multidimensional nonlinear hyperbolic PDEs

We present a numerical method for solving tracking-type optimal control problems subject to scalar nonlinear hyperbolic balance laws in one and two space dimensions. Our approach is based on the formal optimality system and requires numerical solutions of the hyperbolic balance law forward in time and its nonconservative adjoint equation backward in time. To this end, we develop a hybrid method, which utilizes advantages of both the Eulerian finite-volume central-upwind scheme (for solving the balance law) and the Lagrangian discrete characteristics method (for solving the adjoint transport equation). Experimental convergence rates as well as numerical results for optimization problems with both linear and nonlinear constraints and a duct design problem are presented.     

Approximate Gradient Projection Method with Runge-Kutta Schemes for Optimal Control Problems

We consider an optimal control problem for systems governed by ordinary differential equations with control constraints. The state equation is discretized by the explicit fourth order Runge-Kutta scheme and the controls are approximated by discontinuous piecewise affine ones. We then propose an approximate gradient projection method that generates sequences of discrete controls and progressively refines the discretization during the iterations. Instead of using the exact discrete directional derivative, which is difficult to calculate, we use an approximate derivative of the cost functional defined by discretizing the continuous adjoint equation by the same Runge-Kutta scheme and the integral involved by Simpson's integration rule, both involving intermediate approximations. The main result is that accumulation points, if they exist, of sequences constructed by this method satisfy the weak necessary conditions for optimality for the continuous problem. Finally, numerical examples are given.     

Approximation of weak adjoints by reverse automatic differentiation of BDF methods

We shed light on the relation between the discrete adjoints of multistep backward differentiation formula (BDF) methods and the solution of the adjoint differential equation. To this end, we develop a functional-analytic framework based on a constrained variational problem and introduce the notion of weak adjoint solutions of ordinary differential equations. We devise a Petrov-Galerkin finite element (FE) interpretation of the BDF method and its discrete adjoint scheme obtained by reverse internal numerical differentiation. We show how the FE approximation of the weak adjoint is computed by the discrete adjoint scheme and prove its convergence in the space of normalized functions of bounded variation. We also show convergence of the discrete adjoints to the classical adjoints on the inner time interval. Finally, we give numerical results for non-adaptive and fully adaptive BDF schemes. The presented framework opens the way to carry over techniques on global error estimation from FE methods to BDF methods.     

Optimality conditions for stochastic boundary control problems governed by semilinear parabolic equations

We study the boundary control problems for stochastic parabolic equations with Neumann boundary conditions. Imposing super-parabolic conditions, we establish the existence and uniqueness of the solution of state and adjoint equations with non-homogeneous boundary conditions by the Galerkin approximations method. We also find that, in this case, the adjoint equation (BSPDE) has two boundary conditions (one is non-homogeneous, the other is homogeneous). By these results we derive necessary optimality conditions for the control systems under convex state constraints by the convex perturbation method.     

State Trajectory Compression in Optimal Control

In optimal control problems with nonlinear time-dependent 3D PDEs, the computation of the reduced gradient by adjoint methods requires one solve of the state equation forward in time, and one backward solve of the adjoint equation. Since the state enters into the adjoint equation, the storage of a 4D discretization is necessary. We propose a lossy compression algorithm using a cheap predictor for the state data, with additional entropy coding of prediction errors. Analytical and numerical results indicate that compression factors around 30 can be obtained without exceeding the FE discretization error. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical analysis of an inverse problem for the eikonal equation

We are concerned with the inverse problem for an eikonal equation of determining the speed function using observations of the arrival time on a fixed surface. This is formulated as an optimisation problem for a quadratic functional with the state equation being the eikonal equation coupled to the so-called Soner boundary condition. The state equation is discretised by a suitable finite difference scheme for which we obtain existence, uniqueness and an error bound. We set up an approximate optimisation problem and show that a subsequence of the discrete mimina converges to a solution of the continuous optimisation problem as the mesh size goes to zero. The derivative of the discrete functional is calculated with the help of an adjoint equation which can be solved efficiently by using fast marching techniques. Finally we describe some numerical results.     

Residual-based a posteriori error control for Space-time finite element approximations of parabolic optimal control problems

In this paper we investigate a space-time finite element approximation of parabolic optimal control problems. The first order optimality conditions are transformed into an elliptic equation of fourth order in space and second order in time involving only the state or the adjoint state in the space-time domain. We derive a priori and a posteriori error estimates for the time discretization of the state and the adjoint state. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Homogenization of periodic optimal control problems via multi-scale convergence

The aim of this paper is to provide an alternate treatment of the homogenization of an optimal control problem in the framework of two-scale (multi-scale) convergence in the periodic case. The main advantage of this method is that we are able to show the convergence of cost functionals directly without going through the adjoint equation. We use a corrector result for the solution of the state equation to achieve this.     

Numerical analysis and applications of Fokker-Planck equations for stochastic dynamical systems with multiplicative α-stable noises

In this paper, we study the nonlocal Fokker-Planck equations (FPEs) associated with Lévy-driven scalar stochastic dynamical systems. We first derive the Fokker-Planck equation for the case of multiplicative symmetric α-stable noises, by the adjoint operator method. Then we construct a finite difference scheme to simulate the nonlocal FPE on either bounded or infinite domain. It is shown that the semi-discrete scheme satisfies the discrete maximum principle and converges. Some experiments are conducted to validate the numerical method. Finally, we extend the results to the asymmetric case and present an application to the nonlinear filtering problem.     

Shape Optimal Design Problem with Convective and Radiative Heat Transfer: Analysis and Implementation

We present a study of an optimal design problem for a coupled system, governed by a steady-state potential flow equation and a thermal equation taking into account radiative phenomena with multiple reflections. The state equation is a nonlinear integro-differential system. We seek to minimize a cost function, depending on the temperature, with respect to the domain of the equations. First, we consider an optimal design problem in an abstract framework and, with the help of the classical adjoint state method, give an expression of the cost function differential. Then, we apply this result in the two-dimensional case to the nonlinear integro-differential system considered. We prove the differentiability of the cost function, introduce the adjoint state equation, and give an expression of its exact differential. Then, we discretize the equations by a finite-element method and use a gradient-type algorithm to decrease the cost function. We present numerical results as applied to the automotive industry.     

Analysis of Nonlinear Quadratic Control in Infinite Time

This study refers to minimization of quadratic functionals in infinite time. The coefficients of the quadratic form are quadratic matrices, function of the state variable. Dynamic constraints are represented by a bilinear differential systems of the form. The necessary extremum conditions determine the adjoint variables λ and the control variables u as functions of state variable, respectively the adjoint system corresponding to those functions. Thus it will be obtained a matrix differential equation where the solution representing the positive defined symmetric matrix P ( x ), verifies the Riccati algebraic equation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A method of adjoint operators for solving boundary value problems for second-order ordinary differential equations

In this paper, for a linear boundary value problem we propose a method that reduces the differential problem to a discrete (difference) problem. The difference equations, which are an exact analog of the differential equation, are constructed by an adjoint operator method. The adjoint equations are solved by a factorization method.     

Discrete-analytical methods for the implementation of variational principles in environmental applications

A new method of constructing numerical schemes on the base of a variational principle for models including convection-diffusion operators is proposed. An original element is the use of analytical solutions of local adjoint problems formulated for the operators of convection-diffusion within the framework of the splitting technique. This results in numerical schemes which are absolutely stable, monotonic, transportive, and differentiable with respect to the state functions and parameters of the model. Artificial numerical diffusion is avoided due to the analytical solutions. The variational technique provides strong consistency between the numerical schemes of the main and adjoint problems. A theoretical study of the new class of schemes is given. The quality of the numerical approximations is demonstrated by an example of the non-linear Burgers equation. These new schemes enhance our variational methodology of environmental modelling. As one of the environmental applications, an inverse problem of risk assessment for Lake Baikal is presented.     

Solving optimal control problems for the unsteady Burgers equation in COMSOL Multiphysics

The optimal control of unsteady Burgers equation without constraints and with control constraints are solved using the high-level modelling and simulation package COMSOL Multiphysics. Using the first-order optimality conditions, projection and semi-smooth Newton methods are applied for solving the optimality system. The optimality system is solved numerically using the classical iterative approach by integrating the state equation forward in time and the adjoint equation backward in time using the gradient method and considering the optimality system in the space-time cylinder as an elliptic equation and solving it adaptively. The equivalence of the optimality system to the elliptic partial differential equation (PDE) is shown by transforming the Burgers equation by the Cole-Hopf transformation to a linear diffusion type equation. Numerical results obtained with adaptive and nonadaptive elliptic solvers of COMSOL Multiphysics are presented both for the unconstrained and the control constrained case.     

Efficient topology optimization of thermo-elasticity problems using coupled field adjoint sensitivity analysis method

We develop a unified and efficient adjoint design sensitivity analysis (DSA) method for weakly coupled thermo-elasticity problems. Design sensitivity expressions with respect to thermal conductivity and Young's modulus are derived. Besides the temperature and displacement adjoint equations, a coupled field adjoint equation is defined regarding the obtained adjoint displacement field as the adjoint load in the temperature field. Thus, the computing cost is significantly reduced compared to other sensitivity analysis methods. The developed DSA method is further extended to a topology design optimization method. For the topology design optimization, the design variables are parameterized using a bulk material density function. Numerical examples show that the DSA method developed is extremely efficient and the optimal topology varies significantly depending on the ratio of mechanical and thermal loadings.     

Adjoint equation and Lyapunov regularity for linear stochastic differential algebraic equations of index 1

We introduce a concept of adjoint equation and Lyapunov regularity of a stochastic differential algebraic Equation (SDAE) of index 1. The notion of adjoint SDAE is introduced in a similar way as in the deterministic differential algebraic equation case. We prove a multiplicative ergodic theorem for the adjoint SDAE and the adjoint Lyapunov spectrum. Employing the notion of adjoint equation and Lyapunov spectrum of an SDAE, we are able to define Lyapunov regularity of SDAEs. Some properties and an example of a metal oxide semiconductor field-effect transistor ring oscillator under thermal noise are discussed.