Method For Optimal Control Problems Research Articles

Adaptive Stochastic Meshfree Methods for Optimal Control Problem Governed By Random Elliptic Equations

Adaptive Stochastic Meshfree Methods for Optimal Control Problem Governed By Random Elliptic Equations

The nonconforming virtual element method for optimal control problem governed by Stokes equations

The nonconforming virtual element method for optimal control problem governed by Stokes equations

Non-Overlapping Domain Decomposition for 1D Optimal Control Problems Governed by Time-Fractional Diffusion Equations on Coupled Domains: Optimality System and Virtual Controls

We consider a non-overlapping domain decomposition method for optimal control problems of the tracking type governed by time-fractional diffusion equations in one space dimension, where the fractional time derivative is considered in the Caputo sense. We concentrate on a transmission problem defined on two adjacent intervals, where at the interface we introduce an iterative non-overlapping domain decomposition in the spirit of P.L. Lions for the corresponding first-order optimality system, such that the optimality system corresponding to the optimal control problem on the entire domain is iteratively decomposed into two systems on the respective sub-domains; this approach can be framed as first optimize, then decompose. We show that the iteration involving the states and adjoint states converges in the appropriate spaces. Moreover, we show that the decomposed systems on the sub-domain can in turn be interpreted as optimality systems of so-called virtual control problems on the sub-domains. Using this property, we are able to solve the original optimal control problem by an iterative solution of optimal control problems on the sub-domains. This approach can be framed as first decompose, then optimize. We provide a mathematical analysis of the problems as well as a numerical finite difference discretization using the L1-method with respect to the Caputo derivative, along with two examples in order to verify the method.

Retracted: Efficient Stochastic Galerkin Spectral Methods for Optimal Control Problems Constrained by Fractional PDEs with Uncertain Inputs

Retracted: Efficient Stochastic Galerkin Spectral Methods for Optimal Control Problems Constrained by Fractional PDEs with Uncertain Inputs

Parallel-in-time multiple shooting for optimal control problems governed by the Navier–Stokes equations

In the past decades, multiple shooting methods have proven to be a promising direction to speed up the optimization process, especially in the context of ODE-based optimization. Very recently, Fang et al. (Journal of Computational Physics, vol. 452, 110926, 2022) proposed a multiple shooting algorithm for large-scale PDE-constrained optimization. The current paper continues this line of work and explores the potential of multiple shooting methods for optimal control problems governed by the three-dimensional Navier–Stokes equations. Similar to Fang et al., the augmented Lagrangian (AL) method is used to solve the resulting equality-constrained optimization problem, and we employ the classical limited-memory BFGS method for the unconstrained subproblems inside the AL loop. In the current work, we exploit the multiple shooting paradigm in full by processing the shooting windows parallel-in-time, allowing for significant parallel speed-ups compared to single shooting. The proposed method is validated on a velocity tracking case, using up to 100 windows. Our analysis shows that the multiple shooting algorithm allows for considerable algorithmic and parallel speed-ups. While algorithmic speed-up depends on the exact tracking case and initialization of the shooting windows, the multiple shooting algorithm always outperforms single shooting in terms of computational time (if the number of windows is sufficiently high) due to the parallel-in-time implementation. For a given amount of resources, we also show that the proposed parallel-in-time strategy can outperform spatial parallelization alone, especially when the spatial parallelization is saturated.

A posteriori error estimates of a DG method for optimal control problems governed by the transport equation

A posteriori error estimates of a DG method for optimal control problems governed by the transport equation

Adaptive Virtual Element Method for Optimal Control Problem Governed by Stokes Equations

Adaptive Virtual Element Method for Optimal Control Problem Governed by Stokes Equations

Optimization over the Pareto front of nonconvex multi-objective optimal control problems

Simultaneous optimization of multiple objective functions results in a set of trade-off, or Pareto, solutions. Choosing a, in some sense, best solution in this set is in general a challenging task: In the case of three or more objectives the Pareto front is usually difficult to view, if not impossible, and even in the case of just two objectives constructing the whole Pareto front so as to visually inspect it might be very costly. Therefore, optimization over the Pareto (or efficient) set has been an active area of research. Although there is a wealth of literature involving finite dimensional optimization problems in this area, there is a lack of problem formulation and numerical methods for optimal control problems, except for the convex case. In this paper, we formulate the problem of optimizing over the Pareto front of nonconvex constrained and time-delayed optimal control problems as a bi-level optimization problem. Motivated by existing solution differentiability results, we propose an algorithm incorporating (i) the Chebyshev scalarization, (ii) a concept of the essential interval of weights, and (iii) the simple but effective bisection method, for optimal control problems with two objectives. We illustrate the working of the algorithm on two example problems involving an electric circuit and treatment of tuberculosis and discuss future lines of research for new computational methods.

Lagrangian methods for optimal control problems governed by multivalued quasi-hemivariational inequalities

The primary objective of this paper is to develop a general augmented Lagrangian approach for optimal control problems governed by multivalued quasi-hemivariational inequalities. By imposing general coercivity and monotonicity-type conditions, we establish multiple existence results for the optimal control problems. Furthermore, we establish a comprehensive characterization of the zero duality gap property for the primal-dual problems, utilizing the concept of lower semicontinuity of new perturbation functions. Our results represent a significant advancement over recent literature on the subject.

Automatic scenario generation for efficient solution of robust optimal control problems

AbstractExisting methods for nonlinear robust control often use scenario‐based approaches to formulate the control problem as large nonlinear optimization problems. The optimization problems are challenging to solve due to their size, especially if the control problems include time‐varying uncertainty. This paper draws from local reduction methods used in semi‐infinite optimization to solve robust optimal control problems with parametric and time‐varying uncertainty. By iteratively adding interim worst‐case scenarios to the problem, methods based on local reduction provide a way to manage the total number of scenarios. We show that the local reduction method for optimal control problems consists of solving a series of simplified optimal control problems to find worst‐case constraint violations. In particular, we present examples where local reduction methods find worst‐case scenarios that are not on the boundary of the uncertainty set. We also provide bounds on the error if local solvers are used. The proposed approach is illustrated with two case studies with parametric and additive time‐varying uncertainty. In the first case study, the number of scenarios obtained from local reduction is 101, smaller than in the case when all extreme scenarios are considered. In the second case study, the number of scenarios obtained from the local reduction is two compared to 512 extreme scenarios. Our approach was able to satisfy the constraints both for parametric uncertainty and time‐varying disturbances, whereas approaches from literature either violated the constraints or became computationally expensive.

A space–time variational method for optimal control problems: well-posedness, stability and numerical solution

We consider an optimal control problem constrained by a parabolic partial differential equation with Robin boundary conditions. We use a space–time variational formulation in Lebesgue–Bochner spaces yielding a boundedly invertible solution operator. The abstract formulation of the optimal control problem yields the Lagrange function and Karush–Kuhn–Tucker conditions in a natural manner. This results in space–time variational formulations of the adjoint and gradient equation in Lebesgue–Bochner spaces, which are proven to be boundedly invertible. Necessary and sufficient optimality conditions are formulated and the optimality system is shown to be boundedly invertible. Next, we introduce a conforming uniformly stable simultaneous space–time (tensorproduct) discretization of the optimality system in these Lebesgue–Bochner spaces. Using finite elements of appropriate orders in space and time for trial and test spaces, this setting is known to be equivalent to a Crank–Nicolson time-stepping scheme for parabolic problems. Comparisons with existing methods are detailed. We show numerical comparisons with time-stepping methods. The space–time method shows good stability properties and requires fewer degrees of freedom in time to reach the same accuracy.

POD-Based reduced order methods for optimal control problems governed by parametric partial differential equation with varying boundary control

In this work we propose tailored model order reduction for varying boundary optimal control problems governed by parametric partial differential equations. With varying boundary control, we mean that a specific parameter changes where the boundary control acts on the system. This peculiar formulation might benefit from model order reduction. Indeed, fast and reliable simulations of this model can be of utmost usefulness in many applied fields, such as geophysics and energy engineering. However, varying boundary control features very complicated and diversified parametric behaviour for the state and adjoint variables. The state solution, for example, changing the boundary control parameter, might feature transport phenomena. Moreover, the problem loses its affine structure. It is well known that classical model order reduction techniques fail in this setting, both in accuracy and in efficiency. Thus, we propose reduced approaches inspired by the ones used when dealing with wave-like phenomena. Indeed, we compare standard proper orthogonal decomposition with two tailored strategies: geometric recasting and local proper orthogonal decomposition. Geometric recasting solves the optimization system in a reference domain simplifying the problem at hand avoiding hyper-reduction, while local proper orthogonal decomposition builds local bases to increase the accuracy of the reduced solution in very general settings (where geometric recasting is unfeasible). We compare the various approaches on two different numerical experiments based on geometries of increasing complexity.

A Priori Error Analysis of Mixed Virtual Element Methods for Optimal Control Problems Governed by Darcy Equation

A Priori Error Analysis of Mixed Virtual Element Methods for Optimal Control Problems Governed by Darcy Equation

Stochastic perturbation method for optimal control problem governed by parabolic PDEs with small uncertainties

In this paper, we investigate the first-order and second-order perturbation approximation schemes for an optimal control problem governed by parabolic PDEs with small uncertainties. First, we propose the finite dimensional noise assumption for the random coefficient, and expand the state and co-state variables up to a certain order with respect to a small parameter by using the perturbation technique. Then, we insert all the expansions into the known deterministic parametric optimality system, and obtain the first-order and second-order optimality systems. These two systems are discretized by the finite element method in space and the backward Euler method in time to establish the first-order and second-order approximate schemes. Further, a priori error estimates are derived for the state, co-state and control variables of the first-order and second-order approximation, respectively. Finally, some numerical experiments are presented to verify the theoretical results.

A priori and a posteriori error analysis of hp spectral element discretization for optimal control problems with elliptic equations

. A priori and a posteriori error estimates of hp spectral element method for an elliptic optimal control problem with L2-norm control constraint are investigated in this paper. We then present the optimality conditions and set up hp spectral element discretization scheme. Based on some important interpolation operators and suitable immediate variable, a priori error estimates for this problem in hp spectral element discretization are established carefully. Furthermore, using some interpolation operators, a rigorous posteriori error estimates of hp spectral element approximation are also proved for control and the coupled state approximation in L2−H1-norm and L2−L2-norm, respectively. Such estimators can be used to construct reliable adaptive spectral element methods for optimal control problems. Finally, the error analysis results are confirmed by numerical results.

Efficient Stochastic Galerkin Spectral Methods for Optimal Control Problems Constrained by Fractional PDEs with Uncertain Inputs

This paper is devoted to designing fast solvers and efficient preconditioners for the optimal control problems (OCPs) constrained by stochastic fractional elliptic equations. We first prove the existence and uniqueness of the stochastic optimal control solution and then derive the stochastic optimality system. For the numerical approximation, we use the stochastic Galerkin spectral methods, which apply the stochastic Galerkin method for the discretization of random variables and employ the spectral-Galerkin approach for the approximation of spatial variables. To solve the large coupled saddle-point system resulted from discretization, we adopt the most commonly used MINRES method and a more effective PPCG method in low-rank matrix iteration format. Specially, we develop the efficient preconditioners based on the matrix decomposition method and the virtual variable method. We also study the eigenvalue distribution of the corresponding preconditioned matrix. Besides, for the approximation of the discretized state equation, we use the mean-based approximation and the Ullmann approximation to handle different values of the variance of the random inputs. Finally, we present numerical experiments to demonstrate the effectiveness of our solvers and preconditioners.

Tractable global solutions to chance-constrained Bayesian optimal experiment design for arbitrary prior and noise distributions

Optimal experiment design (OED) aims to optimize the information content of experimental observations by designing the experimental conditions. In Bayesian OED for parameter estimation, the design selection is based on an expected utility metric that accounts for the joint probability distribution of the uncertain parameters and the observations. This work presents solution methods for two approximate formulations of the Bayesian OED problem based on Kullback–Leibler divergence for the particular case of Gaussian prior and observation noise distributions and the general case of arbitrary prior distributions and arbitrary observation noise distributions when the observation noise corresponds to arbitrary functions of the states and random variables with an arbitrary multivariate distribution. The proposed methods also allow satisfying path constraints with a specified probability. The solution approach relies on the reformulation of the approximate Bayesian OED problem as an optimal control problem (OCP), for which a parsimonious input parameterization is adopted to reduce the number of decision variables. An efficient global solution method for OCPs via sum-of-squares polynomials and parallel computing is then applied, which is based on approximating the cost of the OCP by a polynomial function of the decision variables and solving the resulting polynomial optimization problem to global optimality in a tractable way via semidefinite programming. It is established that the difference between the cost obtained by solving the polynomial optimization problem and the globally optimal cost of the OCP is bounded and depends on the polynomial approximation error.

Residual-based a posteriori error estimates for the [formula omitted] version of the finite element discretization of the elliptic Robin boundary control problem

Optimal control problems governed by partial differential equations have become a very active and successful research area. So, in this paper, we analyzed a priori and a posteriori error estimates for the hp finite element discretization of elliptic Robin boundary control problems. With the discrete and continuous optimality conditions of the problem, we constructed the error estimators. Based on the residual of the model equations for the coupled state and control approximations, the upper error bound is proved using Scott–Zhang-type quasi interpolation estimates. In order to provide the optimality, lower error bound is shown using some polynomial inverse estimates in weighted Sobolev spaces. Such estimators can be used to construct reliable adaptive methods for optimal control problems.

An Adaptive Moving Mesh Method for Solving Optimal Control Problems in Viscous Incompressible Fluid

An adaptive moving mesh method for optimal control problems in viscous incompressible fluid is proposed with the incompressible Navier–Stokes system used to describe the motion of the fluid. The moving distance of nodes in the adopted mesh moving strategy is found by solving a diffusion equation with source terms, and an algorithm that fully considers the characteristics of the control problem is given with symmetry reduction to the incompressible Navier–Stokes equations. Numerical examples are provided to show that the proposed algorithm can solve the optimal control problem stably and efficiently on the premise of ensuring high precision.

A spectral Petrov-Galerkin method for optimal control problem governed by a fractional ordinary differential equation

In this paper, a spectral Petrov-Galerkin method is investigated for an optimal control problem with a fractional ordinary differential equation constraint. With the study of its well-posedness and the optimality condition in standard Sobolev space, the regularity for the optimal control problem is established in the framework of weighted Sobolev space, which is superior to the counterpart analyzed in standard Sobolev space. Based on the obtained regularity results, the error estimate of the spectral Petrov-Galerkin method is presented, and optimal convergence orders in Lω−α,02 -norm or Lω0,−α2 -norm are determined. Finally, numerical examples with both smooth inputs and rough inputs are given to verify the theoretical prediction.