Introduction to numerical linear algebra and optimisation
Introduction to numerical linear algebra and optimisation
Algorithm 755: ADOL-C: a package for the automatic differentiation of algorithms written in C/C++
ACM Transactions on Mathematical Software (TOMS)
Recipes for adjoint code construction
ACM Transactions on Mathematical Software (TOMS)
Control of the Burgers equation by a reduced-order approach using proper orthogonal decomposition
Journal of Optimization Theory and Applications
Evaluating derivatives: principles and techniques of algorithmic differentiation
Evaluating derivatives: principles and techniques of algorithmic differentiation
On the Lagrange--Newton--SQP Method for the Optimal Control of Semilinear Parabolic Equations
SIAM Journal on Control and Optimization
ACM Transactions on Mathematical Software (TOMS)
Using automatic differentiation for second-order matrix-free methods in PDE-constrained optimization
Automatic differentiation of algorithms
Analysis of instantaneous control for the Burgers equation
Nonlinear Analysis: Theory, Methods & Applications
SIAM Journal on Scientific Computing
Solving optimal control problems for the unsteady Burgers equation in COMSOL Multiphysics
Journal of Computational and Applied Mathematics
An all-at-once approach for the optimal control of the unsteady Burgers equation
Journal of Computational and Applied Mathematics
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Adjoint techniques are a common tool in the numerical treatment of optimal control problems. They are used for efficient evaluations of the gradient of the objective in gradient-based optimization algorithms. Different adjoint techniques for the optimal control of Burgers equation with Neumann boundary control are studied. The methods differ in the point in the numerical algorithm at which the adjoints are incorporated. Discretization methods for the continuous adjoint are discussed and compared with methods applying the application of the discrete adjoint. At the example of the implicit Euler method and the Crank Nicolson method it is shown that a discretization for the adjoint problem that is adjoint to the discretized optimal control problem avoids additional errors in gradient-based optimization algorithms. The approach of discrete adjoints coincides with that of automatic differentiation tools (AD) which provide exact gradient calculations on the discrete level.