Showing papers by "Andrea Walther published in 2007"
TL;DR: This paper presents the integration schemes that are automatically generated when differentiating the discretization of the state equation using Automatic Differentiation (AD), and shows that they can be seen as discretized methods for the and adjoint differential equation of the underlying control problem.
Abstract: 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.
74 citations
01 Jan 2007
TL;DR: In this paper, an automatic differentiation tool is used to derive adjoint flow solvers in an automated fashion by the use of so-called automatic differentiation tools, and an RAE2822 airfoil is chosen under transonic flight conditions for drag reduction.
Abstract: Detailed numerical shape optimization will play a strategic role for future aircraft design. It offers the possibility of designing or improving aircraft components with respect to a pre-specified figure of merit. Optimization methods based on exact derivatives of the cost function with respect to the design variables still suffer from the high computational costs if many design variables are used. However, these gradients can be efficiently obtained by the solution of so-called adjoint flow equations. Furthermore, it is possible to derive these adjoint solvers in an automated fashion by the use of so-called automatic differentiation (AD) tools. In the present paper the efficient and automated differentiation of an entire design chain, including the flow solver, is presented. As test case for AD-generated adjoint sensitivity calculations an inviscid RAE2822 airfoil is chosen under transonic flight conditions for drag reduction.
40 citations
TL;DR: 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.
Abstract: 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.
19 citations
TL;DR: A QR-based nullspace factorization of KKT matrices and maintains the reduced Hessian positive definite, so that the resulting quasi-Newton steps in the primal and dual variables are downhill for suitably weighted merit functions.
Abstract: For use in a total quasi-Newton NLP code [Griewank, A. and Walther, A., 2002, On constrained optimization by adjoint based quasi-Newton methods. Optimization Methods and Software, 17, 869-889.], we describe a QR-based nullspace factorization of KKT matrices. We illustrate the linear algebra in detail and present a theory for maintaining factorized matrices after low-rank updates. Each update of the whole system is incorporated with a computational effort that grows only quadratically with respect to the number of variables and active constraints. Furthermore, our method is special in making use of quasi-Newton updates for the constraint Jacobian approximation, instead of the usual way of using the exact derivative or divided differences. To avoid singularity or blow-up of the KKT matrix, we limit the variations of its determinant to a certain factor and dampen or augment the updates if necessary. We maintain the reduced Hessian positive definite, so that the resulting quasi-Newton steps in the primal and dual variables are downhill for suitably weighted merit functions.
17 citations
TL;DR: The tapeless forward mode of ADOL-C as discussed by the authors enables the joint computation of function and derivative values directly from main memory within one sweep, and shorter runtimes are achieved due to the avoidance of tape handling and a more effective, joint optimization for function and derivatives.
Abstract: Sensitivity information is required by numerous applications such as, for example, optimization algorithms, parameter estimations or real time control. Sensitivities can be computed with working accuracy using the forward mode of automatic differentiation (AD). ADOL-C is an AD-tool for programs written in C or C++. Originally, when applying ADOL-C, tapes for values, operations and locations are written during the function evaluation to generate an internal function representation. Subsequently, these tapes are evaluated to compute the derivatives, sparsity patterns etc., using the forward or reverse mode of AD. The generation of the tapes can be completely avoided by applying the recently implemented tapeless variant of the forward mode for scalar and vector calculations. The tapeless forward mode enables the joint computation of function and derivative values directly from main memory within one sweep. Compared to the original approach shorter runtimes are achieved due to the avoidance of tape handling and a more effective, joint optimization for function and derivative code. Advantages and disadvantages of the tapeless forward mode provided by ADOL-C will be discussed. Furthermore, runtime comparisons for two implemented variants of the tapeless forward mode are presented. The results are based on two numerical examples that require the computation of sensitivity information.
3 citations
16 May 2007
TL;DR: The more common multi-level checkpointing as well as the less known binomial checkpointing are presented and the checkpointing approaches are compared with respect to the number of time steps the adjoint of which can be calculated, the run-time needed for the adjointed calculation and the memory requirement.
Abstract: Checkpointing techniques become more and necessary for the computation of adjoints. This paper presents the more common multi-level checkpointing as well as the less known binomial checkpointing. The checkpointing approaches are compared with respect to the number of time steps the adjoint of which can be calculated, the run-time needed for the adjoint calculation and the memory requirement. Some examples illustrate the shown results
1 citations