Showing papers by "Andrea Walther published in 2012"

Recent Advances in Algorithmic Differentiation

Abstract: The proceedings represent the state of knowledge in the area of algorithmic differentiation (AD). The 31 contributed papers presented at the AD2012 conference cover the application of AD to many areas in science and engineering as well as aspects of AD theory and its implementation in tools. For all papers the referees, selected from the program committee and the greater community, as well as the editors have emphasized accessibility of the presented ideas also to non-AD experts. In the AD tools arena new implementations are introduced covering, for example, Java and graphical modeling environments or join the set of existing tools for Fortran. New developments in AD algorithms target the efficiency of matrix-operation derivatives, detection and exploitation of sparsity, partial separability, the treatment of nonsmooth functions, and other high-level mathematical aspects of the numerical computations to be differentiated. Applications stem from the Earth sciences, nuclear engineering, fluid dynamics, and chemistry, to name just a few. In many cases the applications in a given area of science or engineering share characteristics that require specific approaches to enable AD capabilities or provide an opportunity for efficiency gains in the derivative computation. The description of these characteristics and of the techniques for successfully using AD should make the proceedings a valuable source of information for users of AD tools.

On the numerical stability of algorithmic differentiation

Abstract: In contrast to integration, the differentiation of a function is an ill-conditioned process, if only an oracle is available for its pointwise evaluation. That is, unrelated small variations in the value of the composite function are allowed at nearly identical arguments. In contrast, we show here that, if the function is defined by an evaluation procedure as a composition of arithmetic operations and elementary functions, then automatic, or algorithmic differentiation is backward stable in the sense of Wilkinson. More specifically, the derivative values obtained are exact for a perturbation of the elementary components at the level of the machine precision. We also provide a forward error analysis for both the forward and reverse mode. The theoretical analysis is confirmed by numerical experiments.

A first-order convergence analysis of trust-region methods with inexact Jacobians and inequality constraints

Abstract: A class of trust-region algorithms is developed and analyzed for the solution of minimization problems with nonlinear inequality constraints. Based on composite-step trust-region methods with barrier functions, the resulting algorithm also does not require the computation of exact Jacobians; only Jacobian vector products are used along with approximate Jacobian matrices. Therefore, the proposed method is targeted on small or medium size problems with dense Jacobians of the constraints. As demonstrated on small numerical examples, this feature has significant potential benefits for problems where Jacobian calculations are expensive.

Efficient aerodynamic shape optimization by structure exploitation

Abstract: In this paper, we consider an optimization problem for the complete design chain of an airfoil. Starting with a parameter vector, one has to perform a three step procedure to evaluate the desired objective: Generate a grid around the airfoil, compute the flow around the airfoil, and compute the objective. Applying a gradient-based optimization method, one has to provide derivatives for this complex process. In the present paper, we propose the advanced use of automatic differentiation to compute the required gradient information. We report numerical results together with a mesh independency study and an analysis of the optimization process for an inviscid RAE2822 airfoil under transonic flight conditions.

On the Efficient Computation of Sparsity Patterns for Hessians

Abstract: The exploitation of sparsity forms an important ingredient for the efficient solution of large-scale problems. For this purpose, this paper discusses two algorithms to detect the sparsity pattern of Hessians: An approach for the computation of exact sparsity patterns and a second one for the overestimation of sparsity patterns. For both algorithms, corresponding complexity results are stated. Subsequently, new data structures and set operations are presented yielding a new complexity result together with an alternative implementation of the exact approach. For several test problems, the obtained runtimes confirm the new theoretical result, i.e., a significant reduction in the runtime needed by the exact approach. A comparison with the runtime required for the overestimation of the sparsity pattern is included together with a corresponding discussion. Finally, possible directions for future research are stated.

Exploiting Sparsity in Automatic Differentiation on Multicore Architectures

Abstract: We discuss the design, implementation and performance of algorithms suitable for the efficient computation of sparse Jacobian and Hessian matrices using Automatic Differentiation via operator overloading on multicore architectures. The procedure for exploiting sparsity (for runtime and memory efficiency) in serial computation involves a number of steps. Using nonlinear optimization problems as test cases, we show that the algorithms involved in the various steps can be adapted to multithreaded computations.

Tailoring the high-harmonic emission in two-level systems and semiconductors by pulse shaping

Abstract: The generation of specific high harmonics for an optical two-level system is elucidated. The desired emitted radiation can be induced by a carefully designed excitation pulse, which is found by a multiparameter optimization procedure. The presented mechanism can also be applied to semiconductor structures for which the calculations result in much higher emission frequencies. The optimization procedure is either performed using a genetic algorithm or a rigorous mathematical optimization technique.

An Exploratory Line Search for Piecewise Differentiable Objective Functions based on Algorithmic Differentiation

Abstract: Nonsmoothness is a typical characteristic of numerous objective functions in optimisation that arises from applications. The standard approach in algorithmic differentiation (AD) is to consider only differentiable functions that are defined by an evaluation program. We extend this functionality by allowing also the functions abs(), min() and max() during the function evaluation yielding piecewise differential nonlinear functions. We will define an evaluation procedure for these functions and employ ADOL-C in an adapted gradient based optimisation method that was adjusted to the special properties of the objective functions considered here. First numerical results will be presented. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

Computing roots for the modelling of waves in acoustic waveguides

Abstract: Computer aided simulation of guided acoustic waves in single- or multilayered waveguides is an essential tool for several applications of acoustics and ultrasonics (i.e. pipe inspection, noise reduction). To simulate wave propagation in geometrically simple waveguides (plates or rods), one may employ the analytical Global Matrix Method [3]. This requires the computation of all roots of the determinate of a certain submatrix. The evaluation of all real or even complex roots is actually the methods most concerning restriction. Previous approaches based on so called mode-tracers which use the physical phenomenon that solutions (roots) appear in a certain pattern (waveguide modes) and thus use known solutions to limit the root finding algorithms search space with respect to consecutive solutions. As the limitation of the search space might be unstable in some cases, we propose to replace the mode-tracer with a suitable version of an interval Newton method based on INTLAB [4]. To apply this interval based method, we extended the interval and derivative computation provided by INTLAB such that corresponding information is also available for Bessel functions used in the circular model (rods) of acoustic waveguides. We present numerical results of a simple acoustic waveguide and discuss extensions required for more realistic scenarios. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

Engineering high harmonic generation in semiconductors via pulse shaping

Abstract: High harmonic generation is investigated for a two-band model of a semiconductor nanostructure. Similar to an atomic two-level system, the semiconductor emits high harmonic radiation. We show how one can specifically enhance the emission for a given frequency by applying a non-trivially shaped laser pulse. Therefore, the semiconductor Bloch equations including the interband and additionally the intraband dynamics are solved numerically and the spectral shape of the input pulse is computed via an optimization algorithm. It is demonstrated that desired emission frequencies can be favored even though the overall input power is kept constant. We also suggest special metallic nano geometries to achieve enhanced localized optical fields. They are found by geometric optimization.

Structure Exploiting Adjoints for Finite Element Discretizations

Abstract: This paper presents some details for the development, analysis, and implementation of efficient numerical optimization algorithms using algorithmic differentiation (AD) in the context of partial differential equation (PDE) constrained optimization. This includes an error analysis for the discrete adjoints computed with AD and a systematic structure exploitation including efficient checkpointing routines, especially multistage and online checkpointing approaches.