The leapfrog scheme is a commonly used second-order difference scheme for solving differential equations. If Z(t) denotes the state of the system at time t, the leapfrog scheme computes the state at the next time step as Z(t + 1) = H(Z(t); Z(t \Gamma 1); W ), where H is the nonlinear timestepping operator and W are parameters that are not time dependent. In this article, we show how the associativity of the chain rule of differential calculus can be used to compute a so-called adjoint x T \Delta (dZ(t)=d[Z(0);W ]) efficiently in a parallel fashion. To this end, we (1) employ the reverse mode of automatic differentiation at the outermost level, (2) use a sparsity-exploiting incarnation of the forward mode of automatic differentiation to compute derivatives of H at every time step, and (3) exploit chain rule associativity to compute derivatives at individual time steps in parallel. We report on experimental results with a 2-D shallow-water equation model problem on an IBM SP parallel...

this paper, we discuss how AD can be used to enhance a compressible Navier-Stokes solver. Section 2 describes the twodimensional Navier-Stokes model and solver used in our studies. Section 3 gives a brief introduction to source transformation tools for automatic differentiation. Section 4 discusses how derivatives computed using AD can be used for shape optimization. Section 5 explains how an explicit solver can be transformed into an implicit solver using a Jacobian computed using AD. Section 6 briefly describes how AD might be used in optimal control. We conclude with a summary of our results and a discussion of how insight into the high-level mathematics of a computation can greatly reduce the cost of derivative computations using AD.

Many applications require the derivatives of functions defined by computer programs. Automatic differentiation (AD) is a means of developing code to compute the derivatives of complicated functions accurately and efficiently, without the difficulties associated with developing correct code by hand. We discuss some of the issues involved in developing automatic differentiation tools for parallel programming environments. 1. Introduction Derivatives of functions are used in a variety of applications, ranging from optimization to sensitivity analysis of computer models. Automatic differentiation (AD) provides a mechanism for computing the derivatives of a complicated function---expressed in the form of a program---accurately and efficiently, without the difficulty of developing correct code by hand or the potentially exponential time and space required by traditional symbolic manipulation. Many programs are being developed on or ported to parallel computing platforms. Thus, there is a n...

This paper provides a survey of shape parameterization techniques for multidisciplinary

This paper provides a step-by-step guide to solving dynamic stochastic games using the homotopy method. The homotopy method facilitates exploring the equilibrium correspondence in a systematic fashion; it is especially useful in games that have multiple equilibria. We discuss the theory of the homotopy method and its implementation and present two detailed examples of dynamic stochastic games that are solved using this method.

We describe the development of a differentiated version of PETSc, an objectoriented toolkit for the parallel solution of scientific problems modeled by partial differential equations. Traditionally, automatic differentiation tools are applied to scientific applications to produce derivative-augmented code, which can then be used for sensitivity analysis, optimization, or parameter estimation. Scientific toolkits play an increasingly important role in developing large-scale scientific applications. By differentiating PETSc, we provide accurate derivative computations in applications implemented using the toolkit. In addition to using automatic differentiation to generate a derivative enhanced version of PETSc, we exploit the component-based organization of the toolkit, applying high-level mathematical insight to increase the accuracy and efficiency of derivative computations. 1 Introduction In complex computational models of physical phenomena, it is often necessary or desir...

Many aerospace applications require parallel implicit solution strategies and software. We consider the use of two computational tools, PETSc and ADIFOR, to implement a Newton-Krylov-Schwarz method with pseudo-transient continuation for a particular application, namely, a steady-state, fully implicit, three-dimensional compressible Euler model of flow over an M6 wing. We describe how automatic differentiation (AD) can be used within the PETSc framework to compute the required derivatives. We present performance data demonstrating the suitability of AD and PETSc for this problem. We conclude with a synopsis of our results and a description of opportunities for future work. Key words: Compressible Euler, PETSc, Nonlinear PDEs, Automatic Differentiation 1 Introduction Parallel implicit solution strategies are important in aerodynamic applications modeled by PDEs with disparate temporal and spatial scales. Within this family of techniques, Newton-Krylov methods have been shown to be wi...

The NEOS Server provides access to a variety of optimization packages via the Internet. The new Kestrel interface to the NEOS Server extends the server's capabilities by permitting local programs to request optimization services and retrieve results. As a result, a locally running modeling system can have much the same access to remote NEOS solvers as to those installed locally.

This paper concerns on-going work; it compares several implementations of backward AD, describes a simple operator-overloading implementation specialized for gradient computations, and compares the implementations on some mesh-optimization examples. Ideas from the specialized implementation could be used in fully general source-to-source translators for C and C++

Often the most robust and efficient algorithms for the solution of large-scale problems involving nonlinear PDEs and optimization require the computation of derivative quantities. We examine the use of automatic differentiation (AD) to provide code for computing first and second derivatives in conjunction with two parallel numerical toolkits, the Portable, Extensible Toolkit for Scientific Computing (PETSc) and the Toolkit for Advanced Optimization (TAO). We discuss how the use of mathematical abstractions for vectors and matrices in these libraries facilitates the use of AD to automatically generate derivative codes and present performance data demonstrating the suitability of this approach. 1 Introduction As the complexity of advanced computational science applications has increased, the use of object-oriented software methods for the development of both applications and numerical toolkits has also increased. The migration toward this approach can be attributed in part to ...