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Generating efficient derivative code with TAF: Adjoint and tangent linear Euler flow around an airfoil
, 2004
"... FastOpt's new automatic differentiation tool TAF is applied to the twodimensional NavierStokes solver NSC2KE. For a configuration that simulates the Euler flow around a NACA airfoil, TAF has generated the tangent linear and adjoint models as well as the second derivative (Hessian) code. Owing ..."
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Cited by 12 (1 self)
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FastOpt's new automatic differentiation tool TAF is applied to the twodimensional NavierStokes solver NSC2KE. For a configuration that simulates the Euler flow around a NACA airfoil, TAF has generated the tangent linear and adjoint models as well as the second derivative (Hessian) code. Owing to TAF's capability of generating efficient adjoints of iterative solvers, the derivative code has a high performance: Running both the solver and its adjoint requires 3.4 times as long as running the solver only. Further examples of highly efficient tangent linear, adjoint, and Hessian codes for large and complex threedimensional Fortran 7790 climate models are listed. These examples suggest that the performance of the NSC2KE adjoint may well be generalised to more complex threedimensional CFD codes. We also sketch how TAF can improve the adjoint's performance by exploiting selfadjointness, which is a common feature of CFD codes.
A userâ€™s guide to solving dynamic stochastic games using the homotopy method
, 2008
"... This paper provides a stepbystep 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 homot ..."
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Cited by 11 (6 self)
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This paper provides a stepbystep 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.
Automatic differentiation of numerical integration algorithms
 Mathematics of Computation
, 1996
"... Abstract. Automatic differentiation (AD) is a technique for automatically augmenting computer programs with statements for the computation of derivatives. This article discusses the application of automatic differentiation to numerical integration algorithms for ordinary differential equations (ODEs ..."
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Cited by 10 (4 self)
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Abstract. Automatic differentiation (AD) is a technique for automatically augmenting computer programs with statements for the computation of derivatives. This article discusses the application of automatic differentiation to numerical integration algorithms for ordinary differential equations (ODEs), in particular, the ramifications of the fact that AD is applied not only to the solution of such an algorithm, but to the solution procedure itself. This subtle issue can lead to surprising results when AD tools are applied to variablestepsize, variableorder ODE integrators. The computation of the final time step plays a special role in determining the computed derivatives. We investigate these issues using various integrators and suggest constructive approaches for obtaining the desired derivatives. 1.
TimeParallel Computation of PseudoAdjoints for a Leapfrog Scheme
 Preprint ANL/MCSP6390197, Mathematics and Computer Science Division, Argonne National Laboratory
, 1997
"... The leapfrog scheme is a commonly used secondorder 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 ..."
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Cited by 10 (5 self)
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The leapfrog scheme is a commonly used secondorder 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 socalled 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 sparsityexploiting 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 2D shallowwater equation model problem on an IBM SP parallel...
Kestrel: An Interface from Modeling Systems to the NEOS Server. Published ahead of print
 in INFORMS Journal on Computing
, 2008
"... ..."
A Survey Of Shape Parameterization Techniques
, 1999
"... This paper provides a survey of shape parameterization techniques for multidisciplinary ..."
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Cited by 8 (1 self)
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This paper provides a survey of shape parameterization techniques for multidisciplinary
Automatic Differentiation and NavierStokes Computations
"... this paper, we discuss how AD can be used to enhance a compressible NavierStokes solver. Section 2 describes the twodimensional NavierStokes model and solver used in our studies. Section 3 gives a brief introduction to source transformation tools for automatic differentiation. Section 4 discusses ..."
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Cited by 8 (0 self)
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this paper, we discuss how AD can be used to enhance a compressible NavierStokes solver. Section 2 describes the twodimensional NavierStokes 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 highlevel mathematics of a computation can greatly reduce the cost of derivative computations using AD.
Automatic Differentiation for MessagePassing Parallel Programs
, 1998
"... 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. ..."
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Cited by 6 (4 self)
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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 functionexpressed in the form of a programaccurately 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...
Parallel Simulation of Compressible Flow Using Automatic Differentiation and PETSc
"... Many aerospace applications require parallel implicit solution strategies and software. We consider the use of two computational tools, PETSc and ADIFOR, to implement a NewtonKrylovSchwarz method with pseudotransient continuation for a particular application, namely, a steadystate, fully implici ..."
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Cited by 6 (1 self)
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Many aerospace applications require parallel implicit solution strategies and software. We consider the use of two computational tools, PETSc and ADIFOR, to implement a NewtonKrylovSchwarz method with pseudotransient continuation for a particular application, namely, a steadystate, fully implicit, threedimensional 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, NewtonKrylov methods have been shown to be wi...
Developing a DerivativeEnhanced ObjectOriented Toolkit for Scientific Computations
, 1998
"... 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 derivativeaugme ..."
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Cited by 6 (5 self)
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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 derivativeaugmented code, which can then be used for sensitivity analysis, optimization, or parameter estimation. Scientific toolkits play an increasingly important role in developing largescale 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 componentbased organization of the toolkit, applying highlevel mathematical insight to increase the accuracy and efficiency of derivative computations.