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Increasing memory locality by executing several model instances simultaneously 1

引用

6th International Conference on Automatic Differentiation, AD 2012

作者： Giering, Ralf Voßbeck, Michael FastOpt GmbH Lerchenstrasse 28a Hamburg 22767 Germany

ISBN: (数字)9783642300233

ISBN: (纸本)9783642300226

We present a new source-to-source transformation which generates code to compute several model instances simultaneously. Due to the increased memory locality of memory accesses this speeds up the computation on processors using a cache hierarchy to overcome the relative slow memory access. The speedup depends on the model code, the processor, the compiler, and on the number of instances. © 2012 Springer-Verlag.

关键词： Bandpass filters

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Application of automatic differentiation to an incompressible URANS solver 1

引用

6th International Conference on Automatic Differentiation, AD 2012

作者： Özkaya, Emre Nemili, Anil Gauger, Nicolas R. Computational Mathematics Group CCES RWTH Aachen University Schinkelstr. 2 Aachen 52062 Germany

ISBN: (数字)9783642300233

ISBN: (纸本)9783642300226

This paper deals with the task of generating a discrete adjoint solver from a given primal Unsteady Reynolds Averaged Navier-Stokes (URANS) solver for incompressible flows. This adjoint solver is to be employed in active flow control problems to enhance the performance of aerodynamic configurations. We discuss on how the development of such a code can be eased through the use of the reverse mode of Automatic/Algorithmic Differentiation (AD). If AD is applied in a black-box fashion then the resulting adjoint URANS solver will have prohibitively expensive memory requirements. We present several strategies to circumvent the excessive memory demands. We also address the parallelization of the adjoint code and the adjoint counterparts of the MPI directives that are used in the primal solver. The adjoint code is validated by applying it to the standard test case of a rotating cylinder by active flow control. The sensitivities based on the adjoint code are compared with the values obtained from finite differences and forward mode AD code. © 2012 Springer-Verlag.

关键词： Navier Stokes equations

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Evaluating an element of the Clarke generalized Jacobian of a piecewise differentiable function 1

引用

6th International Conference on Automatic Differentiation, AD 2012

作者： Khan, Kamil A. Barton, Paul I. Process Systems Engineering Laboratory Department of Chemical Engineering Massachusetts Institute of Technology Cambridge MA 02139 United States

ISBN: (数字)9783642300233

ISBN: (纸本)9783642300226

The (Clarke) generalized Jacobian of a locally Lipschitz continuous function is a derivative-like set-valued mapping that contains slope information. Several methods for optimization and equation solving require evaluation of generalized Jacobian elements. However, since the generalized Jacobian does not satisfy calculus rules sharply, this evaluation can be difficult. In this work, a method is presented for evaluating generalized Jacobian elements of a nonsmooth function that is expressed as a finite composition of absolute value functions and continuously differentiable functions. The method makes use of the principles of automatic differentiation and the theory of piecewise differentiable functions, and is guaranteed to be computationally tractable relative to the cost of a function evaluation. © 2012 Springer-Verlag.

关键词： Function evaluation

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Exploiting sparsity in automatic differentiation on multicore architectures 1

引用

6th International Conference on Automatic Differentiation, AD 2012

作者： Letschert, Benjamin Kulshreshtha, Kshitij Walther, Andrea Nguyen, Duc Gebremedhin, Assefaw Pothen, Alex Institut für Mathematik Universität Paderborn Paderborn Germany Department of Computer Science Purdue University West Lafayette IN United States

ISBN: (数字)9783642300233

ISBN: (纸本)9783642300226

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. © 2012 Springer-Verlag.

关键词： computational efficiency

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Storing versus recomputation on multiple DAGs 1

引用

6th International Conference on Automatic Differentiation, AD 2012

作者： Cole-Mullen, Heather Lyons, Andrew Utke, Jean Argonne National Laboratory University of Chicago Chicago IL United States Dartmouth College Hanover NH United States

ISBN: (数字)9783642300233

ISBN: (纸本)9783642300226

Recomputation and storing are typically seen as tradeoffs for checkpointing schemes in the context of adjoint computations. At finer granularity during the adjoint sweep, in practice, only the store-all or recompute-all approaches are fully automated. This paper considers a heuristic approach for exploiting finer granularity recomputations to reduce the storage requirements and thereby improve the overall adjoint efficiency without the need for manual intervention. © 2012 Springer-Verlag.

关键词： Heuristic methods

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Adjoint mode computation of subgradients for McCormick relaxations 1

引用

6th International Conference on Automatic Differentiation, AD 2012

作者： Beckers, Markus Mosenkis, Viktor Naumann, Uwe RWTH Aachen University Aachen D-52062 Germany STCE RWTH Aachen University Aachen D-52062 Germany

ISBN: (数字)9783642300233

ISBN: (纸本)9783642300226

In Mitsos et al. (SIAM Journal on Optimization 20(2):573-601, 2009), a method similar to Algorithmic Differentiation (AD) is presented which allows the propagation of, in general nondifferentiable, McCormick relaxations (McCormick, Mathematical Programming 10(2):147-175, 1976;Steihaug, Twelfth Euro AD Workshop, Berlin, 2011) of factorable functions and of the corresponding subgradients in tangent-linear mode. Subgradients are natural extensions of "usual" derivatives which allow the application of derivative-based methods to possibly nondifferentiable convex and concave functions. The software package libMC (Mitsos et al. SIAM Journal on Optimization 20(2):573-601, 2009) performs the automatic propagation of the relaxation and of corresponding subgradients based on the principles of tangent-linear mode AD by overloading. Similar ideas have been ported to Fortran yielding modMC as part of our ongoing collaboration with the authors of Mitsos et al. (SIAM Journal on Optimization 20(2):573-601, 2009). In this article an adjoint method for the computation of subgradients for McCormick relaxations is presented. A corresponding implementation by overloading in Fortran is provided in the form of amodMC. The calculated subgradients are used in a deterministic global optimization algorithm based on a branch-and-bound method. The superiority of adjoint over tangent-linear mode is illustrated by two examples. © 2012 Springer-Verlag.

关键词： Branch and bound method

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Connections between power series methods and automatic differentiation 1

引用

6th International Conference on Automatic Differentiation, AD 2012

作者： Carothers, David C. Lucas, Stephen K. Parker, G. Edgar Rudmin, Joseph D. Sochacki, James S. Thelwell, Roger J. Tongen, Anthony Warne, Paul G. James Madison University Harrisonburg VA 22807 United States

ISBN: (数字)9783642300233

ISBN: (纸本)9783642300226

There is a large overlap in the work of the Automatic Differentiation community and those whose use Power Series Methods. Automatic Differentiation is predominately applied to problems involving differentiation, and Power series began as a tool in the ODE setting. Three examples are presented to highlight this overlap, and several interesting results are presented. © 2012 Springer-Verlag.

关键词： Artificial intelligence

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Efficient automatic differentiation of matrix functions 1

引用

6th International Conference on Automatic Differentiation, AD 2012

作者： Olsen, Peder A. Rennie, Steven J. Goel, Vaibhava IBM TJ Watson Research Center Yorktown Heights NY United States

ISBN: (数字)9783642300233

ISBN: (纸本)9783642300226

Forward and reverse mode automatic differentiation methods for functions that take a vector argument make derivative computation efficient. However, the determinant and inverse of a matrix are not readily expressed in the language of vectors. The derivative of a function f(X) for a d x d matrix X is itself a d x d matrix. The second derivative, or Hessian, is a d 2 x d 2 matrix, and so computing and storing the Hessian can be very costly. In this paper, we present a new calculus for matrix differentiation, and introduce a new matrix operation, the box product, to accomplish this. The box product can be used to elegantly and efficiently compute both the first and second order matrix derivatives of any function that can be expressed solely in terms of arithmetic, transposition, trace and log determinant operations. The Hessian of such a function can be implicitly represented as a sum of Kronecker, outer, and box products, which allows us to compute the Newton step efficiently. Whereas the direct computation requires O(d4) storage and O(d 6) operations, the indirect representation of the Hessian allows the storage to be reduced to O(kd2), where k is the number of times the variable X occurs in the expression for the derivative. Likewise, the cost of computing the Newton direction is reduced to O(d5) in general, and O(d3) for k = 1 and k = 2. © 2012 Springer-Verlag.

关键词： Calculations

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Automatic differentiation through the use of hyper-dual numbers for second derivatives 1

引用

6th International Conference on Automatic Differentiation, AD 2012

作者： Fike, Jeffrey A. Alonso, Juan J. Department of Aeronautics and Astronautics Stanford University Stanford CA 94305 United States

ISBN: (数字)9783642300233

ISBN: (纸本)9783642300226

Automatic Differentiation techniques are typically derived based on the chain rule of differentiation. Other methods can be derived based on the inherent mathematical properties of generalized complex numbers that enable first-derivative information to be carried in the non-real part of the number. These methods are capable of producing effectively exact derivative values. However, when second-derivative information is desired, generalized complex numbers are not sufficient. Higher-dimensional extensions of generalized complex numbers, with multiple non-real parts, can produce accurate second-derivative information provided that multiplication is commutative. One particular number system is developed, termed hyper-dual numbers, which produces exact first- and second-derivative information. The accuracy of these calculations is demonstrated on an unstructured, parallel, unsteady Reynolds-Averaged Navier-Stokes solver. © 2012 Springer-Verlag.

关键词： Numbering systems

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Sparse Jacobian construction for mapped grid visco-resistive magnetohydrodynamics 1

引用

6th International Conference on Automatic Differentiation, AD 2012

作者： Reynolds, Daniel R. Samtaney, Ravi Department of Mathematics Southern Methodist University Dallas TX 75275 United States Department of Mechanical Engineering Division of Physical Science and Engineering King Abdullah University of Science and Technology Thuwal Saudi Arabia

ISBN: (数字)9783642300233

ISBN: (纸本)9783642300226

We apply the automatic differentiation tool OpenAD toward constructing a preconditioner for fully implicit simulations of mapped grid visco-resistive magnetohydrodynamics (MHD), used in modeling tokamak fusion devices. Our simulation framework employs a fully implicit formulation in time, and a mapped finite volume spatial discretization. We solve this model using inexact Newton-Krylov methods. Of critical importance in these iterative solvers is the development of an effective preconditioner, which typically requires knowledge of the Jacobian of the nonlinear residual function. However, due to significant nonlinearity within our PDE system, our mapped spatial discretization, and stencil adaptivity at physical boundaries, analytical derivation of these Jacobian entries is highly nontrivial. This paper therefore focuses on Jacobian construction using automatic differentiation. In particular, we discuss applying OpenAD to the case of a spatially-adaptive stencil patch that automatically handles differences between the domain interior and boundary, and configuring AD for reduced stencil approximations to the Jacobian. We investigate both scalar and vector tangent mode differentiation, along with simple finite difference approaches, to compare the resulting accuracy and efficiency of Jacobian construction in this application. © 2012 Springer-Verlag.

关键词： Iterative methods

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