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

References in zbMATH (referenced in 487 articles , 1 standard article )

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  1. Saman, M. Y.; Evans, D. J.: Distributed computing on cluster systems (2001)
  2. Schumacher, Michael: Objective coordination in multi-agent system engineering. Design and implementation (2001)
  3. Skjellum, Anthony; Wooley, Diane G.; Lu, Ziyang; Wolf, Michael; Bangalore, Purushotham V.; Lumsdaine, Andrew; Squyres, Jeffrey M.; McCandless, Brian: Object-oriented analysis and design of the message passing interface (2001)
  4. Tryggvason, G.; Bunner, B.; Esmaeeli, A.; Juric, D.; Al-Rawahi, N.; Tauber, W.; Han, J.; Nas, S.; Jan, Y.-J.: A front-tracking method for the computations of multiphase flow. (2001)
  5. Watson, Layne T.; Baker, Chuck A.: A fully-distributed parallel global search algorithm (2001)
  6. Badía, J. M.; Vidal, A. M.: Inverse Toeplitz eigenproblem on personal computer networks (2000)
  7. Boisvert, Ronald F.: Mathematical software: Past, present, and future (2000)
  8. Bova, S. W.; Carey, G. F.: A distributed memory parallel element-by-element scheme for semiconductor device simulation (2000)
  9. Goldfield, E. M.: Parallel strategies for four-atom quantum dynamics calculations (2000)
  10. Grosz, Lutz: How to vectorize the algebraic multilevel iteration (2000)
  11. Kalia, Rajiv K.; Campbell, Timothy J.; Chatterjee, Alok; Nakano, Aiichiro; Vashishta, Priya; Ogata, Shuji: Multiresolution algorithms for massively parallel molecular dynamics simulations of nanostructured materials (2000)
  12. Knottenbelt, W. J.; Harrison, P. G.; Mestern, M. A.; Kritzinger, P. S.: A probabilistic dynamic technique for the distributed generation of very large state spaces (2000)
  13. Koradi, R.; Billeter, M.; Güntert, P.: Point-centered domain decomposition for parallel molecular dynamics simulation (2000)
  14. Laganà, Antonio; Crocchianti, Stefano; Bolloni, Alessandro; Piermarini, Valentina; Baraglia, Ranieri; Ferrini, Renato; Laforenza, Domenico: Computational granularity and parallel models to scale up reactive scattering calculations (2000)
  15. Qiang, Ji; Ryne, Robert D.; Habib, Salman; Decyk, Viktor: An object-oriented parallel particle-in-cell code for beam dynamics simulation in linear accelerators. (2000)
  16. Rendleman, Charles A.; Beckner, Vincent E.; Lijewski, Mike; Crutchfield, William; Bell, John B.: Parallelization of structured, hierarchical adaptive mesh refinement algorithms (2000)
  17. Sarkis, Marcus; Koobus, Bruno: A scaled and minimum overlap restricted additive Schwarz method with application to aerodynamics (2000)
  18. Soni, Bharat K.; Koomullil, Roy; Thompson, David S.; Thornburg, Hugh: Solution adaptive grid strategies based on point redistribution (2000)
  19. Stankovic, Nenad: An open Java system for SPMD programming (2000)
  20. Stankovic, Nenad; Zhang, Kang: An evaluation of Java implementations of message-passing (2000)

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