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. Creech, Angus C. W.; Jackson, Adrian; Maddison, James R.: Adapting and optimising fluidity for high-fidelity coastal modelling (2018)
  2. Desai, Ajit; Khalil, Mohammad; Pettit, Chris; Poirel, Dominique; Sarkar, Abhijit: Scalable domain decomposition solvers for stochastic PDEs in high performance computing (2018)
  3. Fohrmeister, Volker; Bartels, Alexander; Mosler, Jörn: Variational updates for thermomechanically coupled gradient-enhanced elastoplasticity -- implementation based on hyper-dual numbers (2018)
  4. Gorobets, A.; Soukov, S.; Bogdanov, P.: Multilevel parallelization for simulating compressible turbulent flows on most kinds of hybrid supercomputers (2018)
  5. Jordan, Charles; Joswig, Michael; Kastner, Lars: Parallel enumeration of triangulations (2018)
  6. Kalantzis, Vassilis; Xi, Yuanzhe; Saad, Yousef: Beyond automated multilevel substructuring: domain decomposition with rational filtering (2018)
  7. Lorenz, Eric; Sivadasan, Vishnu; Bonn, Daniel; Hoekstra, Alfons G.: Combined lattice-Boltzmann and rigid-body method for simulations of shear-thickening dense suspensions of hard particles (2018)
  8. Lunet, Thibaut; Bodart, Julien; Gratton, Serge; Vasseur, Xavier: Time-parallel simulation of the decay of homogeneous turbulence using Parareal with spatial coarsening (2018)
  9. Luo, Xiangyu; Wei, Yingying: Nonparametric Bayesian learning of heterogeneous dynamic transcription factor networks (2018)
  10. Morra, Gabriele: Pythonic geodynamics. Implementations for fast computing (2018)
  11. Neukirchen, Helmut: Elephant against Goliath: performance of big data versus high-performance computing DBSCAN clustering implementations (2018)
  12. Polyakov, S. V.; Podryga, V. O.; Puzyrkov, D. V.: High performance computing in multiscale problems of gas dynamics (2018)
  13. Rovenskaya, O. I.; Croce, G.: Heat transfer in rough microchannels under rarefied flow conditions (2018)
  14. Sassi Hidri, Minyar; Zoghlami, Mohamed Ali; Ben Ayed, Rahma: Speeding up the large-scale consensus fuzzy clustering for handling big data (2018)
  15. Sawant, Saurabh S.; Tumuklu, Ozgur; Jambunathan, Revathi; Levin, Deborah A.: Application of adaptively refined unstructured grids in DSMC to shock wave simulations (2018)
  16. Towara, M.; Naumann, U.: SIMPLE adjoint message passing (2018)
  17. Van Straalen, Brian; Trebotich, David; Ovsyannikov, Andrey; Graves, Daniel T.: Scalable structured adaptive mesh refinement with complex geometry (2018)
  18. Afzal, Asif; Ansari, Zahid; Rimaz Faizabadi, Ahmed; Ramis, M. K.: Parallelization strategies for computational fluid dynamics software: state of the art review (2017)
  19. Andrianov, A. N.; Anikin, Anton S.; Bychkov, I. V.; Gornov, A. Yu.: Numerical solution of huge-scale quasiseparable optimization problems (2017)
  20. Berger-Vergiat, Luc; Waisman, Haim: An overlapping domain decomposition preconditioning method for monolithic solution of shear bands (2017)

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