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.

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  1. Che, Yonggang; Xu, Chuanfu; Wang, Tongtong: Parallelizing and optimizing a detonation combustion simulation application on heterogeneous platform (2018)
  2. Creech, Angus C. W.; Jackson, Adrian; Maddison, James R.: Adapting and optimising fluidity for high-fidelity coastal modelling (2018)
  3. Jordan, Charles; Joswig, Michael; Kastner, Lars: Parallel enumeration of triangulations (2018)
  4. Kalantzis, Vassilis; Xi, Yuanzhe; Saad, Yousef: Beyond automated multilevel substructuring: domain decomposition with rational filtering (2018)
  5. Lunet, Thibaut; Bodart, Julien; Gratton, Serge; Vasseur, Xavier: Time-parallel simulation of the decay of homogeneous turbulence using Parareal with spatial coarsening (2018)
  6. Morra, Gabriele: Pythonic geodynamics. Implementations for fast computing (2018)
  7. Rovenskaya, O. I.; Croce, G.: Heat transfer in rough microchannels under rarefied flow conditions (2018)
  8. Sassi Hidri, Minyar; Zoghlami, Mohamed Ali; Ben Ayed, Rahma: Speeding up the large-scale consensus fuzzy clustering for handling big data (2018)
  9. Towara, M.; Naumann, U.: SIMPLE adjoint message passing (2018)
  10. Van Straalen, Brian; Trebotich, David; Ovsyannikov, Andrey; Graves, Daniel T.: Scalable structured adaptive mesh refinement with complex geometry (2018)
  11. Afzal, Asif; Ansari, Zahid; Rimaz Faizabadi, Ahmed; Ramis, M. K.: Parallelization strategies for computational fluid dynamics software: state of the art review (2017)
  12. Andrianov, A. N.; Anikin, Anton S.; Bychkov, I. V.; Gornov, A. Yu.: Numerical solution of huge-scale quasiseparable optimization problems (2017)
  13. Botbol, Vincent; Chailloux, Emmanuel; Le Gall, Tristan: Static analysis of communicating processes using symbolic transducers (2017)
  14. Calandra, H.; Gratton, S.; Vasseur, X.: A geometric multigrid preconditioner for the solution of the Helmholtz equation in three-dimensional heterogeneous media on massively parallel computers (2017)
  15. Hofer, Christoph: Parallelization of continuous and discontinuous Galerkin dual-primal isogeometric tearing and interconnecting methods (2017)
  16. Jambunathan, Revathi; Levin, Deborah A.: Advanced parallelization strategies using hybrid MPI-CUDA octree DSMC method for modeling flow through porous media (2017)
  17. Jeong, Jena; Ramézani, Hamidréza; Leklou, Nordine: Porous-micro-dilatation theory for random crystallization: Monte Carlo simulation for delayed ettringite formation (2017)
  18. Munguía, Lluís-Miquel; Ahmed, Shabbir; Bader, David A.; Nemhauser, George L.; Goel, Vikas; Shao, Yufen: A parallel local search framework for the fixed-charge multicommodity network flow problem (2017)
  19. Schutte, Aaron D.: A nilpotent algebra approach to Lagrangian mechanics and constrained motion (2017)
  20. Briggs, J. P.; Pennycook, S. J.; Fergusson, J. R.; Jäykkä, J.; Shellard, E. P. S.: Separable projection integrals for higher-order correlators of the cosmic microwave sky: acceleration by factors exceeding 100 (2016)

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