MPI

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 507 articles , 1 standard article )

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  1. Badri, M. A.; Rastiello, G.; Foerster, E.: Preconditioning strategies for vectorial finite element linear systems arising from phase-field models for fracture mechanics (2021)
  2. Iavernaro, F.; Mazzia, F.; Mukhametzhanov, M. S.; Sergeyev, Ya. D.: Computation of higher order Lie derivatives on the infinity computer (2021)
  3. Larios-Cárdenas, Luis Ángel; Gibou, Frederic: A deep learning approach for the computation of curvature in the level-set method (2021)
  4. Sukov, S. A.: Parallelization for unstructured adaptive mesh CFD algorithm (2021)
  5. Akimova, Elena N.; Misilov, Vladimir E.: Efficient numerical algorithm for solving the gravimetry problem of finding a lateral density in a layer: parallel implementation (2020)
  6. Ballard, Grey; Klinvex, Alicia; Kolda, Tamara G.: TuckerMPI: a parallel C++/MPI software package for large-scale data compression via the Tucker tensor decomposition (2020)
  7. Demidov, D.; Rossi, R.: Subdomain deflation combined with local AMG: a case study using AMGCL library (2020)
  8. Junttila, Tommi; Karppa, Matti; Kaski, Petteri; Kohonen, Jukka: An adaptive prefix-assignment technique for symmetry reduction (2020)
  9. Kalantzis, Vassilis: A domain decomposition Rayleigh-Ritz algorithm for symmetric generalized eigenvalue problems (2020)
  10. Kelly, Conlain; Olsen, Nicholas; Negrut, Dan: Billion degree of freedom granular dynamics simulation on commodity hardware via heterogeneous data-type representation (2020)
  11. Moratilla-Vega, M. A.; Lackhove, K.; Janicka, J.; Xia, H.; Page, G. J.: Jet noise analysis using an efficient LES/high-order acoustic coupling method (2020)
  12. Popovici, Doru Thom; Schatz, Martin D.; Franchetti, Franz; Low, Tze Meng: A flexible framework for multidimensional DFTs (2020)
  13. Utrera, Gladys; Gil, Marisa; Martorell, Xavier: A methodology approach to compare performance of parallel programming models for shared-memory architectures (2020)
  14. Andonovski, Nemanja; Moglie, Franco; Lenci, Stefano: Introduction to scientific computing technologies for global analysis of multidimensional nonlinear dynamical systems (2019)
  15. Chen, Jian; Takeyama, Tomohide; O-Tani, Hideyuki; Fujita, Kohei; Motoyama, Hiroki; Hori, Muneo: Using high performance computing for liquefaction hazard assessment with statistical soil models (2019)
  16. Demidov, D.: AMGCL: an efficient, flexible, and extensible algebraic multigrid implementation (2019)
  17. Lee, Chang-Ock; Nam, Changmin; Park, Jongho: Domain decomposition methods using dual conversion for the total variation minimization with (L^1) fidelity term (2019)
  18. Lee, Jungki; Jeong, Hogwan: Parallel volume integral equation method for two-dimensional elastodynamic analysis (2019)
  19. Lupo Pasini, Massimiliano: Convergence analysis of Anderson-type acceleration of Richardson’s iteration. (2019)
  20. Myatt, Jason F.; Shaw, John G.; Follett, Russell K.; Edgell, Dana H.; Froula, Dustin H.; Palastro, John P.; Goncharov, Valeri N.: \textitLPSE: a 3-D wave-based model of cross-beam energy transfer in laser-irradiated plasmas (2019)

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