Local linearization method: Difference between revisions

Revision as of 17:13, 8 December 2020

Revision as of 17:15, 8 December 2020

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== Historical notes ==

Below is a time line of the main developments of the Local Linearization (LL) method.

* Pope D.A. (1963) introduces the LL discretization for ODEs and the LL scheme based on Taylor expansion. <ref name=″:18″> Pope, D. A. (1963). An exponential method of numerical integration of ordinary differential equations. Communications of the ACM, 6(8), 491-493. [https://doi.org/10.1145%2F366707.367592 doi:10.1145/366707.367592]</ref>

* Ozaki T. (1985) introduces the LL method for the integration and estimation of SDEs. The term "Local Linearization" (LL) is used for first time. <ref name=″:19″> Ozaki, T. (1985). 2 Non-linear time series models and dynamical systems. Handbook of statistics, 5, 25-83.[https://doi.org/10.1016/S0169-7161(85)05004-0 <small>doi:10.1016/S0169-7161(85)05004-0</small>]</ref>

* Biscay R. et al. (1996) reformulate the strong LL method for SDEs.<ref name=″:20″> Biscay, R., Jimenez, J. C., Riera, J. J., & Valdes, P. A. (1996). Local linearization method for the numerical solution of stochastic differential equations. Annals of the Institute of Statistical Mathematics, 48(4), 631-644.[[doi:10.1007/BF00052324|<small>doi:10.1007/BF00052324</small>]] </ref>

* Shoji I. and Ozaki T. (1997) reformulate the weak LL method for SDEs.<ref name=″:21″> Shoji, I., & Ozaki, T. (1997). Comparative study of estimation methods for continuous time stochastic processes. Journal of time series analysis, 18(5), 485-506.[[doi:10.1111/1467-9892.00064|<small>doi: 10.1111/1467-9892.00064</small>]]</ref>

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* Carbonell F.M. et al. (2005) introduce the LL method for RDEs. <ref name=″:24″>Carbonell, F., Jimenez, J. C., Biscay, R. J., & De La Cruz, H. (2005). The local linearization method for numerical integration of random differential equations. BIT Numerical Mathematics, 45(1), 1-14. [https://doi.org/10.1007%2FS10543-005-2645-9 doi:10.1007/s10543-005-2645-9]</ref>

* Jimenez J.C. et al. (2006) introduce the LL method for DDEs. <ref name=":13" />

* De la Cruz H. et al. (2006,2007) and Tokman M. (2006) introduce the two classes of HOLL integrators for ODEs: Thethe integrator-based <ref name=":1" /> and the quadrature-based.<ref name=":17" /><ref name=":2" />

* De la Cruz H. et al. (2010) introduce strong HOLL method for SDEs. <ref name=":4" />