On the Complexity of Parametric ODEs and Related Problems

Stefan Heinrich ()
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Stefan Heinrich: University of Kaiserslautern, Department of Computer Science

A chapter in Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan, 2018, pp 567-595 from Springer

Abstract: Abstract We present an iterative Monte Carlo procedure to solve initial value problems for systems of ordinary differential equations depending on a parameter. It is based on a multilevel Monte Carlo algorithm for parametric indefinite integration. As an application, we also obtain a respective method for solving almost linear first order partial differential equations. We also consider deterministic algorithms. We study the convergence and, in the framework of information-based complexity, the minimal errors and show that the developed algorithms are of optimal order (in some limit cases up to logarithmic factors). In this way we extend recent complexity results on parametric ordinary differential equations. Moreover, we obtain the complexity of almost linear first-order partial differential equations, which has not been analyzed before.

Date: 2018
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-72456-0_26

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DOI: 10.1007/978-3-319-72456-0_26

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