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On the Complexity of Parametric ODEs and Related Problems

In: Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan

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  • Stefan Heinrich

    (University of Kaiserslautern, Department of Computer Science)

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.

Suggested Citation

  • Stefan Heinrich, 2018. "On the Complexity of Parametric ODEs and Related Problems," Springer Books, in: Josef Dick & Frances Y. Kuo & Henryk Woźniakowski (ed.), Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan, pages 567-595, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-72456-0_26
    DOI: 10.1007/978-3-319-72456-0_26
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