Abstract
The computational complexity of the solution h to the ordinary differential equation h(0) = 0, h′(t) = g(t, h(t)) under various assumptions on the function g has been investigated in hope of understanding the intrinsic hardness of solving the equation numerically. Kawamura showed in 2010 that the solution h can be PSPACE-hard even if g is assumed to be Lipschitz continuous and polynomial-time computable. We place further requirements on the smoothness of g and obtain the following results: the solution h can still be PSPACE-hard if g is assumed to be of class C 1; for each k ≥ 2, the solution h can be hard for the counting hierarchy if g is of class C k.
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Kawamura, A., Ota, H., Rösnick, C., Ziegler, M. (2012). Computational Complexity of Smooth Differential Equations. In: Rovan, B., Sassone, V., Widmayer, P. (eds) Mathematical Foundations of Computer Science 2012. MFCS 2012. Lecture Notes in Computer Science, vol 7464. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32589-2_51
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DOI: https://doi.org/10.1007/978-3-642-32589-2_51
Publisher Name: Springer, Berlin, Heidelberg
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