87,002 research outputs found

    Computational Complexity of Smooth Differential Equations

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    The computational complexity of the solutions hh to the ordinary differential equation h(0)=0h(0)=0, h(t)=g(t,h(t))h'(t) = g(t, h(t)) under various assumptions on the function gg has been investigated. Kawamura showed in 2010 that the solution hh can be PSPACE-hard even if gg is assumed to be Lipschitz continuous and polynomial-time computable. We place further requirements on the smoothness of gg and obtain the following results: the solution hh can still be PSPACE-hard if gg is assumed to be of class C1C^1; for each k2k\ge2, the solution hh can be hard for the counting hierarchy even if gg is of class CkC^k.Comment: 15 pages, 3 figure

    Computational Complexity of Smooth Differential Equations

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    Parameterized Uniform Complexity in Numerics: from Smooth to Analytic, from NP-hard to Polytime

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    The synthesis of classical Computational Complexity Theory with Recursive Analysis provides a quantitative foundation to reliable numerics. Here the operators of maximization, integration, and solving ordinary differential equations are known to map (even high-order differentiable) polynomial-time computable functions to instances which are `hard' for classical complexity classes NP, #P, and CH; but, restricted to analytic functions, map polynomial-time computable ones to polynomial-time computable ones -- non-uniformly! We investigate the uniform parameterized complexity of the above operators in the setting of Weihrauch's TTE and its second-order extension due to Kawamura&Cook (2010). That is, we explore which (both continuous and discrete, first and second order) information and parameters on some given f is sufficient to obtain similar data on Max(f) and int(f); and within what running time, in terms of these parameters and the guaranteed output precision 2^(-n). It turns out that Gevrey's hierarchy of functions climbing from analytic to smooth corresponds to the computational complexity of maximization growing from polytime to NP-hard. Proof techniques involve mainly the Theory of (discrete) Computation, Hard Analysis, and Information-Based Complexity

    A Survey on Continuous Time Computations

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    We provide an overview of theories of continuous time computation. These theories allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuous time analog models. We survey the existing models, summarizing results, and point to relevant references in the literature
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