Computational Complexity of Smooth Differential Equations

Kawamura, Akitoshi; Ota, Hiroyuki; Rösnick, Carsten; Ziegler, Martin

doi:10.2168/lmcs-10(1:6)2014

Cited by 17 publications

(10 citation statements)

References 18 publications

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“…Recently Kawamura and Cook [11] developed a framework applicable to the space C[0, 1] (which is not σ-compact), using higher-order complexity theory and in particular second-order polynomials. In particular their theory enables them to prove uniform versions of older results about the complexity of solving differential equations, as well as new results [12,13].…”

Section: Introductionmentioning

confidence: 99%

Analytical properties of resource-bounded real functionals

Férée¹,

Gomaa²,

Hoyrup³

2014

Journal of Complexity

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Computable analysis is an extension of classical discrete computability by enhancing the normal Turing machine model. It investigates mathematical analysis from the computability perspective. Though it is well developed on the computability level, it is still under developed on the complexity perspective, that is, when bounding the available computational resources. Recently Kawamura and Cook developed a framework to define the computational complexity of operators arising in analysis. Our goal is to understand the effects of complexity restrictions on the analytical properties of the operator. We focus on the case of norms over C[0,1] and introduce the notion of dependence of a norm on a point and relate it to the query complexity of the norm. We show that the dependence of almost every point is of the order of the query complexity of the norm. A norm with small complexity depends on a few points but, as compensation, highly depends on them. We briefly show how to obtain similar results for non-deterministic time complexity. We characterize the functionals that are computable using one oracle call only and discuss the uniformity of that characterization. This paper is a significant revision and expansion of an earlier conference version [1].

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Section: Introductionmentioning

confidence: 99%

Analytical properties of resource-bounded real functionals

Férée¹,

Gomaa²,

Hoyrup³

2014

Journal of Complexity

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“…On another different note, in this work the computational complexity of the IVP is examined at the "macro level" (rational t → ∞), but what is left aside is the "micro level" (real t), where the precision of t plays an important role [1]. Another work is planned devoted to that case.…”

Section: Discussionmentioning

confidence: 99%

“…In the text that follows, t in the IVP is treated as rational, while x 0 is a real vector. The complexity analysis of another special case of IVP, where t ∈ R, is given in [1].…”

Section: Is Unique;mentioning

confidence: 99%

“…There exists a huge number of algorithms and program packages for obtaining numerical solutions of systems of differential equations originating from math, physics, celestial mechanics and engineering. However, there is little available research in the area of computational complexity of the initial value problem itself (some results are obtained in [1] and [2]). In most other works, complexity of particular algorithms is analyzed, in terms of either the number of basic arithmetical operations performed on each step, or the number of calls to first or higher-order derivatives.…”

Section: Introductionmentioning

confidence: 99%

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The Computational Complexity of the Initial Value Problem for the Three Body Problem

Vasiliev

Pavlov

2017

J Math Sci

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The paper is concerned with the computational complexity of the initial value problem (IVP) for a system of ordinary dynamical equations. Formal problem statement is given, containing a Turing machine with an oracle for getting the initial values as real numbers. It is proven that the computational complexity of the IVP for the three body problem is not bounded by a polynomial. The proof is based on the analysis of oscillatory solutions of the Sitnikov problem that have complex dynamical behavior. These solutions contradict the existence of an algorithm that solves the IVP in polynomial time.

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“…Recently Kawamura and Cook [KC10] developed a framework applicable to the space C[0, 1] (which is not σ-compact), using higher-order complexity theory and in particular second-order polynomials. In particular their theory enables them to prove uniform versions of older results about the complexity of solving differential equations, as well as new results [Kaw10,KORZ12].…”

Section: Introductionmentioning

confidence: 99%

On the Query Complexity of Real Functionals

Férée

Hoyrup

Gomaa

2013

2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science

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Recently Kawamura and Cook developed a framework to define the computational complexity of operators arising in analysis. Our goal is to understand the effects of complexity restrictions on the analytical properties of the operator. We focus on the case of norms over C[0, 1] and introduce the notion of dependence of a norm on a point and relate it to the query complexity of the norm. We show that the dependence of almost every point is of the order of the query complexity of the norm. A norm with small complexity depends on a few points but, as compensation, highly depends on them. We characterize the functionals that are computable using one oracle call only and discuss the uniformity of that characterization.

show abstract

Computational Complexity of Smooth Differential Equations

Cited by 17 publications

References 18 publications

Analytical properties of resource-bounded real functionals

Analytical properties of resource-bounded real functionals

The Computational Complexity of the Initial Value Problem for the Three Body Problem

On the Query Complexity of Real Functionals

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