Complexity of differential and integral equations

Werschulz, Arthur G.

doi:10.1016/0885-064x(85)90013-5

Cited by 26 publications

(24 citation statements)

References 18 publications

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“…The main results of those papers are that the asymptotic cost of the numerical solution of such initial value problems is polynomial in the number of digits of accuracy, B = log 2 (1/ε) . These results improved on the results of the standard theory of information based complexity (see, e.g., [36]), which predicted exponential cost of solving IVP for ODE. We believe that our model, assuming more smoothness of the problem and solution, is more realistic.…”

Section: Introductionsupporting

confidence: 79%

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Polynomial cost for solving IVP for high-index DAE

Corless

Ilie

2008

Bit Numer Math

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Abstract.We show that the cost of solving initial value problems for high-index differential algebraic equations is polynomial in the number of digits of accuracy requested. The algorithm analyzed is built on a Taylor series method developed by Pryce for solving a general class of differential algebraic equations. The problem may be fully implicit, of arbitrarily high fixed index and contain derivatives of any order. We give estimates of the residual which are needed to design practical error control algorithms for differential algebraic equations. We shown that adaptive meshes are always more efficient than non-adaptive meshes. Finally, we construct sufficiently smooth interpolants of the discrete solution.AMS subject classification (2000): 34A09, 65L80, 68Q25.

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Section: Introductionsupporting

confidence: 79%

“…Following the standard theory of computational complexity [36], we ignore memory hierarchy, overheads and interpolation costs.…”

Section: Polynomial Cost Of Pryce's Algorithmmentioning

confidence: 99%

Polynomial cost for solving IVP for high-index DAE

Corless

Ilie

2008

Bit Numer Math

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“…The bottleneck is the advection process. The computational cost for solving a ordinary differential equation for the advection with accuracy e is exponential in ln(e) (Werschulz, 1991). Roughly we can say, that the run time depends linearly on the input size, the size of the high-dimensional space, the dimension of the manifold and the number of steps during the advection, which is dependent on the curvature and the length of the advection path.…”

Section: Resultsmentioning

confidence: 99%

Robust non-linear dimensionality reduction using successive 1-dimensional Laplacian Eigenmaps

Gerber

Taşdizen

Whitaker

2007

Proceedings of the 24th International Conference on Machine Learning

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Non-linear dimensionality reduction of noisy data is a challenging problem encountered in a variety of data analysis applications. Recent results in the literature show that spectral decomposition, as used for example by the Laplacian Eigenmaps algorithm, provides a powerful tool for non-linear dimensionality reduction and manifold learning. In this paper, we discuss a significant shortcoming of these approaches, which we refer to as the repeated eigendirections problem. We propose a novel approach that combines successive 1-dimensional spectral embeddings with a data advection scheme that allows us to address this problem. The proposed method does not depend on a non-linear optimization scheme; hence, it is not prone to local minima. Experiments with artificial and real data illustrate the advantages of the proposed method over existing approaches. We also demonstrate that the approach is capable of correctly learning manifolds corrupted by significant amounts of noise.

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“…Equations of this form arise in rational expectations theories of asset pricing, see, for example, Lucas (1978) or Tauchen and Hussey (1991). Much is known about the complexity of computing approximate solutions to Fredholm integral equations with general kernels, see Werschulz (1991) and Heinrich (1998) for surveys of deterministic and stochastic complexity bounds for this problem. In the case of general kernels, the results in Werschulz (1991) show that the problem is intractable in the worst case using deterministic algorithms.…”

Section: Fredholm Integral Equationsmentioning

confidence: 99%

“…Much is known about the complexity of computing approximate solutions to Fredholm integral equations with general kernels, see Werschulz (1991) and Heinrich (1998) for surveys of deterministic and stochastic complexity bounds for this problem. In the case of general kernels, the results in Werschulz (1991) show that the problem is intractable in the worst case using deterministic algorithms. By exploiting the special structure of the kernel for the class of Fredholm integral equations combined with additional special structure on the functions π i and p i (to be defined shortly) we will be able to show that the fixed point problem, and the associated Fredholm problem, is strongly tractable.…”

Section: Fredholm Integral Equationsmentioning

confidence: 99%

Is There a Curse of Dimensionality for Contraction Fixed Points in the Worst Case?

Rust¹,

Traub²,

Woźniakowski³

2002

Econometrica

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This paper analyzes the complexity of the contraction fixed point problem: compute an ε-approximation to the fixed point V * = Γ(V * ) of a contraction mapping Γ that maps a Banach space B d of continuous functions of d variables into itself. We focus on quasi linear contractions where Γ is a nonlinear functional of a finite number of conditional expectation operators. This class includes contractive Fredholm integral equations that arise in asset pricing applications and the contractive Bellman equation from dynamic programming. In the absence of further restrictions on the domain of Γ, the quasi linear fixed point problem is subject to the curse of dimensionality, i.e., in the worst case the minimal number of function evaluations and arithmetic operations required to compute an ε-approximation to a fixed point V * ∈ B d increases exponentially in d. We show that the curse of dimensionality disappears if the domain of Γ has additional special structure. We identify a particular type of special structure for which the problem is strongly tractable even in the worst case, i.e., the number of function evaluations and arithmetic operations needed to compute an ε-approximation of V * is bounded by Cε −p where C and p are constants independent of d. We present examples of economic problems that have this type of special structure including a class of rational expectations asset pricing problems for which the optimal exponent p = 1 is nearly achieved.

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Complexity of differential and integral equations

Cited by 26 publications

References 18 publications

Polynomial cost for solving IVP for high-index DAE

Polynomial cost for solving IVP for high-index DAE

Robust non-linear dimensionality reduction using successive 1-dimensional Laplacian Eigenmaps

Is There a Curse of Dimensionality for Contraction Fixed Points in the Worst Case?

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