Automatic integration using asymptotically optimal adaptive Simpson quadrature

Plaskota, Leszek

doi:10.1007/s00211-014-0684-3

Cited by 12 publications

(14 citation statements)

References 13 publications

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“…and the computational cost for minimization is significantly smaller than that for function approximation. The minimization problem (MIN) for functions in the whole Sobolev space W 2,∞ has a similar lower complexity bound as (19) for the function approximation problem by a similar proof. However, for functions only in the cone C, we have not yet derived a lower bound on the complexity of the minimization problem (MIN) for functions in C.…”

Section: The Computational Cost Of Mmentioning

confidence: 89%

Local adaption for approximation and minimization of univariate functions

Choi

Ding

Hickernell

et al. 2017

Journal of Complexity

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Most commonly used adaptive algorithms for univariate real-valued function approximation and global minimization lack theoretical guarantees. Our new locally adaptive algorithms are guaranteed to provide answers that satisfy a user-specified absolute error tolerance for a cone, C, of non-spiky input functions in the Sobolev space W 2,∞ [a, b]. Our algorithms automatically determine where to sample the function-sampling more densely where the second derivative is larger. The computational cost of our algorithm for approximating a univariate function f on a bounded interval with L ∞ -error no greater than ε is O f 1 2 /ε as ε → 0. This is the same order as that of the best function approximation algorithm for functions in C. The computational cost of our global minimization algorithm is of the same order and the cost can be substantially less if f significantly exceeds its minimum over much of the domain. Our Guaranteed Automatic Integration Library (GAIL) contains these new algorithms. We provide numerical experiments to illustrate their superior performance.

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Section: The Computational Cost Of Mmentioning

confidence: 89%

Local adaption for approximation and minimization of univariate functions

Choi

Ding

Hickernell

et al. 2017

Journal of Complexity

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“…where L * m,r f denotes the approximation corresponding to (8), are equal. Observe that such a partition exists since the local errors continuously depend on the points x i .…”

Section: Optimal Partitionmentioning

confidence: 99%

“…On the other hand, if the function is smooth in the whole domain, then adaptive algorithms can improve the error only by a constant compared to nonadaptive algorithms. The exact asymptotic constants for quadratures of degree of exactness r − 1 and for functions f ∈ C r ([a, b]) with f (r) > 0 were obtained in [8] for r = 4 and in [2,3] for arbitrary r. Procedures corresponding to the optimal strategies for automatic integration were also proposed.…”

Section: Introductionmentioning

confidence: 99%

Automatic approximation using asymptotically optimal adaptive interpolation

Plaskota

Samoraj

2021

Numer Algor

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We present an asymptotic analysis of adaptive methods for Lp approximation of functions f ∈ Cr([a, b]), where $1\le p\le +\infty $ 1 ≤ p ≤ + ∞ . The methods rely on piecewise polynomial interpolation of degree r − 1 with adaptive strategy of selecting m subintervals. The optimal speed of convergence is in this case of order m−r and it is already achieved by the uniform (nonadaptive) subdivision of the initial interval; however, the asymptotic constant crucially depends on the chosen strategy. We derive asymptotically best adaptive strategies and show their applicability to automatic Lp approximation with a given accuracy ε.

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“…In particular, we rigorously discuss the accuracy and cost of an adaptive process for a well precised class of problems, not only for a number of computational examples. For the integration of scalar C 4 functions similar questions have recently been addressed for the Simpson rule in [9], where it is shown that the adaptive mesh selection allows us to reduce the error by reducing the asymptotic constant of the method. Adaptive mesh points for the approximation of univariate W 2,∞ functions is discussed in [1] .…”

Section: Introductionmentioning

confidence: 95%

Adaptive mesh point selection for the efficient solution of scalar IVPs

Kacewicz¹

2017

Numer Algor

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We discuss adaptive mesh point selection for the solution of scalar IVPs. We consider a method that is optimal in the sense of the speed of convergence, and aim at minimizing the local errors. Although the speed of convergence cannot be improved by using the adaptive mesh points compared to the equidistant points, we show that the factor in the error expression can be significantly reduced. We obtain formulas specifying the gain achieved in terms of the number of discretization subintervals, as well as in terms of the prescribed level of the local error. Both nonconstructive and constructive versions of the adaptive mesh selection are shown, and a numerical example is given.Comment: 21 pages. Numerical Algorithms final versio

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Automatic integration using asymptotically optimal adaptive Simpson quadrature

Cited by 12 publications

References 13 publications

Local adaption for approximation and minimization of univariate functions

Local adaption for approximation and minimization of univariate functions

Automatic approximation using asymptotically optimal adaptive interpolation

Adaptive mesh point selection for the efficient solution of scalar IVPs

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