The local integro splines with optimized knots
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(2019) 38:156
The local integro splines with optimized knots Xiao Guo1
· Xuli Han2 · Yali Zhang2
Received: 20 April 2019 / Revised: 3 August 2019 / Accepted: 24 September 2019 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019
Abstract This paper presents a new approach to optimize knots of local integro splines that approximate given function integrals. The problem of computing knots is reformulated as L 1 convex optimization problem, in which both the number of knots and their locations can be automatically determined at the same time. The numerical examples show that approximation integro splines obtained by our method have few knots, while satisfying the fitting precision. Keywords Integro cubic splines · Knot optimization · Spline fitting Mathematics Subject Classification 65D07
1 Introduction In many practical problems, using the integral values of function to construct approximation function is rather than using the function values. Such application arises in mathematical statistics, financial pricing, climatology, oceanography and so on, see Delhez (2003), Huang and Chen (2005) and Killworth (1996). Behforooz (2006) first introduce integro cubic splines to approximate the function. Recently, more and more authors have studied the construction of integro splines. In Zhanlav and Mijiddorj (2010), Zhanlav proposed an approach to obtain local integro spline using B-representation of splines. After that, integro quartic and sextic splines are constructed in Lang and Xu (2012) and Wu and Zhang (2013). The shapepreserving prosperities of integro splines were given in Zhanlav and Mijiddorj (2017). Boujraf et al. (2015) provided an effective method to generate integro spline quasi-interpolants, but they did not discuss how to optimize the integro spline knots. Fink et al. (2009) present a task of encoding an approximated probability density function, which is as close as possible to the initial distribution, by variable-width bars with the required number of intervals. They specified the number must be smaller than the size of the initial distribution. We consider to construct to approximated integro spline defined on
Communicated by Antonio José Silva Neto.
B
Xiao Guo [email protected]
1
School of Mathematical Sciences, Changsha Normal University, Changsha 410010, China
2
School of Mathematics and Statistics, Central South University, Changsha 410083, China
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fewer intervals than the number of given subintervals. This approximation helps to generate an approximating probability density function in Fink et al. (2009), and also serves as lossy data compression, since it allows converting a probability distribution based on a large point set into the integro splines defined on sparse intervals. This paper is organized as follows. Section 2 is devoted to introduce integro cubic splines and properties of B-spline. In Sect. 3, we will propose a convex optimization method to determine the number and values of spline knots. Numerical experiments are
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