Spline Fitting

2011 ◽  
pp. 166-171
Author(s):  
Michael Wodny
Keyword(s):  
Optimization Problems ◽  
Smoothing Splines ◽  
Cubic Spline ◽  
Smoothing Parameter ◽  
Spline Functions ◽  
System Of Equations ◽  
Linear System Of Equations ◽  
Spline Fitting ◽  
Natural Cubic Spline ◽  
The Given

Given are the m points (xi,yi), i=1,2,…,m. Spline functions are introduced, and it is noticed that the interpolation task in the case of natural splines has a unique solution. The interpolating natural cubic spline is constructed. For the construction of smoothing splines, different optimization problems are formulated. A selected problem is looked at in detail. The construction of the solution is carried out in two steps. In the first step the unknown Di=s(xi) are calculated via a linear system of equations. The second step is the construction of the interpolating natural cubic spline with respect to these (xi,Di), i=1,2,…,m. Every optimization problem contains a smoothing parameter. A method of estimation of the smoothing parameter from the given data is motivated briefly.

scholarly journals Two-piece Cubic Spline Functions

2003 ◽  
Vol 2 (1) ◽  
pp. 25-33
Author(s):  
Hannah Vijayakumar
Keyword(s):  
Cubic Spline ◽  
Oscillatory Behavior ◽  
Spline Functions ◽  
Entire Interval ◽  
Natural Spline ◽  
Cubic Polynomial ◽  
Natural Cubic Spline ◽  
Cubic Spline Function ◽  
High Degree ◽  
The Given

.M Prenter defines a cubic Spline function in an interval [a, b] as a piecewise cubic polynomial which is twice continuously differentiable in the entire interval [a, b]. The smooth cubic spline functions fitting the given data are the most popular spline functions and when used for interpolation, they do not have the oscillatory behavior which characterized high-degree polynomials. The natural spline has been shown to be unique function possessing the minimum curvature property of all functions interpolating the data and having square integrable second derivative. In this sense, the natural cubic spline is the smoothest function which interpolates the data. Here Two-piece Natural Cubic Spline functions have been defined. An approximation with no indication of its accuracy is utterly valueless. Where an approximation is intended for the general use, one must , of course, go for the trouble of estimating the error as precisely as possible. In this section, an attempt has been made to derive closed form expressions for the error-functions in the case of Two-piece Spline Functions.

A Computer System to Derive Developable Hull Surfaces and Tables of Offsets

1981 ◽  
Vol 18 (03) ◽  
pp. 227-233
Author(s):  
John C. Clements
Keyword(s):  
Computer System ◽  
Tangent Plane ◽  
Cubic Spline ◽  
Ruled Surface ◽  
Spline Functions ◽  
Simple Method ◽  
The Given

The fairbody, chine and sheer lines of a proposed vessel are represented by cubic spline functions. Between each pair of chine lines that ruled surface is generated which has the same tangent plane at all points of each generator or ruling line. A procedure based on the multiconic development of a surface is used to modify the given chine lines to ensure that no ruling lines intersect at a point within the surface. The result is a developable hull surface. A simple method is suggested for fairing the modified chine lines and the steps necessary to generate tables of offsets from ruling-line intercepts are outlined briefly.

ON THE COMPUTATION OF GRAMIAN MATRICES FOR REFINABLE BASES ON THE INTERVAL

2008 ◽  
Vol 06 (03) ◽  
pp. 459-479 ◽  
Author(s):  
MIRIAM PRIMBS
Keyword(s):  
Linear System ◽  
Multiresolution Analysis ◽  
System Of Equations ◽  
Simple Method ◽  
Linear System Of Equations ◽  
Boundary Functions ◽  
Spline Basis ◽  
Compactly Supported Functions ◽  
The Given ◽  
Schoenberg Spline

In this paper, we discuss a simple method for the numerical computation of Gramian matrices, that appears within the construction of multiresolution analysis on the interval. The presented approach covers all, the orthogonal, the semiorthogonal and the biorthogonal cases. We consider constructions, which are based on translates of a known pair of compactly supported functions ϕ,[Formula: see text] inside the interval, and supplemented by boundary functions. Using the given two-scale-coefficients of these boundary functions, we can reduce the problem to a linear system of equations. Furthermore, we discuss conditions providing that these systems are uniquely solvable. In particular, no integrals have to be computed numerically. Finally, using the Schoenberg spline basis on the interval as an example, we show how to apply the method to a well-known problem.

Algorithm for the Calculation of the Two Variables Cubic Spline Function

2013 ◽  
Vol 59 (1) ◽  
pp. 149-161
Author(s):  
Ion Lixandru
Keyword(s):  
Existence And Uniqueness ◽  
Spline Function ◽  
Global Solutions ◽  
Cubic Spline ◽  
Practical Calculation ◽  
System Of Equations ◽  
Interpolation Spline ◽  
Linear System Of Equations ◽  
Uniqueness Of The Solution ◽  
Cubic Spline Function

Abstract When having just one variable, the existence and uniqueness of the interpolation spline function reduces to studying the solutions of an algebrical system of equations. This allows us to find a practical way of calculating the interpolation spline function. Also in the case of two variables spline functions, we can construct a linear system of equations determined by the continuity conditions of the spline function and of its partial derivatives on the edge of each division rectangle. The existence and uniqueness of the solution of the obtained system ensure the existence and uniqueness of the two variables interpolation spline function and offers a practical calculation method. This can be used to determine approximate global solutions, of some partial differential equations, solutions whose values can be determined at any point of their domain of definition and can provide information on derivatives approximate of solutions. After calculating the two variable cubic spline function, we must assess the rest of the approximation.

Opposition-based learning Multi-Verse Optimizer with disruption operator for optimization problems

2021 ◽  
Author(s):  
Mohammad Shehab ◽  
Laith Abualigah
Keyword(s):  
Real World ◽  
Optimization Problems ◽  
Second Stage ◽  
Opposition Based Learning ◽  
High Fitness ◽  
Fitness Value ◽  
Speed Up ◽  
Disruption Operator ◽  
The Given ◽  
Real World Problems

Abstract Multi-Verse Optimizer (MVO) algorithm is one of the recent metaheuristic algorithms used to solve various problems in different fields. However, MVO suffers from a lack of diversity which may trapping of local minima, and premature convergence. This paper introduces two steps of improving the basic MVO algorithm. The first step using Opposition-based learning (OBL) in MVO, called OMVO. The OBL aids to speed up the searching and improving the learning technique for selecting a better generation of candidate solutions of basic MVO. The second stage, called OMVOD, combines the disturbance operator (DO) and OMVO to improve the consistency of the chosen solution by providing a chance to solve the given problem with a high fitness value and increase diversity. To test the performance of the proposed models, fifteen CEC 2015 benchmark functions problems, thirty CEC 2017 benchmark functions problems, and seven CEC 2011 real-world problems were used in both phases of the enhancement. The second step, known as OMVOD, incorporates the disruption operator (DO) and OMVO to improve the accuracy of the chosen solution by giving a chance to solve the given problem with a high fitness value while also increasing variety. Fifteen CEC 2015 benchmark functions problems, thirty CEC 2017 benchmark functions problems and seven CEC 2011 real-world problems were used in both phases of the upgrade to assess the accuracy of the proposed models.

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