Some computational aspects of smooth approximation
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Some computational aspects of smooth approximation Karel Segeth
Received: 21 September 2012 / Accepted: 3 December 2012 / Published online: 13 December 2012 © Springer-Verlag Wien 2012
Abstract The paper is devoted to the problem of smooth approximation of data. We are concerned with the exact interpolation of the data at nodes and, at the same time, with the smoothness of the interpolating curve and its derivatives. The same procedure is applied to the fitting of data (smoothing of data), too. The approximating curve is defined as the solution of a variational problem with constraints (interpolation conditions at nodes, data fitting property of the curve) the existence of whose unique solution we prove. Except for the constraints, the formulation of the problem of data approximation is not unique in general as our requirements on the behavior of the approximating curve between the nodes can be very subjective. The smooth approximation of this kind can lead, e.g., to the cubic spline interpolation.We discuss the choice of basis systems for this way of approximation and present results of several 1D numerical examples that show some advantages and drawbacks of smooth approximation. Keywords Smooth approximation · Approximation of data · Interpolation of data · Fitting of data · Smoothing of data Mathematics Subject Classification (2000)
65D05 · 65D10 · 41A05 · 41A63
1 Introduction In many branches of science and technology, measurements of the values of a continuous function of one, two, or three independent variables are carried out. We always
K. Segeth (B) Technical University of Liberec, Studentská 2, Liberec 461 17, Czech Republic e-mail: [email protected]
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get a finite number of function values measured at a finite number of nodes but we are interested also in the values corresponding to other points in some domain. This is the well-known problem of data approximation. In the paper, we are concerned with the smoothness of the approximating curve and its derivatives, and also with some constrains, e.g. the exact interpolation of the data at nodes. The interpolating or fitting curve is thus defined as the solution of a constrained variational problem. The problem of data approximation does not have a unique solution. In addition to the properties of a function that are formulated by mathematical means (e.g. interpolation conditions at nodes or data fitting property of the curve) there are also requirements on the subjective perception of the form of the approximating curve or surface between nodes. The visual assessment of an appropriate approximating function is mostly based on the imagination that can hardly be formalized (cf. [6]). In the paper, we are concerned with the exact interpolation of the data given at nodes on one hand and with the fitting (smoothing) of the data, too. A possible way to smooth approximation is to minimize the L 2 norm of the approximating function. A more sophisticated criterion is to minimize, with some weights chosen, the L 2 norm of some (or possibly all
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