Abstract
In this chapter the RBF mathematical concepts are exposed considering firstly the interpolation problem with the RBF function defined by known values at source points; a first hands-on example is provided showing how RBF work. Further topics of RBF theory are then introduced considering the differentiation of RBF, the fitting of an RBF with known values at locations different from the source points, the use of regression instead of exact interpolation and finally the management of noisy datasets. The chapter contents will not provide the details of mathematical theory but they will be mainly focused on the relevant results suitable for a wise and aware practical application of RBF.
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Notes
- 1.
Mathcad software allows to use live equations and to introduce short piece of code suitable for setting up and demonstrating algorithms. Functions and constants can be defined in any place of the worksheet and are evaluated in the definition order. Vector and matrixes can be zero based or one based. In this book the one base convention is adopted. Plotting features are available and used in the book.
- 2.
Mathcad variables, constants and functions can be assigned using the := operator.
- 3.
In Mathcad a program block works as other assignments (in this case a constant) but allows to have more freedom to internally specify how a constant or a function is structured. Local variable assignments are possible using the operator ←, the variable \(n_{edge}\) is an example (it is local and it is not exposed outside the block). Global variables of the worksheet (a and b) can be used inside a block. Mathcad programming block convention is to output the last variable (or the result of last function).
- 4.
The range variable i is inserted using the .. operator which allows to define a list using the first value and the last one (if the increment is 1); it accepts 3 arguments (the second value of the list after a comma) when an increment different than 1 has to be prescribed. “1 .. 5” results in “1 2 3 4 5”, “1, 3 .. 7” results in “1 3 5 7”, “1, 1.1 .. 1.5” results in “1 1.1 1.2 1.3 1.4 1.5”.
- 5.
Mathcad accepts nested vectors. In this example a cloud of point is stored in the vector P; each element of P is a two components vector. Direct assignment of vectors and matrixes components can be performed with the subscript operator.
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Biancolini, M.E. (2017). Radial Basis Functions. In: Fast Radial Basis Functions for Engineering Applications. Springer, Cham. https://doi.org/10.1007/978-3-319-75011-8_2
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DOI: https://doi.org/10.1007/978-3-319-75011-8_2
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