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# MultiquadricInterpolation - CE 3101 Fall 2011 A Brief...

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CE 3101 Fall 2011 A Brief Introduction to Multiquadric Interpolation Background Given a set of n distinct, but otherwise arbitrary, observation locations in < 2 , { ( x i , y i ) : i = 1, 2, ..., n } and given the values of a real function at these locations { z i = z ( x i , y i ) : i = 1, 2, ..., n } the construction of a function F ( x , y ) mapping < 2 < and satisfying F ( x i , y i ) = z i i = 1,2,... n (1) is called scattered data interpolation . An important class 1 of scattered data interpolation methods is known as radial basis function interpolation (e.g. Madych and Nelson [1988] and Powell [1990]). The general class of radial basis function interpolation includes kriging (e.g. Isaaks and Srivastava [1989]), thin plate splines (Duchon [1978] ), and multiquadric interpolation (Hardy [1990] ), to name a few popular specific cases. Furthermore, radial basis function methods have been found to be pragmatically superior to many other published methods of scattered data interpolation (e.g. Franke [1982] ). Multiquadric Interpolation Multiquadric interpolation is one of the more popular methods 2 of radial basis function interpolation. Mul- tiquadric interpolants take the form F ( x , y ) = μ + n X i = 1 w i Φ ( d i ( x , y )) (2) μ is an unknown coefficient 3 computed by applying (1). (This is discussed in more detail below.) w i is an unknown coefficient associated with observation i , which is computed by applying (1). (This is discussed in more detail below.) Φ

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• Spring '11
• Barnes