G. E. Fasshauer, Solving differential equations with radial basis functions: multilevel methods and smoothing, Advances in Comp. Math., 11, (1999), 139--159.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....be solved. 3.1 Compactly Supported Radial Basis Functions Compactly supported radial basis functions (CSRBF) was firstly introduced by Wu [5] and later expanded by Wendland [4] in the mid 1990s. Floater and Iske [6] first adopted the CSRBFs for multi step scattered data interpolation. Fasshauer [7, 8] then introduced a multilevel approximation scheme incorporated with CSRBF for the solution of boundary value problems and linear partial differential equations. The principle idea of CSRBFs is to use a polynomial as a function of Euclidean distance r with support on [0; 1] CSRBFs must be ....

Gregory E. Fasshauer, "Solving differential equations with radial basis functions: Multilevel methods nd smoothing", to appear at the J. Adv. in Comput. Math. 10

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G. E. Fasshauer, Solving differential equations with radial basis functions: multilevel methods and smoothing, Advances in Comp. Math., 11, (1999), 139--159.

....equations. This special issue is a much needed effort to collect the state of the art in this field. RBFs are usually grouped into two categories: globally supported and locally supported functions. The advantages of one category over the other have been discussed in many papers (see, e.g. [4, 17] and references therein) In this paper we will concentrate on the use of globally supported functions, and multiquadrics in particular. Table 1 lists some of the classical globally supported RBFs. As usual we use r = k Delta k (the Euclidean norm) and c is a parameter to be set by the user. ....

....can be achieved. Thus, Newton iteration with smoothing is described by u k = u k Gamma1 Gamma S t k T hk (u k Gamma1 )F (u k Gamma1 ) k 1; 5) 3 where S t k is an appropriate smoothing operator with smoothing parameter t k . The smoothed version (5) was referred to in our earlier papers [4, 7, 9] as Nash iteration and in [7] a number of different interpretations and implementations of this general concept were studied. The algorithms used in the present paper correspond to the so called simple algorithm of [7, Alg. 2.1] and a slight modification thereof in which the computational meshes ....

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Fasshauer, G. E., Solving differential equations with radial basis functions: Multilevel methods and smoothing, Adv. in Comput. Math. 11 (1999), 139-- 159.

.... and applied) has focussed on the use of RBFs in approximation theory (i.e. scattered data approximation) More recently, however, some authors have also considered using RBFs for the numerical solution of partial differential equations (PDEs) An overview of some of these approaches is given in [4]. In this paper we will show how RBF collocation (briefly explained in Sect. 3) can be applied to the solution of nonlinear PDEs. More specifically, we will begin in Sect. 2 by reviewing an operator Newton framework discussed in detail in [7] and then couple this approach with compactly ....

Fasshauer, G. E., Solving differential equations with radial basis functions: Multilevel methods and smoothing, Advances in Comp. Math. 11 (1999), 139--159.

....of L. The superscript (2) indicates that L acts on OE as a function of the second variable, i.e. the center location. Expansion (2) is motivated by a connection to scattered Hermite interpolation which ensures the invertibility of the related collocation matrix (for more details see Fasshauer[2,4] or Franke Schaback[7] The remainder of the paper is organized as follows. In Section 2 we outline our algorithmic approach which is a Newton like multilevel method with interlaced smoothing operations, and then discuss our two examples in the following two sections. The last section is ....

.... is offset by their modest approximation properties (see Example 2 in Section 4) To obtain the best results with compactly supported functions (for large data sets) it is recommended that a multilevel algorithm be employed (a more detailed discussion of this issue can be found in Fasshauer[4] or in Schaback[12] The basic multilevel algorithm for the solution of the problem Lu = F on Omega ae IR d by collocation can be described as follows: Multilevel Collocation Algorithm. 1. Generate a nested sequence of computational grids X 1 ae Delta Delta Delta ae XK ae Omega , and ....

Fasshauer, G. E., Solving differential equations with radial basis functions: multilevel methods and smoothing, Adv. in Comput. Math., to appear.

....a certain iteration, or it may even be possible to take the next mesh with a meshsize that is less than two times as fine. The first algorithm is a direct analog of the Floater Iske method which we have applied previously to scattered data approximation and linear differential equations (see [4] [5], 6] 7] The correction v is computed with one single computational step on the next finer grid Xk . Thus the simple algorithm can be given as Algorithm 2.1 ( Simple Algorithm ) 1. Let u0 = 0. 2. For k = 1; 2; 3; a) Solve Lu k Gamma1 v = f Gamma Luk Gamma1 on Xk . b) Smooth the ....

.... s(x; 0) v(x) The first of these is the classical heat equation in one spatial dimension, for which the time dependent Green s function (or heat kernel ) is precisely the Gauss Weierstrass kernel, here convolved with the odd periodic extension of v (as utilized in the smoothing context in [5] and [7] The second problem above gives a similar evolution but for the biharmonic operator. Approximations to the actions of these smoothers can be efficiently realized using finite differences and explicit time stepping. In the earlier notation, As = sxx in the harmonic case, and As = Gammas ....

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G.E. Fasshauer. Solving differential equations with radial basis functions: multilevel methods and smoothing. To appear in: Advances in Comp. Math.

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