John C. Mason

Detecting and approximating fault lines from randomly scattered data

Abstract Discretely defined surfaces that exhibit vertical displacements across unknown fault lines can be difficult to approximate accurately unless a representation of the faults is known. Accurate representations of these faults enable the construction of constrained approximation models that can successfully overcome common problems such as over-smoothing. In this paper we review an existing method for detecting fault lines and present a new detection approach based on data triangulations and discrete Gaussian curvature (DGC). Furthermore, we show that if the fault line can be described non-parametrically, then accurate support vector machine (SVM) models can be constructed that are independent of the type of triangulation used in the detection algorithms. We shall also see that SVM models are particularly effective when the data produced by the detection algorithms are noisy. We compare the performances of the various new and established models.

Laurent-Padé approximants to four kinds of Chebyshev polynomial expansions. Part I. Maehly type approximants

Laurent Padé-Chebyshev rational approximants,Am(z,z−1)/Bn(z, z−1), whose Laurent series expansions match that of a given functionf(z,z−1) up to as high a degree inz, z−1 as possible, were introduced for first kind Chebyshev polynomials by Clenshaw and Lord [2] and, using Laurent series, by Gragg and Johnson [4]. Further real and complex extensions, based mainly on trigonometric expansions, were discussed by Chisholm and Common [1]. All of these methods require knowledge of Chebyshev coefficients off up to degreem+n. Earlier, Maehly [5] introduced Padé approximants of the same form, which matched expansions betweenf(z,z−1)Bn(z, z−1)). The derivation was relatively simple but required knowledge of Chebyshev coefficients off up to degreem+2n. In the present paper, Padé-Chebyshev approximants are developed not only to first, but also to second, third and fourth kind Chebyshev polynomial series, based throughout on Laurent series representations of the Maehly type. The procedures for developing the Padé-Chebyshev coefficients are similar to that for a traditional Padé approximant based on power series [8] but with essential modifications. By equating series coefficients and combining equations appropriately, a linear system of equations is successfully developed into two sub-systems, one for determining the denominator coefficients only and one for explicitly defining the numerator coefficients in terms of the denominator coefficients. In all cases, a type (m, n) Padé-Chebyshev approximant, of degreem in the numerator andn in the denominator, is matched to the Chebyshev series up to terms of degreem+n, based on knowledge of the Chebyshev coefficients up to degreem+2n. Numerical tests are carried out on all four Padé-Chebyshev approximants, and results are outstanding, with some formidable improvements being achieved over partial sums of Laurent-Chebyshev series on a variety of functions. In part II of this paper [7] Padé-Chebyshev approximants of Clenshaw-Lord type will be developed for the four kinds of Chebyshev series and compared with those of the Maehly type.

Constrained near-minimax approximation by weighted expansion and interpolation using Chebyshev polynomials of the second, third, and fourth kinds

Well known results on near-minimax approximation using Chebyshev polynomials of the first kind are here extended to Chebyshev polynomials of the second, third, and fourth kinds. Specifically, polynomial approximations of degreen weighted by (1−x2)1/2, (1+x)1/2 or (1−x)1/2 are obtained as partial sums of weighted expansions in Chebyshev polynomials of the second, third, or fourth kinds, respectively, to a functionf continuous on [−1, 1] and constrained to vanish, respectively, at ±1, −1 or +1. In each case a formula for the norm of the resulting projection is determined and shown to be asymptotic to 4π−2logn +A +o(1), and this provides in each case and explicit bound on the relative closeness of a minimax approximation. The constantA that occurs for the second kind polynomial is markedly smaller, by about 0.27, than that for the third and fourth kind, while the latterA is identical to that for the first kind, where the projection norm is the classical Lebesgue constant λn. The results on the third and fourth kind polynomials are shown to relate very closely to previous work of P.V. Galkin and of L. Brutman.Analogous approximations are also obtained by interpolation at zeros of second, third, or fourth kind polynomials of degreen+1, and the norms of resulting projections are obtained explicitly. These are all observed to be asymptotic to 2π−1logn +B +o(1), and so near-minimax approximations are again obtained. The norms for first, third, and fourth kind projections appear to be converging to each other. However, for the second kind projection, we prove that the constantB is smaller by a quantity asymptotic to 2π−1log2, based on a conjecture about the point of attainment of the supremum defining the projection norm, and we demonstrate that the projection itself is remarkably close to a minimal (weighted) interpolation projection.All four kinds of Chebyshev polynomials possess a weighted minimax property, and, in consequence, all the eight approximations discussed here coincide with minimax approximations when the functionf is a suitably weighted polynomial of degreen+1.

A Chebyshev polynomial method for line integrals with singularities

In this paper we consider a Chebyshev polynomial method for the calculation of line integrals along curves with Cauchy principal value or Hadamard finite part singularities. The major point we address is how to reconstruct the value of the integral when the parametrization of the curve is unknown and only empirical data are available at some discrete set of nodes.We replace the curve by a near‐minimax parametric polynomial approximation, and express the integrand by means of a sum of Chebyshev polynomials. We make use of a mapping property of the Hadamard finite part operator to calculate the value of the integral.

The minimality properties of Chebyshev polynomials and their lacunary series

By considering four kinds of Chebyshev polynomials, an extended set of (real) results are given for Chebyshev polynomial minimality in suitably weighted Hölder norms on [−1, 1], as well as (L∞ minimax properties, and bestL1 sufficiency requirements based on Chebyshev interpolation. Finally we establish bestLp,L∞ andL1 approximation by partial sums of lacunary Chebyshev series of the form ∑1=0∞aiϕbi(x) whereϕx(x) is a Chebyshev polynomial andb is an odd integer ≥3. A complete set of proofs is provided.

Approximation on an Infinite Range to Ordinary Differential Equations Solutions by a Function of a Radial Basis function

School of Computing and Engineering, University of Huddersfield, Huddersfield HD1 3DH, UK, {d.p.jenkinson,j.c.mason}@hud.ac.uk Summary. A function y = g(L) of a linear form L(x) = ∑n j=1 cjφj(x) has already been adopted in the approximation of a variety of smooth functions, especially those that behave like a power of x as x → ∞. In particular, Mason [7] in his thesis considers g(L) = L for approximating a decaying function, where R is a power of 2 and L(x) is a polynomial. Mason and Upton [8] use g(L) = L and g(L) = e, and in the latter case adopt a basis of Gaussian radial basis functions. Also, Crampton et al. [3] discuss “additive” linear iteration algorithms which are in general convergent to near-best approximations, and Dunham and Williams [5] discuss the existence of best approximations of the form g(L), especially L. In the present study, we find that approximations of the form g(L(x)), where L is a radial basis function (RBF) of the cubic, multiquadric or inverse-multiquadric form, are effective for approximating functions that behave on [0,∞) like x for small x and like x for large x, where α, β are known and finite. Numerical methods, based on weighted least squares, are adopted for the same selection of (nonlinear) ordinary differential equation (ODE) solutions as that considered by rational approximation in [7] (namely the Thomas-Fermi equation, the Blasius equation and Dawson’s integral), and RBF sums perform with similar, if slightly less accurate, versatility. Accuracy of 2 to 4 decimals, by comparison with known solutions, is readily achievable, without the need to adopt high degrees in the basis.