An algorithm for selecting a good value for the parameter c in radial basis function interpolation

The accuracy of many schemes for interpolating scattered data with radial basis functions depends on a shape parameter c of the radial basis function. In this paper we study the effect of c on the quality of fit of the multiquadric, inverse multiquadric and Gaussian interpolants. We show, numerically, that the value of the optimal c (the value of c that minimizes the interpolation error) depends on the number and distribution of data points, on the data vector, and on the precision of the computation. We present an algorithm for selecting a good value for c that implicitly takes all the above considerations into account. The algorithm selects c by minimizing a cost function that imitates the error between the radial interpolant and the (unknown) function from which the data vector was sampled. The cost function is defined by taking some norm of the error vector E = (E ₁, ... , E_N)^T where E _k = E_k = f_k - S^k x_k) and S _k is the interpolant to a reduced data set obtained by removing the point x _k and the corresponding data value f _k from the original data set. The cost function can be defined for any radial basis function and any dimension. We present the results of many numerical experiments involving interpolation of two dimensional data sets by the multiquadric, inverse multiquadric and Gaussian interpolants and we show that our algorithm consistently produces good values for the parameter c. This revised version was published online in June 2006 with corrections to the Cover Date.     

Univariate multiquadric approximation: Quasi-interpolation to scattered data

The univariate multiquadric function with centerx _j ∈R has the form {? _j (x)=[(x?x _j )²+c ²]^1/2, x∈R} wherec is a positive constant. We consider three approximations, namely, ? _A f, ?_? f, and ? _C f, to a function {f(x),x ₀≤x≤x _N } from the space that is spanned by the multiquadrics {? _j :j=0, 1, ...,N} and by linear polynomials, the centers {x _j :j=0, 1,...,N} being given distinct points of the interval [x ₀,x _N ]. The coefficients of ? _A f and ?_? f depend just on the function values {f(x _j ):j=0, 1,...,N}. while ? _A f, ? _C f also depends on the extreme derivativesf′(x ₀) andf′(x _N ). These approximations are defined by quasi-interpolation formulas that are shown to give good accuracy even if the distribution of the centers in [x ₀,x _N ] is very irregular. Whenf is smooth andc=O(h), whereh is the maximum distance between adjacent centers, we find that the error of each quasi-interpolant isO(h ²|logh|) away from the ends of the rangex ₀≤x≤x _N. Near the ends of the range, however, the accuracy of ? _A f and ?_? f is onlyO(h), because the polynomial terms of these approximations are zero and a constant, respectively. Thus, some of the known accuracy properties of quasiinterpolation when there is an infinite regular grid of centers {x _j =jh:j ∈F} given by Buhmann (1988), are preserved in the case of a finite rangex ₀≤x≤x _N , and there is no need for the centers {x _j :j=0, 1, ...,N} to be equally spaced.     

Explicit solutions for a two-phase unidimensional Lamé-Clapeyron-Stefan problem with source terms in both phases

A two-phase Stefan problem with heat source terms of a general similarity type in both liquid and solid phases for a semi-infinite phase-change material is studied. We assume the initial temperature is a negative constant and we consider two different boundary conditions at the fixed face x=0, a constant temperature or a heat flux of the form (q₀>0). The internal heat source functions are given by (j=1 solid phase; j=2 liquid phase) where βj=βj(η) are functions with appropriate regularity properties, ρ is the mass density, l is the fusion latent heat by unit of mass, is the diffusion coefficient, x is the spatial variable and t is the temporal variable. We obtain for both problems explicit solutions with a restriction for data only for the second boundary conditions on x=0. Moreover, the equivalence of the two free boundary problems is also proved. We generalize the solution obtained in [J.L. Menaldi, D.A. Tarzia, Generalized Lamé-Clapeyron solution for a one-phase source Stefan problem, Comput. Appl. Math. 12 (2) (1993) 123-142] for the one-phase Stefan problem. Finally, a particular case where βj (j=1,2) are of exponential type given by βj(x)=exp(−²(x+dj)) with x and dj∈R is also studied in details for both boundary temperature conditions at x=0. This type of heat source terms is important through the use of microwave energy following [E.P. Scott, An analytical solution and sensitivity study of sublimation-dehydration within a porous medium with volumetric heating, J. Heat Transfer 116 (1994) 686-693]. We obtain a unique solution of the similarity type for any data when a temperature boundary condition at the fixed face x=0 is considered; a similar result is obtained for a heat flux condition imposed on x=0 if an inequality for parameter q₀ is satisfied.     

Power-series unification and reversion

In analysis, it is sometimes necessary to unite a pair of power series into a single power series. If x = x(z) = Σ_ja_jz^j and y = y(z) = Σ_jb_jz^j are given power series, then by eliminating the common parameter z, the power-series unification is obtained: y = y(x) = Σ_kc_kx^k, where the coefficients c_k are to be determined in terms of the given power-series coefficients a_j and b_j. In a special case that y = z, the power-series reversion is obtained: z = z(x) = Σ_kd_kx^k, where the coefficients d_k are to be expressed in terms of the original power-series coefficients a_j. In this paper, explicit and recurrent formulas for the desired coefficients are derived. A simple technique of matrix formulation is developed for simplicity of computation. Finally, a complete computer program with a typical example is presented.     

Decomposition and approximation of multivariate functions on the cube

In this paper, we present a decomposition method of multivariate functions. This method shows that any multivariate function f on [0, 1]d is a finite sum of the form ∑jφjψj , where each φj can be extended to a smooth periodic function, each ψj is an algebraic polynomial, and each φjψj is a product of separated variable type and its smoothness is same as f . Since any smooth periodic function can be approximated well by trigonometric polynomials, using our decomposition method, we find that any smooth multivariate function on [0, 1]d can be approximated well by a combination of algebraic polynomials and trigonometric polynomials. Meanwhile, we give a precise estimate of the approximation error.     

Generalized prolate spheroidal wave functions

The following equation $$(1 - x^2 )d^2 y/dx^2 + [(\beta - \alpha - (\alpha + \beta + 2)x]dy/dx + (\chi (c) - c^2 x^2 )y = 0$$ has been solved wherex(c) a separation constant is the characteristic value and is a function ofc. This solution is a generalization of spheroidal wave function into the series form ofP _n ^α;β (x),α andβ both separately are greater than ?1. The finite transform and its properties have been defined and a boundary value problem has been solved applying these tools.     

Trigonometric wavelet method for some elliptic boundary value problems

In this paper, we apply trigonometric wavelet method to investigate the numerical solution of elliptic boundary value problem in an exterior unit disk. The simple computation formulae of the entries in the stiffness matrix are obtained. It shows that we only need to compute 2(j²+1) elements of one ²j+2ײj+2 stiffness matrix. Moreover, the error estimates of the approximation solutions are given and some test examples are presented.     

On a theorem of J. L. Krivine concerning block finite representability of lp in general Banach spaces

A simplified proof and generalizations are given for the following remarkable theorem of J. L. Krivine: Let (x_j) be a sequence in a Banach space with infinite-dimensional linear span. Then either there is a 1 ? p < ∞ so that l^p is block finitely represented in (x_j) or c₀ is block finitely represented in some permutation of (x_j).     

Solutions de type multi-soliton des équations de KdV généralisées

We consider the generalized Korteweg–de Vries equations in the subcritical and critical cases. Let R_j(t,x)=Q_cj(x?c_jt?x_j) be N soliton solutions of this equation, with corresponding speeds 0<c₁<c₂<?<c_N. In this Note, we give a sketch of the proof of the following result. Given $\text{{c}{}_{\mathrm{j}}\text{},}\text{{x}{}_{\mathrm{j}}\text{}}$, there exists one and only one solution ? of the generalized KdV equation such that 6?(t)?∑R_j(t)6_H¹→0 as t→+∞. Complete proofs will appear later. To cite this article: Y. Martel, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

A multiscale Galerkin method for the hypersingular integral equation reduced by the harmonic equation

The aim of this paper is to investigate the numerical solution of the hypersingular integral equation reduced by the harmonic equation. First, we transform the hypersingular integral equation into 2π-periodic hypersingular integral equation with the map x=cot(θ/2). Second, we initiate the study of the multiscale Galerkin method for the 2π-periodic hypersingular integral equation. The trigonometric wavelets are used as trial functions. Consequently, the 2j+1 × 2j+1 stiffness matrix Kj can be partitioned j×j block matrices. Furthermore, these block matrices are zeros except main diagonal block matrices. These main diagonal block matrices are symmetrical and circulant matrices, and hence the solution of the associated linear algebraic system can be solved with the fast Fourier transform and the inverse fast Fourier transform instead of the inverse matrix. Finally, we provide several numerical examples to demonstrate our method has good accuracy even though the exact solutions are multi-peak and almost singular.     

A new form of trigonometric orthogonality and Gaussian-type quadrature

Gauss's (2n+1)-point trigonometric interpolation formula, based upon f(x_i), i = 1(1)2n+1, gives a trigonometric sum of the n^th order, S_2n+1(x = a₀ + ∑_jⁿ = 1^(a_jcos jx + b_jsin jx), which may be integrated to provide formulas for either direct quadrature or stepwise integration of differential equations having periodic (or near-periodic) solutions. An “orthogonal” trigonometric sum S_2r+1(x) is one that satisfies

$\text{∫}{}_{\mathrm{a}}{}^{\mathrm{b}}\text{S}{}_{\mathrm{2r+1}}\text{(x)S}{}_{\mathrm{2r\prime +1}}\text{(x)dx=0, r′<r}$

and two other arbitrarily imposable conditions needed to make S_2r1(x) unique. Two proofs are given of a fundamental factor theorem for any S_2n+1(x) (somewhat different from that for polynomials) from which we derive 2r-point Gaussian-type quadrature formulas, r = [n/2] + 1, which are exact for any S_4r?1(x). We have

$\text{∫}{}_{\mathrm{a}}{}^{\mathrm{b}}\text{S}{}_{\mathrm{4r?1(x)}}\text{dx=∑}{}_{\mathrm{j=1}}{}^{\mathrm{2r}}\text{A}{}_{\mathrm{j}}\text{S}{}_{\mathrm{4r?1}}\text{(x}{}_{\mathrm{j}}\text{)}$

where the nodes x_j, j = 1(1)2r, are the zeros of the orthogonal S_2r+1(x). It is proven that A_j > 0 and that 2r-1 of the nodes must lie within the interval [a,b], and the remaining node (which may or may not be in [a,b]) must be real. Unlike Legendre polynomials, any [a′,b′] other than a translation of [a,b], requires different and unrelated sets of nodes and weights. Gaussian-type quadrature formulas are applicable to the numerical integration of the Gauss (2n+1)-point interpolation formulas, with extra efficiency when the latter are expressed in barycentric form. S_2r+1(x), x_jandA_j, j = 1(1)2r, were calculated for [a,b] = [0, π/4], 2r = 2 and 4, to single-precision accuracy.     

Некоторые вопросы теории приближения в пространствах L p(x) (E)

Some aspects of approximation theory are studied in the paper for the spaces of integrable functions with variable exponent. In particular, necessary and suffcient conditions on the variable exponent are established that guarantee the basis property of the trigonometric system in the corresponding normed spaces. Namely, these conditions require that the periodic variable exponent p(x) > 1 satisfy the Dini-Lipschitz condition.     

A singular boundary value problem for odd-order differential equations

The odd-order differential equation (−1)ⁿx⁽²ⁿ⁺¹⁾=f(t,x,…,x⁽²ⁿ⁾) together with the Lidstone boundary conditions x^(2j)(0)=x^(2j)(T)=0, 0?j?n−1, and the next condition x⁽²ⁿ⁾(0)=0 is discussed. Here f satisfying the local Carathéodory conditions can have singularities at the value zero of all its phase variables. Existence result for the above problem is proved by the general existence principle for singular boundary value problems.     

On <Emphasis Type="Italic">β</Emphasis>-expansions of unity for rational and transcendental numbers <Emphasis Type="Italic">β</Emphasis>

We investigate the sequence of integers x ₁, x ₂, x ₃, … lying in {0, 1, …, [β]} in a so-called Rényi β-expansion of unity 1 = \(\sum\limits_{j = 1}^\infty {x_j \beta ^{ - j} } \) for rational and transcendental numbers β > 1. In particular, we obtain an upper bound for two strings of consecutive zeros in the β-expansion of unity for rational β. For transcendental numbers β which are badly approximable by algebraic numbers of every large degree and bounded height, we obtain an upper bound for the Diophantine exponent of the sequence X = (xj) _j=1 ^∞ in terms of β.     

On the Asymptotics of Fekete-Type Points for Univariate Radial Basis Interpolation

Suppose that K ^d is compact and that we are given a function fC(K) together with distinct points x_iK, 1in. Radial basis interpolation consists of choosing a fixed (basis) function g : ⁺→ and looking for a linear combination of the translates g(|x−x_j|) which interpolates f at the given points. Specifically, we look for coefficients c_j such that has the property that F(x_i)=f(x_i), 1in. The Fekete-type points of this process are those for which the associated interpolation matrix [g(|x_i−x_j|)]_1i,jn has determinant as large as possible (in absolute value). In this work, we show that, in the univariate case, for a broad class of functions g, among all point sequences which are (strongly) asymptotically distributed according to a weight function, the equally spaced points give the asymptotically largest determinant. This gives strong evidence that the Fekete points themselves are indeed asymptotically equally spaced.     

Construction of the asymptotics of the solutions of the one-dimensional Schrödinger equation with rapidly oscillating potential

We obtain asymptotic formulas for the solutions of the one-dimensional Schrödinger equation ? y″ +q(x)y = 0 with oscillating potential q(x)=x ^β P(x ^1+α)+cx ^?2 as x→ +∞. The real parameters α and β satisfy the inequalities β ? α ≥ ?1, 2α ? β > 0 and c is an arbitrary real constant. The real function P(x) is either periodic with period T, or a trigonometric polynomial. To construct the asymptotics, we apply the ideas of the averaging method and use Levinson’s fundamental theorem.     

Coloring some classes of mixed graphs

We consider the coloring problem for mixed graphs, that is, for graphs containing edges and arcs. A mixed coloring c is a coloring such that for every edge [x_i,x_j], c(x_i)≠c(x_j) and for every arc (x_p,x_q), c(x_p)<c(x_q). We will analyse the complexity status of this problem for some special classes of graphs.     

On multiple zeros of a partial theta function

We consider the partial theta function θ(q, x) := ∑_j=0^∞q^j(j+1)/2x^j, where x ∈ ? is a variable and q ∈ ?, 0 < |q| < 1, is a parameter. We show that, for any fixed q, if ζ is a multiple zero of the function θ(q, · ), then |ζ| ≤ 8¹¹.     

An application of the maximum principle to describe the layer behavior of large solutions and related problems

This work is devoted to the analysis of the asymptotic behavior of positive solutions to some problems of variable exponent reaction-diffusion equations, when the boundary condition goes to infinity (large solutions). Specifically, we deal with the equations ??u = u ^p(x), ??u = ?m(x)u?+?a(x)u ^p(x) where a(x)??? a ₀ >?0, p(x)??? 1 in ??, and ??u = e ^p(x) where p(x)??? 0 in ??. In the first two cases p is allowed to take the value 1 in a whole subdomain ${\Omega_c\subset \Omega}$ , while in the last case p can vanish in a whole subdomain ${\Omega_c\subset \Omega}$ . Special emphasis is put in the layer behavior of solutions on the interphase ?? _i :?= ??? _c ???. A similar study of the development of singularities in the solutions of several logistic equations is also performed. For example, we consider ???u = ?? m(x)u?a(x) u ^p(x) in ??, u = 0 on ???, being a(x) and p(x) as in the first problem. Positive solutions are shown to exist only when the parameter ?? lies in certain intervals: bifurcation from zero and from infinity arises when ?? approaches the boundary of those intervals. Such bifurcations together with the associated limit profiles are analyzed in detail. For the study of the layer behavior of solutions the introduction of a suitable variant of the well-known maximum principle is crucial.