Solutions to a nonlinear Neumann problem in three-dimensional exterior domains

We prove the existence of multipeak solutions to a nonlinear elliptic Neumann problem involving nearly critical Sobolev exponent, in three-dimensional exterior domains.     

A Neumann problem in exterior domain

We are concerned with the existence of radial solutions for the following Neumann problem

Numerical differentiation by radial basis functions approximation

Based on radial basis functions approximation, we develop in this paper a new com-putational algorithm for numerical differentiation. Under an a priori and an a posteriori choice rules for the regularization parameter, we also give a proof on the convergence error estimate in reconstructing the unknown partial derivatives from scattered noisy data in multi-dimension. Numerical examples verify that the proposed regularization strategy with the a posteriori choice rule is effective and stable to solve the numerical differential problem. *The work described in this paper was partially supported by a grant from CityU (Project No. 7001646) and partially supported by the National Natural Science Foundation of China (No. 10571079).     

Diffraction echo signal generated by the incidence of a plane wave on a prolate spheroid

The problem of the diffraction of a plane wave by a prolate spheroid placed in a fluid is considered in the high-frequency approximation. The spheroid is hereby either a solid body or a shell. The contribution to the echo signal, i.e., the back-reflected field, of diffraction waves of creep and break-off type is investigated. Computational formulas for the geometric divergences of surface rays and also of creep and break-off waves are obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituts im. V. A. Steklova AN SSSR, Vol. 140, pp. 18–35, 1984.     

On the integrability of radial basis functions

In this paper we investigate the integrability of certain radial basis functions. From the following forms of function σ, $$\varphi \left( r \right) = \left\{ \begin{gathered} \sum\limits_{k = 0}^{d + [a]} {c_k r^{a - k} + g(r) } r > A, \hfill \\ \sum\limits_{k = 0}^{d + [a]} {c_k r^{a - k} \ln r + g(r), } r > A. \hfill \\ \end{gathered} \right.$$ where A≧0 and $g \circ || \circ || \in L^1 \left( {R^d } \right)$ , we construct the function $$\psi (t) = \sum\limits_{j \in J} {a_j \varphi \left( {||t - t_j ||} \right),} $$ where J is a finite index set, $\left\{ {a_j } \right\}_{j \in J} \subseteq R$ and $\left\{ {t_j } \right\}_{j \in J} \subseteq R^d $ . We show that if $\hat \psi $ is continuous at the origin, the ψ is integrable in R^d.     

Solution of the Neumann problem for the Laplace equation

For fairly general open sets it is shown that we can express a solution of the Neumann problem for the Laplace equation in the form of a single layer potential of a signed measure which is given by a concrete series. If the open set is simply connected and bounded then the solution of the Dirichlet problem is the double layer potential with a density given by a similar series.     

On the decay of solutions to the 2D Neumann exterior problem for the wave equation

We consider the exterior problem in the plane for the wave equation with a Neumann boundary condition and study the asymptotic behavior of the solution for large times. For possible application we are interested to show a decay estimate which does not involve weighted norms of the initial data. In the paper we prove such an estimate, by a combination of the estimate of the local energy decay and decay estimates for the free space solution.     

Approximation by radial basis functions with finitely many centers

Interpolation by translates of “radial” basis functions Φ is optimal in the sense that it minimizes the pointwise error functional among all comparable quasiinterpolants on a certain “native” space of functions $\mathcal{F}_\Phi $ . Since these spaces are rather small for cases where Φ is smooth, we study the behavior of interpolants on larger spaces of the form $\mathcal{F}_{\Phi _0 } $ for less smooth functions Φ₀. It turns out that interpolation by translates of Φ to mollifications of functionsf from $\mathcal{F}_{\Phi _0 } $ yields approximations tof that attain the same asymptotic error bounds as (optimal) interpolation off by translates of Φ₀ on $\mathcal{F}_{\Phi _0 } $ .     

Limit problems for interpolation by analytic radial basis functions

Interpolation problems for analytic radial basis functions like the Gaussian and inverse multiquadrics can degenerate in two ways: the radial basis functions can be scaled to become increasingly flat, or the data points coalesce in the limit while the radial basis functions stay fixed. Both cases call for a careful regularization, which, if carried out explicitly, yields a preconditioning technique for the degenerating linear systems behind these interpolation problems. This paper deals with both cases. For the increasingly flat limit, we recover results by Larsson and Fornberg together with Lee, Yoon, and Yoon concerning convergence of interpolants towards polynomials. With slight modifications, the same technique can also handle scenarios with coalescing data points for fixed radial basis functions. The results show that the degenerating local Lagrange interpolation problems converge towards certain Hermite–Birkhoff problems. This is an important prerequisite for dealing with approximation by radial basis functions adaptively, using freely varying data sites.     

Convergence properties of radial basis functions

The multivariate interpolation problem is that of choosing a functions fromR toR that satisfies the interpolation conditions

Hermite-Birkhoff interpolation of scattered data by radial basis functions

For Hermite-Birkhoff interpolation of scattered multidimensional data by radial basis function φ, existence and characterization theorems and a variational principle are proved. Examples include φ(r)=r^b, Duchon’s thin-plate splines, Hardy’s multiquadrics, and inverse multiquadrics.     

On quasi-interpolation by radial basis functions with scattered centres

Approximation by radial basis functions with “quasi-uniformly” distributed centres inR ^d is discussed. A construction of new polynomially decaying functions that span the approximation space is presented and the properties of the quasi-interpolation operator with these functions are investigated. It is shown that the quasi-interpolant reproduces polynomials and gives approximation orders identical to those in the uniform square-grid case.     

A numerical model of turbulent shear flow behind a prolate spheroid

A finite element based computer model is developed to simulate turbulent shear flow. The results obtained from the model are compared with numerical finite difference based calculations and experimentally determined values associated with flow behind a prolate spheroid. The results obtained utilizing a velocity-pressure formulation are, in the region immediately downstream of the spheroid, markedly better than those obtained under the standard assumption of zero pressure gradient boundary layer flow.     

Using radial basis functions to construct local volatility surfaces

This article presents a new method for constructing a volatility surface for use in local volatility option pricing models. It builds on previous work focussing on non-parametric regression approaches using a set of radial basis functions, specifically thin plate splines. Optimal parameters are found using a trust region optimisation approach. While there is still much work to be done, the results are encouraging and show that the method is relatively tractable, stable and accurate.     

A nonresonance condition for radial solutions of a nonlinear Neumann elliptic problem

We prove an existence result for radial solutions of a Neumann elliptic problem whose nonlinearity asymptotically lies between the first two eigenvalues. To this aim, we introduce an alternative nonresonance condition with respect to the second eigenvalue which, in the scalar case, generalizes the classical one, in the spirit of Fonda et al. (1991) [2]. Our approach also applies for nonlinearities which do not necessarily satisfy a subcritical growth assumption.     

Multiscale methods with compactly supported radial basis functions for the Stokes problem on bounded domains

In this paper, we investigate the application of radial basis functions (RBFs) for the approximation with collocation of the Stokes problem. The approximate solution is constructed in a multi-level fashion, each level using compactly supported radial basis functions with decreasing scaling factors. We use symmetric collocation and give sufficient conditions for convergence and consider stability analysis. Numerical experiments support the theoretical results.     

A finite element method for the double-layer potential solutions of the Neumann exterior problem

A variational formulation for the integral equation used for the double layer potential solution of the Neumann exterior problem in the Laplace equation was proposed in [4]. This formulation allows the use of a finite element method which we describe and experiment here.     

Generalization of the Neumann problem to harmonic functions outside cuts on the plane

We consider a boundary value problem for harmonic functions outside cuts on the plane. The jump of the normal derivative and a linear combination of the normal derivative on one side with the jump of the unknown function are given on each cut. The problem is considered with three conditions at infinity, which lead to distinct results on the existence and number of solutions. We obtain an integral representation of the solution in the form of potentials whose density satisfies a uniquely solvable Fredholm integral equation of the second kind.     

The 2-D Neumann problem in an exterior domain bounded by closed and open curves

The Neumann problem for the Laplace equation in an exterior connected plane region bounded by closed and open curves is studied. The existence of classical solution is proved by potential theory. The problem is reduced to the Fredholm equation of the second kind, which is uniquely solvable. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.