Error saturation in Gaussian radial basis functions on a finite interval

AbstractRadial basis function (RBF) interpolation is a “meshless” strategy with great promise for adaptive approximation. One restriction is “error saturation” which occurs for many types of RBFs including Gaussian RBFs of the form ϕ(x;α,h)=exp(−α2(x/h)2): in the limit h→0 for fixed α, the error does not converge to zero, but rather to ES(α). Previous studies have theoretically determined the saturation error for Gaussian RBF on an infinite, uniform interval and for the same with a single point omitted. (The gap enormously increases ES(α).) We show experimentally that the saturation error on the unit interval, x∈[−1,1], is about 0.06exp(−0.47/α2)‖f‖∞ — huge compared to the O(2π/α2)exp(−π2/[4α2]) saturation error for a grid with one point omitted. We show that the reason the saturation is so large on a finite interval is that it is equivalent to an infinite grid which is uniform except for a gap of many points. The saturation error can be avoided by choosing α≪1, the “flat limit”, but the condition number of the interpolation matrix explodes as O(exp(π2/[4α2])). The best strategy is to choose the largest α which yields an acceptably small saturation error: If the user chooses an error tolerance δ, then αoptimum(δ)=1/−2log(δ/0.06)

How fast do radial basis function interpolants of analytic functions converge?

The question in the title is answered using tools of potential theory. Convergence and divergence rates of interpolants of analytic functions on the unit interval are analyzed. The starting point is a complex variable contour integral formula for the remainder in RBF interpolation. We study a generalized Runge phenomenon and explore how the location of centers and affects convergence. Special attention is given to Gaussian and inverse quadratic radial functions, but some of the results can be extended to other smooth basis functions. Among other things, we prove that, under mild conditions, inverse quadratic RBF interpolants of functions that are analytic inside the strip $|Im(z)| < (1/2\epsilon)$, where $\epsilon$ is the shape parameter, converge exponentially

Nonlinear Approximation Using Gaussian Kernels

It is well-known that non-linear approximation has an advantage over linear schemes in the sense that it provides comparable approximation rates to those of the linear schemes, but to a larger class of approximands. This was established for spline approximations and for wavelet approximations, and more recently by DeVore and Ron for homogeneous radial basis function (surface spline) approximations. However, no such results are known for the Gaussian function, the preferred kernel in machine learning and several engineering problems. We introduce and analyze in this paper a new algorithm for approximating functions using translates of Gaussian functions with varying tension parameters. At heart it employs the strategy for nonlinear approximation of DeVore and Ron, but it selects kernels by a method that is not straightforward. The crux of the difficulty lies in the necessity to vary the tension parameter in the Gaussian function spatially according to local information about the approximand: error analysis of Gaussian approximation schemes with varying tension are, by and large, an elusive target for approximators. We show that our algorithm is suitably optimal in the sense that it provides approximation rates similar to other established nonlinear methodologies like spline and wavelet approximations. As expected and desired, the approximation rates can be as high as needed and are essentially saturated only by the smoothness of the approximand.Comment: 15 Pages; corrected typos; to appear in J. Funct. Ana

RHIC-tested predictions for low-pTp_T and high-pTp_T hadron spectra in nearly central Pb+Pb collisions at the LHC

We study the hadron spectra in nearly central $A$+$A$ collisions at RHIC and LHC in a broad transverse momentum range. We cover the low-$p_T$ spectra using longitudinally boost-invariant hydrodynamics with initial energy and net-baryon number densities from the perturbative QCD (pQCD)+saturation model. Build-up of the transverse flow and sensitivity of the spectra to a single decoupling temperature \Tdec are studied. Comparison with RHIC data at \ssNN=130 and 200 GeV suggests a rather high value \Tdec=150 MeV. The high-$p_T$ spectra are computed using factorized pQCD cross sections, nuclear parton distributions, fragmentation functions, and describing partonic energy loss in the quark-gluon plasma by quenching weights. Overall normalization is fixed on the basis of p+$\bar{\rm p}$(p) data and the strength of energy loss is determined from RHIC Au+Au data. Uncertainties are discussed. With constraints from RHIC data, we predict the $p_T$ spectra of hadrons in 5 % most central Pb+Pb collisions at the LHC energy \ssNN=5500 GeV. Due to the closed framework for primary production, we can also predict the net-baryon number at midrapidity, as well as the strength of partonic energy losses at the LHC. Both at the LHC and RHIC, we recognize a rather narrow crossover region in the $p_T$ spectra, where the hydrodynamic and pQCD fragmentation components become of equal size. We argue that in this crossover region the two contributions are to a good approximation mutually independent. In particular, our results suggest a wider $p_T$-region of applicability for hydrodynamical models at the LHC than at RHIC.Comment: 45 pages, 16 eps-figure

Stable and accurate least squares radial basis function approximations on bounded domains

The computation of global radial basis function (RBF) approximations requires the solution of a linear system which, depending on the choice of RBF parameters, may be ill-conditioned. We study the stability and accuracy of approximation methods using the Gaussian RBF in all scaling regimes of the associated shape parameter. The approximation is based on discrete least squares with function samples on a bounded domain, using RBF centers both inside and outside the domain. This results in a rectangular linear system. We show for one-dimensional approximations that linear scaling of the shape parameter with the degrees of freedom is optimal, resulting in constant overlap between neighbouring RBF's regardless of their number, and we propose an explicit suitable choice of the proportionality constant. We show numerically that highly accurate approximations to smooth functions can also be obtained on bounded domains in several dimensions, using a linear scaling with the degrees of freedom per dimension. We extend the least squares approach to a collocation-based method for the solution of elliptic boundary value problems and illustrate that the combination of centers outside the domain, oversampling and optimal scaling can result in accuracy close to machine precision in spite of having to solve very ill-conditioned linear systems

The Canonical Function Method and its applications in Quantum Physics

The Canonical Function Method (CFM) is a powerful method that solves the radial Schr\"{o}dinger equation for the eigenvalues directly without having to evaluate the eigenfunctions. It is applied to various quantum mechanical problems in Atomic and Molecular physics with presence of regular or singular potentials. It has also been developed to handle single and multiple channel scattering problems where the phaseshift is required for the evaluation of the scattering cross-section. Its controllable accuracy makes it a valuable tool for the evaluation of vibrational levels of cold molecules, a sensitive test of Bohr correspondance principle and a powerful method to tackle local and non-local spin dependent problems.Comment: 30 pages, 12 figures- To submit to Reviews of Modern Physic

Flame front propagation IV: Random Noise and Pole-Dynamics in Unstable Front Propagation II

The current paper is a corrected version of our previous paper arXiv:adap-org/9608001. Similarly to previous version we investigate the problem of flame propagation. This problem is studied as an example of unstable fronts that wrinkle on many scales. The analytic tool of pole expansion in the complex plane is employed to address the interaction of the unstable growth process with random initial conditions and perturbations. We argue that the effect of random noise is immense and that it can never be neglected in sufficiently large systems. We present simulations that lead to scaling laws for the velocity and acceleration of the front as a function of the system size and the level of noise, and analytic arguments that explain these results in terms of the noisy pole dynamics.This version corrects some very critical errors made in arXiv:adap-org/9608001 and makes more detailed description of excess number of poles in system, number of poles that appear in the system in unit of time, life time of pole. It allows us to understand more correctly dependence of the system parameters on noise than in arXiv:adap-org/9608001Comment: 23 pages, 4 figures,revised, version accepted for publication in journal "Combustion, Explosion and Shock Waves". arXiv admin note: substantial text overlap with arXiv:nlin/0302021, arXiv:adap-org/9608001, arXiv:nlin/030201

A Guide to RBF-Generated Finite Differences for Nonlinear Transport: Shallow Water Simulations on a Sphere

The current paper establishes the computational efficiency and accuracy of the RBF-FD method for large-scale geoscience modeling with comparisons to state-of-the-art methods as high-order discontinuous Galerkin and spherical harmonics, the latter using expansions with close to 300,000 bases. The test cases are demanding fluid flow problems on the sphere that exhibit numerical challenges, such as Gibbs phenomena, sharp gradients, and complex vortical dynamics with rapid energy transfer from large to small scales over short time periods. The computations were possible as well as very competitive due to the implementation of hyperviscosity on large RBF stencil sizes (corresponding roughly to 6th to 9th order methods) with up to O(105) nodes on the sphere. The RBF-FD method scaled as O(N) per time step, where N is the total number of nodes on the sphere. In Appendix A, guidelines are given on how to chose parameters when using RBF-FD to solve hyperbolic PDEs