On the convergence of the rescaled localized radial basis function method

The rescaled localized RBF method was introduced in Deparis, Forti, and Quarteroni (2014) for scattered data interpolation. It is a rational approximation method based on interpolation with compactly supported radial basis functions. It requires the solution of two linear systems with the same sparse matrix, which has a small condition number, due to the scaling of the basis function. Hence, it can be computed using an unpreconditioned conjugate gradient method in linear time. Numerical evidence provided in Deparis, Forti, and Quarteroni (2014) shows that the method produces good approximations for many examples but no theoretical results were provided. In this paper, we discuss the convergence of the rescaled localized RBF method in the case of quasi-uniform data and stationary scaling. As the method is not only interpolatory but also reproduces constants exactly, linear convergence is expected. We can show this linear convergence up to a certain conjecture

Rescaled localized radial basis functions and fast decaying polynomial reproduction

openApproximating a set of data can be a difficult task but it is very useful in applications. Through a linear combination of basis functions we want to reconstruct an unknown quantity from partial information. We study radial basis functions (RBFs) to obtain an approximation method that is meshless, provides a data dependent approximation space and generalization to larger dimensions is not an obstacle. We analyze a rational approximation method with compactly supported radial basis functions (Rescaled localized radial basis function method). The method reproduces exactly the constants and the density of the interpolation nodes influences the support of the RBFs. There is a proof of the convergence in a quasi-uniform setting up to a conjecture: we can determine a lower bound for the approximant of the constant function 1 uniformly with respect to the size of the support of the kernel. We investigate the statement of the conjecture and bring some practical and theoretical results to support it. We study the Runge phenomenon on the approximant and obtain uniform estimates on the cardinal functions. We extend the distinguishing features of the method reproducing exactly larger polynomial spaces. We replace local polynomial reproduction with basis functions that decrease rapidly and approximate exactly a polynomial space. This change releases the basis functions from the compactness of the support and guarantees the same convergence rate (the oversampling problem does not appear). The rescaled localized radial basis function method can be interpreted in this new framework because the cardinal functions have global support even if the kernel has compact support. The decay of the basis functions undertake convergence and stability. In this analysis the smoothness of the approximant is not important, what matters is the "locality" provided by the fast decay. With a moving least squares approach we provide an example of a smooth quasi-interpolant. We continue trying to improve the performance of the method even when the weight functions do not have compact support. All the new theoretical results introduced in this work are also supported by numerical evidence.Approximating a set of data can be a difficult task but it is very useful in applications. Through a linear combination of basis functions we want to reconstruct an unknown quantity from partial information. We study radial basis functions (RBFs) to obtain an approximation method that is meshless, provides a data dependent approximation space and generalization to larger dimensions is not an obstacle. We analyze a rational approximation method with compactly supported radial basis functions (Rescaled localized radial basis function method). The method reproduces exactly the constants and the density of the interpolation nodes influences the support of the RBFs. There is a proof of the convergence in a quasi-uniform setting up to a conjecture: we can determine a lower bound for the approximant of the constant function 1 uniformly with respect to the size of the support of the kernel. We investigate the statement of the conjecture and bring some practical and theoretical results to support it. We study the Runge phenomenon on the approximant and obtain uniform estimates on the cardinal functions. We extend the distinguishing features of the method reproducing exactly larger polynomial spaces. We replace local polynomial reproduction with basis functions that decrease rapidly and approximate exactly a polynomial space. This change releases the basis functions from the compactness of the support and guarantees the same convergence rate (the oversampling problem does not appear). The rescaled localized radial basis function method can be interpreted in this new framework because the cardinal functions have global support even if the kernel has compact support. The decay of the basis functions undertake convergence and stability. In this analysis the smoothness of the approximant is not important, what matters is the "locality" provided by the fast decay. With a moving least squares approach we provide an example of a smooth quasi-interpolant. We continue trying to improve the performance of the method even when the weight functions do not have compact support. All the new theoretical results introduced in this work are also supported by numerical evidence

Approximation Theory XV: San Antonio 2016

These proceedings are based on papers presented at the international conference Approximation Theory XV, which was held May 22\u201325, 2016 in San Antonio, Texas. The conference was the fifteenth in a series of meetings in Approximation Theory held at various locations in the United States, and was attended by 146 participants. The book contains longer survey papers by some of the invited speakers covering topics such as compressive sensing, isogeometric analysis, and scaling limits of polynomials and entire functions of exponential type. The book also includes papers on a variety of current topics in Approximation Theory drawn from areas such as advances in kernel approximation with applications, approximation theory and algebraic geometry, multivariate splines for applications, practical function approximation, approximation of PDEs, wavelets and framelets with applications, approximation theory in signal processing, compressive sensing, rational interpolation, spline approximation in isogeometric analysis, approximation of fractional differential equations, numerical integration formulas, and trigonometric polynomial approximation

Mode coupling in the nonlinear response of black holes

We study the properties of the outgoing gravitational wave produced when a non-spinning black hole is excited by an ingoing gravitational wave. Simulations using a numerical code for solving Einstein's equations allow the study to be extended from the linearized approximation, where the system is treated as a perturbed Schwarzschild black hole, to the fully nonlinear regime. Several nonlinear features are found which bear importance to the data analysis of gravitational waves. When compared to the results obtained in the linearized approximation, we observe large phase shifts, a stronger than linear generation of gravitational wave output and considerable generation of radiation in polarization states which are not found in the linearized approximation. In terms of a spherical harmonic decomposition, the nonlinear properties of the harmonic amplitudes have simple scaling properties which offer an economical way to catalog the details of the waves produced in such black hole processes.Comment: 17 pages, 20 figures, abstract and introduction re-writte

Numerical solution of gravitational dynamics in asymptotically anti-de Sitter spacetimes

A variety of gravitational dynamics problems in asymptotically anti-de Sitter (AdS) spacetime are amenable to efficient numerical solution using a common approach involving a null slicing of spacetime based on infalling geodesics, convenient exploitation of the residual diffeomorphism freedom, and use of spectral methods for discretizing and solving the resulting differential equations. Relevant issues and choices leading to this approach are discussed in detail. Three examples, motivated by applications to non-equilibrium dynamics in strongly coupled gauge theories, are discussed as instructive test cases. These are gravitational descriptions of homogeneous isotropization, collisions of planar shocks, and turbulent fluid flows in two spatial dimensions.Comment: 70 pages, 19 figures; v4: fixed minus sign typo in last term of eqn. (3.47

Far-from-equilibrium dynamics of a strongly coupled non-Abelian plasma with non-zero charge density or external magnetic field

Using holography, we study the evolution of a spatially homogeneous, far from equilibrium, strongly coupled N=4 supersymmetric Yang-Mills plasma with a non-zero charge density or a background magnetic field. This gauge theory problem corresponds, in the dual gravity description, to an initial value problem in Einstein-Maxwell theory with homogeneous but anisotropic initial conditions. We explore the dependence of the equilibration process on different aspects of the initial departure from equilibrium and, while controlling for these dependencies, examine how the equilibration dynamics are affected by the presence of a non-vanishing charge density or an external magnetic field. The equilibration dynamics are remarkably insensitive to the addition of even large chemical potentials or magnetic fields; the equilibration time is set primarily by the form of the initial departure from equilibrium. For initial deviations from equilibrium which are well localized in scale, we formulate a simple model for equilibration times which agrees quite well with our results.Comment: 54 pages, 18 figures, published version, ref. update

Relativistic Stars in Randall-Sundrum Gravity

The non-linear behaviour of Randall-Sundrum gravity with one brane is examined. Due to the non-compact extra dimension, the perturbation spectrum has no mass gap, and the long wavelength effective theory is only understood perturbatively. The full 5-dimensional Einstein equations are solved numerically for static, spherically symmetric matter localized on the brane, yielding regular geometries in the bulk with axial symmetry. An elliptic relaxation method is used, allowing both the brane and asymptotic radiation boundary conditions to be simultaneously imposed. The same data that specifies stars in 4-dimensional gravity, uniquely constructs a 5-dimensional solution. The algorithm performs best for small stars (radius less than the AdS length) yielding highly non-linear solutions. An upper mass limit is observed for these small stars, and the geometry shows no global pathologies. The geometric perturbation is shown to remain localized near the brane at high densities, the confinement interestingly increasing for both small and large stars as the upper mass limit is approached. Furthermore, the static spatial sections are found to be approximately conformal to those of AdS. We show that the intrinsic geometry of large stars, with radius several times the AdS length, is described by 4-dimensional General Relativity far past the perturbative regime. This indicates that the non-linear long wavelength effective action remains local, even though the perturbation spectrum has no mass gap. The implication is that Randall-Sundrum gravity, with localized brane matter, reproduces relativistic astrophysical solutions, such as neutron stars and massive black holes, consistent with observation.Comment: 57 pages, 26 (colour) figures; minor typos corrected, references added and introduction condense

Scalar field breathers on anti-de Sitter background

We study spatially localized, time-periodic solutions (breathers) of scalar field theories with various self-interacting potentials on Anti-de Sitter (AdS) spacetimes in $D$ dimensions. A detailed numerical study of spherically symmetric configurations in $D=3$ dimensions is carried out, revealing a rich and complex structure of the phase-space (bifurcations, resonances). Scalar breather solutions form one-parameter families parametrized by their amplitude, $\varepsilon$, while their frequency, $\omega=\omega(\varepsilon)$, is a function of the amplitude. The scalar breathers on AdS we find have a small amplitude limit, tending to the eigenfunctions of the linear Klein-Gordon operator on AdS. Importantly most of these breathers appear to be generically stable under time evolution.Comment: 30 pages, 22 figure

Variational treatment of electron-polyatomic molecule scattering calculations using adaptive overset grids

The Complex Kohn variational method for electron-polyatomic molecule scattering is formulated using an overset grid representation of the scattering wave function. The overset grid consists of a central grid and multiple dense, atom-centered subgrids that allow the simultaneous spherical expansions of the wave function about multiple centers. Scattering boundary conditions are enforced by using a basis formed by the repeated application of the free particle Green's function and potential, $\hat{G}^+_0\hat{V}$ on the overset grid in a "Born-Arnoldi" solution of the working equations. The theory is shown to be equivalent to a specific Pad\'e approximant to the $T$-matrix, and has rapid convergence properties, both in the number of numerical basis functions employed and the number of partial waves employed in the spherical expansions. The method is demonstrated in calculations on methane and CF$_4$ in the static-exchange approximation, and compared in detail with calculations performed with the numerical Schwinger variational approach based on single center expansions. An efficient procedure for operating with the free-particle Green's function and exchange operators (to which no approximation is made) is also described