arXiv subjects

Jeremy Levesley

Publications and source records attributed to Jeremy Levesley.

22 records · Page 2Linked to original sources

arXiv2012

A Multiplier Version of the Bernstein Inequality on the Complex Sphere

We prove a multiplier version of the Bernstein inequality on the complex sphere. Included in this is a new result relating a bivariate sum involving Jacobi polynomials and Gegenbauer polynomials, which relates the sum of reproducing kernels on spaces of polynomials irreducibly invariant under the unitary group, with the reproducing kernel of the sum of these spaces, which is irreducibly invariant under the action of the orthogonal group.

math.CA↗

arXiv2012

Approximation on the complex sphere

We develop new elements of harmonic analysis on the complex sphere on the basis of which Bernstein's, Jackson's and Kolmogorov's inequalities are established. We apply these results to get order sharp estimates of $m$-term approximations. The results obtained is a synthesis of new results on classical orthogonal polynomials, harmonic analysis on manifolds and geometric properties of Euclidean spaces.

math.CA↗

arXiv2012

Multilevel Sparse Kernel-Based Interpolation

A multilevel kernel-based interpolation method, suitable for moderately high-dimensional function interpolation problems, is proposed. The method, termed multilevel sparse kernel-based interpolation (MLSKI, for short), uses both level-wise and direction-wise multilevel decomposition of structured (or mildly unstructured) interpolation data sites in conjunction with the application of kernel-based interpolants with different scaling in each direction. The multilevel interpolation algorithm is based on a hierarchical decomposition of the data sites, whereby at each level the detail is added to the interpolant by interpolating the resulting residual of the previous level. On each level, anisotropic radial basis functions are used for solving a number of small interpolation problems, which are subsequently linearly combined to produce the interpolant. MLSKI can be viewed as an extension of $d$-boolean interpolation (which is closely related to ideas in sparse grid and hyperbolic crosses literature) to kernel-based functions, within the hierarchical multilevel framework to achieve accelerated convergence. Numerical experiments suggest that the new algorithm is numerically stable and efficient for the reconstruction of large data in $\mathbb{R}^{d}\times \mathbb{R}$, for $d = 2, 3, 4$, with tens or even hundreds of thousands data points. Also, MLSKI appears to be generally superior over classical radial basis function methods in terms of complexity, run time and convergence at least for large data sets.

math.NA↗

arXiv2010

Time Step Expansions and the Invariant Manifold Approach to Lattice Boltzmann Models

The classical method for deriving the macroscopic dynamics of a lattice Boltzmann system is to use a combination of different approximations and expansions. Usually a Chapman-Enskog analysis is performed, either on the continuous Boltzmann system, or its discrete velocity counterpart. Separately a discrete time approximation is introduced to the discrete velocity Boltzmann system, to achieve a practically useful approximation to the continuous system, for use in computation. Thereafter, with some additional arguments, the dynamics of the Chapman-Enskog expansion are linked to the discrete time system to produce the dynamics of the completely discrete scheme. In this paper we put forward a different route to the macroscopic dynamics. We begin with the system discrete in both velocity space and time. We hypothesize that the alternating steps of advection and relaxation, common to all lattice Boltzmann schemes, give rise to a slow invariant manifold. We perform a time step expansion of the discrete time dynamics using the invariance of the manifold. Finally we calculate the dynamics arising from this system. By choosing the fully discrete scheme as a starting point we avoid mixing approximations and arrive at a general form of the microscopic dynamics up to the second order in the time step. We calculate the macroscopic dynamics of two commonly used lattice schemes up to the first order, and hence find the precise form of the deviation from the Navier-Stokes equations in the dissipative term, arising from the discretization of velocity space. Finally we perform a short wave perturbation on the dynamics of these example systems, to find the necessary conditions for their stability.

cond-mat.stat-mech↗