Jeremy Levesley

Multilevel scattered data approximation by adaptive domain decomposition

Abstract A new multilevel approximation scheme for scattered data is proposed. The scheme relies on an adaptive domain decomposition strategy using quadtree techniques (and their higher-dimensional generalizations). It is shown in the numerical examples that the new method achieves an improvement on the approximation quality of previous well-established multilevel interpolation schemes.

Stability and stabilization of the lattice Boltzmann method

The lattice Boltzmann method (LBM) is known to have stability deficiencies. For example, local blow-ups and spurious oscillations are readily observed when the method is used to model high-Reynolds fluid flow. Beginning from thermodynamic considerations, the LBM can be recognised as a discrete dynamical system generated by entropic involution and free-flight and the stability analysis is more natural. In this paper we solve the stability problem of the LBM on the basis of this thermodynamic point of view. The main instability mechanisms are identified. The simplest and most effective receipt for stabilisation adds no artificial dissipation, preserves the second-order accuracy of the method, and prescribes coupled steps: to start from a local equilibrium, then, after free-flight, perform the overrelaxation collision, and after a second free-flight step go to new local equilibrium. Two other prescriptions add some artificial dissipation locally and prevent the system from loss of positivity and local blow-up. Demonstration of the proposed stable LBMs are provided by the numerical simulation of a 1D shock tube and the unsteady 2D-flow around a square-cylinder up to Reynolds number O(10000).

Nonequilibrium entropy limiters in lattice Boltzmann methods

We construct a system of nonequilibrium entropy limiters for the lattice Boltzmann methods (LBM). These limiters erase spurious oscillations without blurring of shocks, and do not affect smooth solutions. In general, they do the same work for LBM as flux limiters do for finite differences, finite volumes and finite elements methods, but for LBM the main idea behind the construction of nonequilibrium entropy limiter schemes is to transform a field of a scalar quantity — nonequilibrium entropy. There are two families of limiters: (i) based on restriction of nonequilibrium entropy (entropy “trimming”) and (ii) based on filtering of nonequilibrium entropy (entropy filtering). The physical properties of LBM provide some additional benefits: the control of entropy production and accurate estimation of introduced artificial dissipation are possible. The constructed limiters are tested on classical numerical examples: 1D athermal shock tubes with an initial density ratio 1:2 and the 2D lid-driven cavity for Reynolds numbers Re between 2000 and 7500 on a coarse 100×100 grid. All limiter constructions are applicable both for entropic and for non-entropic equilibria.

BENCHOP – The BENCHmarking project in option pricing

The aim of the BENCHOP project is to provide the finance community with a common suite of benchmark problems for option pricing. We provide a detailed description of the six benchmark problems together with methods to compute reference solutions. We have implemented fifteen different numerical methods for these problems, and compare their relative performance. All implementations are available on line and can be used for future development and comparisons.

Estimates of n-widths of Sobolev's classes on compact globally symmetric spaces of rank one

Estimates of Kolmogorovs and linear n-widths of Sobolevs classes on compact globally symmetric spaces of rank 1 (i.e. on Sd, Pd(R), Pd(C), Pd(H), P16(Cay)) are established. It is shown that these estimates have sharp orders in different important cases. New estimates for the (p,q)-norms of multiplier operators Λ={λk}k∈N are given. We apply our results to get sharp orders of best polynomial approximation and n-widths.

The Graduate Student’s Guide to Numerical Analysis ’98

This book contains detailed lecture notes on six topics at the forefront of current research in numerical analysis and applied mathematics. Each set of notes presents a self-contained guide to a current research area and has an extensive bibliography. In addition, most of the notes contain detailed proofs of the key results. The notes start from a level suitable for first year graduate students in applied mathematics, mathematical analysis or numerical analysis, and proceed to current research topics. The reader should therefore be able to gain quickly an insight into the important results and techniques in each area without recourse to the large research literature. Current (unsolved) problems are also described and directions for future research are given. This book is also suitable for professional mathematicians who require a succint and accurate account of recent research in areas parallel to their own, and graduates in mathematical sciences.

Norm estimates of interpolation matrices and their inverses associated with strictly positive definite functions

In this paper, we estimate the norms of the interpolation matrices and their inverses that arise from scattered data interpolation on spheres with strictly positive definite functions. ?

Error bounds for anisotropic RBF interpolation

We present error bounds for the interpolation with anisotropically transformed radial basis functions for both a function and its partial derivatives. The bounds rely on a growth function and do not contain unknown constants. For polyharmonic basic functions in R^2, we show that the anisotropic estimates predict a significant improvement of the approximation error if both the target function and the placement of the centers are anisotropic, and this improvement is confirmed numerically.

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 levelwise and directionwise 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

A neural network approach to the diagnosis of early acute allograft rejection.

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