. This contribution gives a partial survey over the native spaces associated to (not necessarily radial) basis functions. Starting from reproducing kernel Hilbert spaces and invariance properties, the general construction of native spaces is carried out for both the unconditionally and the conditionally positive definite case. The definitions of the latter are based on finitely supported functionals only. Fourier or other transforms are not required. The dependence of native spaces on the domain is studied, and criteria for functions and functionals to be in the native space are given. x1. Introduction and Overview For the numerical treatment of functions of many variables, radial basis functions are useful tools. They have the form OE(kx \Gamma yk 2 ) for vectors x; y 2 IR d with a univariate function OE defined on [0; 1) and the Euclidean norm k \Delta k 2 on IR d . This allows to work efficiently for large dimensions d, because the function boils the multivariate setting down...

The physical phenomena monitored by sensor networks, e.g. forest temperature, water contamination, usually yield sensed data that are strongly correlated in space. With this in mind, researchers have designed a large number of sensor network protocols and algorithms that attempt to exploit such correlations. To carefully study the performance of these algorithms, there is an increasing need to synthetically generate large traces of spatially correlated data representing a wide range of conditions. Further, a mathematical model for generating synthetic traces would provide guidelines for designing more efficient algorithms. These reasons motivate us to obtain a simple and accurate model of spatially correlated sensor network data. The proposed model is Markovian in nature and can capture correlation in data irrespective of the node density, the number of source nodes or the topology. We describe a rigorous mathematical procedure and a simple practical method to extract the model parameters from real traces. We also show how to efficiently generate synthetic traces on a given topology using these parameters. The correctness of the model is verified by statistically comparing synthetic and real data. Further, the model is validated by comparing the performance of algorithms whose behavior depends on the degree of spatial correlation in data, under real and synthetic traces. The real traces are obtained from remote sensing data, publicly available sensor data, and sensor networks that we deploy. We show that the proposed model is more general and accurate than the commonly used jointly Gaussian model. Finally, we create tools that can be easily used by researchers to synthetically generate traces of any size and degree of correlation.

These notes introduce the basic concepts of integral equations and their application in global illumination. Much of the discussion is expressed in the language of linear operators to simplify the notation and to emphasize the algebraic properties of the integral equations. We start by reviewing some facts about linear operators and examining some of the operators that occur in global illumination. Six general methods of solving operator and integral equations are then discussed: the Neumann series, successive approximations, the Nystrom method, collocation, least squares, and the Galerkin method. Finally, we look at some of the steps involved in applying these techniques in the context of global illumination. 1 Introduction The transfer of energy by radiation has a character fundamentally different from the processes of conduction and convection. One reason for this difference is that the radiant energy passing through a point in space cannot be completely described by a single scala...

We propose a new method to analyze and represent data recorded on a domain of gen-eral shape in R d by computing the eigenfunctions of Laplacian defined over there and expanding the data into these eigenfunctions. Instead of directly solving the eigenvalue problem on such a domain via the Helmholtz equation (which can be quite complicated and costly), we find the integral operator commuting with the Laplacian and diagonal-ize that operator. Although our eigenfunctions satisfy neither the Dirichlet nor the Neu-mann boundary condition, computing our eigenfunctions via the integral operator is simple and has a potential to utilize modern fast algorithms to accelerate the compu-tation. We also show that our method is better suited for small sample data than the Karhunen-Loève Transform/Principal Component Analysis. In fact, our eigenfunctions depend only on the shape of the domain, not the statistics of the data. As a further ap-plication, we demonstrate the use of our Laplacian eigenfunctions for solving the heat equation on a complicated domain. Key words: Laplacian eigenfunctions, boundary conditions, Green’s function, spectral decomposition, Karhunen-Loève transform, principal component analysis, heat equation

The physical phenomena monitored by sensor networks, e.g. forest temperature, usually yield sensed data that are strongly correlated in space. We have recently introduced a mathematical model for such data, and used it to generate synthetic traces and study the performance of algorithms whose behavior depends on this spatial correlation [1]. That work studied sensor networks with grid topologies. This work extends our modeling methodology to sensor networks with irregular topologies. We describe a rigorous mathematical procedure and a simple practical method to extract the model parameters from real traces. We also show how to efficiently generate synthetic traces that correspond to sensor networks with arbitrary topologies using the proposed model. The correctness of the model is verified by statistically comparing synthetic and real data. Further, the model is validated by comparing the performance of algorithms whose behavior depends on the degree of spatial correlation in data, under real and synthetic traces. The real traces are obtained from both publically available sensor data, and sensor networks that we deploy. Finally, we augment our existing tracegeneration tool with new functionality suited for sensor networks with irregular topologies.

This contribution extends earlier work [16] on interpolation/approximation by positive definite basis functions in several aspects. First, it works out the relations between various types of kernels in more detail and more generality. Second, it uses the new generality to exhibit the first example of a discontinuous positive definite function. Third, it establishes the first link from (radial) basis function theory to n-widths, and finally it uses this link to prove quasi-optimality results for approximation rates of interpolation processes and decay rates for eigenvalues of integral operators having smooth kernels.

This dissertation considers the behaviour of water waves within a harbour. The problem is motivated by both real world applications as well as being of interest from a purely abstract mathematical perspective. The harbour is modelled as a simple rectangular domain which is partially open on one side to the ocean. Techniques used in previous work on wave diffraction around a breakwater are adopted, and Partial Differential Equation and Integral Equation methods are used to formulate the equations describing the wave field within and outside of the harbour. The governing integral equation that is formed is solved numerically using Galerkin’s method, and graphical representations of the wave field are plotted. The effects of variations in the angle of the incident wave, and the wave number are analysed. In particular the phenomenon of harbour resonance is investigated, focusing on what wave numbers cause a resonant effect, but also examining the effect of changes in the incident wave angle, the size of the opening onto the ocean, and the dimensions of the harbour. i ii

In many computational simulations there are identifiable quantities of physical interest which are weighted integrals of the solution of a boundary or initial value problem. Examples include lift and drag in aerodynamics and well productions in reservoir simulations. Methods that can bound these quantities sharply are therefore of considerable practical importance. Although existing theory can sometimes be used to construct theoretical bounds on these quantities, for example those governed by a self-adjoint operator, it has not hitherto been exploited in a practical context. In this thesis, novel applications of these bounds have been devised and implemented for a number of model problems. In particular, the upscaling problem experienced in the oil industry, which is governed by the steady state diffusion equation, is addressed by finding bounds on the well outflow. Extensions to the theory for problems involving non-self-adjoint operators, which enable computable bounds to be determined for quantities of physical interest, are developed using two approaches. Firstly, semi-discrete approximations are considered for the time-dependent diffusion equations and the advection-diffusion equation, although this

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Apart from omnidirectional, a solid elastic sphere is a natural multimode and multifrequency device for the detection of Gravitational Waves (GW). Motion sensing in a spherical GW detector thus requires a multiple set of transducers attached to