Robert C. Sharpley

An ELLAM Scheme for Advection-Diffusion Equations in Two Dimensions

We develop an Eulerian--Lagrangian localized adjoint method (ELLAM) to solve two-dimensional advection-diffusion equations with all combinations of inflow and outflow Dirichlet, Neumann, and flux boundary conditions. The ELLAM formalism provides a systematic framework for implementation of general boundary conditions, leading to mass-conservative numerical schemes. The computational advantages of the ELLAM approximation have been demonstrated for a number of one-dimensional transport systems; practical implementations of ELLAM schemes in multiple spatial dimensions that require careful algorithm development are discussed in detail in this paper. Extensive numerical results are presented to compare the ELLAM scheme with many widely used numerical methods and to demonstrate the strength of the ELLAM scheme.

Besov-Spaces on Domains in RD

We study Besov spaces 5°(LP(I2)), 0<p,q,a<oo,on do- mains I2 in Rd . We show that there is an extension operator W which is a bounded mapping from B°(LP(U.)) onto B%(Lp(Rd)). This is then used to derive various properties of the Besov spaces such as interpolation theorems for a pair of B%(Lp(il)), atomic decompositions for the elements of 5°(Lp(f2)), and a description of the Besov spaces by means of spline approximation.

FRACTIONAL INTEGRATION IN ORLICZ SPACES

Fractional integration and convolution results are given for Orlicz spaces using an inequality earlier developed for Aa(X~) spaces which generalize Lorentz L(p,q) spaces. The extension problem for convolution operators encountered previously by other authors is almost entirely avoided.

Spaces Λα(X) and interpolation

Abstract For two pairs of rearrangement invariant spaces α = [(X1, Y1), (X2, Y2)] we give necessary and sufficient conditions for pairs (X, Y) to be weak intermediate for σ, i.e., each operator which is of weak types (Xi, Yi), i = 1, 2, also maps X boundedly to Y. Spaces Λα(X) are introduced and are shown to have many of the properties that characterize Lorentz Lpq spaces. Necessary and sufficient conditions in terms of a simple function F(s, t) are given in order that (Λα(X), Λα(Y)) be weak intermediate for σ. Other properties of the function F(s, t) yield sufficient conditions and necessary conditions for interpolation theorems.

Optimized imaging using non-rigid registration.

The extraordinary improvements of modern imaging devices offer access to data with unprecedented information content. However, widely used image processing methodologies fall far short of exploiting the full breadth of information offered by numerous types of scanning probe, optical, and electron microscopies. In many applications, it is necessary to keep measurement intensities below a desired threshold. We propose a methodology for extracting an increased level of information by processing a series of data sets suffering, in particular, from high degree of spatial uncertainty caused by complex multiscale motion during the acquisition process. An important role is played by a non-rigid pixel-wise registration method that can cope with low signal-to-noise ratios. This is accompanied by formulating objective quality measures which replace human intervention and visual inspection in the processing chain. Scanning transmission electron microscopy of siliceous zeolite material exhibits the above-mentioned obstructions and therefore serves as orientation and a test of our procedures.

Weak Interpolation in Banach Spaces

The uses of interpolation theory in various branches of analysis, especially Fourier analysis, are well known. For many operators a weak type interpolation theory is indispensible for accurately describing their mapping properties. Such for example is the case with the Hilbert transform, maximal operators, etc. The early results in interpolation theory were for spaces of measurable functions. Subsequently, a strong type interpolation theory was developed for arbitrary Banach spaces by using various functionalizations of these spaces. The most widely known are the A. P. Calderon complex method [16] and the Lions-Peetre real method [13]. We will show in this paper that it is a straightforward matter to develop a generalized weak type interpolation theory for arbitrary Banach spaces by combining the Peetre functionalization with the maximal operators of Calderon [15]. As would be expected, this generalized weak type theory has many interesting applications.

Second-order characteristic methods for advection-diffusion equations and comparison to other schemes

Abstract We develop two characteristic methods for the solution of the linear advection diffusion equations which use a second order Runge–Kutta approximation of the characteristics within the framework of the Eulerian–Lagrangian localized adjoint method. These methods naturally incorporate all three types of boundary conditions in their formulations, are fully mass conservative, and generate regularly structured systems which are symmetric and positive definite for most combinations of the boundary conditions. Extensive numerical experiments are presented which compare the performance of these two Runge–Kutta methods to many other well perceived and widely used methods which include many Galerkin methods and high resolution methods from fluid dynamics.

Compressed Sensing and Electron Microscopy

Compressed sensing (CS) is a relatively new approach to signal acquisition which has as its goal to minimize the number of measurements needed of the signal in order to guarantee that it is captured to a prescribed accuracy. It is natural to inquire whether this new subject has a role to play in electron microscopy (EM). In this chapter, we shall describe the foundations of CS and then examine which parts of this new theory may be useful in EM.

High-Quality Image Formation by Nonlocal Means Applied to High-Angle Annular Dark-Field Scanning Transmission Electron Microscopy (HAADF–STEM)

Abstract : We outline a new systematic approach to extracting high resolution information from HAADF-STEM images which will be beneficial to the characterization of beam sensitive materials. The idea is to treat several, possibly many low electron dose images with specially adapted digital image processing concepts at a minimum allowable spatial resolution. Our goal is to keep the overall cumulative electron dose as low as possible while still staying close to an acceptable level of physical resolution. We shall present the main conceptual imaging concepts and restoration methods that we believe are suitable for carrying out such a program and, in particular, allow one to correct special acquisition artifacts which result in blurring, aliasing, rastering distortions and noise.

Decomposition of functions into pairs of intrinsic mode functions

The intrinsic mode functions (IMFs) arise as basic modes from the application of the empirical mode decomposition (EMD) to functions or signals. In this procedure, instantaneous frequencies are subsequently extracted from the IMFs by the simple application of the Hilbert transform, thereby providing a multiscale analysis of the signals nonlinear phases. The beauty of this redundant representation method is in its simplicity and extraordinary effectiveness in many important and diverse settings. A fundamental issue in the field is to better understand these demonstrated qualities of the EMD procedures and the elementary modes they produce. For example, it is easily observed that when an EMD procedure is applied to the sum of two arbitrary IMFs, the original modes are rarely reproduced in the generated collection of IMFs. An interesting question from a representation point of view may be stated as follows: for any given sufficiently smooth function and fixed n≥2, when is it possible to represent the function as a sum of (at most) n intrinsic modes? A more interesting question is whether such a decomposition is possible when the extracted modes are constructed from a common formulation of the intrinsic properties of the function being analysed. We provide an answer to these questions for a relaxed version of IMFs, called weak IMFs, which has been shown to be characterized in terms of eigenfunctions of Sturm–Liouville operators. The objective of this study is to further extend that analogy to the relationship between sums of weak IMFs and coupled Sturm–Liouville systems. The construction of this decomposition also provides a guide to an alternate characterization of the instantaneous frequency and bandwidth.