Kullback--Leibler approximation for probability measures on infinite dimensional spaces

In a variety of applications it is important to extract information from a probability measure $\mu$ on an infinite dimensional space. Examples include the Bayesian approach to inverse problems and (possibly conditioned) continuous time Markov processes. It may then be of interest to find a measure $\nu$, from within a simple class of measures, which approximates $\mu$. This problem is studied in the case where the Kullback--Leibler divergence is employed to measure the quality of the approximation. A calculus of variations viewpoint is adopted, and the particular case where $\nu$ is chosen from the set of Gaussian measures is studied in detail. Basic existence and uniqueness theorems are established, together with properties of minimizing sequences. Furthermore, parameterization of the class of Gaussians through the mean and inverse covariance is introduced, the need for regularization is explained, and a regularized minimization is studied in detail. The calculus of variations framework resulting from this work provides the appropriate underpinning for computational algorithms

Kullback--Leibler approximation for probability measures on infinite dimensional spaces

In a variety of applications it is important to extract information from a probability measure $\mu$ on an infinite dimensional space. Examples include the Bayesian approach to inverse problems and (possibly conditioned) continuous time Markov processes. It may then be of interest to find a measure $\nu$, from within a simple class of measures, which approximates $\mu$. This problem is studied in the case where the Kullback--Leibler divergence is employed to measure the quality of the approximation. A calculus of variations viewpoint is adopted, and the particular case where $\nu$ is chosen from the set of Gaussian measures is studied in detail. Basic existence and uniqueness theorems are established, together with properties of minimizing sequences. Furthermore, parameterization of the class of Gaussians through the mean and inverse covariance is introduced, the need for regularization is explained, and a regularized minimization is studied in detail. The calculus of variations framework resulting from this work provides the appropriate underpinning for computational algorithms

Electronic structure and x-ray magnetic dichroism in random substitutional alloys of f-electron elements

The Koringa-Kohn-Rostoker —coherent-potential-approximation method combines multiple-scattering theory and the coherent-potential approximation to calculate the electronic structure of random substitutional alloys of transition metals. In this paper we describe the generalization of this theory to describe f-electron alloys. The theory is illustrated with a calculation of the electronic structure and magnetic dichroism curves for a random substitutional alloy containing rare-earth or actinide elements from first principles

Virtual-crystal approximation that works: Locating a composition phase boundary in Pb(Zr_{1-x}Ti_3)O_3

We present a new method for modeling disordered solid solutions, based on the virtual crystal approximation (VCA). The VCA is a tractable way of studying configurationally disordered systems; traditionally, the potentials which represent atoms of two or more elements are averaged into a composite atomic potential. We have overcome significant shortcomings of the standard VCA by developing a potential which yields averaged atomic properties. We perform the VCA on a ferroelectric oxide, determining the energy differences between the high-temperature rhombohedral, low-temperature rhombohedral and tetragonal phases of Pb(Zr_{1-x}Ti_x)O_3 at x=0.5 and comparing these results to superlattice calculations and experiment. We then use our new method to determine the preferred structural phase at x=0.4. We find that the low-temperature rhombohedral phase becomes the ground state at x=0.4, in agreement with experimental findings.Comment: 5 pages, no figure

A theoretical study of the structural phases of Group 5B - 6B metals and their transport properties

In order to predict the stable and metastable phases of the bcc metals in the block of the Periodic Table defined by groups 5B to 6B and periods 4 to 6, as well as the structure dependence of their transport properties, we have performed full potential computations of the total energies per unit cell as a function of the c/a ratio at constant experimental volume. In all cases, a metastable body centered tetragonal (bct) phase was predicted from the calculations. The total energy differences between the calculated stable and metastable phases ranged from 0.09 eV/cell (vanadium) to 0.39 eV/cell (tungsten). The trends in resistivity as a function of structure and atomic number are discussed in terms of a model of electron transport in metals. Theoretical calculations of the electrical resistivity and other transport properties show that bct phases derived from group 5B elements are more conductive than the corresponding bcc phases, while bct phases formed from group 6B elements are less conductive than the corresponding bcc phases. Special attention is paid to the phases of tantalum where we show that the frequently observed beta phase is not a simple tetragonal distortion of bcc tantalum

The onset of magnetic order in fcc-Fe films on Cu(100)

On the basis of a first-principles electronic structure theory of finite temperature metallic magnetism in layered materials, we investigate the onset of magnetic order in thin (2-8 layers) fcc-Fe films on Cu(100) substrates. The nature of this ordering is altered when the systems are capped with copper. Indeed we find an oscillatory dependence of the Curie temperatures as a function of Cu-cap thickness, in excellent agreement with experimental data. The thermally induced spin-fluctuations are treated within a mean-field disordered local moment (DLM) picture and give rise to layer-dependent `local exchange splittings' in the electronic structure even in the paramagnetic phase. These features determine the magnetic intra- and interlayer interactions which are strongly influenced by the presence and extent of the Cu cap.Comment: 13 pages, 3 figure

Reevaluating electron-phonon coupling strengths: Indium as a test case for ab initio and many-body-theory methods

Using indium as a test case, we investigate the accuracy of the electron-phonon coupling calculated with state-of-the-art ab initio and many-body theory methods. The ab initio calculations -- where electrons are treated in the local-density approximation, and phonons and the electron-phonon interaction are treated within linear response -- predict an electron-phonon spectral function alpha^2 F(omega) which translates into a relative tunneling conductance that agrees with experiment to within one part in 1000. The many-body theory calculations -- where alpha^2 F(omega) is extracted from tunneling data by means of the McMillan-Rowell tunneling inversion method -- provide spectral functions that depend strongly on details of the inversion process. For the the most important moment of alpha^2 F(omega), the mass-renormalization parameter lambda, we report 0.9 +/- 0.1, in contrast to the value 0.805 quoted for nearly three decades in the literature. The ab initio calculations also provide the transport electron-phonon spectral function alpha_{tr}^2 F(omega), from which we calculate the resistivity as a function of temperature in good agreement with experiment.Comment: 16 pages, 5 figure