Turning the nuclear energy density functional method into a proper effective field theory: reflections

Nuclear energy density functionals (EDFs) have a long history of success in reproducing properties of nuclei across the table of the nuclides. They capture quantitatively the emergent features of bound nuclei, such as nuclear saturation and pairing, yet greater accuracy and improved uncertainty quantification are actively sought. Implementations of phenomenological EDFs are suggestive of effective field theory (EFT) formulations and there are hints of an underlying power counting. Multiple paths are possible in trying to turn the nuclear EDF method into a proper EFT. I comment on the current situation and speculate on how to proceed using an effective action formulation.Comment: 14 pages, 2 figures. Contribution to the EPJA topical issue: "The tower of effective (field) theories and the emergence of nuclear phenomena". v2: Corrections and added references based on referee report

Quaternion Matrices : Statistical Properties and Applications to Signal Processing and Wavelets

Similarly to how complex numbers provide a possible framework for extending scalar signal processing techniques to 2-channel signals, the 4-dimensional hypercomplex algebra of quaternions can be used to represent signals with 3 or 4 components. For a quaternion random vector to be suited for quaternion linear processing, it must be (second-order) proper. We consider the likelihood ratio test (LRT) for propriety, and compute the exact distribution for statistics of Box type, which include this LRT. Various approximate distributions are compared. The Wishart distribution of a quaternion sample covariance matrix is derived from first principles. Quaternions are isomorphic to an algebra of structured 4x4 real matrices. This mapping is our main tool, and suggests considering more general real matrix problems as a way of investigating quaternion linear algorithms. A quaternion vector autoregressive (VAR) time-series model is equivalent to a structured real VAR model. We show that generalised least squares (and Gaussian maximum likelihood) estimation of the parameters reduces to ordinary least squares, but only if the innovations are proper. A LRT is suggested to simultaneously test for quaternion structure in the regression coefficients and innovation covariance. Matrix-valued wavelets (MVWs) are generalised (multi)wavelets for vector-valued signals. Quaternion wavelets are equivalent to structured MVWs. Taking into account orthogonal similarity, all MVWs can be constructed from non-trivial MVWs. We show that there are no non-scalar non-trivial MVWs with short support [0,3]. Through symbolic computation we construct the families of shortest non-trivial 2x2 Daubechies MVWs and quaternion Daubechies wavelets.Open Acces

The large deviation approach to statistical mechanics

The theory of large deviations is concerned with the exponential decay of probabilities of large fluctuations in random systems. These probabilities are important in many fields of study, including statistics, finance, and engineering, as they often yield valuable information about the large fluctuations of a random system around its most probable state or trajectory. In the context of equilibrium statistical mechanics, the theory of large deviations provides exponential-order estimates of probabilities that refine and generalize Einstein's theory of fluctuations. This review explores this and other connections between large deviation theory and statistical mechanics, in an effort to show that the mathematical language of statistical mechanics is the language of large deviation theory. The first part of the review presents the basics of large deviation theory, and works out many of its classical applications related to sums of random variables and Markov processes. The second part goes through many problems and results of statistical mechanics, and shows how these can be formulated and derived within the context of large deviation theory. The problems and results treated cover a wide range of physical systems, including equilibrium many-particle systems, noise-perturbed dynamics, nonequilibrium systems, as well as multifractals, disordered systems, and chaotic systems. This review also covers many fundamental aspects of statistical mechanics, such as the derivation of variational principles characterizing equilibrium and nonequilibrium states, the breaking of the Legendre transform for nonconcave entropies, and the characterization of nonequilibrium fluctuations through fluctuation relations.Comment: v1: 89 pages, 18 figures, pdflatex. v2: 95 pages, 20 figures, text, figures and appendices added, many references cut, close to published versio

A Posteriori Bounds for Linear Functional Outputs of Hyperbolic Partial Differential Equations

One of the major difficulties faced in the numerical resolution of the equations of physics is to decide on the right balance between computational cost and solutions accuracy, and to determine how solutions errors affect some given “outputs of interest.” This thesis presents a technique to generate upper and lower bounds for outputs of hyperbolic partial differential equations. The outputs of interest considered are linear functionals of the solutions of the equations. The method is based on the construction of an “augmented” Lagrangian, which includes a formulation of the output as a quadratic form to be minimized and the equilibrium equations as a constraint. The corresponding Lagrange multiplier, or adjoint , is determined by solving a problem involving the adjoint of the operator in the original equations. The bounds are then derived from the underlying unconstrained max-min problem. A predictor is also evaluated as the average value of the bounds. Because the resolution of the max-min problem implies the resolution of the original discrete equations, the adjoint on a fine grid is approximated by a hierarchical procedure that consists of the resolution of the problem on a coarser grid followed by an interpolation on the fine grid. The bounds derived from this approximation are then optimized by the choice of natural boundary conditions for the adjoint and by selecting the value of a stabilization parameter. The Hierarchical Bounds Method is illustrated on three cases. The first one is the convection-diffusion equation, where the bounds obtained are very sharp. The second one is a purely convective problem, discretized using a Taylor-Galerkin approach. The third case is based on the Euler equations for a nozzle flow, which can be reduced to a single nonlinear scalar continuous equation. The resulting discrete nonlinear system of equations is obtained by a Taylor-Galerkin method and is solved by the Newton-Raphson method. The problem is then linearized about the computed solution to obtain a linear system similar to the previous cases and produce the bounds. In a last chapter, the Domain Decomposition is introduced. The domain is decomposed into K subdomains and the problem is solved separately on each of them before continuity at the boundaries is imposed, allowing the computation of the bounds to be parallelized. Because the cost of sparse matrix inversion is of order O(N[third]), Domain Decomposition becomes very useful for two-dimensional problems,where the overall cost is divided by K[squared]

A posteriori bounds for linear functional outputs of hyperbolic partial differential equations

Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 1997.Includes bibliographical references (leaves 88-89).by Hubert J.B. Vailong.M.S

Non-Linear diffusion processes and applications

Diffusion models are useful tools for quantifying the dynamics of continuously evolving processes. Using diffusion models it is possible to formulate compact descriptions for the dynamics of real-world processes in terms of stochastic differential equations. Despite the exibility of these models, they can often be extremely difficult to work with. This is especially true for non-linear and/or time-inhomogeneous diffusion models where even basic statistical properties of the process can be elusive. As such, we explore various techniques for analysing non-linear diffusion models in contexts ranging from conducting inference under discrete observation and solving first passage time problems, to the analysis of jump diffusion processes and highly non-linear diffusion processes. We apply the methodology to a number of real-world ecological and financial problems of interest and demonstrate how non-linear diffusion models can be used to better understand such phenomena. In conjunction with the methodology, we develop a series of software packages that can be used to accurately and efficiently analyse various classes of non-linear diffusion models