On the Spectral Properties of Matrices Associated with Trend Filters

This paper is concerned with the spectral properties of matrices associated with linear filters for the estimation of the underlying trend of a time series. The interest lies in the fact that the eigenvectors can be interpreted as the latent components of any time series that the filter smooths through the corresponding eigenvalues. A difficulty arises because matrices associated with trend filters are finite approximations of Toeplitz operators and therefore very little is known about their eigenstructure, which also depends on the boundary conditions or, equivalently, on the filters for trend estimation at the end of the sample. Assuming reflecting boundary conditions, we derive a time series decomposition in terms of periodic latent components and corresponding smoothing eigenvalues. This decomposition depends on the local polynomial regression estimator chosen for the interior. Otherwise, the eigenvalue distribution is derived with an approximation measured by the size of the perturbation that different boundary conditions apport to the eigenvalues of matrices belonging to algebras with known spectral properties, such as the Circulant or the Cosine. The analytical form of the eigenvectors is then derived with an approximation that involves the extremes only. A further topic investigated in the paper concerns a strategy for a filter design in the time domain. Based on cut-off eigenvalues, new estimators are derived, that are less variable and almost equally biased as the original estimator, based on all the eigenvalues. Empirical examples illustrate the effectiveness of the method

On the evolution of scattering data under perturbations of the Toda lattice

We present the results of an analytical and numerical study of the long-time behavior for certain Fermi-Pasta-Ulam (FPU) lattices viewed as perturbations of the completely integrable Toda lattice. Our main tools are the direct and inverse scattering transforms for doubly-infinite Jacobi matrices, which are well-known to linearize the Toda flow. We focus in particular on the evolution of the associated scattering data under the perturbed vs. the unperturbed equations. We find that the eigenvalues present initially in the scattering data converge to new, slightly perturbed eigenvalues under the perturbed dynamics of the lattice equation. To these eigenvalues correspond solitary waves that emerge from the solitons in the initial data. We also find that new eigenvalues emerge from the continuous spectrum as the lattice system is let to evolve under the perturbed dynamics.Comment: 27 pages, 17 figures. Revised Introduction and Discussion section

Development of a high-order parallel solver for direct and large eddy simulations of turbulent flows

Turbulence is inherent in fluid dynamics, in that laminar flows are rather the exception than the rule, hence the longstanding interest in the subject, both within the academic community and the industrial R&D laboratories. Since 1883, much progress has been made, and statistics applied to turbulence have provided understanding of the scaling laws which are peculiar to several model flows, whereas experiments have given insight on the structure of real-world flows, but, soon enough, numerical approaches to the matter have become the most promising ones, since they lay the ground for the solution of high Reynolds number unsteady Navier-Stokes equations by means of computer systems. Nevertheless, despite the exponential rise in computational capability over the last few decades, the more computer technology advances, the higher the Reynolds number sought for test-cases of industrial interest: there is a natural tendency to perform simulations as large as possible, a habit that leaves no room for wasting resources. Indeed, as the scale separation grows with Re, the reduction of wall clock times for a high-fidelity solution of desired accuracy becomes increasingly important. To achieve this task, a CFD solver should rely on the use of appropriate physical models, consistent numerical methods to discretize the equations, accurate non-dissipative numerical schemes, efficient algorithms to solve the numerics, and fast routines implementing those algorithms. Two archetypal approaches to CFD are direct and large-eddy simulation (DNS and LES respectively), which profoundly differ in several aspects but are both “eddy-resolving” methods, meant to resolve the structures of the flow-field with the highest possible accuracy and putting in as little spurious dissipation as possible. These two requirements of accurate resolution of scales, and energy conservation, should be addressed by any numerical method, since they are essential to many real-world fluid flows of industrial interest. As a consequence, high order numerical schemes, and compact schemes among them, have received much consideration, since they address both goals, at the cost of a lower ease of application of the boundary condition, and a higher computational cost. The latter problem is tackled with parallel computing, which also allows to take advantage of the currently available computer power at the best possible extent. The research activity conducted by the present author has concerned the development, from scratch, of a three-dimensional, unsteady, incompressible Navier-Stokes parallel solver, which uses an advanced algorithm for the process-wise solution of the linear systems arising from the application of high order compact finite difference schemes, and hinges upon a three-dimensional decomposition of the cartesian computational space. The code is written in modern Fortran 2003 — plus a few features which are unique to the 2008 standard — and is parallelized through the use of MPI 3.1 standard’s advanced routines, as implemented by the OpenMPI library project. The coding was carried out with the objective of creating an original CFD high-order parallel solver which is maintainable and extendable, of course within a well-defined range of possibilities. With this main priority being outlined, particular attention was paid to several key concepts: modularity and readability of the source code and, in turn, its reusability; ease of implementation of virtually any new explicit or implicit finite difference scheme; modern programming style and avoidance of deprecated old legacy Fortran constructs and features, so that the world wide web is a reliable and active means to the quick solution of coding problems arising from the implementation of new modules in the code; last but not least, thorough comments, especially in critical sections of the code, explaining motives and possible expected weak links. Design, production, and documentation of a program from scratch is almost never complete. This is certainly true for the present effort. The method and the code are verified against the full three-dimensional Lid-Driven Cavity and Taylor-Green Vortex flows. The latter test is used also for the assessment of scalability and parallel efficiency

Interest rate models with Markov chains

Imperial Users onl

Band-diagonal operators on Banach lattices: matrix dynamics and invariant subspaces

We address the existence of non-trivial closed invariant ideals for positive operators defined on Banach lattices whose order is induced by an unconditional basis. In particular, for band-diagonal positive operators such existence is characterized whenever their matrix representations meet a positiveness criteria. For more general classes of positive operators, sufficient conditions are derived proving, particularly, the sharpness of such results from the standpoint of view of the matrix representations. The whole approach is based on studying the behavior of the dynamics of infinite matrices and the localization of the non-zero entries. Finally, we generalize a theorem of Grivaux regarding the existence of non-trivial closed invariant subspaces for positive tridiagonal operators to a more general class of band-diagonal operators showing, in particular, that a large subclass of them have non-trivial closed invariant subspaces but lack non-trivial closed invariant ideals.Depto. de Análisis Matemático y Matemática AplicadaFac. de Ciencias MatemáticasTRUEMinisterio de Ciencia e Innovacióninpres

Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page