An Extragradient-Based Alternating Direction Method for Convex Minimization

In this paper, we consider the problem of minimizing the sum of two convex functions subject to linear linking constraints. The classical alternating direction type methods usually assume that the two convex functions have relatively easy proximal mappings. However, many problems arising from statistics, image processing and other fields have the structure that while one of the two functions has easy proximal mapping, the other function is smoothly convex but does not have an easy proximal mapping. Therefore, the classical alternating direction methods cannot be applied. To deal with the difficulty, we propose in this paper an alternating direction method based on extragradients. Under the assumption that the smooth function has a Lipschitz continuous gradient, we prove that the proposed method returns an $\epsilon$-optimal solution within $O(1/\epsilon)$ iterations. We apply the proposed method to solve a new statistical model called fused logistic regression. Our numerical experiments show that the proposed method performs very well when solving the test problems. We also test the performance of the proposed method through solving the lasso problem arising from statistics and compare the result with several existing efficient solvers for this problem; the results are very encouraging indeed

Estimating integrated water vapor trends from VLBI, GPS,and numerical weather models: sensitivity totropospheric parameterization

©2018. American Geophysical UnionIn this study, we estimate integrated water vapor (IWV) trends from very long baseline interferometry (VLBI) and global navigation satellite systems (GNSS) data analysis, as well as from numerical weather models (NWMs). We study the impact of modeling and parameterization of the tropospheric delay from VLBI on IWV trends. We address the impact of the meteorological data source utilized to model the hydrostatic delay and the thermal deformation of antennas, as well as the mapping functions employed to project zenith delays to arbitrary directions. To do so, we derive a new mapping function, called Potsdam mapping functions based on NWM data and a new empirical model, GFZ‐PT. GFZ‐PT differs from previous realizations as it describes diurnal and subdiurnal in addition to long‐wavelength variations, it provides harmonic functions of ray tracing‐derived gradients, and it features robustly estimated rates. We find that alternating the mapping functions in VLBI data analysis yields no statistically significant differences in the IWV rates, whereas alternating the meteorological data source distorts the trends significantly. Moreover, we explore methods to extract IWV given a NWM. The rigorously estimated IWV rates from the different VLBI setups, GNSS, and ERA‐Interim are intercompared, and a good agreement is found. We find a quite good agreement comparing ERA‐Interim to VLBI and GNSS, separately, at the level of 75%.DFG, 255986470, GGOS-SIM-2: Simulation des "Global Geodetic Observing System

Evolution of Liouville density of a chaotic system

An area-preserving map of the unit sphere, consisting of alternating twists and turns, is mostly chaotic. A Liouville density on that sphere is specified by means of its expansion into spherical harmonics. That expansion initially necessitates only a finite number of basis functions. As the dynamical mapping proceeds, it is found that the number of non-negligible coefficients increases exponentially with the number of steps. This is to be contrasted with the behavior of a Schr\"odinger wave function which requires, for the analogous quantum system, a basis of fixed size.Comment: LaTeX 4 pages (27 kB) followed by four short PostScript files (2 kB + 2 kB + 1 kB + 4 kB

Algebraic properties of generalized Rijndael-like ciphers

We provide conditions under which the set of Rijndael functions considered as permutations of the state space and based on operations of the finite field \GF (p^k) ($p\geq 2$ a prime number) is not closed under functional composition. These conditions justify using a sequential multiple encryption to strengthen the AES (Rijndael block cipher with specific block sizes) in case AES became practically insecure. In Sparr and Wernsdorf (2008), R. Sparr and R. Wernsdorf provided conditions under which the group generated by the Rijndael-like round functions based on operations of the finite field \GF (2^k) is equal to the alternating group on the state space. In this paper we provide conditions under which the group generated by the Rijndael-like round functions based on operations of the finite field \GF (p^k) ($p\geq 2$) is equal to the symmetric group or the alternating group on the state space.Comment: 22 pages; Prelim0

Nonlinear influence in the frequency domain: alternating series

The nonlinear influence on system output spectrum is studied for a class of nonlinear systems which have Volterra series expansion. It is shown that system output spectrum can be expressed into an alternating series with respect to some model nonlinear parameters under certain conditions. This alternating series has some interesting properties by which system output spectrum can be suppressed easily. The sufficient (and necessary) conditions in which the output spectrum can be transformed into an alternating series are studied. These results reveal a novel characteristic of the nonlinear influence on a system in the frequency domain, and provide a novel insight into the analysis and design of a class of nonlinear systems. Examples are given to illustrate the results