Robust globally divergence-free weak Galerkin finite element methods for natural convection problems

This paper proposes and analyzes a class of weak Galerkin (WG) finite element methods for stationary natural convection problems in two and three dimensions. We use piecewise polynomials of degrees k, k-1, and k(k>=1) for the velocity, pressure, and temperature approximations in the interior of elements, respectively, and piecewise polynomials of degrees l, k, l(l = k-1,k) for the numerical traces of velocity, pressure and temperature on the interfaces of elements. The methods yield globally divergence-free velocity solutions. Well-posedness of the discrete scheme is established, optimal a priori error estimates are derived, and an unconditionally convergent iteration algorithm is presented. Numerical experiments confirm the theoretical results and show the robustness of the methods with respect to Rayleigh number.Comment: 32 pages, 13 figure

Generalized robust shrinkage estimator and its application to STAP detection problem

Recently, in the context of covariance matrix estimation, in order to improve as well as to regularize the performance of the Tyler's estimator [1] also called the Fixed-Point Estimator (FPE) [2], a "shrinkage" fixed-point estimator has been introduced in [3]. First, this work extends the results of [3,4] by giving the general solution of the "shrinkage" fixed-point algorithm. Secondly, by analyzing this solution, called the generalized robust shrinkage estimator, we prove that this solution converges to a unique solution when the shrinkage parameter $\beta$ (losing factor) tends to 0. This solution is exactly the FPE with the trace of its inverse equal to the dimension of the problem. This general result allows one to give another interpretation of the FPE and more generally, on the Maximum Likelihood approach for covariance matrix estimation when constraints are added. Then, some simulations illustrate our theoretical results as well as the way to choose an optimal shrinkage factor. Finally, this work is applied to a Space-Time Adaptive Processing (STAP) detection problem on real STAP data

A New Approach to Forecasting Exchange Rates

Building on purchasing power parity theory, this paper proposes a new approach to forecasting exchange rates using the Big Mac data from The Economist magazine. Our approach is attractive in three aspects. Firstly, it uses easily-available Big Mac prices as input. These prices avoid several serious problems associated with broad price indexes, such as the CPI, that are used in conventional PPP studies. Secondly, this approach provides real-time exchange-rate forecasts at any forecast horizon. Such real-time forecasts can be made on a day-to-day basis if required, so that the forecasts are based on the most up-to-date information set. These high-frequency forecasts could be particularly appealing to decision makers who want up-to-date forecasts of exchange rates. Finally, as our forecasts are obtained through Monte Carlo simulation, estimation uncertainty is made explicit in our framework which provides the entire distribution of exchange rates, not just a single point estimate. A comparison of our forecasts with the random walk model shows that although the random walk is superior for very short horizons, our approach tends to dominate over the medium to longer term.Exchange-rate forecasting, Bic Mac prices, purchasing power parity, Monte Carlo simulation

How Long is the Long Run? Evidence from the Foreign Exchange Market

The aim of this paper is to estimate the length of the long run in the foreign exchange market. We do this by examining the link between exchange rates and relative prices, based on the implications of purchasing power parity (PPP) theory. Using a new approach, we test if the ratios of variances of exchange rates to prices are unity over all horizons, as implied by PPP. Through Monte Carlo simulations, we derive the variance ratios under the null of equal variances and examine the power and size of the test. We find evidence that PPP holds in the long run. While the long run based on the consumer prices appears to be “long”, about five years, the estimate of the long run based on the single good, Big Macs, is shorter (two years).