An adaptive GMsFEM for high-contrast flow problems

In this paper, we derive an a-posteriori error indicator for the Generalized Multiscale Finite Element Method (GMsFEM) framework. This error indicator is further used to develop an adaptive enrichment algorithm for the linear elliptic equation with multiscale high-contrast coefficients. The GMsFEM, which has recently been introduced in [12], allows solving multiscale parameter-dependent problems at a reduced computational cost by constructing a reduced-order representation of the solution on a coarse grid. The main idea of the method consists of (1) the construction of snapshot space, (2) the construction of the offline space, and (3) the construction of the online space (the latter for parameter-dependent problems). In [12], it was shown that the GMsFEM provides a flexible tool to solve multiscale problems with a complex input space by generating appropriate snapshot, offline, and online spaces. In this paper, we study an adaptive enrichment procedure and derive an a-posteriori error indicator which gives an estimate of the local error over coarse grid regions. We consider two kinds of error indicators where one is based on the $L^2$-norm of the local residual and the other is based on the weighted $H^{-1}$-norm of the local residual where the weight is related to the coefficient of the elliptic equation. We show that the use of weighted $H^{-1}$-norm residual gives a more robust error indicator which works well for cases with high contrast media. The convergence analysis of the method is given. In our analysis, we do not consider the error due to the fine-grid discretization of local problems and only study the errors due to the enrichment. Numerical results are presented that demonstrate the robustness of the proposed error indicators.Comment: 26 page

A simulator evaluation of a rate-enhanced instrument landing system display

A piloted simulation study was conducted to evaluate the effect on instrument landing system tracking performance of integrating localizer error rate information with the raw localizer error display. The resulting display was named the pseudo command tracking indicator (PCTI) because it provides an indication of any changes of heading required to track the localizer. Eight instrument-rated pilots each flew five instrument approaches with the PCTI and five instrument approaches with a conventional course deviation indicator. The results show good overall pilot acceptance of the PCTI and a significant reduction in localizer tracking error

An A Posteriori Error Indicator for Discontinuous Galerkin Approximations of Fourth Order Elliptic Problems

We introduce a residual-based a posteriori error indicator for discontinuous Galerkin discretizations of the biharmonic equation with essential boundary conditions. We show that the indicator is both reliable and efficient with respect to the approximation error measured in terms of a natural energy norm, under minimal regularity assumptions. We validate the performance of the indicator within an adaptive mesh refinement procedure and show its asymptotic exactness for a range of test problems

Forecasting Financial Extremes: A Network Degree Measure of Super-exponential Growth

Investors in stock market are usually greedy during bull markets and scared during bear markets. The greed or fear spreads across investors quickly. This is known as the herding effect, and often leads to a fast movement of stock prices. During such market regimes, stock prices change at a super-exponential rate and are normally followed by a trend reversal that corrects the previous over reaction. In this paper, we construct an indicator to measure the magnitude of the super-exponential growth of stock prices, by measuring the degree of the price network, generated from the price time series. Twelve major international stock indices have been investigated. Error diagram tests show that this new indicator has strong predictive power for financial extremes, both peaks and troughs. By varying the parameters used to construct the error diagram, we show the predictive power is very robust. The new indicator has a better performance than the LPPL pattern recognition indicator.Comment: 16 pages, 6 figure

A general aviation simulator evaluation of a rate-enhanced instrument landing system display

A piloted-simulation study was conducted to evaluate the effect on instrument landing system tracking performance of integrating localizer-error rate with raw localizer and glide-slope error. The display was named the pseudocommand tracking indicator (PCTI) because it provides an indication of the change of heading required to track the localizer center line. Eight instrument-rated pilots each flew five instrument approaches with the PCTI and five instrument approaches with a conventional course deviation indicator. The results show good overall pilot acceptance of the display, a significant improvement in localizer tracking error, and no significant changes in glide-slope tracking error or pilot workload

Residual based adaptivity and PWDG methods for the Helmholtz equation

We present a study of two residual a posteriori error indicators for the Plane Wave Discontinuous Galerkin (PWDG) method for the Helmholtz equation. In particular we study the h-version of PWDG in which the number of plane wave directions per element is kept fixed. First we use a slight modification of the appropriate a priori analysis to determine a residual indicator. Numerical tests show that this is reliable but pessimistic in that the ratio between the true error and the indicator increases as the mesh is refined. We therefore introduce a new analysis based on the observation that sufficiently many plane waves can approximate piecewise linear functions as the mesh is refined. Numerical results demonstrate an improvement in the efficiency of the indicators

Losing the error related negativity (ERN): an indicator for willed action

When people make errors in a discrimination task, a negative-going waveform can be observed in scalp-recorded EEG that has been coined the error-related negativity (ERN). We hypothesized that the ERN only occurs with slips, that is unwilled action errors, but not if an error is committed willingly and intentionally. We investigated the occurrence of the ERN in a choice reaction time task that has been shown to produce an ERN and in an error simulation task where subjects had to fake errors while the EEG was recorded. We observed a loss of the ERN when errors were committed in willed actions but not in unwilled actions thus supporting the idea that the production of the ERN is tied to slips in unwilled actions but not mistakes in willed actions

Treatment effect estimation with covariate measurement error

This paper investigates the effect that covariate measurement error has on a conventional treatment effect analysis built on an unconfoundedness restriction that embodies conditional independence restrictions in which there is conditioning on error free covariates. The approach uses small parameter asymptotic methods to obtain the approximate generic effects of measurement error. The approximations can be estimated using data on observed outcomes, the treatment indicator and error contaminated covariates providing an indication of the nature and size of measurement error effects. The approximations can be used in a sensitivity analysis to probe the potential effects of measurement error on the evaluation of treatment effects