A high frequency boundary element method for scattering by convex polygons with impedance boundary conditions

We consider scattering of a time harmonic incident plane wave by a convex polygon with piecewise constant impedance boundary conditions. Standard finite or boundary element methods require the number of degrees of freedom to grow at least linearly with respect to the frequency of the incident wave in order to maintain accuracy. Extending earlier work by Chandler-Wilde and Langdon for the sound soft problem, we propose a novel Galerkin boundary element method, with the approximation space consisting of the products of plane waves with piecewise polynomials supported on a graded mesh with smaller elements closer to the corners of the polygon. Theoretical analysis and numerical results suggest that the number of degrees of freedom required to achieve a prescribed level of accuracy grows only logarithmically with respect to the frequency of the incident wave.

High frequency scattering by convex curvilinear polygons

We consider the scattering of a time-harmonic acoustic incident plane wave by a sound soft convex curvilinear polygon with Lipschitz boundary. For standard boundary or finite element methods, with a piecewise polynomial approximation space, the number of degrees of freedom required to achieve a prescribed level of accuracy grows at least linearly with respect to the frequency of the incident wave. Here we propose a novel Galerkin boundary element method with a hybrid approximation space, consisting of the products of plane wave basis functions with piecewise polynomials supported on several overlapping meshes; a uniform mesh on illuminated sides, and graded meshes refined towards the corners of the polygon on illuminated and shadow sides. Numerical experiments suggest that the number of degrees of freedom required to achieve a prescribed level of accuracy need only grow logarithmically as the frequency of the incident wave increases.

Mechatronic design of a high frequency probe for haptic interaction

Current force feedback, haptic interface devices are generally limited to the display of low frequency, high amplitude spatial data. A typical device consists of a low impedance framework of one or more degrees-of-freedom (dof), allowing a user to explore a pre-defined workspace via an end effector such as a handle, thimble, probe or stylus. The movement of the device is then constrained using high gain positional feedback, thus reducing the apparent dof of the device and conveying the illusion of hard contact to the user. Such devices are, however, limited to a narrow bandwidth of frequencies, typically below 30Hz, and are not well suited to the display of surface properties, such as object texture. This paper details a device to augment an existing force feedback haptic display with a vibrotactile display, thus providing a means of conveying low amplitude, high frequency spatial information of object surface properties. 1. Haptics and Haptic Interfaces Haptics is the study of human touch and interaction with the external environment via touch. Information from the human sense of touch can be classified in to two categories, cutaneous and kinesthetic. Cutaneous information is provided via the mechanoreceptive nerve endings in the glabrous skin of the human hand. It is primarily a means of relaying information regarding small-scale details in the form of skin stretch, compression and vibration.

Detecting intraday periodicities with application to high frequency exchange rates

Many recent papers have documented periodicities in returns, return volatility, bid–ask spreads and trading volume, in both equity and foreign exchange markets. We propose and employ a new test for detecting subtle periodicities in time series data based on a signal coherence function. The technique is applied to a set of seven half-hourly exchange rate series. Overall, we find the signal coherence to be maximal at the 8-h and 12-h frequencies. Retaining only the most coherent frequencies for each series, we implement a trading rule that is based on these observed periodicities. Our results demonstrate in all cases except one that, in gross terms, the rules can generate returns that are considerably greater than those of a buy-and-hold strategy, although they cannot retain their profitability net of transactions costs. We conjecture that this methodology could constitute an important tool for financial market researchers which will enable them to detect, quantify and rank the various periodic components in financial data better.

High frequency variability of the Atlantic meridional overturning circulation

We compare the variability of the Atlantic meridional overturning circulation (AMOC) as simulated by the coupled climate models of the RAPID project, which cover a wide range of resolution and complexity, and observed by the RAPID/MOCHA array at about 26N. We analyse variability on a range of timescales, from five-daily to interannual. In models of all resolutions there is substantial variability on timescales of a few days; in most AOGCMs the amplitude of the variability is of somewhat larger magnitude than that observed by the RAPID array, while the time-mean is within about 10% of the observational estimate. The amplitude of the simulated annual cycle is similar to observations, but the shape of the annual cycle shows a spread among the models. A dynamical decomposition shows that in the models, as in observations, the AMOC is predominantly geostrophic (driven by pressure and sea-level gradients), with both geostrophic and Ekman contributions to variability, the latter being exaggerated and the former underrepresented in models. Other ageostrophic terms, neglected in the observational estimate, are small but not negligible. The time-mean of the western boundary current near the latitude of the RAPID/MOCHA array has a much wider model spread than the AMOC does, indicating large differences among models in the simulation of the wind-driven gyre circulation, and its variability is unrealistically small in the models. In many RAPID models and in models of the Coupled Model Intercomparison Project Phase 3 (CMIP3), interannual variability of the maximum of the AMOC wherever it lies, which is a commonly used model index, is similar to interannual variability in the AMOC at 26N. Annual volume and heat transport timeseries at the same latitude are well-correlated within 15--45N, indicating the climatic importance of the AMOC. In the RAPID and CMIP3 models, we show that the AMOC is correlated over considerable distances in latitude, but not the whole extent of the north Atlantic; consequently interannual variability of the AMOC at 50N, where it is particularly relevant to European climate, is not well-correlated with that of the AMOC at 26N, where it is monitored by the RAPID/MOCHA array.

Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering

In this article we describe recent progress on the design, analysis and implementation of hybrid numerical-asymptotic boundary integral methods for boundary value problems for the Helmholtz equation that model time harmonic acoustic wave scattering in domains exterior to impenetrable obstacles. These hybrid methods combine conventional piecewise polynomial approximations with high-frequency asymptotics to build basis functions suitable for representing the oscillatory solutions. They have the potential to solve scattering problems accurately in a computation time that is (almost) independent of frequency and this has been realized for many model problems. The design and analysis of this class of methods requires new results on the analysis and numerical analysis of highly oscillatory boundary integral operators and on the high-frequency asymptotics of scattering problems. The implementation requires the development of appropriate quadrature rules for highly oscillatory integrals. This article contains a historical account of the development of this currently very active field, a detailed account of recent progress and, in addition, a number of original research results on the design, analysis and implementation of these methods.

Does high-frequency repetitive transcranial magnetic stimulation produce residual and/or cumulative effects within an experimental session?

A common procedure for studying the effects on cognition of repetitive transcranial magnetic stimulation (rTMS) is to deliver rTMS concurrent with task performance, and to compare task performance on these trials versus on trials without rTMS. Recent evidence that TMS can have effects on neural activity that persist longer than the experimental session itself, however, raise questions about the assumption of the transient nature of rTMS that underlies many concurrent (or "online") rTMS designs. To our knowledge, there have been no studies in the cognitive domain examining whether the application of brief trains of rTMS during specific epochs of a complex task may have effects that spill over into subsequent task epochs, and perhaps into subsequent trials. We looked for possible immediate spill-over and longer-term cumulative effects of rTMS in data from two studies of visual short-term delayed recognition. In 54 subjects, 10-Hz rTMS trains were applied to five different brain regions during the 3-s delay period of a spatial task, and in a second group of 15 subjects, electroencephalography (EEG) was recorded while 10-Hz rTMS was applied to two brain areas during the 3-s delay period of both spatial and object tasks. No evidence for immediate effects was found in the comparison of the memory probe-evoked response on trials that were vs. were not preceded by delay-period rTMS. No evidence for cumulative effects was found in analyses of behavioral performance, and of EEG signal, as a function of task block. The implications of these findings, and their relation to the broader literature on acute vs. long-lasting effects of rTMS, are considered.

The signature of ENSO in the Northern Hemisphere midlatitude seasonal mean flow and high-frequency intraseasonal variability

The impact of pronounced positive and negative sea surface temperature (STT) anomalies in the tropical Pacific associated with the El Niño/Southern Oscillation (ENSO) phenomenon on the atmospheric circulation in the Northern Hemisphere extratropics during the boreal winter season is investigated. This includes both the impact on the seasonal mean flow and on the intraseasonal variability on synoptic time scales. Moreover, the interaction between the transient fluctuations on these times scales and the mean circulation is examined. Both data from an ensemble of five simulations with the ECHAM3 atmospheric general circulation model at a horizontal resolution of T42 each covering the period from 1979 through 1992 and operational analyses from ECMWF for the corresponding period are examined. In each of the simulations observed SSTs for the period of investigation are given as lower boundary forcing, but different atmospheric initial conditions are prescribed. The simulations with ECHAM3 reveal a distinct impact of the pronounced SST-anomalies in the tropical Pacific on the atmospheric circulation in the Northern Hemisphere extratropics during El Niño as well as during La Niña events. These changes in the atmospheric circulation, which are found to be highly significant in the Pacific/North American as well as in the Atlantic/European region, are consistent with the essential results obtained from the analyses. The pronounced SST-anomalies in the tropical Pacific lead to changes in the mean circulation, which are characterized by typical circulation patterns. These changes in the mean circulation are accompanied by marked variations of the activity of the transient fluctuations on synoptic time scales, that are changes in both the kinetic energy on these time scales and the atmospheric transports of momentum and heat accomplished by the short baroclinic waves. The synoptic disturbances, on the other hand, play also an important role in controlling the changes in the mean circulation associated with the ENSO phenomenon. They maintain these typical circulation patterns via barotropic, but counteract them via baroclinic processes. The hypothesis of an impact of the ENSO phenomenon in the Atlantic/European region can be supported. As the determining factor the intensification (reduction) of the Aleutian low and the simultaneous reduction (intensification) of the Icelandic low during El Niño and during La Niña events respectively, is identified. The changes in the intensity of the Aleutian low during the ENSO-events are accompanied by an alteration of the transport of momentum caused by the short baroclinic waves over the North American continent in such a way that the changes in the intensity of the Icelandic low during El Niño as well as during La Niña events are maintained.

A high frequency $hp$ boundary element method for scattering by convex polygons

In this paper we propose and analyze a hybrid $hp$ boundary element method for the solution of problems of high frequency acoustic scattering by sound-soft convex polygons, in which the approximation space is enriched with oscillatory basis functions which efficiently capture the high frequency asymptotics of the solution. We demonstrate, both theoretically and via numerical examples, exponential convergence with respect to the order of the polynomials, moreover providing rigorous error estimates for our approximations to the solution and to the far field pattern, in which the dependence on the frequency of all constants is explicit. Importantly, these estimates prove that, to achieve any desired accuracy in the computation of these quantities, it is sufficient to increase the number of degrees of freedom in proportion to the logarithm of the frequency as the frequency increases, in contrast to the at least linear growth required by conventional methods.

Evaluating interval forecasts of high-frequency financial data

A number of methods of evaluating the validity of interval forecasts of financial data are analysed, and illustrated using intraday FTSE100 index futures returns. Some existing interval forecast evaluation techniques, such as the Markov chain approach of Christoffersen (1998), are shown to be inappropriate in the presence of periodic heteroscedasticity. Instead, we consider a regression-based test, and a modified version of Christoffersen's Markov chain test for independence, and analyse their properties when the financial time series exhibit periodic volatility. These approaches lead to different conclusions when interval forecasts of FTSE100 index futures returns generated by various GARCH(1,1) and periodic GARCH(1,1) models are evaluated.

The water quality of the River Enborne, UK: observations from high-frequency Monitoring in a rural, lowland river system

This paper reports the results of a 2-year study of water quality in the River Enborne, a rural river in lowland England. Concentrations of nitrogen and phosphorus species and other chemical determinands were monitored both at high-frequency (hourly), using automated in situ instrumentation, and by manual weekly sampling and laboratory analysis. The catchment land use is largely agricultural, with a population density of 123 persons km−2. The river water is largely derived from calcareous groundwater, and there are high nitrogen and phosphorus concentrations. Agricultural fertiliser is the dominant source of annual loads of both nitrogen and phosphorus. However, the data show that sewage effluent discharges have a disproportionate effect on the river nitrogen and phosphorus dynamics. At least 38% of the catchment population use septic tank systems, but the effects are hard to quantify as only 6% are officially registered, and the characteristics of the others are unknown. Only 4% of the phosphorus input and 9% of the nitrogen input is exported from the catchment by the river, highlighting the importance of catchment process understanding in predicting nutrient concentrations. High-frequency monitoring will be a key to developing this vital process understanding.

Linear and non-linear (non-)forecastability of high-frequency exchange rates

This paper forecasts Daily Sterling exchange rate returns using various naive, linear and non-linear univariate time-series models. The accuracy of the forecasts is evaluated using mean squared error and sign prediction criteria. These show only a very modest improvement over forecasts generated by a random walk model. The Pesaran–Timmerman test and a comparison with forecasts generated artificially shows that even the best models have no evidence of market timing ability.

A high frequency boundary element method for scattering by a class of nonconvex obstacles

In this paper we propose and analyse a hybrid numerical-asymptotic boundary element method for the solution of problems of high frequency acoustic scattering by a class of sound-soft nonconvex polygons. The approximation space is enriched with carefully chosen oscillatory basis functions; these are selected via a study of the high frequency asymptotic behaviour of the solution. We demonstrate via a rigorous error analysis, supported by numerical examples, that to achieve any desired accuracy it is sufficient for the number of degrees of freedom to grow only in proportion to the logarithm of the frequency as the frequency increases, in contrast to the at least linear growth required by conventional methods. This appears to be the first such numerical analysis result for any problem of scattering by a nonconvex obstacle. Our analysis is based on new frequency-explicit bounds on the normal derivative of the solution on the boundary and on its analytic continuation into the complex plane.

Characterising phosphorus and nitrate inputs to a rural river using high-frequency concentration-flow relationships

The total reactive phosphorus (TRP) and nitrate concentrations of the River Enborne, southern England, were monitored at hourly interval between January 2010 and December 2011. The relationships between these high-frequency nutrient concentration signals and flow were used to infer changes in nutrient source and dynamics through the annual cycle and each individual storm event, by studying hysteresis patterns. TRP concentrations exhibited strong dilution patterns with increasing flow, and predominantly clockwise hysteresis through storm events. Despite the Enborne catchment being relatively rural for southern England, TRP inputs were dominated by constant, non-rain-related inputs from sewage treatment works (STW) for the majority of the year, producing the highest phosphorus concentrations through the spring–summer growing season. At higher river flows, the majority of the TRP load was derived from within-channel remobilisation of phosphorus from the bed sediment, much of which was also derived from STW inputs. Therefore, future phosphorus mitigation measures should focus on STW improvements. Agricultural diffuse TRP inputs were only evident during storms in the May of each year, probably relating to manure application to land. The nitrate concentration–flow relationship produced a series of dilution curves, indicating major inputs from groundwater and to a lesser extent STW. Significant diffuse agricultural inputs with anticlockwise hysteresis trajectories were observed during the first major storms of the winter period. The simultaneous investigation of high-frequency time series data, concentration–flow relationships and hysteresis behaviour through multiple storms for both phosphorus and nitrate offers a simple and innovative approach for providing new insights into nutrient sources and dynamics.

Acoustic scattering : high frequency boundary element methods and unified transform methods

We describe some recent advances in the numerical solution of acoustic scattering problems. A major focus of the paper is the efficient solution of high frequency scattering problems via hybrid numerical-asymptotic boundary element methods. We also make connections to the unified transform method due to A. S. Fokas and co-authors, analysing particular instances of this method, proposed by J. A. De-Santo and co-authors, for problems of acoustic scattering by diffraction gratings.