Error analysis of a Monte Carlo algorithm for computing bilinear forms of matrix powers

In this paper we present error analysis for a Monte Carlo algorithm for evaluating bilinear forms of matrix powers. An almost Optimal Monte Carlo (MAO) algorithm for solving this problem is formulated. Results for the structure of the probability error are presented and the construction of robust and interpolation Monte Carlo algorithms are discussed. Results are presented comparing the performance of the Monte Carlo algorithm with that of a corresponding deterministic algorithm. The two algorithms are tested on a well balanced matrix and then the effects of perturbing this matrix, by small and large amounts, is studied.

Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations

In this paper, we extend to the time-harmonic Maxwell equations the p-version analysis technique developed in [R. Hiptmair, A. Moiola and I. Perugia, Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version, SIAM J. Numer. Anal., 49 (2011), 264-284] for Trefftz-discontinuous Galerkin approximations of the Helmholtz problem. While error estimates in a mesh-skeleton norm are derived parallel to the Helmholtz case, the derivation of estimates in a mesh-independent norm requires new twists in the duality argument. The particular case where the local Trefftz approximation spaces are built of vector-valued plane wave functions is considered, and convergence rates are derived.

On aposteriori error analysis of dG schemes approximating hyperbolic conservation laws

This contribution is concerned with aposteriori error analysis of discontinuous Galerkin (dG) schemes approximating hyperbolic conservation laws. In the scalar case the aposteriori analysis is based on the L1 contraction property and the doubling of variables technique. In the system case the appropriate stability framework is in L2, based on relative entropies. It is only applicable if one of the solutions, which are compared to each other, is Lipschitz. For dG schemes approximating hyperbolic conservation laws neither the entropy solution nor the numerical solution need to be Lipschitz. We explain how this obstacle can be overcome using a reconstruction approach which leads to an aposteriori error estimate.

A priori error analysis of space–time Trefftz discontinuous Galerkin methods for wave problems

We present and analyse a space–time discontinuous Galerkin method for wave propagation problems. The special feature of the scheme is that it is a Trefftz method, namely that trial and test functions are solution of the partial differential equation to be discretised in each element of the (space–time) mesh. The method considered is a modification of the discontinuous Galerkin schemes of Kretzschmar et al. (2014) and of Monk & Richter (2005). For Maxwell’s equations in one space dimension, we prove stability of the method, quasi-optimality, best approximation estimates for polynomial Trefftz spaces and (fully explicit) error bounds with high order in the meshwidth and in the polynomial degree. The analysis framework also applies to scalar wave problems and Maxwell’s equations in higher space dimensions. Some numerical experiments demonstrate the theoretical results proved and the faster convergence compared to the non-Trefftz version of the scheme.

Efficient evaluation of highly oscillatory acoustic scattering surface integrals

We consider the approximation of some highly oscillatory weakly singular surface integrals, arising from boundary integral methods with smooth global basis functions for solving problems of high frequency acoustic scattering by three-dimensional convex obstacles, described globally in spherical coordinates. As the frequency of the incident wave increases, the performance of standard quadrature schemes deteriorates. Naive application of asymptotic schemes also fails due to the weak singularity. We propose here a new scheme based on a combination of an asymptotic approach and exact treatment of singularities in an appropriate coordinate system. For the case of a spherical scatterer we demonstrate via error analysis and numerical results that, provided the observation point is sufficiently far from the shadow boundary, a high level of accuracy can be achieved with a minimal computational cost.

Production of relative clauses in monolingual Turkish children

Research on the production of relative clauses (RCs) has shown that in English, although children start using intransitive RCs at an earlier age, more complex, bi-propositional object RCs appear later (Hamburger & Crain, 1982; Diessel and Tomasello, 2005), and children use resumptive pronouns both in acceptable and unacceptable ways (McKee, McDaniel, & Snedeker, 1998; McKee & McDaniel, 2001). To date, it is unclear whether or not the same picture emerges in Turkish, a language with an SOV word-order and overt case marking. Some studies suggested that subject RCs are more frequent in adults and children (Slobin, 1986) and yield a better performance than object RCs (Özcan, 1996), but others reported the opposite pattern (Ekmekçi, 1990). Our study addresses this issue in Turkish children and adults, and uses participants’ errors to account for the emerging asymmetry between subject and object RCs. 37 5-to-8 year old monolingual Turkish children and 23 adult controls participated in a novel elicitation task involving cards, each consisting of four different pictures (see Figure 1). There were two sets of cards, one for the participant and one for the researcher. The former had animals with accessories (e.g., a hat) whereas the latter had no accessories. Participants were instructed to hold their card without showing it to the researcher and describe the animals with particular accessories. This prompted the use of subject and object RCs. The researcher had to identify the animals in her card (see Figure 2). A preliminary repeated measures ANOVA with the factor Group (pre-school, primary-school children) showed no differences between the groups in the use of RCs (p>.1), who were therefore collapsed into one for further analyses. A repeated measures ANOVA with the factors Group (children, adults) and RC-Type (Subject, Object) showed that children used fewer RCs than adults (F(1,58)=7.54, p<.01), and both groups used fewer object than subject RCs (F(1,58)=22.46, p<.001), but there was no Group by RC-Type interaction (see Figure 3). A similar ANOVA on the rate of grammatical RCs showed a main effect of Group (F(1,58)=77.25, p<.001), a main effect of RC-Type (F(1,58)=66.33, p<.001), and an interaction of Group by RC-Type (F(1,58)=64.6, p<.001) (see Figure 4). Children made more errors than adults in object RCs (F(1,58)=87.01, p<.001), and children made more errors in object compared to subject RCs (F(1,36)=106.35, p<.001), but adults did not show this asymmetry. The error analysis revealed that children systematically avoided the object-relativizing morpheme –DIK, which requires possessive agreement with the genitive-marked subject. They also used resumptive pronouns and resumptive full-DPs in the extraction site similarly to English children (see Figure 5). These findings are in line with Slobin (1986) and Özcan (1996). Children’s errors suggest that they avoid morphosyntactic complexity of object RCs and try to preserve the canonical word order by inserting resumptive pronouns in the extraction site. Finally, cross-linguistic similarity in the acquisition of RCs in typologically different languages suggests a higher accessibility of subject RCs both at the structural (Keenan and Comrie, 1977) and conceptual level (Bock and Warren, 1986).

Millimeter-wave measurements of the complex dielectric constant of an advanced thick film UV photoresist

The first measurement of the relative permittivity (εr) and loss tangent (tan δ) of EPON™ SU-8 advanced thick film ultraviolet photoresist is reported at frequencies between 75–110 GHz (W-band). The problems associated with such a measurement are discussed, an error analysis given, and values of εr=1.725±0.08 and tanδ =0.02±0.001 are determined.

Trefftz discontinuous Galerkin methods for acoustic scattering on locally refined meshes

We extend the a priori error analysis of Trefftz-discontinuous Galerkin methods for time-harmonic wave propagation problems developed in previous papers to acoustic scattering problems and locally refined meshes. To this aim, we prove refined regularity and stability results with explicit dependence of the stability constant on the wave number for non convex domains with non connected boundaries. Moreover, we devise a new choice of numerical flux parameters for which we can prove L2-error estimates in the case of locally refined meshes near the scatterer. This is the setting needed to develop a complete hp-convergence analysis.

A twenty-year independent record of sea surface temperature for climate from Along-track Scanning Radiometers

A new record of sea surface temperature (SST) for climate applications is described. This record provides independent corroboration of global variations estimated from SST measurements made in situ. Infrared imagery from Along-Track Scanning Radiometers (ATSRs) is used to create a 20 year time series of SST at 0.1° latitude-longitude resolution, in the ATSR Reprocessing for Climate (ARC) project. A very high degree of independence of in situ measurements is achieved via physics-based techniques. Skin SST and SST estimated for 20 cm depth are provided, with grid cell uncertainty estimates. Comparison with in situ data sets establishes that ARC SSTs generally have bias of order 0.1 K or smaller. The precision of the ARC SSTs is 0.14 K during 2003 to 2009, from three-way error analysis. Over the period 1994 to 2010, ARC SSTs are stable, with better than 95% confidence, to within 0.005 K yr−1(demonstrated for tropical regions). The data set appears useful for cleanly quantifying interannual variability in SST and major SST anomalies. The ARC SST global anomaly time series is compared to the in situ-based Hadley Centre SST data set version 3 (HadSST3). Within known uncertainties in bias adjustments applied to in situ measurements, the independent ARC record and HadSST3 present the same variations in global marine temperature since 1996. Since the in situ observing system evolved significantly in its mix of measurement platforms and techniques over this period, ARC SSTs provide an important corroboration that HadSST3 accurately represents recent variability and change in this essential climate variable.

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.

A frequency-independent boundary element method for scattering by two-dimensional screens and apertures

We propose and analyse a hybrid numerical–asymptotic hp boundary element method (BEM) for time-harmonic scattering of an incident plane wave by an arbitrary collinear array of sound-soft two-dimensional screens. Our method uses an approximation space enriched with oscillatory basis functions, chosen to capture the high-frequency asymptotics of the solution. We provide a rigorous frequency-explicit error analysis which proves that the method converges exponentially as the number of degrees of freedom N increases, and that to achieve any desired accuracy it is sufficient to increase N in proportion to the square of the logarithm of the frequency as the frequency increases (standard BEMs require N to increase at least linearly with frequency to retain accuracy). Our numerical results suggest that fixed accuracy can in fact be achieved at arbitrarily high frequencies with a frequency-independent computational cost, when the oscillatory integrals required for implementation are computed using Filon quadrature. We also show how our method can be applied to the complementary ‘breakwater’ problem of propagation through an aperture in an infinite sound-hard screen.

Spatial analysis of the error in a model of soil nitrogen

Models of the dynamics of nitrogen in soil (soil-N) can be used to aid the fertilizer management of a crop. The predictions of soil-N models can be validated by comparison with observed data. Validation generally involves calculating non-spatial statistics of the observations and predictions, such as their means, their mean squared-difference, and their correlation. However, when the model predictions are spatially distributed across a landscape the model requires validation with spatial statistics. There are three reasons for this: (i) the model may be more or less successful at reproducing the variance of the observations at different spatial scales; (ii) the correlation of the predictions with the observations may be different at different spatial scales; (iii) the spatial pattern of model error may be informative. In this study we used a model, parameterized with spatially variable input information about the soil, to predict the mineral-N content of soil in an arable field, and compared the results with observed data. We validated the performance of the N model spatially with a linear mixed model of the observations and model predictions, estimated by residual maximum likelihood. This novel approach allowed us to describe the joint variation of the observations and predictions as: (i) independent random variation that occurred at a fine spatial scale; (ii) correlated random variation that occurred at a coarse spatial scale; (iii) systematic variation associated with a spatial trend. The linear mixed model revealed that, in general, the performance of the N model changed depending on the spatial scale of interest. At the scales associated with random variation, the N model underestimated the variance of the observations, and the predictions were correlated poorly with the observations. At the scale of the trend, the predictions and observations shared a common surface. The spatial pattern of the error of the N model suggested that the observations were affected by the local soil condition, but this was not accounted for by the N model. In summary, the N model would be well-suited to field-scale management of soil nitrogen, but suited poorly to management at finer spatial scales. This information was not apparent with a non-spatial validation. (c),2007 Elsevier B.V. All rights reserved.

Revisiting youden’s index as a useful measure of the misclassification error in meta-analysis of diagnostic studies

The paper considers meta-analysis of diagnostic studies that use a continuous score for classification of study participants into healthy or diseased groups. Classification is often done on the basis of a threshold or cut-off value, which might vary between studies. Consequently, conventional meta-analysis methodology focusing solely on separate analysis of sensitivity and specificity might be confounded by a potentially unknown variation of the cut-off value. To cope with this phenomena it is suggested to use, instead, an overall estimate of the misclassification error previously suggested and used as Youden’s index and; furthermore, it is argued that this index is less prone to between-study variation of cut-off values. A simple Mantel–Haenszel estimator as a summary measure of the overall misclassification error is suggested, which adjusts for a potential study effect. The measure of the misclassification error based on Youden’s index is advantageous in that it easily allows an extension to a likelihood approach, which is then able to cope with unobserved heterogeneity via a nonparametric mixture model. All methods are illustrated at hand of an example on a diagnostic meta-analysis on duplex doppler ultrasound, with angiography as the standard for stroke prevention.