On the stability and convergence of the finite section method for integral equation formulations of rough surface scattering

We consider the Dirichlet and Robin boundary value problems for the Helmholtz equation in a non-locally perturbed half-plane, modelling time harmonic acoustic scattering of an incident field by, respectively, sound-soft and impedance infinite rough surfaces.Recently proposed novel boundary integral equation formulations of these problems are discussed. It is usual in practical computations to truncate the infinite rough surface, solving a boundary integral equation on a finite section of the boundary, of length 2A, say. In the case of surfaces of small amplitude and slope we prove the stability and convergence as A→∞ of this approximation procedure. For surfaces of arbitrarily large amplitude and/or surface slope we prove stability and convergence of a modified finite section procedure in which the truncated boundary is ‘flattened’ in finite neighbourhoods of its two endpoints. Copyright © 2001 John Wiley & Sons, Ltd.

Some uniform stability and convergence results for integral equations on the real line and projection methods for their solution

The paper considers second kind equations of the form (abbreviated x=y + K2x) in which and the factor z is bounded but otherwise arbitrary so that equations of Wiener-Hopf type are included as a special case. Conditions on a set are obtained such that a generalized Fredholm alternative is valid: if W satisfies these conditions and I − Kz, is injective for each z ε W then I − Kz is invertible for each z ε W and the operators (I − Kz)−1 are uniformly bounded. As a special case some classical results relating to Wiener-Hopf operators are reproduced. A finite section version of the above equation (with the range of integration reduced to [−a, a]) is considered, as are projection and iterated projection methods for its solution. The operators (where denotes the finite section version of Kz) are shown uniformly bounded (in z and a) for all a sufficiently large. Uniform stability and convergence results, for the projection and iterated projection methods, are obtained. The argument generalizes an idea in collectively compact operator theory. Some new results in this theory are obtained and applied to the analysis of projection methods for the above equation when z is compactly supported and k(s − t) replaced by the general kernel k(s,t). A boundary integral equation of the above type, which models outdoor sound propagation over inhomogeneous level terrain, illustrates the application of the theoretical results developed.

The ocean’s role in setting the mean position of the Inter-Tropical Convergence Zone

Through study of observations and coupled climate simulations, it is argued that the mean position of the Inter-Tropical Convergence Zone (ITCZ) north of the equator is a consequence of a northwards heat transport across the equator by ocean circulation. Observations suggest that the hemispheric net radiative forcing of climate at the top of the atmosphere is almost perfectly symmetric about the equator, and so the total (atmosphere plus ocean) heat transport across the equator is small (order 0.2 PW northwards). Due to the Atlantic ocean’s meridional overturning circulation, however, the ocean carries significantly more heat northwards across the equator (order 0.4 PW) than does the coupled system. There are two primary consequences. First, atmospheric heat transport is southwards across the equator to compensate (0.2 PW southwards), resulting in the ITCZ being displaced north of the equator. Second, the atmosphere, and indeed the ocean, is slightly warmer (by perhaps 2 °C) in the northern hemisphere than in the southern hemisphere. This leads to the northern hemisphere emitting slightly more outgoing longwave radiation than the southern hemisphere by virtue of its relative warmth, supporting the small northward heat transport by the coupled system across the equator. To conclude, the coupled nature of the problem is illustrated through study of atmosphere–ocean–ice simulations in the idealized setting of an aquaplanet, resolving the key processes at work.

Optimal convergence estimates for the trace of the polynomial L2-projection operator on a simplex

In this paper we study convergence of the L2-projection onto the space of polynomials up to degree p on a simplex in Rd, d >= 2. Optimal error estimates are established in the case of Sobolev regularity and illustrated on several numerical examples. The proof is based on the collapsed coordinate transform and the expansion into various polynomial bases involving Jacobi polynomials and their antiderivatives. The results of the present paper generalize corresponding estimates for cubes in Rd from [P. Houston, C. Schwab, E. Süli, Discontinuous hp-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 39 (2002), no. 6, 2133-2163].

Forecast convergence score: a forecaster's approach to analysing hydro-meteorological forecast systems.

In this paper the properties of a hydro-meteorological forecasting system for forecasting river flows have been analysed using a probabilistic forecast convergence score (FCS). The focus on fixed event forecasts provides a forecaster's approach to system behaviour and adds an important perspective to the suite of forecast verification tools commonly used in this field. A low FCS indicates a more consistent forecast. It can be demonstrated that the FCS annual maximum decreases over the last 10 years. With lead time, the FCS of the ensemble forecast decreases whereas the control and high resolution forecast increase. The FCS is influenced by the lead time, threshold and catchment size and location. It indicates that one should use seasonality based decision rules to issue flood warnings.

Change in convergence and accommodation after two weeks of eye exercises in typical young adults

Abstract: Introduction Although eye exercises appear to help heterophoria, convergence insufficiency and intermittent strabismus, true treatment effects can be confounded by placebo, practice and encouragement factors. This study assessed objective changes in vergence and accommodation responses in typical naïve young adults after two weeks of exercises compared to control conditions to assess the extent of treatment effects occur above other factors. Methods 156 asymptomatic young adults were randomly assigned to 6 exercise groups or 2 no-treatment groups. Treatment targeted i) accommodation, ii)vergence, iii) both, iv) convergence>accommodation, v)accommodation>convergence, or vi) a placebo. All were re-tested under identical conditions, except for the second control group who were additionally encouraged during testing. Objective accommodation and vergence were assessed to a range of targets moving in depth containing combinations of blur, disparity and proximity/looming cues. Results Response gain improved more for less naturalistic targets where more improvement was possible. Convergence exercises improved vergence for near across all targets (P=.035). Mean accommodation changed similarly,but non-significantly. No other treatment group differed significantly from the non-encouraged control group, while encouraging effort produced significantly increased vergence (P=.004) and accommodation (P=.005) gains in the other control group. Conclusions True treatment effects were small, only significantly better after vergence exercises to a non-accommodative target, and were rarely related to response they were designed to improve. Exercising accommodation without convergence made no difference to accommodation to cues containing detail. Additional effort improved objective responses the most, so should be controlled carefully in research, and considered when auditing treatment.

Approximation by harmonic polynomials in star-shaped domains and exponential convergence of Trefftz hp-dGFEM

We study the approximation of harmonic functions by means of harmonic polynomials in two-dimensional, bounded, star-shaped domains. Assuming that the functions possess analytic extensions to a delta-neighbourhood of the domain, we prove exponential convergence of the approximation error with respect to the degree of the approximating harmonic polynomial. All the constants appearing in the bounds are explicit and depend only on the shape-regularity of the domain and on delta. We apply the obtained estimates to show exponential convergence with rate O(exp(−b square root N)), N being the number of degrees of freedom and b>0, of a hp-dGFEM discretisation of the Laplace equation based on piecewise harmonic polynomials. This result is an improvement over the classical rate O(exp(−b cubic root N )), and is due to the use of harmonic polynomial spaces, as opposed to complete polynomial spaces.

Screening for convergence insufficiency using the CISS is not indicated in young adults

Aim This paper presents Convergence Insufficiency Symptom Survey (CISS) and orthoptic findings in a sample of typical young adults who considered themselves to have normal eyesight apart from weak spectacles. Methods The CISS questionnaire was administered,followed by a full orthoptic evaluation, to 167 university undergraduate and postgraduate students during the recruitment phase of another study. The primary criterion for recruitment to this study was that participants‘feltthey had normal eyesight'. A CISS score of ≥21 was used to define‘significant’symptoms, and convergence insufficiency (CI) was defined as convergence≥8cm from the nose with a fusion range <15Δ base-out with small or no exophoria. Results The group mean CISS score was 15.4. In all, 17(10%) of the participants were diagnosed with CI, but 11(65%) of these did not have significant symptoms. 41(25%) participants returned a‘high’CISS score of ≥21 but only 6 (15%) of these had genuine CI. Sensitivity of the CISS to detect CI in this asymptomatic sample was 38%; specificity 77%; positive predictive value 15%; and negative predictive value 92%. The area under a receiver operating characteristic curve was 0.596 (95% CI 0.46 to 0.73). Conclusions‘Visual symptoms’are common in young adults, but often not related to any clinical defect, while true CI may be asymptomatic. This study suggests that screening for CI is not indicated

Umami: a delicious flavor formed by convergence of taste and olfactory pathways in the human brain

Umami taste is produced by glutamate acting on a fifth taste system. However, glutamate presented alone as a taste stimulus is not highly pleasant, and does not act synergistically with other tastes (sweet, salt, bitter and sour). We show here that when glutamate is given in combination with a consonant, savory, odour (vegetable), the resulting flavor can be much more pleasant. Moreover, we showed using functional brain imaging with fMRI that the glutamate taste and savory odour combination produced much greater activation of the medial orbitofrontal cortex and pregenual cingulate cortex than the sum of the activations by the taste and olfactory components presented separately. Supralinear effects were much less (and significantly less) evident for sodium chloride and vegetable odour. Further, activations in these brain regions were correlated with the pleasantness and fullness of the flavor, and with the consonance of the taste and olfactory components. Supralinear effects of glutamate taste and savory odour were not found in the insular primary taste cortex. We thus propose that glutamate acts by the nonlinear effects it can produce when combined with a consonant odour in multimodal cortical taste-olfactory convergence regions. We propose the concept that umami can be thought of as a rich and delicious flavor that is produced by a combination of glutamate taste and a consonant savory odour. Glutamate is thus a flavor enhancer because of the way that it can combine supralinearly with consonant odours in cortical areas where the taste and olfactory pathways converge far beyond the receptors.

Disparity-driven vs blur-driven models of accommodation and convergence in binocular vision and intermittent strabismus

Background. Current models of concomitant, intermittent strabismus, heterophoria, convergence and accommodation anomalies are either theoretically complex or incomplete. We propose an alternative and more practical way to conceptualize clinical patterns. Methods. In each of three hypothetical scenarios (normal; high AC/A and low CA/C ratios; low AC/A and high CA/C ratios) there can be a disparity-biased or blur-biased “style”, despite identical ratios. We calculated a disparity bias index (DBI) to reflect these biases. We suggest how clinical patterns fit these scenarios and provide early objective data from small illustrative clinical groups. Results. Normal adults and children showed disparity bias (adult DBI 0.43 (95%CI 0.50-0.36), child DBI 0.20 (95%CI 0.31-0.07) (p=0.001). Accommodative esotropes showed less disparity-bias (DBI 0.03). In the high AC/A and low CA/C scenario, early presbyopes had mean DBI of 0.17 (95%CI 0.28-0.06), compared to DBI of -0.31 in convergence excess esotropes. In the low AC/A and high CA/C scenario near exotropes had mean DBI of 0.27, while we predict that non-strabismic, non-amblyopic hyperopes with good vision without spectacles will show lower DBIs. Disparity bias ranged between 1.25 and -1.67. Conclusions. Establishing disparity or blur bias, together with knowing whether convergence to target demand exceeds accommodation or vice versa explains clinical patterns more effectively than AC/A and CA/C ratios alone. Excessive bias or inflexibility in near-cue use increases risk of clinical problems. We suggest clinicians look carefully at details of accommodation and convergence changes induced by lenses, dissociation and prisms and use these to plan treatment in relation to the model.

Noisy Monte Carlo: convergence of Markov chains with approximate transition kernels

Monte Carlo algorithms often aim to draw from a distribution π by simulating a Markov chain with transition kernel P such that π is invariant under P. However, there are many situations for which it is impractical or impossible to draw from the transition kernel P. For instance, this is the case with massive datasets, where is it prohibitively expensive to calculate the likelihood and is also the case for intractable likelihood models arising from, for example, Gibbs random fields, such as those found in spatial statistics and network analysis. A natural approach in these cases is to replace P by an approximation Pˆ. Using theory from the stability of Markov chains we explore a variety of situations where it is possible to quantify how ’close’ the chain given by the transition kernel Pˆ is to the chain given by P . We apply these results to several examples from spatial statistics and network analysis.

Plane wave discontinuous Galerkin methods: exponential convergence of the hp-version

We consider the two-dimensional Helmholtz equation with constant coefficients on a domain with piecewise analytic boundary, modelling the scattering of acoustic waves at a sound-soft obstacle. Our discretisation relies on the Trefftz-discontinuous Galerkin approach with plane wave basis functions on meshes with very general element shapes, geometrically graded towards domain corners. We prove exponential convergence of the discrete solution in terms of number of unknowns.