On the Convergence of Two-Stage Iterative Processes for Solving Linear Equations

This paper considers two-stage iterative processes for solving the linear system $Af = b$. The outer iteration is defined by $Mf^{k + 1} = Nf^k + b$, where $M$ is a nonsingular matrix such that $M - N = A$. At each stage $f^{k + 1} $ is computed approximately using an inner iteration process to solve $Mv = Nf^k + b$ for $v$. At the $k$th outer iteration, $p_k $ inner iterations are performed. It is shown that this procedure converges if $p_k \geqq P$ for some $P$ provided that the inner iteration is convergent and that the outer process would converge if $f^{k + 1} $ were determined exactly at every step. Convergence is also proved under more specialized conditions, and for the procedure where $p_k = p$ for all $k$, an estimate for $p$ is obtained which optimizes the convergence rate. Examples are given for systems arising from the numerical solution of elliptic partial differential equations and numerical results are presented.

Convergence accommodation trumps accommodative convergence

Purpose. This symposium contribution presents research that shows that disparity cues within a near stimulus drive not only vergence but also most of the accommodation. Be-cause blur is a weaker cue, accommodative convergence is therefore only of minor significance for most individuals. Methods. The Infant Vision Laboratory at the University of Reading uses a Power Ref II photorefractor to collect simultaneous accommodation and convergence data from participants fixating targets moving in depth. By manipulating target characteristics, we have been able to test how blur, disparity and proximal cues each contribute to driving responses. Results. Results from a series of studies over the past 12 years have contributed to a coherent body of evidence suggesting that disparity cues override blur and proximity cues in most individuals. Some strabismic patients do use blur as a more strongly weighted cue, and this strategy could contribute to their symptoms, clinical characteristics and response to treatment. Conclusion. Although convergence accommodation is extremely difficult to measure clinically, clinicians should be aware of its importance in binocular vision and strabismus. Although CA/C relationships typically seem more important than AC/A, bo th only partly explain the interplay between convergence and accommodation.

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.

Convergent evolution of floral signals underlies the success of Neotropical orchids

The great majority of plant species in the tropics require animals to achieve pollination, but the exact role of floral signals in attraction of animal pollinators is often debated. Many plants provide a floral reward to attract a guild of pollinators, and it has been proposed that floral signals of non-rewarding species may converge on those of rewarding species to exploit the relationship of the latter with their pollinators. In the orchid family (Orchidaceae), pollination is almost universally animal-mediated, but a third of species provide no floral reward, which suggests that deceptive pollination mechanisms are prevalent. Here, we examine floral colour and shape convergence in Neotropical plant communities, focusing on certain food-deceptive Oncidiinae orchids (e.g. Trichocentrum ascendens and Oncidium nebulosum) and rewarding species of Malpighiaceae. We show that the species from these two distantly related families are often more similar in floral colour and shape than expected by chance and propose that a system of multifarious floral mimicry—a form of Batesian mimicry that involves multiple models and is more complex than a simple one model–one mimic system—operates in these orchids. The same mimetic pollination system has evolved at least 14 times within the species-rich Oncidiinae throughout the Neotropics. These results help explain the extraordinary diversification of Neotropical orchids and highlight the complexity of plant–animal interactions.

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