Monte Carlo numerical treatment of large linear algebra problems

In this paper we deal with performance analysis of Monte Carlo algorithm for large linear algebra problems. We consider applicability and efficiency of the Markov chain Monte Carlo for large problems, i.e., problems involving matrices with a number of non-zero elements ranging between one million and one billion. We are concentrating on analysis of the almost Optimal Monte Carlo (MAO) algorithm for evaluating bilinear forms of matrix powers since they form the so-called Krylov subspaces. Results are presented comparing the performance of the Robust and Non-robust Monte Carlo algorithms. The algorithms are tested on large dense matrices as well as on large unstructured sparse matrices.

Robustness and applicability of Markov chain Monte Carlo algorithms for eigenvalue problems

In this paper we analyse applicability and robustness of Markov chain Monte Carlo algorithms for eigenvalue problems. We restrict our consideration to real symmetric matrices. Almost Optimal Monte Carlo (MAO) algorithms for solving eigenvalue problems are formulated. Results for the structure of both - systematic and probability error are presented. It is shown that the values of both errors can be controlled independently by different algorithmic parameters. The results present how the systematic error depends on the matrix spectrum. The analysis of the probability error is presented. It shows that the close (in some sense) the matrix under consideration is to the stochastic matrix the smaller is this error. Sufficient conditions for constructing robust and interpolation Monte Carlo algorithms are obtained. For stochastic matrices an interpolation Monte Carlo algorithm is constructed. A number of numerical tests for large symmetric dense matrices are performed in order to study experimentally the dependence of the systematic error from the structure of matrix spectrum. We also study how the probability error depends on the balancing of the matrix. (c) 2007 Elsevier Inc. All rights reserved.

Two-point boundary value problems for linear evolution equations

We study boundary value problems for a linear evolution equation with spatial derivatives of arbitrary order, on the domain 0 < x < L, 0 < t < T, with L and T positive nite constants. We present a general method for identifying well-posed problems, as well as for constructing an explicit representation of the solution of such problems. This representation has explicit x and t dependence, and it consists of an integral in the k-complex plane and of a discrete sum. As illustrative examples we solve some two-point boundary value problems for the equations iqt + qxx = 0 and qt + qxxx = 0.

On the spectra of certain integro-differential-delay problems with applications in neurodynamics

We investigate the spectrum of certain integro-differential-delay equations (IDDEs) which arise naturally within spatially distributed, nonlocal, pattern formation problems. Our approach is based on the reformulation of the relevant dispersion relations with the use of the Lambert function. As a particular application of this approach, we consider the case of the Amari delay neural field equation which describes the local activity of a population of neurons taking into consideration the finite propagation speed of the electric signal. We show that if the kernel appearing in this equation is symmetric around some point a= 0 or consists of a sum of such terms, then the relevant dispersion relation yields spectra with an infinite number of branches, as opposed to finite sets of eigenvalues considered in previous works. Also, in earlier works the focus has been on the most rightward part of the spectrum and the possibility of an instability driven pattern formation. Here, we numerically survey the structure of the entire spectra and argue that a detailed knowledge of this structure is important within neurodynamical applications. Indeed, the Amari IDDE acts as a filter with the ability to recognise and respond whenever it is excited in such a way so as to resonate with one of its rightward modes, thereby amplifying such inputs and dampening others. Finally, we discuss how these results can be generalised to the case of systems of IDDEs.

Sex allocation and local mate competition in Old World non-pollinating fig wasps

The populations of many species are structured such that mating is not random and occurs between members of local patches. When patches are founded by a single female and all matings occur between siblings, brothers may compete with each other for matings with their sisters. This local mate competition (LMC) selects for a female-biased sex ratio, especially in species where females have control over offspring sex, as in the parasitic Hymenoptera. Two factors are predicted to decrease the degree of female bias: (1) an increase in the number of foundress females in the patch and (2) an increase in the fraction of individuals mating after dispersal from the natal patch. Pollinating fig wasps are well known as classic examples of species where all matings occur in the local patch. We studied non-pollinating fig wasps, which are more diverse than the pollinating fig wasps and also provide natural experimental groups of species with different male morphologies that are linked to different mating structures. In this group of wasps, species with wingless males mate in the local patch (i.e. the fig fruit) while winged male species mate after dispersal. Species with both kinds of male have a mixture of local and non-local mating. Data from 44 species show that sex ratios (defined as the proportion of males) are in accordance with theoretical predictions: wingless male species < wing-dimorphic male species < winged male species. These results are also supported by a formal comparative analysis that controls for phylogeny. The foundress number is difficult to estimate directly for non-pollinating fig wasps but a robust indirect method leads to the prediction that foundress number, and hence sex ratio, should increase with the proportion of patches occupied in a crop. This result is supported strongly across 19 species with wingless males, but not across 8 species with winged males. The mean sex ratios for species with winged males are not significantly different from 0.5, and the absence of the correlation observed across species with wingless males may reflect weak selection to adjust the sex ratio in species whose population mating structure tends not to be subdivided. The same relationship is also predicted to occur within species if individual females adjust their sex ratios facultatively. This final prediction was not supported by data from a wingless male species, a male wing-dimorphic species or a winged male species.

Learning and natal host influence host preference, handling time and sex allocation behaviour in a pupal parasitoid

The host choice and sex allocation decisions of a foraging female parasitoid will have an enormous influence on the life-history characteristics of her offspring. The pteromalid Pachycrepoideus vindemiae is a generalist idiobiont pupal parasitoid of many species of cyclorrhaphous Diptera. Wasps reared in Musca domestica were larger, had higher attack rates and greater male mating success than those reared in Drosophila melanogaster. In no-choice situations, naive female R vindemiae took significantly less time to accept hosts conspecific with their natal host. Parasitoids that emerged from M. domestica pupae spent similar amounts of time ovipositing in both D. melanogaster and M. domestica. Those parasitoids that had emerged from D. melanogaster spent significantly longer attacking M. domestica pupae. The host choice behaviour of female P. vindemiae was influenced by an interaction between natal host and experience. Female R vindemiae reared in M. domestica only showed a preference among hosts when allowed to gain experience attacking M. domestica, preferentially attacking that species. Similarly, female parasitoids reared on D. melanogaster only showed a preference among hosts when allowed to gain experience attacking D. melanogaster, again preferentially attacking that species. Wasp natal host also influenced sex allocation behaviour. While wasps from both hosts oviposited more females in the larger host, M. domestica, wasps that emerged from M. domestica had significantly more male-biased offspring sex ratios. These results indicate the importance of learning and natal host size in determining R vindemiae attack rates. mating success, host preference and sex allocation behaviour, all critical components of parasitoid fitness.

On a class of nonselfadjoint periodic boundary value problems with discrete real spectrum

In a previous paper (J. of Differential Equations, Vol. 249 (2010), 3081-3098) we examined a family of periodic Sturm-Liouville problems with boundary and interior singularities which are highly non-self-adjoint but have only real eigenvalues. We now establish Schatten class properties of the associated resolvent operator.

Using Doppler radar with a simple explicit microphysics model to diagnose problems with ice sublimation depth-scales in forecast models

Several previous studies have attempted to assess the sublimation depth-scales of ice particles from clouds into clear air. Upon examining the sublimation depth-scales in the Met Office Unified Model (MetUM), it was found that the MetUM has evaporation depth-scales 2–3 times larger than radar observations. Similar results can be seen in the European Centre for Medium-Range Weather Forecasts (ECMWF), Regional Atmospheric Climate Model (RACMO) and Météo-France models. In this study, we use radar simulation (converting model variables into radar observations) and one-dimensional explicit microphysics numerical modelling to test and diagnose the cause of the deep sublimation depth-scales in the forecast model. The MetUM data and parametrization scheme are used to predict terminal velocity, which can be compared with the observed Doppler velocity. This can then be used to test the hypothesis as to why the sublimation depth-scale is too large within the MetUM. Turbulence could lead to dry air entrainment and higher evaporation rates; particle density may be wrong, particle capacitance may be too high and lead to incorrect evaporation rates or the humidity within the sublimating layer may be incorrectly represented. We show that the most likely cause of deep sublimation zones is an incorrect representation of model humidity in the layer. This is tested further by using a one-dimensional explicit microphysics model, which tests the sensitivity of ice sublimation to key atmospheric variables and is capable of including sonde and radar measurements to simulate real cases. Results suggest that the MetUM grid resolution at ice cloud altitudes is not sufficient enough to maintain the sharp drop in humidity that is observed in the sublimation zone.

Solving complex optimal control problems at no cost with PSOPT

This paper introduces PSOPT, an open source optimal control solver written in C++. PSOPT uses pseudospectral and local discretizations, sparse nonlinear programming, automatic differentiation, and it incorporates automatic scaling and mesh refinement facilities. The software is able to solve complex optimal control problems including multiple phases, delayed differential equations, nonlinear path constraints, interior point constraints, integral constraints, and free initial and/or final times. The software does not require any non-free platform to run, not even the operating system, as it is able to run under Linux. Additionally, the software generates plots as well as LATEX code so that its results can easily be included in publications. An illustrative example is provided.

Infant-mother attachment and the growth of externalizing problems across the primary-school years

Background:  Some contend that attachment insecurity increases risk for the development of externalizing behavior problems in children. Method:  Latent-growth curve analyses were applied to data on 1,364 children from the NICHD Study of Early Child Care to evaluate the association between early attachment and teacher-rated externalizing problems across the primary-school years. Results:  Findings indicate that (a) both avoidant and disorganized attachment predict higher levels of externalizing problems but (b) that effects of disorganized attachment are moderated by family cumulative contextual risk, child gender and child age, with disorganized boys from risky social contexts manifesting increases in behavior problems over time. Conclusions:  These findings highlight the potentially conditional role of early attachment in children’s externalizing behavior problems and the need for further research evaluating causation and mediating mechanisms.

Bernoulli free-boundary problems in strip-like domains and a property of permanent waves on water of finite depth

We study weak solutions for a class of free-boundary problems which includes as a special case the classical problem of travelling gravity waves on water of finite depth. We show that such problems are equivalent to problems in fixed domains and study the regularity of their solutions. We also prove that in very general situations the free boundary is necessarily the graph of a function.