On Stochastic Error and Computational Efficiency of the Markov Chain Monte Carlo Method

In Markov Chain Monte Carlo (MCMC) simulations, the thermal equilibria quantities are estimated by ensemble average over a sample set containing a large number of correlated samples. These samples are selected in accordance with the probability distribution function, known from the partition function of equilibrium state. As the stochastic error of the simulation results is significant, it is desirable to understand the variance of the estimation by ensemble average, which depends on the sample size (i.e., the total number of samples in the set) and the sampling interval (i.e., cycle number between two consecutive samples). Although large sample sizes reduce the variance, they increase the computational cost of the simulation. For a given CPU time, the sample size can be reduced greatly by increasing the sampling interval, while having the corresponding increase in variance be negligible if the original sampling interval is very small. In this work, we report a few general rules that relate the variance with the sample size and the sampling interval. These results are observed and confirmed numerically. These variance rules are derived for the MCMC method but are also valid for the correlated samples obtained using other Monte Carlo methods. The main contribution of this work includes the theoretical proof of these numerical observations and the set of assumptions that lead to them

AmelHap: Leveraging drone whole-genome sequence data to create a honey bee HapMap

Honey bee, Apis mellifera, drones are typically haploid, developing from an unfertilized egg, inheriting only their queen’s alleles and none from the many drones she mated with. Thus the ordered combination or ‘phase’ of alleles is known, making drones a valuable haplotype resource. We collated whole-genome sequence data for 1,407 drones, including 45 newly sequenced Scottish drones, collectively representing 19 countries, 8 subspecies and various hybrids. Following alignment to Amel_HAv3.1, variant calling and quality filtering, we retained 17.4 M high quality variants across 1,328 samples with a genotyping rate of 98.7%. We demonstrate the utility of this haplotype resource, AmelHap, for genotype imputation, returning >95% concordance when up to 61% of data is missing in haploids and up to 12% of data is missing in diploids. AmelHap will serve as a useful resource for the community for imputation from low-depth sequencing or SNP chip data, accurate phasing of diploids for association studies, and as a comprehensive reference panel for population genetic and evolutionary analyses.For the purpose of open access, the author has applied a Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript version arising from this submission. This work was supported by a grant from the CB Dennis British Beekeepers’ Research Trust awarded to MB and DW, and through strategic investment funding to the Roslin Institute from the Biotechnology and Biological Sciences Research Council (BBS/E/D/30002276). MP was supported by a Basque Government grant (IT1233-19)

The value of continuity: Refined isogeometric analysis and fast direct solvers

We propose the use of highly continuous finite element spaces interconnected with low continuity hyperplanes to maximize the performance of direct solvers. Starting from a highly continuous Isogeometric Analysis (IGA) discretization, we introduce . C0-separators to reduce the interconnection between degrees of freedom in the mesh. By doing so, both the solution time and best approximation errors are simultaneously improved. We call the resulting method "refined Isogeometric Analysis (rIGA)". To illustrate the impact of the continuity reduction, we analyze the number of Floating Point Operations (FLOPs), computational times, and memory required to solve the linear system obtained by discretizing the Laplace problem with structured meshes and uniform polynomial orders. Theoretical estimates demonstrate that an optimal continuity reduction may decrease the total computational time by a factor between . p2 and . p3, with . p being the polynomial order of the discretization. Numerical results indicate that our proposed refined isogeometric analysis delivers a speed-up factor proportional to . p2. In a . 2D mesh with four million elements and . p=5, the linear system resulting from rIGA is solved 22 times faster than the one from highly continuous IGA. In a . 3D mesh with one million elements and . p=3, the linear system is solved 15 times faster for the refined than the maximum continuity isogeometric analysis

Developing reduced SNP assays from whole-genome sequence data to estimate C-lineage introgression in the Iberian honeybee (Apis mellifera iberiensis)

The honeybee has been subject to a growing number of threats. In Western Europe one such threat is large-scale introductions of commercial strains (C-lineage), which is leading to introgressive hybridization and even the local extinction of native populations (M-lineage). Here, we developed reduced assays of highly informative SNPs from 176 whole genomes to estimate C-lineage introgression in ;M-lineage subspecies Apis mellifera iberiensis. We started by evaluating the effects of sample size and sampling a geographically restricted area on the number of highly informative SNPs. We demonstrated that a bias in the number of fixed SNPs (FST=1) is introduced when the sample size is small (N≤10) and when sampling only captures a small fraction of a population’s genetic diversity. These results underscore the importance of having a representative sample when developing reliable reduced SNP assays for organisms with complex genetic patterns. We used a training dataset to design four independent SNP assays selected from pairwise FST between the Iberian and C-lineage honeybees. The designed assays, which were validated in holdout and simulated hybrid datasets, proved to be highly accurate and can be readily used for monitoring populations not only in the native range of A. m. iberiensis in Iberia but also in the introduced range in the Balearic islands, Macaronesia, and South America, in a time- and cost-effective manner. While our approach used the Iberian honeybee as model system, it has a high value in a wide range of scenarios for the monitoring and conservation of potentially hybridized domestic and wildlife populations.info:eu-repo/semantics/publishedVersio

An Asymptotic Preserving Scheme for the Euler equations in a strong magnetic field

This paper is concerned with the numerical approximation of the isothermal Euler equations for charged particles subject to the Lorentz force. When the magnetic field is large, the so-called drift-fluid approximation is obtained. In this limit, the parallel motion relative to the magnetic field direction splits from perpendicular motion and is given implicitly by the constraint of zero total force along the magnetic field lines. In this paper, we provide a well-posed elliptic equation for the parallel velocity which in turn allows us to construct an Asymptotic-Preserving (AP) scheme for the Euler-Lorentz system. This scheme gives rise to both a consistent approximation of the Euler-Lorentz model when epsilon is finite and a consistent approximation of the drift limit when epsilon tends to 0. Above all, it does not require any constraint on the space and time steps related to the small value of epsilon. Numerical results are presented, which confirm the AP character of the scheme and its Asymptotic Stability

On the thermodynamics of the Swift–Hohenberg theory

We present the microbalance including the microforces, the first- and second-order microstresses for the Swift–Hohenberg equation concomitantly with their constitutive equations, which are consistent with the free-energy imbalance. We provide an explicit form for the microstress structure for a free-energy functional endowed with second-order spatial derivatives. Additionally, we generalize the Swift–Hohenberg theory via a proper constitutive process. Finally, we present one highly resolved three-dimensional numerical simulation to demonstrate the particular form of the resulting microstresses and their interactions in the evolution of the Swift–Hohenberg equation

Numerical approximation of the Euler-Poisson-Boltzmann model in the quasineutral limit

This paper analyzes various schemes for the Euler-Poisson-Boltzmann (EPB) model of plasma physics. This model consists of the pressureless gas dynamics equations coupled with the Poisson equation and where the Boltzmann relation relates the potential to the electron density. If the quasi-neutral assumption is made, the Poisson equation is replaced by the constraint of zero local charge and the model reduces to the Isothermal Compressible Euler (ICE) model. We compare a numerical strategy based on the EPB model to a strategy using a reformulation (called REPB formulation). The REPB scheme captures the quasi-neutral limit more accurately