Paul H. Muir

related: an R package for analysing pairwise relatedness from codominant molecular markers.

Analyses of pairwise relatedness represent a key component to addressing many topics in biology. However, such analyses have been limited because most available programs provide a means to estimate relatedness based on only a single estimator, making comparison across estimators difficult. Second, all programs to date have been platform specific, working only on a specific operating system. This has the undesirable outcome of making choice of relatedness estimator limited by operating system preference, rather than being based on scientific rationale. Here, we present a new R package, called related, that can calculate relatedness based on seven estimators, can account for genotyping errors, missing data and inbreeding, and can estimate 95% confidence intervals. Moreover, simulation functions are provided that allow for easy comparison of the performance of different estimators and for analyses of how much resolution to expect from a given data set. Because this package works in R, it is platform independent. Combined, this functionality should allow for more appropriate analyses and interpretation of pairwise relatedness and will also allow for the integration of relatedness data into larger R workflows.

Runge-Kutta Software with Defect Control four Boundary Value ODEs

A popular approach to the numerical solution of boundary value ODE problems involves the use of collocation methods. Such methods can be naturally implemented so as to provide a continuous approximation to the solution over the entire problem interval. On the other hand, several authors have suggested as an alternative, certain subclasses of the implicit Runge--Kutta formulas, known as mono-implicit Runge--Kutta (MIRK) formulas, which can be implemented at a lower cost per step than the collocation methods. These latter formulas do not have a natural implementation that provides a continuous approximation to the solution; rather, only a discrete approximation at certain points within the problem interval is obtained. However, recent work in the area of initial value problems has demonstrated the possibility of generating inexpensive interpolants for any explicit Runge--Kutta formula. These ideas have recently been extended to develop continuous extensions of the MIRK formulas. In this paper, we describe our investigation of the use of continuous MIRK formulas in the numerical solution of boundary value ODE problems. A primary thrust of this investigation is to consider defect control, based on the continuous MIRK formulas, as an alternative to the standard use of global error control, as the basis for termination and mesh redistribution criteria.

Algorithm 688: EPDCOL: a more efficient PDECOL code

The software package PDECOL [7] is a popular code among scientists wishing to solve systems of nonlinear partial differential equations. The code is based on a method-of-lines approach, with collocation in the space variable to reduce the problem to a system of ordinary differential equations. There are three principal components: the basis functions employed in the collocation; the method used to solve the system of ordinary differential equations; and the linear equation solver which handles the linear algebra. This paper will concentrate on the third component, and will report on the improvement in the performance of PDECOL resulting from replacing the current linear algebra modules of the code by modules which take full advantage of the special structure of the equations which arise. Savings of over 50 percent in total execution time can be realized.

Theory and numerical simulation of nth-order cascaded Raman fiber lasers

Using the classical treatment of the stimulated Raman-scattering process, we use a theoretical model to simulate the operation of an nth-order cascaded Raman fiber laser. We introduce the partial differential equations employed to describe the propagation and time dependence of the forward and reverse-propagating fields of an nth-order cascaded Raman fiber laser. Under steady-state conditions, these equations form the well-known system of first-order, nonlinear boundary-value ordinary differential equations, with separated boundary conditions. We solve this system of equations numerically with the use of mono-implicit Runge–Kutta methods within a defect-control framework. We consider cascaded Raman fiber lasers of orders 2 through 5 and examine the parameters that influence the operation of these devices. We also provide preliminary results on the investigation of a time-dependent model in which the pump power is assumed to vary periodically with time. The associated system of first-order, hyperbolic, partial differential equations is treated by employing a transverse method-of-lines algorithm; the time derivatives are discretized with a finite-difference scheme, yielding a large system of boundary-value ordinary differential equations. We establish that for sinusoidal modulation of the pump the Stokes cavity modes exhibit antiphase dynamics typical of a system of locally coupled nonlinear oscillators.

PMIRKDC: a parallel mono-implicit Runge--Kutta code with defect control for boundary value ODEs

We describe parallel software, PMIRKDC, for solving boundary value ordinary differential equations (BVODEs). This software is based on the package, MIRKDC, which employs mono-implicit Runge-Kutta schemes within a defect control algorithm. The primary computational costs involve the treatment of large, almost block diagonal (ABD) linear systems. The most significant feature of PMIRKDC is the replacement of sequential ABD software, COLROW, with new parallel ABD software, RSCALE, based on a parallel block eigenvalue rescaling algorithm. Other modifications involve parallelization of the setup of the ABD systems and solution interpolants, and defect estimation. Numerical results show almost linear speedups.

Optimal discrete and continuous mono-implicit Runge-Kutta schemes for BVODEs

Recent investigations of discretization schemes for the efficient numerical solution of boundary value ordinary differential equations (BVODEs) have focused on a subclass of the well‐known implicit Runge–Kutta (RK) schemes, called mono‐implicit RK (MIRK) schemes, which have been employed in two software packages for the numerical solution of BVODEs, called TWPBVP and MIRKDC. The latter package also employs continuous MIRK (CMIRK) schemes to provide C1 continuous approximate solutions. The particular schemes implemented in these codes come, in general, from multi‐parameter families and, in some cases, do not represent optimal choices from these families. In this paper, several optimization criteria are identified and applied in the derivation of optimal MIRK and CMIRK schemes for orders 1–6. In some cases the schemes obtained result from the analysis of existent multi‐parameter families; in other cases new families are derived from which specific optimal schemes are then obtained. New MIRK and CMIRK schemes are presented which are superior to those currently available. Numerical examples are provided to demonstrate the practical improvements that can be obtained by employing the optimal schemes.

On the temporal dynamics of spatial stimulus-response transfer between spatial incompatibility and Simon tasks

The Simon effect refers to the performance (response time and accuracy) advantage for responses that spatially correspond to the task-irrelevant location of a stimulus. It has been attributed to a natural tendency to respond toward the source of stimulation. When location is task-relevant, however, and responses are intentionally directed away (incompatible) or toward (compatible) the source of the stimulation, there is also an advantage for spatially compatible responses over spatially incompatible responses. Interestingly, a number of studies have demonstrated a reversed, or reduced, Simon effect following practice with a spatial incompatibility task. One interpretation of this finding is that practicing a spatial incompatibility task disables the natural tendency to respond toward stimuli. Here, the temporal dynamics of this stimulus-response (S-R) transfer were explored with speed-accuracy trade-offs (SATs). All experiments used the mixed-task paradigm in which Simon and spatial compatibility/incompatibility tasks were interleaved across blocks of trials. In general, bidirectional S-R transfer was observed: while the spatial incompatibility task had an influence on the Simon effect, the task-relevant S-R mapping of the Simon task also had a small impact on congruency effects within the spatial compatibility and incompatibility tasks. These effects were generally greater when the task contexts were similar. Moreover, the SAT analysis of performance in the Simon task demonstrated that the tendency to respond to the location of the stimulus was not eliminated because of the spatial incompatibility task. Rather, S-R transfer from the spatial incompatibility task appeared to partially mask the natural tendency to respond to the source of stimulation with a conflicting inclination to respond away from it. These findings support the use of SAT methodology to quantitatively describe rapid response tendencies.

Algorithm 874: BACOLR—spatial and temporal error control software for PDEs based on high-order adaptive collocation

In this article we discuss a new software package, BACOLR, for the numerical solution of a general class of time-dependent 1-D PDEs. This package employs high-order adaptive methods in time and space within a method-of-lines approach and provides tolerance control of the spatial and temporal errors. The DAEs resulting from the spatial discretization (based on B-spline collocation) are handled by a substantially modified version of the Runge-Kutta solver, RADAU5. For each time step, the RADAU5 code computes an estimate of the temporal error and requires it to satisfy the user tolerance. After each time step BACOLR then computes a high-order estimate of the spatial error and requires this error estimate to satisfy the user tolerance. BACOLR was developed through a substantial modification of the adaptive method-of-lines package, BACOL. In this article we introduce the BACOLR package and present numerical results to show that the performance of BACOLR is comparable to and in some cases significantly superior to that of BACOL, which was shown in previous work to be more efficient, reliable and robust than other existing codes, especially for problems with solutions exhibiting narrow spikes or boundary layers.

Mono-implicit Runge-Kutta schemes for the parallel solution of initial value ODEs

Among the numerical techniques commonly considered for the efficient solution of stiff initial value ordinary differential equations are the implicit Runge-Kutta (IRK) schemes. The calculation of the stages of the IRK method involves the solution of a nonlinear system of equations usually employing some variant of Newtons method. Since the costs of the linear algebra associated with the implementation of Newtons method generally dominate the overall cost of the computation, many subclasses of IRK schemes, such as diagonally implicit Runge-Kutta schemes, singly implicit Runge-Kutta schemes, and mono-implicit (MIRK) schemes, have been developed to attempt to reduce these costs. In this paper we are concerned with the design of MIRK schemes that are inherently parallel in that smaller systems of equations are apportioned to concurrent processors. This work builds on that of an earlier investigation in which a special subclass of the MIRK formulas were implemented in parallel. While suitable parallelism was achieved, the formulas were limited to some extent because they all had only stage order 1. This is of some concern since in the application of a Runge-Kutta method to a system of stiff ODEs the phenomenon of order reduction can arise; the IRK method can behave as if its order were only its stage order (or its stage order plus one), regardless of its classical order. The formulas derived in the current paper represent an improvement over the previous investigation in that the full class of MIRK formulas is considered and therefore it is possible to derive efficient, parallel formulas of orders 2, 3, and 4, having stage orders 2 or 3.

A Runge-Kutta BVODE Solver with Global Error and Defect Control

Boundary value ordinary differential equations (BVODEs) are systems of ODEs with boundary conditions imposed at two or more distinct points. The global error (GE) of a numerical solution to a BVODE is the amount by which the numerical solution differs from the exact solution. The defect is the amount by which the numerical solution fails to satisfy the ODEs and boundary conditions. Although GE control is often familiar to users, the defect controlled numerical solution can be interpreted as the exact solution to a perturbation of the original BVODE. Software packages based on GE control and on defect control are in wide use. The defect control solver, BVP_SOLVER, can provide an a posteriori estimate of the GE using Richardson extrapolation. In this article, we consider three more strategies for GE estimation based on (i) the direct use of a higher-order discretization formula (HO), (ii) the use of a higher-order discretization formula within a deferred correction (DC) framework, and (iii) the product of an estimate of the maximum defect and an estimate of the BVODE conditioning constant, and demonstrate that the HO and DC approaches have superior performance. We also modify BVP_SOLVER to introduce GE control.