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He’s healthy, but will he survive the plague? Possible constraints on mate choice for disease resistance

Abstract:

Can females enhance their fitness by choosing a mate based on his disease resistance in addition to his current health and robustness (i.e. male condition)? The complex nature of disease resistance may constrain the evolution of female choice for this trait. Using a mathematical model, we showed that choice for immune function (an element of disease resistance) provided females with a fitness advantage. However, the fitness advantage was often small, much smaller than the fitness advantage females obtained from mating with males in good condition. Females choosing for a combination of male condition and male immune function sometimes showed no fitness advantage compared with females choosing for condition alone, even when condition and immune function were positively correlated. Our results suggest that when condition and immune function are correlated, selection for choice for male immune function may be driven by the fitness advantage that comes from mating with males in the best condition, even if a sexually selected trait correlates with male immune function. Moreover, females choosing for males with maximal immune function produced offspring with immune functions above the level needed for maximal fitness. In some species, females may gain little or no fitness advantage by choosing for male immune function per se in addition to male condition. This may explain why not all studies find evidence for female choice for male immune function.

Authors: Shelley A. Adamo, Raymond J. Spiteri

Download: AdamoSpiteri2009

One-dimensional magnetotelluric inversion with radiation boundary conditions

Abstract:

We present an algebraic method of solving the magnetotelluric inverse problem for the case of one-dimensional conductivity profiles in the class D+. We show that the typically examined Dirichlet boundary conditions are a limiting case of the radiative boundary conditions introduced by Srnka and Crutchfield. By examining the analogous inverse inhomoge-neous string problem studied by Kre ̆ın we demonstrate the usefulness of the conductivity class D+. Results of the inversion procedure are pre- sented, as well as a discussion of the continued fraction expansions re-sulting from the more general boundary conditions. The presentation presupposes no knowledge of magnetotellurics.

Authors: Tyler Helmuth, Raymond Spiteri, Jacek Szmigielski

Download: HelmuthSpiteriSzmgielski2008

A computational model of catalyzed carbon sequestration

Abstract:

This research explores the feasibility of catalysis-based carbon sequestration by efficiently and accurately modeling the underlying chemical reactions and using this model to identify optimal operating conditions. We employ established and novel computational methods to calculate the Arrhenius rate constants required to model the chemical reactions as a coupled system of differential equations and implement this model for carbon sequestration over a palladium catalyst. This approach allows us to explore the behavior of the system for a variety of temperatures, pressures, feed-gas compositions, and catalysts and thereby optimize the amount of carbon sequestered. We discuss trends in the distribution of reaction products as a function of these variables. Preliminary results for this system and previously published results for similar systems indicate that this method can be scaled to accurately predict the efficacy of such systems for carbon sequestration.

Authors: Harrell Sellers, Michael Perrone, Raymond J. Spiteri

Download: 2008CSPG-CSEG-CWLS

A new adaptive folding-up algorithm for information retrieval

Abstract:

Text collections can be represented mathematically as term-document matrices. A term-document matrix can in turn be represented using the matrix factorization method known as the partial (or truncated) singular value decomposition (PSVD). Recomputing the PSVD when changes are made to a text collection is very expensive. Folding-in is one method of approximating the PSVD when new documents are added to a term-document matrix; updating the PSVD of the existing term-document matrix is another method. The folding-in method is computationally inexpensive, but it may cause deterioration in the accuracy of the PSVD. The PSVD-updating method is more expensive than the folding-in method, but it maintains the accuracy of the PSVD. Folding-up is a method that combines folding-in and PSVD-updating. When a text collection expands in small increments, folding-up provides a significant improvement in computation time when compared with either re- computing the PSVD or PSVD-updating, and it reduces the loss of accuracy in the PSVD that can occur with the folding-in method. This paper introduces a new adaptive folding-up method in which a measure of the error in the PSVD is monitored to determine when it is most advantageous to switch from folding-in to updating.

Authors: Jane E. Mason, Raymond J. Spiteri

Download: MasonSpiteri2008

On the performance of implicit-explicit Runge-Kutta methods in models of cardiac electrical activity

Abstract:

Mathematical models of electric activity in cardiac tissue are becoming increasingly powerful tools in the study of cardiac arrhythmias. Considered here are mathematical models based on ordinary differential equations (ODEs) that describe the ionic currents at the myocardial cell level. Generating an efficient numerical solution of these ODEs is a challenging task, and in fact the physiological accuracy of tissue-scale models is often limited by the efficiency of the numerical solution process. In this paper, we examine the efficiency of the numerical solution of 4 cardiac electrophysiological models using implicit-explicit Runge–Kutta (IMEX-RK) splitting methods. We find that variable step-size implementations of IMEX-RK methods (ARK3 and ARK5) that take advantage of Jacobian structure clearly outperform the methods commonly used in practice.

Authors: Raymond J. Spiteri, Ryan C. Dean

Download: tbme2007

A PSE for the numerical solution of nonlinear algebraic equations

Abstract:

Nonlinear algebraic equations (NAEs) occur routinely in many scientific and engineering problems. The process of solving these NAEs involves many challenges, from finding a suitable initial guess to choosing an appropriate convergence criterion. In practice, Newton’s method is the most widely used robust, general-purpose method for solving systems of NAEs. Many variants of Newton’s method exist. However, it is generally impossible to know a priori which variant of Newton’s method will be effective for a given problem. Moreover, the user usually has little control over many aspects of a software library for solving NAEs. For example, the user may not be able to specify easily a particular linear system solver for the Newton direction. This paper describes a problem- solving environment (PSE) called pythNon for solving systems of NAEs. In pythNon, users have direct and convenient access to many aspects of the solution process not ordinarily available in publicly available numerical software libraries. Consequently, the framework provided by pythNon facilitates a much wider exploration of strategies for solving NAEs than is otherwise presently possible. We give some examples to show how pythNon can be used.

Authors: Raymond J. Spiteri and Thian-Peng Ter

Download: pythNon

Observations on the fifth-order WENO method with non-uniform meshes

Abstract:

The weighted essentially non-oscillatory (WENO) methods are a popular high-order spatial discretization for hyperbolic partial differential equations. Typical treatments of WENO methods assume a uniform mesh. In this paper we give explicit formulas for the finite-volume, fifth-order WENO (WENO5) method on non-uniform meshes in a way that is amenable to efficient implementation. We then compare the performance of the non-uniform mesh approach with the classical uniform mesh approach for the finite-volume formulation of the WENO5 method. We find that the numerical results significantly favor the non-uniform mesh approach both in terms of computational efficiency as well as memory usage. We expect this investigation to provide a basis for future work on adaptive mesh methods coupled with the finite-volume WENO methods.

Authors: Rong Wang, Hui Feng, Raymond J. Spiteri

Download: nuweno

Concurrent Programming and Composite Newton Methods

Abstract:

The most widely used, robust, and general-purpose numerical methods for approximating the solution to systems of nonlinear algebraic equations (NAEs) are based on Newton’s method. Many variants of Newton’s method exist in order to take advantage of problem structure. However, no Newton variant converges quickly for all problems and initial guesses. It is generally impossible to know a priori which variant of Newton’s method will be effective for a given problem: some variants and initial guesses may not lead to convergence at all, or if they do, the convergence may be extremely slow. New multi-core computer architectures allow the use of multiple Newton variants in parallel to potentially enhance the overall convergence for a given problem. For example, by sharing intermediate results each variant can make use of the best iterate generated thus far. This results in a sequential combination of Newton variants that we call a composite Newton method. In this paper, we survey concurrent pro- gramming techniques, describe an implementation of composite Newton methods, and give some experimental results.

Author: Craig Thompson

Advisor: Raymond J. Spiteri

Download: cthompson_bsc_report

Translating Parameter Estimation Problems from EASY-FIT to SOCS

Abstract:

Mathematical models often involve unknown parameters that must be fit to experimental data. These so-called parameter estimation problems have many applications that may involve differential equations, optimization, and control theory. EASY-FIT and SOCS are two software packages that solve parameter estimation problems [15], [7]. In this thesis, we discuss the design and implementation of a source-to-source translator called EF2SOCS used to translate EASY-FIT input into SOCS input. This makes it possible to test SOCS on a large number of parameter estimation problems available in the EASY-FIT problem database that vary both in size and difficulty.

Parameter estimation problems typically have many locally optimal solutions, and the solution obtained often depends critically on the initial guess for the solution. A 3-stage approach is followed to enhance the convergence of solutions in SOCS. The stages are designed to use an initial guess that is progressively closer to the optimal solution found by EASY-FIT. Using this approach we run EF2SOCS on all translatable problems (691) from the EASY-FIT database. We find that all but 7 problems produce converged solutions in SOCS. We describe the reasons that SOCS was not able solve these problems, compare the solutions found by SOCS and EASY-FIT, and suggest possible improvements to both EF2SOCS and SOCS.

Author: Matthew W. Donaldson

Advisor: Raymond J. Spiteri

Download: mdonaldson_msc_thesis

Linear instability of the fifth-order WENO method

Abstract:

The weighted essentially nonoscillatory (WENO) methods are popular spatial discretization methods for hyperbolic partial differential equations. In this paper we show that the combination of the widely used fifth-order WENO spatial discretization (WENO5) and the forward Euler time integration method is linearly unstable when numerically integrating hyperbolic conservation laws. Consequently it is not convergent. Furthermore we show that all two-stage, second-order explicit Runge–Kutta (ERK) methods are linearly unstable (and hence do not converge) when coupled with WENO5. We also show that all optimal first- and second-order strong-stability-preserving (SSP) ERK methods are linearly unstable when coupled with WENO5. Moreover the popular three-stage, third-order SSP(3,3) ERK method offers no linear stability advantage over non-SSP ERK methods, including ones with negative coefficients, when coupled with WENO5. We give new linear stability criteria for combinations of WENO5 with general ERK methods of any order. We find that a sufficient condition for the combination of an ERK method and WENO5 to be linearly stable is that the linear stability region of the ERK method should include the part of the imaginary axis of the form [−ιμ,ιμ] for some μ > 0. The linear stability analysis also provides insight into the behavior of ERK methods applied to nonlinear problems and problems with discontinuous solutions. We confirm the assertions of our analysis by means of numerical tests.

Authors: Rong Wang, Raymond J. Spiteri

Download: 63786-gg