409 research outputs found
Populations of models, Experimental Designs and coverage of parameter space by Latin Hypercube and Orthogonal Sampling
In this paper we have used simulations to make a conjecture about the
coverage of a dimensional subspace of a dimensional parameter space of
size when performing trials of Latin Hypercube sampling. This takes the
form . We suggest that this coverage formula is
independent of and this allows us to make connections between building
Populations of Models and Experimental Designs. We also show that Orthogonal
sampling is superior to Latin Hypercube sampling in terms of allowing a more
uniform coverage of the dimensional subspace at the sub-block size level.Comment: 9 pages, 5 figure
On the analysis of mixed-index time fractional differential equation systems
In this paper we study the class of mixed-index time fractional differential
equations in which different components of the problem have different time
fractional derivatives on the left hand side. We prove a theorem on the
solution of the linear system of equations, which collapses to the well-known
Mittag-Leffler solution in the case the indices are the same, and also
generalises the solution of the so-called linear sequential class of time
fractional problems. We also investigate the asymptotic stability properties of
this class of problems using Laplace transforms and show how Laplace transforms
can be used to write solutions as linear combinations of generalised
Mittag-Leffler functions in some cases. Finally we illustrate our results with
some numerical simulations.Comment: 21 pages, 6 figures (some are made up of sub-figures - there are 15
figures or sub-figures
Inferring diffusion in single live cells at the single molecule level
The movement of molecules inside living cells is a fundamental feature of
biological processes. The ability to both observe and analyse the details of
molecular diffusion in vivo at the single molecule and single cell level can
add significant insight into understanding molecular architectures of diffusing
molecules and the nanoscale environment in which the molecules diffuse. The
tool of choice for monitoring dynamic molecular localization in live cells is
fluorescence microscopy, especially so combining total internal reflection
fluorescence (TIRF) with the use of fluorescent protein (FP) reporters in
offering exceptional imaging contrast for dynamic processes in the cell
membrane under relatively physiological conditions compared to competing single
molecule techniques. There exist several different complex modes of diffusion,
and discriminating these from each other is challenging at the molecular level
due to underlying stochastic behaviour. Analysis is traditionally performed
using mean square displacements of tracked particles, however, this generally
requires more data points than is typical for single FP tracks due to
photophysical instability. Presented here is a novel approach allowing robust
Bayesian ranking of diffusion processes (BARD) to discriminate multiple complex
modes probabilistically. It is a computational approach which biologists can
use to understand single molecule features in live cells.Comment: combined ms (1-37 pages, 8 figures) and SI (38-55, 3 figures
Impact of tissue microstructure on a model of cardiac electromechanics based on MRI data
Cardiac motion is a vital process as it sustains the pumping of blood in the body. For this reason motion abnormalities are often associated with severe cardiac pathologies. Clinical imaging techniques, such as MRI, are powerful in assessing motion abnormalities but their connection with pathology often remains unknown.

Computational models of cardiac motion, integrating imaging data, would thus be of great help in linking tissue structure (i.e. cells organisation into fibres and sheets) to motion abnormalities and to pathology. Current models, though, are not able yet to correctly predict realistic cardiac motion in the healthy or diseased heart.

Our hypothesis is that a more realistic description of tissue structure within an electromechanical model of the heart, with structural information extracted from data rather than mathematically defined, and a more careful definition of tissue material properties, would better represent the high heterogeneity of cardiac tissue, thus improving the predictive power of the model
Efficient simulation of stochastic chemical kinetics with the Stochastic Bulirsch-Stoer extrapolation method
BackgroundBiochemical systems with relatively low numbers of components must be simulated stochastically in order to capture their inherent noise. Although there has recently been considerable work on discrete stochastic solvers, there is still a need for numerical methods that are both fast and accurate. The Bulirsch-Stoer method is an established method for solving ordinary differential equations that possesses both of these qualities.ResultsIn this paper, we present the Stochastic Bulirsch-Stoer method, a new numerical method for simulating discrete chemical reaction systems, inspired by its deterministic counterpart. It is able to achieve an excellent efficiency due to the fact that it is based on an approach with high deterministic order, allowing for larger stepsizes and leading to fast simulations. We compare it to the Euler ?-leap, as well as two more recent ?-leap methods, on a number of example problems, and find that as well as being very accurate, our method is the most robust, in terms of efficiency, of all the methods considered in this paper. The problems it is most suited for are those with increased populations that would be too slow to simulate using Gillespie’s stochastic simulation algorithm. For such problems, it is likely to achieve higher weak order in the moments.ConclusionsThe Stochastic Bulirsch-Stoer method is a novel stochastic solver that can be used for fast and accurate simulations. Crucially, compared to other similar methods, it better retains its high accuracy when the timesteps are increased. Thus the Stochastic Bulirsch-Stoer method is both computationally efficient and robust. These are key properties for any stochastic numerical method, as they must typically run many thousands of simulations
A discrete least squares collocation method for two-dimensional nonlinear time-dependent partial differential equations
In this paper, we develop regularized discrete least squares collocation and
finite volume methods for solving two-dimensional nonlinear time-dependent
partial differential equations on irregular domains. The solution is
approximated using tensor product cubic spline basis functions defined on a
background rectangular (interpolation) mesh, which leads to high spatial
accuracy and straightforward implementation, and establishes a solid base for
extending the computational framework to three-dimensional problems. A
semi-implicit time-stepping method is employed to transform the nonlinear
partial differential equation into a linear boundary value problem. A key
finding of our study is that the newly proposed mesh-free finite volume method
based on circular control volumes reduces to the collocation method as the
radius limits to zero. Both methods produce a large constrained least-squares
problem that must be solved at each time step in the advancement of the
solution. We have found that regularization yields a relatively
well-conditioned system that can be solved accurately using QR factorization.
An extensive numerical investigation is performed to illustrate the
effectiveness of the present methods, including the application of the new
method to a coupled system of time-fractional partial differential equations
having different fractional indices in different (irregularly shaped) regions
of the solution domain
A Stochastic Method for All Seasons
It is well known that the numerical solution of stochastic ordinary differential equations leads to a step size reduction when explicit methods are used. This has led to a plethora of implicit methods with a wide variety of stability properties. However, for stochastic problems whose eigenvalues lie near the negative real axis, explicit methods with extended stability regions can be very effective. In this paper we extend these ideas to the stochastic realm and present a family of weak order two explicit stochastic Runge-Kutta methods with extended stability intervals that can be used to solve a variety of non-stiff and stiff problems
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