1,203 research outputs found
Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations
Relatively little is known about the ability of numerical methods for stochastic differential equations (SDEs) to reproduce almost sure and small-moment stability. Here, we focus on these stability properties in the limit as the timestep tends to zero. Our analysis is motivated by an example of an exponentially almost surely stable nonlinear SDE for which the Euler-Maruyama (EM)method fails to reproduce this behavior for any nonzero timestep. We begin by showing that EM correctly reproduces almost sure and small-moment exponential stability for sufficiently small timesteps on scalar linear SDEs. We then generalize our results to multidimensional nonlinear SDEs. We show that when the SDE obeys a linear growth condition, EM recovers almost surely exponential stability very well. Under the less restrictive condition that the drift coefficient of the SDE obeys a one-sided Lipschitz condition, where EM may break down, we show that the backward Euler method maintains almost surely exponential stability
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
Simulation of cell movement through evolving environment: a fictitious domain approach
A numerical method for simulating the movement of unicellular organisms which respond to chemical signals is presented. Cells are modelled as objects of finite size while the extracellular space is described by reaction-diffusion partial differential equations. This modular simulation allows the implementation of different models at the different scales encountered in cell biology and couples them in one single framework. The global computational cost is contained thanks to the use of the fictitious domain method for finite elements, allowing the efficient solve of partial differential equations in moving domains. Finally, a mixed formulation is adopted in order to better monitor the flux of chemicals, specifically at the interface between the cells and the extracellular domain
Complexity measures for classes of sequences and cryptographic apllications
Pseudo-random sequences are a crucial component of cryptography, particularly
in stream cipher design. In this thesis we will investigate several measures of
randomness for certain classes of finitely generated sequences.
We will present a heuristic algorithm for calculating the k-error linear complexity
of a general sequence, of either finite or infinite length, and results on the
closeness of the approximation generated.
We will present an linear time algorithm for determining the linear complexity
of a sequence whose characteristic polynomial is a power of an irreducible element,
again presenting variations for both finite and infinite sequences. This algorithm
allows the linear complexity of such sequences to be determined faster than was
previously possible.
Finally we investigate the stability of m-sequences, in terms of both k-error
linear complexity and k-error period. We show that such sequences are inherently
stable, but show that some are more stable than others
Stochastic B-series analysis of iterated Taylor methods
For stochastic implicit Taylor methods that use an iterative scheme to
compute their numerical solution, stochastic B--series and corresponding growth
functions are constructed. From these, convergence results based on the order
of the underlying Taylor method, the choice of the iteration method, the
predictor and the number of iterations, for It\^o and Stratonovich SDEs, and
for weak as well as strong convergence are derived. As special case, also the
application of Taylor methods to ODEs is considered. The theory is supported by
numerical experiments
Shining Light on Modifications of Gravity
Many modifications of gravity introduce new scalar degrees of freedom, and in
such theories matter fields typically couple to an effective metric that
depends on both the true metric of spacetime and on the scalar field and its
derivatives. Scalar field contributions to the effective metric can be
classified as conformal and disformal. Disformal terms introduce gradient
couplings between scalar fields and the energy momentum tensor of other matter
fields, and cannot be constrained by fifth force experiments because the
effects of these terms are trivial around static non-relativistic sources. The
use of high-precision, low-energy photon experiments to search for conformally
coupled scalar fields, called axion-like particles, is well known. In this
article we show that these experiments are also constraining for disformal
scalar field theories, and are particularly important because of the difficulty
of constraining these couplings with other laboratory experiments.Comment: 20 pages, 10 figures. v2: Matches version accepted by JCAP;
additional discussion of the strong coupling scale. Conclusions unchange
Using atom interferometry to detect dark energy
We review the tantalising prospect that the first evidence for the dark energy driving the observed acceleration of the Universe on giga-parsec scales may be found through metre scale laboratory based atom interferometry experiments. To do that, we first introduce the idea that scalar fields could be responsible for dark energy and show that in order to be compatible with fifth force constraints these fields must have a screening mechanism which hides their effects from us within the solar system. Particular emphasis is placed on one such screening mechanism known as the chameleon effect where the field's mass becomes dependent on the environment. The way the field behaves in the presence of a spherical source is determined and we then go on to show how in the presence of the kind of high vacuum associated with atom interferometry experiments, and when the test particle is an atom, it is possible to use the associated interference pattern to place constraints on the acceleration due to the fifth force of the chameleon field - this has already been used to rule out large regions of the chameleon parameter space and maybe one day will be able to detect the force due to the dark energy field in the laboratory
A stochastic model of jaguar abundance in the Peruvian Amazon under climate variation scenarios
The jaguar (Panthera onca) is the dominant predator in Central and South America, but is now considered near-threatened. Estimating jaguar population size is difficult, due to uncertainty in the underlying dynamical processes as well as highly variable and sparse data. We develop a stochastic temporal model of jaguar abundance in the Peruvian Amazon, taking into account prey availability, under various climate change scenarios. The model is calibrated against existing data sets and an elicitation study in Pacaya Samiria. In order to account for uncertainty and variability, we construct a population of models over four key parameters, namely three scaling parameters for aquatic, small land, and large land animals and a hunting index. We then use this population of models to construct probabilistic evaluations of jaguar populations under various climate change scenarios characterized by increasingly severe flood and drought events and discuss the implications on jaguar numbers. Results imply that jaguar populations exhibit some robustness to extreme drought and flood, but that repeated exposure to these events over short periods can result in rapid decline. However, jaguar numbers could return to stabilityāalbeit at lower numbersāif there are periods of benign climate patterns and other relevant factors are conducive
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