exponential timestepping | nonlinear dielectrics | SDEs | SPDEs | lasers | noise-controlled dynamics | Numerical methods for stochastic differential equationsNewton's second law of motion relates force to acceleration. Consequently, second-order differential equations are common in scientific applications, in the guise of ``Langevin'', ``Monte Carlo'', ``molecular'' or ``dissipative particle'' dynamics, and the study of methods for second-order ordinary differential equations is one of the most mature branches of numerical analysis. Numerical methods that preserve geometric properties of a the flow of a differential equation are increasingly useful in deterministic dynamics when it is important that the numerics reproduce the dynamics as accurately as possible over long times. In particular, when solving separable Hamiltonian problems, explicit partitioned Runge-Kutta (PRK) methods can be constructed that are symplectic. In stochastic dynamics, an ensemble of different trajectories originates even from a single initial condition, corresponding to different realizations of the stochastic process. The geometric property of the flow defined by a bundle of neighbouring initial conditions is not relevant. There is, however, an appropriate long-term quantity: the stationary density. We construct explicit PRK methods that approximate the stationary density with high-order accuracy. Numerical methods for second-order for stochastic differential equations.Kevin Burrage, Ian Lennane and Grant LytheSIAM Journal on Scientific Computing 29 245-264 (2007) Accurate Stationary Densities with Partitioned Numerical Methods for Stochastic Differential EquationsSIAM Journal on Numerical Analysis 471601-1618 (2009)Kevin Burrage and Grant Lythe Accurate stationary densities with partitioned numerical methods for stochastic partial differential equationsStochastic Partial Differential Equations: Analysis and Computations 2 262-280 (2014)Kevin Burrage and Grant Lythe Kink stochasticsScientists in many fields face a similar challenge: to understand a large system, driven by many noisy and nonlinear influences, containing persistent identifiable structures. The challenge is being met with a two-pronged approach. Firstly, using the most powerful computers available, perform numerical simulations of the full model on the largest domain and with the highest spatial resolution possible, for as long a time as possible. Secondly, develop theoretical methods that efficiently identify the structures of interest and predict their number, structure, dynamics and interactions. Noise, fluctuations or randomness is part of every real phenomenon. It provides an extra fascinating element for the nonlinear-dynamicist to study. Just as differential equations were invented to describe deterministic dynamics, stochastic differential equations are the tool needed to describe dynamics that have a random, or stochastic, component. In practice, solving a nonlinear stochastic differential equation is like solving an nonlinear ordinary differential equation: exact analytical solutions are seldom available, but paths can be generated in a few seconds on a computer and analysed using tools from pure and applied mathematics. A new area is the study of stochastic partial differential equations, which describe spatially extended systems with randomness, such as vortices in a fluctuating environment. Even though the theory of such equations is still in its infancy, the recent rapid increase in the computing power is starting to make it possible to explore their dynamics. The focus of my research in stochastic partial differential equations is on the statistics and dynamics of coherent structures that maintain their identity as they move and are buffeted by local fluctuations. In their simplest manifestation they are known as kinks and a systematic study is possible, which will serve as a stepping stone towards the study of more complicated structures such as vortices.
More on kinksNucleationMany spatially-extended systems in physics, chemistry or biology exhibit two locally-stable states or phases. The two states coexist in the sense that, at any one time, different parts of the system are in different states. Fluctuations cause transitions of part of the system from one state to another, called nucleation events. We consider a string of particles, each coupled to its two neighbours, subject to noise and to the double-well potential: Click on the image below to see the system in real time (java applet) Diffusion-limited reactionReaction rates controlled by collisions between diffusing particles depend on the distribution of distances between particles as well as on the density of particles. In particular, as Noyes stated in 1961 Any rigorous treatment of chemical kinetics in solution must consider concentration gradients that are established by the existence of the reaction itselfWe study the dynamics of point particles in one dimension, nucleated at random positions and times then diffusing until colliding and annihilating with another particle. Competition between nucleation and annihilation produces a statistically steady state with a well-defined mean density of particles and distribution of distances between particles.
r_{0} ∼ Γ^{1/2} (paired)versus r_{0} ∼ Q^{1/3} (unpaired).We exhibit the crossover between these two cases in terms of the following dimensionless quantity: ε = (2Γ/D)^{1/3} b. Diffusion-limited reaction in one dimensionGrant Lythe Physica D 222 159-163 (2006)Diffusion-limited reaction in one dimension: Paired and unpaired nucleationSalman Habib, Katja Lindenberg, Grant Lythe and Carmen Molina-París.Journal of Chemical Physics 115 73 (2001) The exponential timestepping method for stochastic differential equations
Efficient numerical solution of stochastic differential equations using exponential timesteppingJournal of Statistical Physics 100 1097 (2000)Exponential timestepping with boundary test for stochastic differential equations.SIAM Journal of Scientific Computing 24 1809 (2003)Multidimensional exponential timestepping with boundary test.SIAM Journal of Scientific Computing 27 793-808 (2005)Noise and nonlinear dielectricsNonlinear dielectrics, such as SrTiO3 (STO), present unique opportunities to develop practical electrically tunable devices, and also to explore novel scientific and technological concepts that exploit strong nonlinearities.
Transmission, reflection and second-harmonic generation in a nonlinear waveguideRoberto Camassa, Alp T. Findikoglu and Grant LytheSIAM Journal of Applied Mathematics 66 1-28 (2005) Dielectric nonlinearity and stochastic effects in strontium titanateAlp Findikoglu, Roberto Camassa, Grant Lythe and Q.X. Jia Applied Physics Letters 80 3391 (2002)New potential applications of nonlinear dielectricsAlp Findikoglu et al Integrated Ferroelectrics 22 259-268Stochastic differential equationsDifferential equations have long been used to model the dynamics of physical systems. With the availability of computers, the tendency to focus only on analysis of linear equations is being replaced by a methodology that profits from a judicious mixture of numerical generation of paths, bifurcation theory and asymptotic analysis. However, when random perturbations (i.e. noise) play an important role, this new spirit is not so widespread. One reason is that the mathematical tools appropriate for describing stochastic paths are not sufficiently well-known. It is partly as a result of this that there is a widespread misconception that noise acts only to smear out deterministic dynamics. Solving a stochastic differential equation (SDE) is akin to solving an ordinary differential equation: exact analytical solutions are seldom available, but paths can be generated in a matter of seconds on a computer. Analysis is based on Ito calculus, that permits computation of experimentally accessible quantities inaccessible to traditional methods. I perform numerics and analysis of nonlinear stochastic differential equations, stochastic partial differential equations and differential delay equations. Stochastic calculus: application to dynamic bifurcations and threshold crossingsKalvis Jansons and Grant Lythe Journal of Statistical Physics 90 227-251 (1998)Stochastic partial differential equations(.ps file)Stochastic partial differential equations describe continuum systems with noise. The noise can be white in space and time, described solely by an amplitude, or can have space or time scales of its own. The numerical and analytical tools for solving SPDEs are generalisations of the powerful stochastic analysis used for ODEs with noise. Because they focus on one realisation at a time, stochastic DEs are natural tools when noise is an active part of the dynamics, not merely an agent that rattles the system around in a pre-determined potential. In a simple example of a non-equilibrium spatially extended system, microscopic white noise produces a characteristic macroscopic domain size that is a function the rate of change of the parameter and the amplitude of the noise. Domain formation in transitions with noise and a time-dependent bifurcation parameterGrant Lythe Physical Review E 53 R4271-4274 (1996)Dynamics of defect formationEsteban Moro and Grant Lythe Physical Review E 59 R1303-1306 (1999)Stochastic PDEs: convergence to the continuum?Salman Habib and Grant Lythe. Computer Physics Communications 142 29 (2001)Semiconductor laser dynamicsSemiconductor lasers have a wide range of applications because they are of relatively small size, they can be massively produced at low cost, and they are easy to operate. Despite their successful technology, semiconductor lasers are very sensitive to any external perturbation. A small amount of optical feedback resulting from the reflection from an optical disk or from the end of an optical fiber is sufficient to generate pulsating instabilities. Systematic experimental studies of semiconductor lasers, in particular time series analysis, is not possible because the timescale of the intensity pulsations is typically in the picosecond regime. Most of the progress in understanding these bifurcations comes from extensive numerical studies of simple models and their comparison to the experimentally obtained Fourier spectra.Coexisting periodic attractors in injection-locked diode lasersA. Gavrielides, V. Kovanis, P.M. Farangis, T. Erneux and G. LytheQuantum and semiclassical optics 9 785 (1997) Slowly passing through resonance strongly depends on noiseJ.C. Celet, D. Dangoisse, P. Glorieux, G. Lythe and T. ErneuxPhysical Review Letters 81 975-978 (1998)
The low pump limit of the bifurcation to periodic intensities in a semiconductor laser subject to external optical feedbackG.D. Lythe and T. Erneux, A. Gavrielides and V. KovanisPhysical Review A 55 4443-4448, (1997) Noise-controlled dynamicsSmall amounts of noise can dramatically change and simplify non-equilibrium dynamics. Examples include slow-fast dynamics, when long quiescent phases are occassionally interrupted by bursts of activity; and spatially extended systems undergoing pattern-forming transitions such as annealing and the formation of convective patterns. Macroscopic length- and time-scales are found to emerge from microscopic fluctuations.In noise-controlled dynamics, length and time scales depend on the product of a slow inverse time and the logarithm of the noise level. Examples are described in ordinary and partial differential equations modelling laser and fluid systems. The simplest example of the disproportionate and simplifying effect of noise is a dynamic bifurcation. In the corresponding situation for a spatially extended system, microscopic noise produces a characteristic macroscopic domain size.
Stochastic Stokes Drift
Stochastic Stokes' driftKalvis Jansons and Grant Lythe Physical Review Letters 81 3136-3139Sensory detection in noisy environmentsMany animals manage to detect a weak signal against a noisy background. The key to understanding this may lie in the concept of stochastic resonance, a mechanism by which noise actually assists, rather than hinders, detection of signals. Stochastic resonance has been demonstrated in a wide variety of mathematical models, from simple bistable systems to full-scale models of auditory systems and of neural cell networks. The challenge is to establish whether it is indeed used by living animals. An example that is being actively investigated at Leeds is found in moths, which have tympanic organs sensitive to ultrasound that enable them to detect the echolocation calls of bat predators and take evasive action. Detection involves the identification of a periodic train of bat echolocation calls against a background of biotic and abiotic noise. Resolving the puzzle of how the two-celled moth organ is capable of performing this detection will require integrating experiment and theory from beginning to end, including stochastic modelling and statistical analysis of time series. A systems approach is needed to understand the throughput information from the sensory cells to the behavioural consequences, with input from neurophysiology, signal processing, behavioural ecology and evolutionary modelling. |