Accurate stationary densities with partitioned numerical methods for stochastic partial differential equations

We consider the numerical solution, by finite differences, of second-order-in-time stochastic partial differential equations (SPDEs) in one space dimension. New timestepping methods are introduced by generalising recently-introduced methods for second-order-in-time stochastic differential equations to multidimensional systems. These stochastic methods, based on leapfrog and Runge–Kutta methods, are designed to give good approximations to the stationary variances and the correlations in the position and velocity variables. In particular, we introduce the reverse leapfrog method and stochastic Runge–Kutta Leapfrog methods, analyse their performance applied to linear SPDEs and perform numerical experiments to examine their accuracy applied to a type of nonlinear SPDE

Numerical approximation of SDEs and stochastic Swift-Hohenberg equation

We consider the numerical approximation of stochastic di®erential equations inter- preted both in the It^o and Stratonovich sense and develop three stochastic time- integration techniques based on the deterministic exponential time di®erencing schemes. Two of the numerical schemes are suited for the simulations of It^o stochastic ordinary di®erential equations (SODEs) and they are referred to as the stochastic exponential time di®erencing schemes, SETD0 and SETD1. The third numerical scheme is a new numerical method we propose for the simulations of Stratonovich SODEs. We call this scheme, the Exponential Stratonovich Integrator (ESI). We investigate numerically the convergence of these three numerical methods, in ad- dition to three standard approximation schemes and also compare the accuracy and e±ciency of these schemes. The e®ect of small noise is also studied. We study the theoretical convergence of the stochastic exponential time di®erencing scheme (SETD0) for parabolic stochastic partial di®erential equations (SPDEs) with in¯nite-dimensional additive noise and one-dimensional multiplicative noise. We ob- tain a strong error temporal estimate of O(¢tµ + ²¢tµ + ²2¢t1=2) for SPDEs forced with a one-dimensional multiplicative noise and also obtain a strong error temporal estimate of O(¢tµ + ²2¢t) for SPDEs forced with an in¯nite-dimensional additive noise. We examine convergence for second-order and fourth-order SPDEs. We consider the e®ects of spatially correlated and uncorrelated noise on bifurcations for SPDEs. In particular, we study a fourth-order SPDE, the Swift-Hohenberg equa- tion, and allow the control parameter to °uctuate. Numerical simulations show a shift in the pinning region with multiplicative noise

Numerical simulations of the spiking activity and the related first exit time of stochastic neural systems

The aim of this thesis was to study, using numerical simulation techniques, the possible effects of an additive noise on the firing properties of stochastic neural models, and the related first exit time problems. The research is divided into three main investigations. First, using SDELab, mathematical software for solving stochastic differential equations within MATLAB, we examine the influence of an additive noise on the output spike trains for the space-clamped Hodgkin Huxley (HH) model and the spatially-extended FitzHugh Nagumo (FHN) system. We find that a suitable amount of additive noise can enhance the regularity of the repetitive spiking of the space-clamped HH model. Meanwhile, we find the FHN system to be sensitive to noise, requiring that very small values of noise are chosen, in order to produce regular spikes. Second, under additive noise, we use fixed and exponential time-stepping Euler algorithms, with boundary tests, to calculate the mean first exit times (MFET) for one-dimensional neural diffusion models, represented by a stochastic space-clamped FHN system and the Ornstein-Uhlenbeck (OU) model. The strategies and theory behind these numerical methods and their convergence rates in the MFET are also considered. We find that, for different values of noise, these methods with boundary tests can improve the rate of convergence from order one half to order one, which coincides with previous studies. Finally, we look at spatially-extended systems, represented by the Barkley system with additive noise that is white in time and correlated in space, calculating mean nucleation times and mean lifetimes of traveling waves, using an efficient numerical simulation. A simple model of the dynamics of the underlying Barkley model is introduced, in order to compute the mean lifetimes, particulary for interacting waves. The reduced model is easy to use and allows us to explore the full dynamics of the kinks and antikinks, in particular over long periods. One application of the reduced model is to calculate the mean number of kinks at a given time and use this to obtain the probability that the system is excitable at a given position. With these three investigations into the effects of additive noise on stochastic neural models, we have demonstrated some of the interesting results that can be achieved using numerical techniques. We hope to extend this work, in the future, to include the effects of multiplicative noise.EThOS - Electronic Theses Online ServiceGBUnited Kingdo