A modified semi--implict Euler-Maruyama Scheme for finite element discretization of SPDEs with additive noise

We consider the numerical approximation of a general second order semi--linear parabolic stochastic partial differential equation (SPDE) driven by additive space-time noise. We introduce a new modified scheme using a linear functional of the noise with a semi--implicit Euler--Maruyama method in time and in space we analyse a finite element method (although extension to finite differences or finite volumes would be possible). We prove convergence in the root mean square $L^{2}$ norm for a diffusion reaction equation and diffusion advection reaction equation. We present numerical results for a linear reaction diffusion equation in two dimensions as well as a nonlinear example of two-dimensional stochastic advection diffusion reaction equation. We see from both the analysis and numerics that the proposed scheme has better convergence properties than the standard semi--implicit Euler--Maruyama method

Dendritic cable with active spines: a modeling study in the spike-diffuse-spike framework

The spike-diffuse-spike (SDS) model describes a passive dendritic tree with active dendritic spines. Spine-head dynamics is modelled with a simple integrate-and-fire process, whilst communication between spines is mediated by the cable equation. Here we develop a computational framework that allows the study of multiple spiking events in a network of such spines embedded in a simple one-dimensional cable. This system is shown to support saltatory waves as a result of the discrete distribution of spines. Moreover, we demonstrate one of the ways to incorporate noise into the spine-head whilst retaining computational tractability of the model. The SDS model sustains a variety of propagating patterns

Selection of quasi-stationary states in the stochastically forced Navier-Stokes equation on the torus

The stochastically forced vorticity equation associated with the two dimensional incompressible Navier-Stokes equation on $D_\delta:=[0,2\pi\delta]\times [0,2\pi]$ is considered for $\delta\approx 1$, periodic boundary conditions, and viscocity $0<\nu\ll 1$. An explicit family of quasi-stationary states of the deterministic vorticity equation is known to play an important role in the long-time evolution of solutions both in the presence of and without noise. Recent results show the parameter $\delta$ plays a central role in selecting which of the quasi-stationary states is most important. In this paper, we aim to develop a finite dimensional model that captures this selection mechanism for the stochastic vorticity equation. This is done by projecting the vorticity equation in Fourier space onto a center manifold corresponding to the lowest eight Fourier modes. Through Monte Carlo simulation, the vorticity equation and the model are shown to be in agreement regarding key aspects of the long-time dynamics. Following this comparison, perturbation analysis is performed on the model via averaging and homogenization techniques to determine the leading order dynamics for statistics of interest for $\delta\approx1$.Comment: 23 pages, 27 figure

Numerical variational methods applied to cylinder buckling

We review and compare different computational variational methods applied to a system of fourth order equations that arises as a model of cylinder buckling. We describe both the discretization and implementation, in particular how to deal with a 1 dimensional null space. We show that we can construct many different solutions from a complex energy surface. We examine numerically convergence in the spatial discretization and in the domain size. Finally we give a physical interpretation of some of the solutions found.Comment: 23 pages, 12 figures, 6 table

Faith and Justice: A Platonic Reading of Pauline Justification

Abstract This dissertation is a broad work of philosophical theology that examines the Christian Doctrine of Justification by Faith in light of the Platonic framework for constructing ethics. Prescinding from specific Post-Reformation debates on justification, it seeks to position the philosophic problem of justification in terms of Platonism’s preoccupation with human assimilation to the Divine. It sets out the rich background of Platonism in the Christian tradition, including the heavily monotheistic Middle-Platonism. Relying much on the work of Lloyd Gerson, it lays down a schema of how understand Platonic ethics and Platonism more generally; and draws on the work of George Van Kooten in discerning Platonic motifs in Paul. Through a reading of key Platonic dialogues, especially the Phaedrus, Symposium, and the Republic, the work discerns a schema of Platonic ethics as it relates to justification: the end of all human beings is likeness unto/harmonization with God, but this cannot happen without divine aid. Those who receive this divine aid, or ‘divine gift’ are philosophers, but the philosopher is no mere intellectual, but a lover of God, who lives by a kind of ‘faith that works by love’ (Gal 5:6). The philosopher is finally reconciled to God by the justification of his soul, which consists in the harmonization of his soul after the pattern of divine justice. This schema for Platonic ethics is used as a heuristic tool for exegesis of Romans 1-6, and Galatians 2-3, 5. In Romans, the attempt is made to reconcile language in Romans 2, which speaks of every man being rewarded for his works, who ‘perseveres in doing good’ (2:7), and in Romans 3, which commends justification by faith ‘apart from the law’ (3:21). Platonic concepts concerning the nature of the just soul are used to help clarify the meaning of Christian justification. Paul’s critique of the Law in Galatians is understood against the background of Platonic themes on the inadequacy of written law to provide a complete moral guide. In Galatians, it is understood that Pauline justification is never through faith alone, but specifically by a faith, given by the Spirit, that works by Love (Gal 5-6). The primacy which Platonic ethics gives to divine gift, as the primary author of our love of God, and our striving to see Him, is therefore shown to prefigure the Pauline doctrine of Justification, which, nevertheless, can only be fully understood in light of Christian revelation

Weak Convergence Of Tamed Exponential Integrators for Stochastic Differential Equations

We prove weak convergence of order one for a class of exponential based integrators for SDEs with non-globally Lipschtiz drift. Our analysis covers tamed versions of Geometric Brownian Motion (GBM) based methods as well as the standard exponential schemes. The numerical performance of both the GBM and exponential tamed methods through four different multi-level Monte Carlo techniques are compared. We observe that for linear noise the standard exponential tamed method requires severe restrictions on the stepsize unlike the GBM tamed method.Comment: 24 pages, 3 figure