arXiv subjects

Gabriel J. Lord

Publications and source records attributed to Gabriel J. Lord.

17 recordsLinked to original sources

arXiv2026

State and Parameter Estimation for a Neural Model of Local Field Potentials

The study of cortical dynamics during different states such as decision making, sleep and movement, is an important topic in Neuroscience. Modelling efforts aim to relate the neural rhythms present in cortical recordings to the underlying dynamics responsible for their emergence. We present an effort to characterize the neural activity from the cortex of a mouse during natural sleep, captured through local field potential measurements. Our approach relies on using a discretized Wilson--Cowan Amari neural field model for neural activity, along with a data assimilation method that allows the Bayesian joint estimation of the state and parameters. We demonstrate the feasibility of our approach on synthetic measurements before applying it to a dataset available in literature. Our findings suggest the potential of our approach to characterize the stimulus received by the cortex from other brain regions, while simultaneously inferring a state that aligns with the observed signal.

q-bio.NC↗

arXiv2026

On a class of constrained particle filters for continuous-discrete state space models

Particle filters (PFs) are recursive Monte Carlo algorithms for Bayesian tracking and prediction in state space models. This paper addresses continuous-discrete filtering problems, where the hidden state evolves as an Itô stochastic differential equation (SDE) and observations arrive at discrete times. We propose a novel class of constrained PFs that enforce compact support on the state at each observation instant, thereby limiting exploration to plausible regions of the state space. Unlike earlier approaches that truncate the likelihood, the proposed method constrains the dynamics directly, yielding improved numerical stability. Under standard regularity assumptions, we prove convergence of the constrained filter, derive uniform-in-time error estimates, and extend the analysis to account for discretisation errors arising from numerical SDE solvers. A numerical study on a stochastic Lorenz-96 system demonstrates the practical application of the methodology when the constraint is implemented via barrier functions.

stat.CO↗

arXiv2026

Numerical approximation of SDEs driven by fractional Brownian motion for all $H\in(0,1)$ using WIS integration

We examine the numerical approximation of a quasilinear stochastic differential equation (SDE) with multiplicative fractional Brownian motion. The stochastic integral is interpreted in the Wick-Itô-Skorohod (WIS) sense that is well defined and centered for all $H\in(0,1)$. We give an introduction to the theory of WIS integration before we examine existence and uniqueness of a solution to the SDE. We then introduce our numerical method which is based on previous theoretical results for $H\geq \frac{1}{2}$. We construct explicitly a translation operator required for the practical implementation of the method and are not aware of any other implementation of a numerical method for the WIS SDE. We then prove a strong convergence result that gives, in the autonomous case, an error of $O(Δt^H)$ and in the non-autonomous case $O(Δt^{\min(H,ζ)})$, where $ζ$ is a time-Hölder continuity parameter. We present some numerical experiments and conjecture that the theoretical results may not be optimal since we observe numerically a rate of $\min(H+\frac{1}{2},1)$ in the autonomous case. This work opens up the possibility to efficiently simulate SDEs for all $H$ values, including small values of $H$ when the stochastic integral is interpreted in the WIS sense.

math.NA↗

arXiv2026

On a class of constrained Bayesian filters and their numerical implementation in high-dimensional state-space Markov models

Bayesian filtering is a key tool in many problems that involve the online processing of data, including data assimilation, optimal control, nonlinear tracking and others. Unfortunately, the implementation of filters for nonlinear, possibly high-dimensional, dynamical systems is far from straightforward, as computational methods have to meet a delicate trade-off involving stability, accuracy and computational cost. In this paper we investigate the design, and theoretical features, of constrained Bayesian filters for state space models. The constraint on the filter is given by a sequence of compact subsets of the state space that determines the sources and targets of the Markov transition kernels in the dynamical model. Subject to such constraints, we provide sufficient conditions for filter stability and approximation error rates with respect to the original (unconstrained) Bayesian filter. Then, we look specifically into the implementation of constrained filters in a continuous-discrete setting where the state of the system is a continuous-time stochastic Itô process but data are collected sequentially over a time grid. We propose an implementation of the constraint that relies on a data-driven modification of the drift of the Itô process using barrier functions, and discuss the relation of this scheme with methods based on the Doob $h$-transform. Finally, we illustrate the theoretical results and the performance of the proposed methods in computer experiments for a partially-observed stochastic Lorenz 96 model.

stat.ME↗

arXiv2025

Neural Field Equations with random data

We study neural field equations, which are prototypical models of large-scale cortical activity, subject to random data. We view this spatially-extended, nonlocal evolution equation as a Cauchy problem on abstract Banach spaces, with randomness in the synaptic kernel, firing rate function, external stimuli, and initial conditions. We determine conditions on the random data that guarantee existence, uniqueness, and measurability of the solution in an appropriate Banach space, and examine the regularity of the solution in relation to the regularity of the inputs. We present results for linear and nonlinear neural fields, and for the two most common functional setups in the numerical analysis of this problem. In addition to the continuous problem, we analyse in abstract form neural fields that have been spatially discretised, setting the foundations for analysing uncertainty quantification (UQ) schemes.

math.NA↗

arXiv2025

Stochastic collocation schemes for Neural Field Equations with random data

We develop and analyse numerical schemes for uncertainty quantification in neural field equations subject to random parametric data in the synaptic kernel, firing rate, external stimulus, and initial conditions. The schemes combine a generic projection method for spatial discretisation to a stochastic collocation scheme for the random variables. We study the problem in operator form, and derive estimates for the total error of the schemes, in terms of the spatial projector. We give conditions on the projected random data which guarantee analyticity of the semi-discrete solution as a Banach-valued function. We illustrate how to verify hypotheses starting from analytic random data and a choice of spatial projection. We provide evidence that the predicted convergence rates are found in various numerical experiments for linear and nonlinear neural field problems.

math.NA↗

arXiv2024

Strong Convergence of a Splitting Method for the Stochastic Complex Ginzburg-Landau Equation

We consider the numerical approximation of the stochastic complex Ginzburg-Landau equation with additive noise on the one dimensional torus. The complex nature of the equation means that many of the standard approaches developed for stochastic partial differential equations can not be directly applied. We use an energy approach to prove an existence and uniqueness result as well to obtain moment bounds on the stochastic PDE before introducing our numerical discretization. For such a well studied deterministic equation it is perhaps surprising that its numerical approximation in the stochastic setting has not been considered before. Our method is based on a spectral discretization in space and a Lie-Trotter splitting method in time. We obtain moment bounds for the numerical method before proving our main result: strong convergence on a set of arbitrarily large probability. From this we obtain a result on convergence in probability. We conclude with some numerical experiments that illustrate the effectiveness of our method.

math.NA↗

arXiv2024

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.

math.NA↗

arXiv2023

An adaptive splitting method for the Cox-Ingersoll-Ross process

We propose a new splitting method for strong numerical solution of the Cox-Ingersoll-Ross model. For this method, applied over both deterministic and adaptive random meshes, we prove a uniform moment bound and strong error results of order $1/4$ in $L_1$ and $L_2$ for the parameter regime $κθ>σ^2$. We then extend the new method to cover all parameter values by introducing a \emph{soft zero} region (where the deterministic flow determines the approximation) giving a hybrid type method to deal with the reflecting boundary. From numerical simulations we observe a rate of order $1$ when $κθ>σ^2$ rather than $1/4$. Asymptotically, for large noise, we observe that the rates of convergence decrease similarly to those of other schemes but that the proposed method making use of adaptive timestepping displays smaller error constants.

math.NA↗

arXiv2021

Strong Convergence of a GBM Based Tamed Integrator for SDEs and an Adaptive Implementation

We introduce a tamed exponential time integrator which exploits linear terms in both the drift and diffusion for Stochastic Differential Equations (SDEs) with a one sided globally Lipschitz drift term. Strong convergence of the proposed scheme is proved, exploiting the boundedness of the geometric Brownian motion (GBM) and we establish order 1 convergence for linear diffusion terms. In our implementation we illustrate the efficiency of the proposed scheme compared to existing fixed step methods and utilize it in an adaptive time stepping scheme. Furthermore we extend the method to nonlinear diffusion terms and show it remains competitive. The efficiency of these GBM based approaches are illustrated by considering some well-known SDE models.

math.NA↗

arXiv2019

Local time stepping methods and discontinuous Galerkin methods applied to diffusion advection reaction equations

This paper is focussed on the numerical resolution of diffusion advection and reaction equations (DAREs) with special features (such as fractures, walls, corners, obstacles or point loads) which globally, as well as locally, have important effects on the solution. We introduce a multilevel and local time solver of DAREs based on the discontinuous Galerkin (DG) method for the spatial discreization and time stepping methods such as exponential time differencing (ETD), exponential Rosenbrock (EXPR) and implicit Euler (Impl) methods. The efficiency of our solvers is shown with several experiments on cyclic voltammetry models and fluid flows through domains with fractures.

math.NA↗

arXiv2016

Adaptive timestepping strategies for nonlinear stochastic systems

We introduce a class of adaptive timestepping strategies for stochastic differential equations with non-Lipschitz drift coefficients. These strategies work by controlling potential unbounded growth in solutions of a numerical scheme due to the drift. We prove that the Euler-Maruyama scheme with an adaptive timestepping strategy in this class is strongly convergent. Specific strategies falling into this class are presented and demonstrated on a selection of numerical test problems. We observe that this approach is broadly applicable, can provide more dynamically accurate solutions than a drift-tamed scheme with fixed stepsize, and can improve MLMC simulations.

math.NA↗

arXiv2016

A New Class of Exponential Integrators for Stochastic Differential Equations With Multiplicative Noise

In this paper, we present new types of exponential integrators for Stochastic Differential Equations (SDEs) that take the advantage of the exact solution of (generalised) geometric Brownian motion. We examine both Euler and Milstein versions of the scheme and prove strong convergence. For the special case of linear noise we obtain an improved rate of convergence for the Euler version over standard integration methods. We investigate the efficiency of the methods compared with other exponential integrators and show that by introducing a suitable homotopy parameter these schemes are competitive not only when the noise is linear but also in the presence of nonlinear noise terms.

math.NA↗

arXiv2011

Effects of noise on models of spiny dendrites

We study the effects of noise in two models of spiny dendrites. Through the introduction of different types of noise to both the Spike-diffuse-spike (SDS) and Baer-Rinzel (BR) models we investigate the change in behaviour of the travelling wave solutions present in the deterministic systems, as noise intensity increases. We show that the speed of wave propagation in the SDS and BR models respectively decreases and increases as the noise intensity in the spine heads increases. Interestingly the discrepancy between the models does not seem to arise from the type of active spine head dynamics employed by the model but rather by the form of the spine density used. In contrast the cable is very robust to noise and as such the speed shows very little variation from the deterministic system. We look at the effect of the noise interpretation used to evaluate the stochastic integral; Ito or Statonovich and discuss which may be appropriate. We also show that the correlation time and length scales of the noise can enhance propagation of travelling wave solutions where the white noise dominates the signal and produces noise induced phenomena.

math.DS↗

arXiv2010

Stochastic Exponential Integrators for a Finite Element Discretization of SPDEs

We consider the numerical approximation of general semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive space-time noise. In contrast to the standard time stepping methods which uses basic increments of the noise and the approximation of the exponential function by a rational fraction, we introduce a new scheme, designed for finite elements, finite volumes or finite differences space discretization, similar to the schemes in \cite{Jentzen3,Jentzen4} for spectral methods and \cite{GTambue} for finite element methods. We use the projection operator, the smoothing effect of the positive definite self-adjoint operator and linear functionals of the noise in Fourier space to obtain higher order approximations. We consider noise that is white in time and either in $H^1$ or $H^2$ in space and give convergence proofs in the mean square $L^{2}$ norm for a diffusion reaction equation and in mean square $ H^{1}$ norm in the presence of an advection term. For the exponential integrator we rely on computing the exponential of a non-diagonal matrix. In our numerical results we use two different efficient techniques: the real fast \Leja points and Krylov subspace techniques. We present results for a linear reaction diffusion equation in two dimensions as well as a nonlinear example of two-dimensional stochastic advection diffusion reaction equation motivated from realistic porous media flow.

math.NA↗

arXiv2006

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.

math.AP↗

arXiv2005

Cylinder Buckling: The Mountain Pass as an Organizing Center

We revisit the classical problem of the buckling of a long thin axially compressed cylindrical shell. By examining the energy landscape of the perfect cylinder we deduce an estimate of the sensitivity of the shell to imperfections. Key to obtaining this is the existence of a mountain pass point for the system. We prove the existence on bounded domains of such solutions for all most all loads and then numerically compute example mountain pass solutions. Numerically the mountain pass solution with lowest energy has the form of a single dimple. We interpret these results and validate the lower bound against some experimental results available in the literature.

math.AP↗