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A walk outside spheres for the fractional Laplacian
Tony Shardlow
Mar 14, 2019
The solution of the exterior-value problem for the fractional Laplacian can be computed by a Walk Outside Spheres algorithm. This involves sampling α-stable Levy processes on their exit from maximally inscribed balls and...
math.NA Arnoldi algorithm Eigenvalue problems Exterior-value problems Levy processes multilevel Monte Carlo Numerical solution of PDEs Fractional Laplacian Walk on spheres /dk/atira/pure/subjectarea/asjc/2600/2605 name=Computational Mathematics /dk/atira/pure/subjectarea/asjc/2600/2604 name=Applied Mathematics /dk/atira/pure/subjectarea/asjc/2600/2602 name=Algebra and Number Theory
A walk outside spheres for the fractional Laplacian
Tony Shardlow
Mar 14, 2019
The solution of the exterior-value problem for the fractional Laplacian can be computed by a Walk Outside Spheres algorithm. This involves sampling α-stable Levy processes on their exit from maximally inscribed balls and...
math.NA Arnoldi algorithm Eigenvalue problems Exterior-value problems Levy processes multilevel Monte Carlo Numerical solution of PDEs Fractional Laplacian Walk on spheres /dk/atira/pure/subjectarea/asjc/2600/2605 name=Computational Mathematics /dk/atira/pure/subjectarea/asjc/2600/2604 name=Applied Mathematics /dk/atira/pure/subjectarea/asjc/2600/2602 name=Algebra and Number Theory
A coupled Cahn–Hilliard particle system
Tony Shardlow
Aug 13, 2002
A Cahn-Hilliard equation is coupled to a system of stochastic differential equations to model a random growth process. We show the model is well posed and analyze the model asymptotically (in the limit of the interfacial...
Modified equations for stochastic differential equations
Tony Shardlow
Mar 31, 2006
We describe a backward error analysis for stochastic differential equations with respect to weak convergence. Modified equations are provided for forward and backward Euler approximations to Itô SDEs with additive noise, and...
Periodic orbits and unstable manifolds
Tony Shardlow
Jan 01, 1996
Consider the unstable manifold of a hyperbolic periodic orbit of an ordinary differential equation under C1 perturbations of the vector field and under approximation by a one-step numerical method, which is at least first order....
Splitting for dissipative particle dynamics
Tony Shardlow
Dec 31, 2003
We study numerical methods for dissipative particle dynamics, a system of stochastic differential equations for simulating particles interacting pairwise according to a soft potential at constant temperature where the total...
Nucleation of waves in excitable media by noise
Tony Shardlow
Dec 31, 2004
We are interested in reaction-diffusion equations that model excitable media under the influence of an additive noise. In many models of this type, the homogeneous zero state is stable, and interesting dynamics are observed only...
Langevin equations for landmark image registration with uncertainty
Registration of images parameterized by landmarks provides a useful method of describing shape variations by computing the minimum-energy time-dependent deformation field that flows from one landmark set to the other. This is...
Analysis of the geodesic interpolating spline
We study the geodesic interpolating spline with a biharmonic regulariser for solving the landmark image registration problem. We show existence of solutions, discuss uniqueness and show how the problem can be efficiently solved...
Geometric ergodicity for dissipative particle dynamics
Dissipative particle dynamics is a model of multi-phase fluid flows described by a system of stochastic differential equations. We consider the problem of N particles evolving on the one-dimensional periodic domain of length L...
A perturbation theory for ergodic properties of Markov chains
Perturbations to Markov chains and Markov processes are considered. The unperturbed problem is assumed to be geometrically ergodic in the sense usually established through the use of Foster--Lyapunov drift conditions. The...
The Milstein scheme for stochastic delay differential equations without using anticipative calculus
The Milstein scheme is the simplest nontrivial numerical scheme for
stochastic differential equations with a strong order of convergence one.
The scheme has been extended to the stochastic delay differential...
stochastic differential equations with a strong order of convergence one.
The scheme has been extended to the stochastic delay differential...
Modified equations for stochastic differential equations
Tony Shardlow
Mar 31, 2006
We describe a backward error analysis for stochastic differential equations with respect to weak convergence. Modified equations are provided for forward and backward Euler approximations to Itô SDEs with additive noise, and...
Postprocessing for stochastic parabolic partial differential equations
We investigate the strong approximation of stochastic parabolic partial differential equations with additive noise. We introduce postprocessing in the context of a standard Galerkin approximation, although other spatial...
Nucleation of waves in excitable media by noise
Tony Shardlow
Dec 31, 2004
We are interested in reaction-diffusion equations that model excitable media under the influence of an additive noise. In many models of this type, the homogeneous zero state is stable, and interesting dynamics are observed only...
The Milstein scheme for stochastic delay differential equations without using anticipative calculus
The Milstein scheme is the simplest nontrivial numerical scheme for
stochastic differential equations with a strong order of convergence one.
The scheme has been extended to the stochastic delay differential...
stochastic differential equations with a strong order of convergence one.
The scheme has been extended to the stochastic delay differential...
SDELab
We introduce SDELab, a package for solving stochastic differential equations (SDEs) within MATLAB. SDELab features explicit and implicit integrators for a general class of Itô and Stratonovich SDEs, including Milstein's method...
Periodic orbits and unstable manifolds
Tony Shardlow
Jan 01, 1996
Consider the unstable manifold of a hyperbolic periodic orbit of an ordinary differential equation under C1 perturbations of the vector field and under approximation by a one-step numerical method, which is at least first order....
A coupled Cahn–Hilliard particle system
Tony Shardlow
Aug 13, 2002
A Cahn-Hilliard equation is coupled to a system of stochastic differential equations to model a random growth process. We show the model is well posed and analyze the model asymptotically (in the limit of the interfacial...
From weakly interacting particles to a regularised Dean–Kawasaki model
The evolution of finitely many particles obeying Langevin dynamics is described by Dean-Kawasaki equations, a class of stochastic equations featuring a non-Lipschitz multiplicative noise in divergence form. We derive a...
Use of Bayesian statistics to calculate transient heat fluxes on compressor disks
Future gas turbine engines require improved understanding of the heat transfer between compressor disks and air in compressor cavities under transient operating conditions. Calculation of transient heat fluxes from...
/dk/atira/pure/subjectarea/asjc/2200/2206 name=Computational Mechanics /dk/atira/pure/subjectarea/asjc/3100/3104 name=Condensed Matter Physics /dk/atira/pure/subjectarea/asjc/2200/2211 name=Mechanics of Materials /dk/atira/pure/subjectarea/asjc/2200/2210 name=Mechanical Engineering /dk/atira/pure/subjectarea/asjc/1500/1507 name=Fluid Flow and Transfer Processes
Improving multilevel Monte Carlo for stochastic differential equations with application to the Langevin equation
This paper applies several well-known tricks from the numerical treatment of deterministic differential equations to improve the efficiency of the multilevel Monte Carlo (MLMC) method for stochastic diffe- rential equations...
A regularised Dean-Kawasaki model
The Dean–Kawasaki model consists of a nonlinear stochastic partial differential equation featuring a conservative, multiplicative, stochastic term with non-Lipschitz coefficient, driven by space-time white noise; this...
Dean–Kawasaki model Langevin dynamics Mild solutions Spatial regularization of space-time white noise Stochastic wave equation /dk/atira/pure/subjectarea/asjc/2600/2603 name=Analysis /dk/atira/pure/subjectarea/asjc/2600/2605 name=Computational Mathematics /dk/atira/pure/subjectarea/asjc/2600/2604 name=Applied Mathematics
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