49 research outputs found
Strong convergence for split-step methods in stochastic jump kinetics
Mesoscopic models in the reaction-diffusion framework have gained recognition
as a viable approach to describing chemical processes in cell biology. The
resulting computational problem is a continuous-time Markov chain on a discrete
and typically very large state space. Due to the many temporal and spatial
scales involved many different types of computationally more effective
multiscale models have been proposed, typically coupling different types of
descriptions within the Markov chain framework.
In this work we look at the strong convergence properties of the basic first
order Strang, or Lie-Trotter, split-step method, which is formed by decoupling
the dynamics in finite time-steps. Thanks to its simplicity and flexibility,
this approach has been tried in many different combinations.
We develop explicit sufficient conditions for path-wise well-posedness and
convergence of the method, including error estimates, and we illustrate our
findings with numerical examples. In doing so, we also suggest a certain
partition of unity representation for the split-step method, which in turn
implies a concrete simulation algorithm under which trajectories may be
compared in a path-wise sense
Multilevel Monte Carlo methods
The author's presentation of multilevel Monte Carlo path simulation at the
MCQMC 2006 conference stimulated a lot of research into multilevel Monte Carlo
methods. This paper reviews the progress since then, emphasising the
simplicity, flexibility and generality of the multilevel Monte Carlo approach.
It also offers a few original ideas and suggests areas for future research
Coarse-graining of non-reversible stochastic differential equations: quantitative results and connections to averaging
This work is concerned with model reduction of stochastic differential equations and builds on the idea of replacing drift and noise coefficients of preselected relevant, e.g. slow variables by their conditional expectations. We extend recent results by Legoll & Lelièvre [Nonlinearity 23, 2131, 2010] and Duong et al. [Nonlinearity 31, 4517, 2018] on effective reversible dynamics by conditional expectations to the setting of general non-reversible processes with non-constant diffusion coefficient. We prove relative entropy and Wasserstein error estimates for the difference between the time marginals of the effective and original dynamics as well as an entropy error bound for the corresponding path space measures. A comparison with the averaging principle for systems with time-scale separation reveals that, unlike in the reversible setting, the effective dynamics for a non-reversible system need not agree with the averaged equations. We present a thorough comparison for the Ornstein-Uhlenbeck process and make a conjecture about necessary and sufficient conditions for when averaged and effective dynamics agree for nonlinear non-reversible processes. The theoretical results are illustrated with suitable numerical examples
Mini-Workshop: Dynamics of Stochastic Systems and their Approximation
The aim of this workshop was to bring together specialists in the area of stochastic dynamical systems and stochastic numerical analysis to exchange their ideas about the state of the art of approximations of stochastic dynamics. Here approximations are considered in the analytical sense in terms of deriving reduced dynamical systems, which are less complex, as well as in the numerical sense via appropriate simulation methods. The main theme is concerned with the efficient treatment of stochastic dynamical systems via both approaches assuming that ideas and methods from one ansatz may prove beneficial for the other. A particular goal was to systematically identify open problems and challenges in this area
Coupling sample paths to the thermodynamic limit in Monte Carlo estimators with applications to gene expression
Many biochemical systems appearing in applications have a multiscale structure so that they converge to piecewise deterministic Markov processes in a thermodynamic limit. The statistics of the piecewise deterministic process can be obtained much more efficiently than those of the exact process. We explore the possibility of coupling sample paths of the exact model to the piecewise deterministic process in order to reduce the variance of their difference. We then apply this coupling to reduce the computational complexity of a Monte Carlo estimator. Motivated by the rigorous results in [1], we show how this method can be applied to realistic biological models with nontrivial scalings
The heterogeneous multiscale method
The heterogeneous multiscale method (HMM), a general framework for designing multiscale algorithms, is reviewed. Emphasis is given to the error analysis that comes naturally with the framework. Examples of finite element and finite difference HMM are presented. Applications to dynamical systems and stochastic simulation algorithms with multiple time scales, spall fracture and heat conduction in microprocessors are discusse