907 research outputs found
A technique for studying strong and weak local errors of splitting stochastic integrators
We present a technique, based on so-called word series, to write down in a systematic way expansions of the strong and weak local errors of splitting algorithms for the integration of Stratonovich stochastic differential equations. Those expansions immediately lead to the corresponding order conditions. Word series are similar to, but simpler than, the B-series used to analyze Runge--Kutta and other one-step integrators. The suggested approach makes it unnecessary to use the Baker--Campbell--Hausdorff formula. As an application, we compare two splitting algorithms recently considered by Leimkuhler and Matthews to integrate the Langevin equations. The word series method clearly bears out reasons for the advantages of one algorithm over the other.Ministerio de EconomĆa, Industria y Competitividad (Project MTM2013-46553-C3-1-P)
Exotic aromatic B-series for the study of long time integrators for a class of ergodic SDEs
We introduce a new algebraic framework based on a modification (called
exotic) of aromatic Butcher-series for the systematic study of the accuracy of
numerical integrators for the invariant measure of a class of ergodic
stochastic differential equations (SDEs) with additive noise. The proposed
analysis covers Runge-Kutta type schemes including the cases of partitioned
methods and postprocessed methods. We also show that the introduced exotic
aromatic B-series satisfy an isometric equivariance property.Comment: 33 page
Non-intrusive and structure preserving multiscale integration of stiff ODEs, SDEs and Hamiltonian systems with hidden slow dynamics via flow averaging
We introduce a new class of integrators for stiff ODEs as well as SDEs. These
integrators are (i) {\it Multiscale}: they are based on flow averaging and so
do not fully resolve the fast variables and have a computational cost
determined by slow variables (ii) {\it Versatile}: the method is based on
averaging the flows of the given dynamical system (which may have hidden slow
and fast processes) instead of averaging the instantaneous drift of assumed
separated slow and fast processes. This bypasses the need for identifying
explicitly (or numerically) the slow or fast variables (iii) {\it
Nonintrusive}: A pre-existing numerical scheme resolving the microscopic time
scale can be used as a black box and easily turned into one of the integrators
in this paper by turning the large coefficients on over a microscopic timescale
and off during a mesoscopic timescale (iv) {\it Convergent over two scales}:
strongly over slow processes and in the sense of measures over fast ones. We
introduce the related notion of two-scale flow convergence and analyze the
convergence of these integrators under the induced topology (v) {\it Structure
preserving}: for stiff Hamiltonian systems (possibly on manifolds), they can be
made to be symplectic, time-reversible, and symmetry preserving (symmetries are
group actions that leave the system invariant) in all variables. They are
explicit and applicable to arbitrary stiff potentials (that need not be
quadratic). Their application to the Fermi-Pasta-Ulam problems shows accuracy
and stability over four orders of magnitude of time scales. For stiff Langevin
equations, they are symmetry preserving, time-reversible and Boltzmann-Gibbs
reversible, quasi-symplectic on all variables and conformally symplectic with
isotropic friction.Comment: 69 pages, 21 figure
Word series for the numerical integration of stochastic differential equations
El presente trabajo tiene dos objetivos bien diferenciados. Por una parte, busca establecer un marco algebraico que permita describir la combinatoria que ataƱe a los integradores splitting de ecuaciones diferenciales y, por otra, busca probar la eficacia del citado enfoque a travĆ©s de la obtenciĆ³n de resultados teĆ³ricos referentes a dichos integradores, como son, por ejemplo, las condiciones de orden o las ecuaciones modificadas. Las llamadas word series, series de palabras, son sin duda un novedoso instrumento de anĆ”lisis extremadamente Ćŗtil a la hora de analizar integradores splitting para ecuaciones diferenciales estocĆ”sticas en el sentido de Ito y en el de StratonovichDepartamento de MatemĆ”tica AplicadaDoctorado en MatemĆ”tica
Analysis of a splitting scheme for a class of nonlinear stochastic Schr\uf6dinger equations
We analyze the qualitative properties and the order of convergence of a splitting scheme for a class of nonlinear stochastic Schr\uf6dinger equations driven by additive It\uf4 noise. The class of nonlinearities of interest includes nonlocal interaction cubic nonlinearities. We show that the numerical solution is symplectic and preserves the expected mass for all times. On top of that, for the convergence analysis, some exponential moment bounds for the exact and numerical solutions are proved. This enables us to provide strong orders of convergence as well as orders of convergence in probability and almost surely. Finally, extensive numerical experiments illustrate the performance of the proposed numerical scheme
Numerical solution of conservative finite-dimensional stochastic Schrodinger equations
The paper deals with the numerical solution of the nonlinear Ito stochastic
differential equations (SDEs) appearing in the unravelling of quantum master
equations. We first develop an exponential scheme of weak order 1 for general
globally Lipschitz SDEs governed by Brownian motions. Then, we proceed to study
the numerical integration of a class of locally Lipschitz SDEs. More precisely,
we adapt the exponential scheme obtained in the first part of the work to the
characteristics of certain finite-dimensional nonlinear stochastic Schrodinger
equations. This yields a numerical method for the simulation of the mean value
of quantum observables. We address the rate of convergence arising in this
computation. Finally, an experiment with a representative quantum master
equation illustrates the good performance of the new scheme.Comment: Published at http://dx.doi.org/10.1214/105051605000000403 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Wasserstein distance estimates for the distributions of numerical approximations to ergodic stochastic differential equations
We present a framework that allows for the non-asymptotic study of the 2
-Wasserstein distance between the invariant distribution of an ergodic stochastic differential equation and the distribution of its numerical approximation in the strongly log-concave case. This allows us to study in a unified way a number of different integrators proposed in the literature for the overdamped and underdamped Langevin dynamics. In addition, we analyze a novel splitting method for the underdamped Langevin dynamics which only requires one gradient evaluation per time step. Under an additional smoothness assumption on a d
--dimensional strongly log-concave distribution with condition number Īŗ
, the algorithm is shown to produce with an O(Īŗ5/4d1/4Ļµā1/2)
complexity samples from a distribution that, in Wasserstein distance, is at most Ļµ>0
away from the target distribution
Wasserstein distance estimates for the distributions of numerical approximations to ergodic stochastic differential equations
We present a framework that allows for the non-asymptotic study of the 2
-Wasserstein distance between the invariant distribution of an ergodic stochastic differential equation and the distribution of its numerical approximation in the strongly log-concave case. This allows us to study in a unified way a number of different integrators proposed in the literature for the overdamped and underdamped Langevin dynamics. In addition, we analyze a novel splitting method for the underdamped Langevin dynamics which only requires one gradient evaluation per time step. Under an additional smoothness assumption on a d
--dimensional strongly log-concave distribution with condition number Īŗ
, the algorithm is shown to produce with an O(Īŗ5/4d1/4Ļµā1/2)
complexity samples from a distribution that, in Wasserstein distance, is at most Ļµ>0
away from the target distribution
Least-biased correction of extended dynamical systems using observational data
We consider dynamical systems evolving near an equilibrium statistical state
where the interest is in modelling long term behavior that is consistent with
thermodynamic constraints. We adjust the distribution using an
entropy-optimizing formulation that can be computed on-the- fly, making
possible partial corrections using incomplete information, for example measured
data or data computed from a different model (or the same model at a different
scale). We employ a thermostatting technique to sample the target distribution
with the aim of capturing relavant statistical features while introducing mild
dynamical perturbation (thermostats). The method is tested for a point vortex
fluid model on the sphere, and we demonstrate both convergence of equilibrium
quantities and the ability of the formulation to balance stationary and
transient- regime errors.Comment: 27 page
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