Stochastic Lie Group Integrators

نویسندگان

  • Simon J. A. Malham
  • Anke Wiese
چکیده

We present Lie group integrators for nonlinear stochastic differential equations with non-commutative vector fields whose solution evolves on a smooth finite dimensional manifold. Given a Lie group action that generates transport along the manifold, we pull back the stochastic flow on the manifold to the Lie group via the action, and subsequently pull back the flow to the corresponding Lie algebra via the exponential map. We construct an approximation to the stochastic flow in the Lie algebra via closed operations and then push back to the Lie group and then to the manifold, thus ensuring our approximation lies in the manifold. We call such schemes stochastic Munthe-Kaas methods after their deterministic counterparts. We also present stochastic Lie group integration schemes based on Castell–Gaines methods. These involve using an underlying ordinary differential integrator to approximate the flow generated by a truncated stochastic exponential Lie series. They become stochastic Lie group integrator schemes if we use Munthe-Kaas methods as the underlying ordinary differential integrator. Further, we show that some Castell–Gaines methods are uniformly more accurate than the corresponding stochastic Taylor schemes. Lastly we demonstrate our methods by simulating the dynamics of a free rigid body such as a satellite and an autonomous underwater vehicle both perturbed by two independent multiplicative stochastic noise processes.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

ar X iv : 0 70 4 . 00 22 v 2 [ m at h . N A ] 1 6 O ct 2 00 7 STOCHASTIC LIE GROUP INTEGRATORS

We present Lie group integrators for nonlinear stochastic differential equations with non-commutative vector fields whose solution evolves on a smooth finite dimensional manifold. Given a Lie group action that generates transport along the manifold, we pull back the stochastic flow on the manifold to the Lie group via the action, and subsequently pull back the flow to the corresponding Lie alge...

متن کامل

An introduction to Lie group integrators

The significance of the geometry of differential equations was well understood already in the nineteenth century, and in the last few decades such aspects have played an increasing role in numerical methods for differential equations. Nowadays, there is a rich selection of integrators which preserve properties like symplecticity, reversibility, phase volume and first integrals, either exactly o...

متن کامل

An Overview of Lie Group Variational Integrators and Their Applications to Optimal Control

We introduce a general framework for the construction of variational integrators of arbitrarily high-order that incorporate Lie group techniques to automatically remain on a Lie group, while retaining the geometric structure-preserving properties characteristic of variational integrators, including symplecticity, momentum-preservation, and good long-time energy behavior. This is achieved by con...

متن کامل

Lie Group Variational Integrators for the Full Body Problem

We develop the equations of motion for full body models that describe the dynamics of rigid bodies, acting under their mutual gravity. The equations are derived using a variational approach where variations are defined on the Lie group of rigid body configurations. Both continuous equations of motion and variational integrators are developed in Lagrangian and Hamiltonian forms, and the reductio...

متن کامل

Generalized polar coordinates on Lie groups and numerical integrators

Motivated by recent developments in numerical Lie group integrators, we introduce a family of local coordinates on Lie groups denoted generalized polar coordinates. Fast algorithms are derived for the computation of the coordinate maps, their tangent maps and the inverse tangent maps. In particular we discuss algorithms for all the classical matrix Lie groups and optimal complexity integrators ...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:
  • SIAM J. Scientific Computing

دوره 30  شماره 

صفحات  -

تاریخ انتشار 2008