27,152 research outputs found
A symmetry-adapted numerical scheme for SDEs
We propose a geometric numerical analysis of SDEs admitting Lie symmetries
which allows us to individuate a symmetry adapted coordinates system where the
given SDE has notable invariant properties. An approximation scheme preserving
the symmetry properties of the equation is introduced. Our algorithmic
procedure is applied to the family of general linear SDEs for which two
theoretical estimates of the numerical forward error are established.Comment: A numerical example adde
The Magnus expansion and some of its applications
Approximate resolution of linear systems of differential equations with
varying coefficients is a recurrent problem shared by a number of scientific
and engineering areas, ranging from Quantum Mechanics to Control Theory. When
formulated in operator or matrix form, the Magnus expansion furnishes an
elegant setting to built up approximate exponential representations of the
solution of the system. It provides a power series expansion for the
corresponding exponent and is sometimes referred to as Time-Dependent
Exponential Perturbation Theory. Every Magnus approximant corresponds in
Perturbation Theory to a partial re-summation of infinite terms with the
important additional property of preserving at any order certain symmetries of
the exact solution. The goal of this review is threefold. First, to collect a
number of developments scattered through half a century of scientific
literature on Magnus expansion. They concern the methods for the generation of
terms in the expansion, estimates of the radius of convergence of the series,
generalizations and related non-perturbative expansions. Second, to provide a
bridge with its implementation as generator of especial purpose numerical
integration methods, a field of intense activity during the last decade. Third,
to illustrate with examples the kind of results one can expect from Magnus
expansion in comparison with those from both perturbative schemes and standard
numerical integrators. We buttress this issue with a revision of the wide range
of physical applications found by Magnus expansion in the literature.Comment: Report on the Magnus expansion for differential equations and its
applications to several physical problem
Structure-Preserving Discretization of Incompressible Fluids
The geometric nature of Euler fluids has been clearly identified and
extensively studied over the years, culminating with Lagrangian and Hamiltonian
descriptions of fluid dynamics where the configuration space is defined as the
volume-preserving diffeomorphisms, and Kelvin's circulation theorem is viewed
as a consequence of Noether's theorem associated with the particle relabeling
symmetry of fluid mechanics. However computational approaches to fluid
mechanics have been largely derived from a numerical-analytic point of view,
and are rarely designed with structure preservation in mind, and often suffer
from spurious numerical artifacts such as energy and circulation drift. In
contrast, this paper geometrically derives discrete equations of motion for
fluid dynamics from first principles in a purely Eulerian form. Our approach
approximates the group of volume-preserving diffeomorphisms using a finite
dimensional Lie group, and associated discrete Euler equations are derived from
a variational principle with non-holonomic constraints. The resulting discrete
equations of motion yield a structure-preserving time integrator with good
long-term energy behavior and for which an exact discrete Kelvin's circulation
theorem holds
Stochastic Variational Integrators
This paper presents a continuous and discrete Lagrangian theory for
stochastic Hamiltonian systems on manifolds. The main result is to derive
stochastic governing equations for such systems from a critical point of a
stochastic action. Using this result the paper derives Langevin-type equations
for constrained mechanical systems and implements a stochastic analog of
Lagrangian reduction. These are easy consequences of the fact that the
stochastic action is intrinsically defined. Stochastic variational integrators
(SVIs) are developed using a discretized stochastic variational principle. The
paper shows that the discrete flow of an SVI is a.s. symplectic and in the
presence of symmetry a.s. momentum-map preserving. A first-order mean-square
convergent SVI for mechanical systems on Lie groups is introduced. As an
application of the theory, SVIs are exhibited for multiple, randomly forced and
torqued rigid-bodies interacting via a potential.Comment: 21 pages, 8 figure
Low dimensional manifolds for exact representation of open quantum systems
Weakly nonlinear degrees of freedom in dissipative quantum systems tend to
localize near manifolds of quasi-classical states. We present a family of
analytical and computational methods for deriving optimal unitary model
transformations based on representations of finite dimensional Lie groups. The
transformations are optimal in that they minimize the quantum relative entropy
distance between a given state and the quasi-classical manifold. This naturally
splits the description of quantum states into quasi-classical coordinates that
specify the nearest quasi-classical state and a transformed quantum state that
can be represented in fewer basis levels. We derive coupled equations of motion
for the coordinates and the transformed state and demonstrate how this can be
exploited for efficient numerical simulation. Our optimization objective
naturally quantifies the non-classicality of states occurring in some given
open system dynamics. This allows us to compare the intrinsic complexity of
different open quantum systems.Comment: Added section on semi-classical SR-latch, added summary of method,
revised structure of manuscrip
Study of the risk-adjusted pricing methodology model with methods of Geometrical Analysis
Families of exact solutions are found to a nonlinear modification of the
Black-Scholes equation. This risk-adjusted pricing methodology model (RAPM)
incorporates both transaction costs and the risk from a volatile portfolio.
Using the Lie group analysis we obtain the Lie algebra admitted by the RAPM
equation. It gives us the possibility to describe an optimal system of
subalgebras and correspondingly the set of invariant solutions to the model. In
this way we can describe the complete set of possible reductions of the
nonlinear RAPM model. Reductions are given in the form of different second
order ordinary differential equations. In all cases we provide solutions to
these equations in an exact or parametric form. We discuss the properties of
these reductions and the corresponding invariant solutions.Comment: larger version with exact solutions, corrected typos, 13 pages,
Symposium on Optimal Stopping in Abo/Turku 200
Algebraic Structures and Stochastic Differential Equations driven by Levy processes
We construct an efficient integrator for stochastic differential systems
driven by Levy processes. An efficient integrator is a strong approximation
that is more accurate than the corresponding stochastic Taylor approximation,
to all orders and independent of the governing vector fields. This holds
provided the driving processes possess moments of all orders and the vector
fields are sufficiently smooth. Moreover the efficient integrator in question
is optimal within a broad class of perturbations for half-integer global root
mean-square orders of convergence. We obtain these results using the
quasi-shuffle algebra of multiple iterated integrals of independent Levy
processes.Comment: 41 pages, 11 figure
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