21,349 research outputs found
Strong convergence rates of probabilistic integrators for ordinary differential equations
Probabilistic integration of a continuous dynamical system is a way of
systematically introducing model error, at scales no larger than errors
introduced by standard numerical discretisation, in order to enable thorough
exploration of possible responses of the system to inputs. It is thus a
potentially useful approach in a number of applications such as forward
uncertainty quantification, inverse problems, and data assimilation. We extend
the convergence analysis of probabilistic integrators for deterministic
ordinary differential equations, as proposed by Conrad et al.\ (\textit{Stat.\
Comput.}, 2017), to establish mean-square convergence in the uniform norm on
discrete- or continuous-time solutions under relaxed regularity assumptions on
the driving vector fields and their induced flows. Specifically, we show that
randomised high-order integrators for globally Lipschitz flows and randomised
Euler integrators for dissipative vector fields with polynomially-bounded local
Lipschitz constants all have the same mean-square convergence rate as their
deterministic counterparts, provided that the variance of the integration noise
is not of higher order than the corresponding deterministic integrator. These
and similar results are proven for probabilistic integrators where the random
perturbations may be state-dependent, non-Gaussian, or non-centred random
variables.Comment: 25 page
The physics of symplectic integrators: perihelion advances and symplectic corrector algorithms
Symplectic integrators evolve dynamical systems according to modified
Hamiltonians whose error terms are also well-defined Hamiltonians. The error of
the algorithm is the sum of each error Hamiltonian's perturbation on the exact
solution. When symplectic integrators are applied to the Kepler problem, these
error terms cause the orbit to precess. In this work, by developing a general
method of computing the perihelion advance via the Laplace-Runge-Lenz vector
even for non-separable Hamiltonians, I show that the precession error in
symplectic integrators can be computed analytically. It is found that at each
order, each paired error Hamiltonians cause the orbit to precess oppositely by
exactly the same amount after each period. Hence, symplectic corrector, or
process integrators, which have equal coefficients for these paired error
terms, will have their precession errors exactly cancel after each period. Thus
the physics of symplectic integrators determines the optimal algorithm for
integrating long time periodic motions.Comment: 18 pages, 5 figures, 1 tabl
Algebraic structure of stochastic expansions and efficient simulation
We investigate the algebraic structure underlying the stochastic Taylor
solution expansion for stochastic differential systems.Our motivation is to
construct efficient integrators. These are approximations that generate strong
numerical integration schemes that are more accurate than the corresponding
stochastic Taylor approximation, independent of the governing vector fields and
to all orders. The sinhlog integrator introduced by Malham & Wiese (2009) is
one example. Herein we: show that the natural context to study stochastic
integrators and their properties is the convolution shuffle algebra of
endomorphisms; establish a new whole class of efficient integrators; and then
prove that, within this class, the sinhlog integrator generates the optimal
efficient stochastic integrator at all orders.Comment: 19 page
Collective symplectic integrators
We construct symplectic integrators for Lie-Poisson systems. The integrators
are standard symplectic (partitioned) Runge--Kutta methods. Their phase space
is a symplectic vector space with a Hamiltonian action with momentum map
whose range is the target Lie--Poisson manifold, and their Hamiltonian is
collective, that is, it is the target Hamiltonian pulled back by . The
method yields, for example, a symplectic midpoint rule expressed in 4 variables
for arbitrary Hamiltonians on . The method specializes in
the case that a sufficiently large symmetry group acts on the fibres of ,
and generalizes to the case that the vector space carries a bifoliation.
Examples involving many classical groups are presented
A study of accuracy in selected numerical-analysis integration techniques
Report discusses several methods of performing numerical integration with computer. When data can be expressed as state vector that is dependent variable in a differential equation, self-starting integrators can be used to predict future data
Design of quasi-symplectic propagators for Langevin dynamics
A vector field splitting approach is discussed for the systematic derivation
of numerical propagators for deterministic dynamics. Based on the formalism, a
class of numerical integrators for Langevin dynamics are presented for single
and multiple timestep algorithms
Convergence analysis of domain decomposition based time integrators for degenerate parabolic equations
Domain decomposition based time integrators allow the usage of parallel and
distributed hardware, making them well-suited for the temporal discretization
of parabolic systems, in general, and degenerate parabolic problems, in
particular. The latter is due to the degenerate equations' finite speed of
propagation. In this study, a rigours convergence analysis is given for such
integrators without assuming any restrictive regularity on the solutions or the
domains. The analysis is conducted by first deriving a new variational
framework for the domain decomposition, which is applicable to the two standard
degenerate examples. That is, the -Laplace and the porous medium type vector
fields. Secondly, the decomposed vector fields are restricted to the underlying
pivot space and the time integration of the parabolic problem can then be
interpreted as an operators splitting applied to a dissipative evolution
equation. The convergence results then follow by employing elements of the
approximation theory for nonlinear semigroups
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