1,684 research outputs found
An efficient shooting algorithm for Evans function calculations in large systems
In Evans function computations of the spectra of asymptotically
constant-coefficient linear operators, a basic issue is the efficient and
numerically stable computation of subspaces evolving according to the
associated eigenvalue ODE. For small systems, a fast, shooting algorithm may be
obtained by representing subspaces as single exterior products
\cite{AS,Br.1,Br.2,BrZ,BDG}. For large systems, however, the dimension of the
exterior-product space quickly becomes prohibitive, growing as ,
where is the dimension of the system written as a first-order ODE and
(typically ) is the dimension of the subspace. We resolve this
difficulty by the introduction of a simple polar coordinate algorithm
representing ``pure'' (monomial) products as scalar multiples of orthonormal
bases, for which the angular equation is a numerically optimized version of the
continuous orthogonalization method of Drury--Davey \cite{Da,Dr} and the radial
equation is evaluable by quadrature. Notably, the polar-coordinate method
preserves the important property of analyticity with respect to parameters.Comment: 21 pp., two figure
Domains of analyticity of Lindstedt expansions of KAM tori in dissipative perturbations of Hamiltonian systems
Many problems in Physics are described by dynamical systems that are
conformally symplectic (e.g., mechanical systems with a friction proportional
to the velocity, variational problems with a small discount or thermostated
systems). Conformally symplectic systems are characterized by the property that
they transform a symplectic form into a multiple of itself. The limit of small
dissipation, which is the object of the present study, is particularly
interesting.
We provide all details for maps, but we present also the modifications needed
to obtain a direct proof for the case of differential equations. We consider a
family of conformally symplectic maps defined on a
-dimensional symplectic manifold with exact symplectic form
; we assume that satisfies
. We assume that the family
depends on a -dimensional parameter (called drift) and also on a small
scalar parameter . Furthermore, we assume that the conformal factor
depends on , in such a way that for we have
(the symplectic case).
We study the domains of analyticity in near of
perturbative expansions (Lindstedt series) of the parameterization of the
quasi--periodic orbits of frequency (assumed to be Diophantine) and of
the parameter . Notice that this is a singular perturbation, since any
friction (no matter how small) reduces the set of quasi-periodic solutions in
the system. We prove that the Lindstedt series are analytic in a domain in the
complex plane, which is obtained by taking from a ball centered at
zero a sequence of smaller balls with center along smooth lines going through
the origin. The radii of the excluded balls decrease faster than any power of
the distance of the center to the origin
A Dynamically Adaptive Sparse Grid Method for Quasi-Optimal Interpolation of Multidimensional Analytic Functions
In this work we develop a dynamically adaptive sparse grids (SG) method for
quasi-optimal interpolation of multidimensional analytic functions defined over
a product of one dimensional bounded domains. The goal of such approach is to
construct an interpolant in space that corresponds to the "best -terms"
based on sharp a priori estimate of polynomial coefficients. In the past, SG
methods have been successful in achieving this, with a traditional construction
that relies on the solution to a Knapsack problem: only the most profitable
hierarchical surpluses are added to the SG. However, this approach requires
additional sharp estimates related to the size of the analytic region and the
norm of the interpolation operator, i.e., the Lebesgue constant. Instead, we
present an iterative SG procedure that adaptively refines an estimate of the
region and accounts for the effects of the Lebesgue constant. Our approach does
not require any a priori knowledge of the analyticity or operator norm, is
easily generalized to both affine and non-affine analytic functions, and can be
applied to sparse grids build from one dimensional rules with arbitrary growth
of the number of nodes. In several numerical examples, we utilize our
dynamically adaptive SG to interpolate quantities of interest related to the
solutions of parametrized elliptic and hyperbolic PDEs, and compare the
performance of our quasi-optimal interpolant to several alternative SG schemes
Efficient numerical stability analysis of detonation waves in ZND
As described in the classic works of Lee--Stewart and Short--Stewart, the
numerical evaluation of linear stability of planar detonation waves is a
computationally intensive problem of considerable interest in applications.
Reexamining this problem from a modern numerical Evans function point of view,
we derive a new algorithm for their stability analysis, related to a much older
method of Erpenbeck, that, while equally simple and easy to implement as the
standard method introduced by Lee--Stewart, appears to be potentially faster
and more stable
Conformal Maps to Multiply-Slit Domains and Applications
By exploiting conformal maps to vertically slit regions in the complex plane, a recently developed rational spectral method [Tee and Trefethen, 2006] is able to solve PDEs with interior layer-like behaviour using significantly fewer collocation points than traditional spectral methods. The conformal maps are chosen to 'enlarge the region of analyticity' in the solution: an idea which can be extended to other numerical methods based upon global polynomial interpolation. Here we show how such maps can be rapidly computed in both periodic and nonperiodic geometries, and apply them to some challenging differential equations
Iterative structure of finite loop integrals
In this paper we develop further and refine the method of differential
equations for computing Feynman integrals. In particular, we show that an
additional iterative structure emerges for finite loop integrals. As a concrete
non-trivial example we study planar master integrals of light-by-light
scattering to three loops, and derive analytic results for all values of the
Mandelstam variables and and the mass . We start with a recent
proposal for defining a basis of loop integrals having uniform transcendental
weight properties and use this approach to compute all planar two-loop master
integrals in dimensional regularization. We then show how this approach can be
further simplified when computing finite loop integrals. This allows us to
discuss precisely the subset of integrals that are relevant to the problem. We
find that this leads to a block triangular structure of the differential
equations, where the blocks correspond to integrals of different weight. We
explain how this block triangular form is found in an algorithmic way. Another
advantage of working in four dimensions is that integrals of different loop
orders are interconnected and can be seamlessly discussed within the same
formalism. We use this method to compute all finite master integrals needed up
to three loops. Finally, we remark that all integrals have simple Mandelstam
representations.Comment: 26 pages plus appendices, 5 figure
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