Special Algorithms for Exponentially Small Phenomena

The standard Sobolev space [Formula: see text], with arbitrary positive integers [Formula: see text] and [Formula: see text] for which [Formula: see text], has the reproducing kernel [Formula: see text] for all [Formula: see text], where [Formula: see text] are components of [Formula: see text]-variate [Formula: see text], and [Formula: see text] with non-negative integers [Formula: see text]. We obtain a more explicit form for the reproducing kernel [Formula: see text] and find a closed form for the kernel [Formula: see text]. Knowing the form of [Formula: see text], we present applications on the best embedding constants between the Sobolev space [Formula: see text] and [Formula: see text], and on strong polynomial tractability of integration with an arbitrary probability density. We prove that the best embedding constants are exponentially small in [Formula: see text], whereas worst case integration errors of algorithms using [Formula: see text] function values are also exponentially small in [Formula: see text] and decay at least like [Formula: see text]. This yields strong polynomial tractability in the worst case setting for the absolute error criterion.

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Metastability for a generalized Burgers equation with applications to propagating flame fronts

European Journal of Applied Mathematics ◽

10.1017/s0956792598003623 ◽

1999 ◽

Vol 10 (1) ◽

pp. 27-53 ◽

Cited By ~ 21

Author(s):

X. SUN ◽

M. J. WARD

Keyword(s):

Flame Front ◽

Principal Eigenvalue ◽

Vertical Channel ◽

Front Propagation ◽

Slow Motion ◽

Diffusion Limit ◽

Asymptotic Results ◽

Small Diffusion ◽

Generalized Burgers Equation ◽

Exponentially Small

In the small diffusion limit ε→0, metastable dynamics is studied for the generalized Burgers problemformula hereHere u=u(x, t) and f(u) is smooth, convex, and satisfies f(0)=f′(0)=0. The choice f(u)=u2/2 has been shown previously to arise in connection with the physical problem of upward flame-front propagation in a vertical channel in a particular parameter regime. In this context, the shape y=y(x, t) of the flame-front interface satisfies u=−yx. For this problem, it is shown that the principal eigenvalue associated with the linearization around an equilibrium solution corresponding to a parabolic-shaped flame-front interface is exponentially small. This exponentially small eigenvalue then leads to a metastable behaviour for the time- dependent problem. This behaviour is studied quantitatively by deriving an asymptotic ordinary differential equation characterizing the slow motion of the tip location of a parabolic-shaped interface. Similar metastability results are obtained for more general f(u). These asymptotic results are shown to compare very favourably with full numerical computations.

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Perturbation Theory and Boundary Conditions: Analogous Treatments of Anharmonic Oscillators and Double-Wells and Similarly Related Potentials and the Calculation of Exponentially Small Contributions to Eigenvalues

Fortschritte der Physik/Progress of Physics ◽

10.1002/prop.2190380202 ◽

1990 ◽

Vol 38 (2) ◽

pp. 77-164 ◽

Cited By ~ 14

Author(s):

P. Achuthan ◽

H. J. W. Müller-Kirsten ◽

A. Wiedemann

Keyword(s):

Perturbation Theory ◽

Boundary Conditions ◽

Anharmonic Oscillators ◽

Related Potentials ◽

Double Wells ◽

Exponentially Small

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Analysis of the buildup of spatiotemporal correlations and their bounds outside of the light cone

10.21468/scipostphys.5.5.052 ◽

2018 ◽

Vol 5 (5) ◽

Cited By ~ 1

Author(s):

Nils O. Abeling ◽

Lorenzo Cevolani ◽

Stefan Kehrein

Keyword(s):

Correlation Function ◽

Power Law ◽

Light Cone ◽

Relativistic Quantum ◽

Initial State ◽

Quantum Theories ◽

Luttinger Model ◽

Power Law Decay ◽

Exactly Solvable ◽

Exponentially Small

In non-relativistic quantum theories the Lieb-Robinson bound defines an effective light cone with exponentially small tails outside of it. In this work we use it to derive a bound for the correlation function of two local disjoint observables at different times if the initial state has a power-law decay. We show that the exponent of the power-law of the bound is identical to the initial (equilibrium) decay. We explicitly verify this result by studying the full dynamics of the susceptibilities and correlations in the exactly solvable Luttinger model after a sudden quench from the non-interacting to the interacting model.

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Hyperasymptotics

Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences ◽

10.1098/rspa.1990.0111 ◽

1990 ◽

Vol 430 (1880) ◽

pp. 653-668 ◽

Cited By ~ 99

Keyword(s):

Asymptotic Expansion ◽

Differential Equations ◽

Ordinary Differential Equations ◽

Transition Point ◽

Asymptotic Series ◽

Stokes Phenomenon ◽

Numerical Computations ◽

Borel Summation ◽

Nested Sequence ◽

Exponentially Small

We develop a technique for systematically reducing the exponentially small (‘superasymptotic’) remainder of an asymptotic expansion truncated near its least term, for solutions of ordinary differential equations of Schrödinger type where one transition point dominates. This is achieved by repeatedly applying Borel summation to a resurgence formula discovered by Dingle, relating the late to the early terms of the original expansion. The improvements form a nested sequence of asymptotic series truncated at their least terms. Each such ‘hyperseries’ involves the terms of the original asymptotic series for the particular function being approximated, together with terminating integrals that are universal in form, and is half the length of its predecessor. The hyperasymptotic sequence is therefore finite, and leads to an ultimate approximation whose error is less than the square of the original superasymptotic remainder. The Stokes phenomenon is automatically and exactly incorporated into the scheme. Numerical computations confirm the efficacy of the technique.

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Loss probabilities in a simple circuit-switched network

Advances in Applied Probability ◽

10.2307/1427446 ◽

1994 ◽

Vol 26 (2) ◽

pp. 456-473 ◽

Cited By ~ 6

Author(s):

J. A. Morrison

Keyword(s):

Fixed Point ◽

Parameter Space ◽

Asynchronous Transfer Mode ◽

Practical Interest ◽

Integral Representations ◽

Cellular Systems ◽

Holding Period ◽

Loss Probabilities ◽

Erlang Loss ◽

Exponentially Small

In this paper a particular loss network consisting of two links with C1 and C2 circuits, respectively, and two fixed routes, is investigated. A call on route 1 uses a circuit from both links, and a call on route 2 uses a circuit from only the second link. Calls requesting routes 1 and 2 arrive as independent Poisson streams. A call requesting route 1 is blocked and lost if there are no free circuits on either link, and a call requesting route 2 is blocked and lost if there is no free circuit on the second link. Otherwise the call is connected and holds a circuit from each link on its route for the holding period of the call.The case in which the capacities C1, and C2, and the traffic intensities v1, and v2, all become large of O(N) where N » 1, but with their ratios fixed, is considered. The loss probabilities L1 and L2 for calls requesting routes 1 and 2, respectively, are investigated. The asymptotic behavior of L1 and L2 as N→ ∞ is determined with the help of double contour integral representations and saddlepoint approximations. The results differ in various regions of the parameter space (C1, C2, v1, v2). In some of these results the loss probabilities are given in terms of the Erlang loss function, with appropriate arguments, to within an exponentially small relative error. The results provide new information when the loss probabilities are exponentially small in N. This situation is of practical interest, e.g. in cellular systems, and in asynchronous transfer mode networks, where very small loss probabilities are desired.The accuracy of the Erlang fixed-point approximations to the loss probabilities is also investigated. In particular, it is shown that the fixed-point approximation E2 to L2 is inaccurate in a certain region of the parameter space, since L2 « E2 there. On the other hand, in some regions of the parameter space the fixed-point approximations to both L1 and L2 are accurate to within an exponentially small relative error.

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