95 research outputs found
A spectral order method for inverting sectorial Laplace transforms
Laplace transforms which admit a holomorphic extension to some sector
strictly containing the right half plane and exhibiting a potential behavior
are considered. A spectral order, parallelizable method for their numerical
inversion is proposed. The method takes into account the available information
about the errors arising in the evaluations. Several numerical illustrations
are provided.Comment: 17 pages 11 figure
Parabolic and Hyperbolic Contours for Computing the Bromwich Integral
Some of the most effective methods for the numerical inversion of the Laplace transform are based on the approximation of the Bromwich contour integral. The accuracy of these methods often hinges on a good choice of contour, and several such contours have been proposed in the literature. Here we analyze two recently proposed contours, namely a parabola and a hyperbola. Using a representative model problem, we determine estimates for the optimal parameters that define these contours. An application to a fractional diffusion equation is presented.\ud
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JACW was supported by the National Research Foundation in South Africa under grant FA200503230001
Efficient sum-of-exponentials approximations for the heat kernel and their applications
In this paper, we show that efficient separated sum-of-exponentials
approximations can be constructed for the heat kernel in any dimension. In one
space dimension, the heat kernel admits an approximation involving a number of
terms that is of the order for any x\in\bbR and
, where is the desired precision. In all
higher dimensions, the corresponding heat kernel admits an approximation
involving only terms for fixed accuracy
. These approximations can be used to accelerate integral
equation-based methods for boundary value problems governed by the heat
equation in complex geometry. The resulting algorithms are nearly optimal. For
points in the spatial discretization and time steps, the cost is
in terms of both memory and CPU time for
fixed accuracy . The algorithms can be parallelized in a
straightforward manner. Several numerical examples are presented to illustrate
the accuracy and stability of these approximations.Comment: 23 pages, 5 figures, 3 table
Fine scales of decay of operator semigroups
Motivated by potential applications to partial differential equations, we
develop a theory of fine scales of decay rates for operator semigroups. The
theory contains, unifies, and extends several notable results in the literature
on decay of operator semigroups and yields a number of new ones. Its core is a
new operator-theoretical method of deriving rates of decay combining
ingredients from functional calculus, and complex, real and harmonic analysis.
It also leads to several results of independent interest.Comment: Version 2 includes numerous minor corrections, and is the authors'
final version. The pape will be published in the Journal of the European
Mathematical Society in April 201
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