270 research outputs found
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
Quadrature integration techniques for random hyperbolic PDE problems
In this paper, we consider random hyperbolic partial differential equation (PDE) problems following the mean square approach and Laplace transform technique. Randomness requires not only the computation of the approximating stochastic processes, but also its statistical moments. Hence, appropriate numerical methods should allow for the efficient computation of the expectation and variance. Here, we analyse different numerical methods around the inverse Laplace transform and its evaluation by using several integration techniques, including midpoint quadrature rule, Gauss?Laguerre quadrature and its extensions, and the Talbot algorithm. Simulations, numerical convergence, and computational process time with experiments are shown.This research has been funded by the Spanish Ministerio de Economía, Industria y Competitividad (MINECO), the Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER UE) grant MTM2017-89664-
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