9,803 research outputs found
O(1) Computation of Legendre polynomials and Gauss-Legendre nodes and weights for parallel computing
A self-contained set of algorithms is proposed for the fast evaluation of Legendre polynomials of arbitrary degree and argument is an element of [-1, 1]. More specifically the time required to evaluate any Legendre polynomial, regardless of argument and degree, is bounded by a constant; i.e., the complexity is O(1). The proposed algorithm also immediately yields an O(1) algorithm for computing an arbitrary Gauss-Legendre quadrature node. Such a capability is crucial for efficiently performing certain parallel computations with high order Legendre polynomials, such as computing an integral in parallel by means of Gauss-Legendre quadrature and the parallel evaluation of Legendre series. In order to achieve the O(1) complexity, novel efficient asymptotic expansions are derived and used alongside known results. A C++ implementation is available from the authors that includes the evaluation routines of the Legendre polynomials and Gauss-Legendre quadrature rules
Fourier Based Fast Multipole Method for the Helmholtz Equation
The fast multipole method (FMM) has had great success in reducing the
computational complexity of solving the boundary integral form of the Helmholtz
equation. We present a formulation of the Helmholtz FMM that uses Fourier basis
functions rather than spherical harmonics. By modifying the transfer function
in the precomputation stage of the FMM, time-critical stages of the algorithm
are accelerated by causing the interpolation operators to become
straightforward applications of fast Fourier transforms, retaining the
diagonality of the transfer function, and providing a simplified error
analysis. Using Fourier analysis, constructive algorithms are derived to a
priori determine an integration quadrature for a given error tolerance. Sharp
error bounds are derived and verified numerically. Various optimizations are
considered to reduce the number of quadrature points and reduce the cost of
computing the transfer function.Comment: 24 pages, 13 figure
Hot new directions for quasi-Monte Carlo research in step with applications
This article provides an overview of some interfaces between the theory of
quasi-Monte Carlo (QMC) methods and applications. We summarize three QMC
theoretical settings: first order QMC methods in the unit cube and in
, and higher order QMC methods in the unit cube. One important
feature is that their error bounds can be independent of the dimension
under appropriate conditions on the function spaces. Another important feature
is that good parameters for these QMC methods can be obtained by fast efficient
algorithms even when is large. We outline three different applications and
explain how they can tap into the different QMC theory. We also discuss three
cost saving strategies that can be combined with QMC in these applications.
Many of these recent QMC theory and methods are developed not in isolation, but
in close connection with applications
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