133 research outputs found
Rank-1 lattice rules for multivariate integration in spaces of permutation-invariant functions: Error bounds and tractability
We study multivariate integration of functions that are invariant under
permutations (of subsets) of their arguments. We find an upper bound for the
th minimal worst case error and show that under certain conditions, it can
be bounded independent of the number of dimensions. In particular, we study the
application of unshifted and randomly shifted rank- lattice rules in such a
problem setting. We derive conditions under which multivariate integration is
polynomially or strongly polynomially tractable with the Monte Carlo rate of
convergence . Furthermore, we prove that those tractability
results can be achieved with shifted lattice rules and that the shifts are
indeed necessary. Finally, we show the existence of rank- lattice rules
whose worst case error on the permutation- and shift-invariant spaces converge
with (almost) optimal rate. That is, we derive error bounds of the form
for all , where denotes
the smoothness of the spaces.
Keywords: Numerical integration, Quadrature, Cubature, Quasi-Monte Carlo
methods, Rank-1 lattice rules.Comment: 26 pages; minor changes due to reviewer's comments; the final
publication is available at link.springer.co
Construction of quasi-Monte Carlo rules for multivariate integration in spaces of permutation-invariant functions
We study multivariate integration of functions that are invariant under the
permutation (of a subset) of their arguments. Recently, in Nuyens,
Suryanarayana, and Weimar (Adv. Comput. Math. (2016), 42(1):55--84), the
authors derived an upper estimate for the th minimal worst case error for
such problems, and showed that under certain conditions this upper bound only
weakly depends on the dimension. We extend these results by proposing two
(semi-) explicit construction schemes. We develop a component-by-component
algorithm to find the generating vector for a shifted rank- lattice rule
that obtains a rate of convergence arbitrarily close to
, where denotes the smoothness of our
function space and is the number of cubature nodes. Further, we develop a
semi-constructive algorithm that builds on point sets which can be used to
approximate the integrands of interest with a small error; the cubature error
is then bounded by the error of approximation. Here the same rate of
convergence is achieved while the dependence of the error bounds on the
dimension is significantly improved
Strang splitting in combination with rank- and rank- lattices for the time-dependent Schr\"odinger equation
We approximate the solution for the time dependent Schr\"odinger equation
(TDSE) in two steps. We first use a pseudo-spectral collocation method that
uses samples of functions on rank-1 or rank-r lattice points with unitary
Fourier transforms. We then get a system of ordinary differential equations in
time, which we solve approximately by stepping in time using the Strang
splitting method. We prove that the numerical scheme proposed converges
quadratically with respect to the time step size, given that the potential is
in a Korobov space with the smoothness parameter greater than .
Particularly, we prove that the required degree of smoothness is independent of
the dimension of the problem. We demonstrate our new method by comparing with
results using sparse grids from [12], with several numerical examples showing
large advantage for our new method and pushing the examples to higher
dimensionality. The proposed method has two distinctive features from a
numerical perspective: (i) numerical results show the error convergence of time
discretization is consistent even for higher-dimensional problems; (ii) by
using the rank- lattice points, the solution can be efficiently computed
(and further time stepped) using only -dimensional Fast Fourier Transforms.Comment: Modified. 40pages, 5 figures. The proof of Lemma 1 is updated after
the paper is publishe
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