2,156 research outputs found
Optimal Summation and Integration by Deterministic, Randomized, and Quantum Algorithms
We survey old and new results about optimal algorithms for summation of
finite sequences and for integration of functions from Hoelder or Sobolev
spaces. First we discuss optimal deterministic and randomized algorithms. Then
we add a new aspect, which has not been covered before on conferences about
(quasi-) Monte Carlo methods: quantum computation. We give a short introduction
into this setting and present recent results of the authors on optimal quantum
algorithms for summation and integration. We discuss comparisons between the
three settings. The most interesting case for Monte Carlo and quantum
integration is that of moderate smoothness k and large dimension d which, in
fact, occurs in a number of important applied problems. In that case the
deterministic exponent is negligible, so the n^{-1/2} Monte Carlo and the
n^{-1} quantum speedup essentially constitute the entire convergence rate. We
observe that -- there is an exponential speed-up of quantum algorithms over
deterministic (classical) algorithms, if k/d tends to zero; -- there is a
(roughly) quadratic speed-up of quantum algorithms over randomized classical
algorithms, if k/d is small.Comment: 13 pages, contribution to the 4th International Conference on Monte
Carlo and Quasi-Monte Carlo Methods, Hong Kong 200
Quantum Integration in Sobolev Classes
We study high dimensional integration in the quantum model of computation. We
develop quantum algorithms for integration of functions from Sobolev classes
and analyze their convergence rates. We also prove lower
bounds which show that the proposed algorithms are, in many cases, optimal
within the setting of quantum computing. This extends recent results of Novak
on integration of functions from H\"older classes.Comment: Paper submitted to the Journal of Complexity. 28 page
Some Results on the Complexity of Numerical Integration
This is a survey (21 pages, 124 references) written for the MCQMC 2014
conference in Leuven, April 2014. We start with the seminal paper of Bakhvalov
(1959) and end with new results on the curse of dimension and on the complexity
of oscillatory integrals. Some small errors of earlier versions are corrected
On a Problem in Quantum Summation
We consider the computation of the mean of sequences in the quantum model of
computation. We determine the query complexity in the case of sequences which
satisfy a -summability condition for . This settles a problem left
open in Heinrich (2001).Comment: 21 pages, paper submitted to the Journal of Complexit
Quantum Approximation II. Sobolev Embeddings
A basic problem of approximation theory, the approximation of functions from
the Sobolev space W_p^r([0,1]^d) in the norm of L_q([0,1]^d), is considered
from the point of view of quantum computation. We determine the quantum query
complexity of this problem (up to logarithmic factors). It turns out that in
certain regions of the domain of parameters p,q,r,d quantum computation can
reach a speedup of roughly squaring the rate of convergence of classical
deterministic or randomized approximation methods. There are other regions were
the best possible rates coincide for all three settings.Comment: 23 pages, paper submitted to the Journal of Complexit
On the Use of Multipole Expansion in Time Evolution of Non-linear Dynamical Systems and Some Surprises Related to Superradiance
A new numerical method is introduced to study the problem of time evolution
of generic non-linear dynamical systems in four-dimensional spacetimes. It is
assumed that the time level surfaces are foliated by a one-parameter family of
codimension two compact surfaces with no boundary and which are conformal to a
Riemannian manifold C. The method is based on the use of a multipole expansion
determined uniquely by the induced metric structure on C. The approach is fully
spectral in the angular directions. The dynamics in the complementary 1+1
Lorentzian spacetime is followed by making use of a fourth order finite
differencing scheme with adaptive mesh refinement.
In checking the reliability of the introduced new method the evolution of a
massless scalar field on a fixed Kerr spacetime is investigated. In particular,
the angular distribution of the evolving field in to be superradiant scattering
is studied. The primary aim was to check the validity of some of the recent
arguments claiming that the Penrose process, or its field theoretical
correspondence---superradiance---does play crucial role in jet formation in
black hole spacetimes while matter accretes onto the central object. Our
findings appear to be on contrary to these claims as the angular dependence of
a to be superradiant scattering of a massless scalar field does not show any
preference of the axis of rotation. In addition, the process of superradiance,
in case of a massless scalar field, was also investigated. On contrary to the
general expectations no energy extraction from black hole was found even though
the incident wave packets was fine tuned to be maximally superradiant. Instead
of energy extraction the to be superradiant part of the incident wave packet
fails to reach the ergoregion rather it suffers a total reflection which
appears to be a new phenomenon.Comment: 49 pages, 11 figure
- …