578 research outputs found
Discontinuous information in the worst case and randomized settings
We believe that discontinuous linear information is never more powerful than
continuous linear information for approximating continuous operators. We prove
such a result in the worst case setting. In the randomized setting we consider
compact linear operators defined between Hilbert spaces. In this case, the use
of discontinuous linear information in the randomized setting cannot be much
more powerful than continuous linear information in the worst case setting.
These results can be applied when function evaluations are used even if
function values are defined only almost everywhere
Adaptive approximation of monotone functions
We study the classical problem of approximating a non-decreasing function in norm by sequentially querying its
values, for known compact real intervals , and a
known probability measure on \cX. For any function~ we characterize
the minimum number of evaluations of that algorithms need to guarantee an
approximation with an error below after
stopping. Unlike worst-case results that hold uniformly over all , our
complexity measure is dependent on each specific function . To address this
problem, we introduce GreedyBox, a generalization of an algorithm originally
proposed by Novak (1992) for numerical integration. We prove that GreedyBox
achieves an optimal sample complexity for any function , up to logarithmic
factors. Additionally, we uncover results regarding piecewise-smooth functions.
Perhaps as expected, the error of GreedyBox decreases much faster
for piecewise- functions than predicted by the algorithm (without any
knowledge on the smoothness of ). A simple modification even achieves
optimal minimax approximation rates for such functions, which we compute
explicitly. In particular, our findings highlight multiple performance gaps
between adaptive and non-adaptive algorithms, smooth and piecewise-smooth
functions, as well as monotone or non-monotone functions. Finally, we provide
numerical experiments to support our theoretical results
An Analysis of the Quasicontinuum Method
The aim of this paper is to present a streamlined and fully three-dimensional
version of the quasicontinuum (QC) theory of Tadmor et al. and to analyze its
accuracy and convergence characteristics. Specifically, we assess the effect of
the summation rules on accuracy; we determine the rate of convergence of the
method in the presence of strong singularities, such as point loads; and we
assess the effect of the refinement tolerance, which controls the rate at which
new nodes are inserted in the model, on the development of dislocation
microstructures.Comment: 30 pages, 16 figures. To appear in Jornal of the Mechanics and
Physics of Solid
On sequential and parallel solution of initial value problems
AbstractWe deal with the solution of systems z′(x) = f(x, z(x)), x ϵ [0, 1], z(0) = η, where the function ƒ [0, 1] × Rs → Rs has r continuous bounded partial derivatives. We assume that available information about the problem consists of evaluations of n linear functionals at ƒ. If an adaptive choice of these functionals is allowed (which is suitable for sequential processing), then the minimal error of an algorithm is of order n−(r+1), for any dimension s. We show that if nonadaptive information (well-suited for parallel computation) is used, then the minimal error cannot be essentially less than n−(r+1)(s+1). Thus, adaption is significantly better, and the advantage of using it grows with s. This yields that the ε-complexity in sequential computation is smaller for adaptive information. For parallel computation, nonadaptive information is more efficient only if the number of processors is very large, depending exponentially on the dimension s. We conclude that using parallelism by computing the information nonadaptively is not feasible
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
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