3,653 research outputs found
On Integration Methods Based on Scrambled Nets of Arbitrary Size
We consider the problem of evaluating for a function . In situations where
can be approximated by an estimate of the form
, with a point set in
, it is now well known that the Monte Carlo
convergence rate can be improved by taking for the first
points, , of a scrambled
-sequence in base . In this paper we derive a bound for the
variance of scrambled net quadrature rules which is of order
without any restriction on . As a corollary, this bound allows us to provide
simple conditions to get, for any pattern of , an integration error of size
for functions that depend on the quadrature size . Notably,
we establish that sequential quasi-Monte Carlo (M. Gerber and N. Chopin, 2015,
\emph{J. R. Statist. Soc. B, to appear.}) reaches the
convergence rate for any values of . In a numerical study, we show that for
scrambled net quadrature rules we can relax the constraint on without any
loss of efficiency when the integrand is a discontinuous function
while, for sequential quasi-Monte Carlo, taking may only
provide moderate gains.Comment: 27 pages, 2 figures (final version, to appear in The Journal of
Complexity
Remarks on numerical integration, discrepancy, and diaphony
The goal of this paper is twofold. First, we present a unified way of
formulating numerical integration problems from both approximation theory and
discrepancy theory. Second, we show how techniques, developed in approximation
theory, work in proving lower bounds for recently developed new type of
discrepancy -- the smooth discrepancy
Quadrature rules and distribution of points on manifolds
We study the error in quadrature rules on a compact manifold. As in the
Koksma-Hlawka inequality, we consider a discrepancy of the sampling points and
a generalized variation of the function. In particular, we give sharp
quantitative estimates for quadrature rules of functions in Sobolev classes
Construction of interlaced scrambled polynomial lattice rules of arbitrary high order
Higher order scrambled digital nets are randomized quasi-Monte Carlo rules
which have recently been introduced in [J. Dick, Ann. Statist., 39 (2011),
1372--1398] and shown to achieve the optimal rate of convergence of the root
mean square error for numerical integration of smooth functions defined on the
-dimensional unit cube. The key ingredient there is a digit interlacing
function applied to the components of a randomly scrambled digital net whose
number of components is , where the integer is the so-called
interlacing factor. In this paper, we replace the randomly scrambled digital
nets by randomly scrambled polynomial lattice point sets, which allows us to
obtain a better dependence on the dimension while still achieving the optimal
rate of convergence. Our results apply to Owen's full scrambling scheme as well
as the simplifications studied by Hickernell, Matou\v{s}ek and Owen. We
consider weighted function spaces with general weights, whose elements have
square integrable partial mixed derivatives of order up to , and
derive an upper bound on the variance of the estimator for higher order
scrambled polynomial lattice rules. Employing our obtained bound as a quality
criterion, we prove that the component-by-component construction can be used to
obtain explicit constructions of good polynomial lattice point sets. By first
constructing classical polynomial lattice point sets in base and dimension
, to which we then apply the interlacing scheme of order , we obtain a
construction cost of the algorithm of order operations
using memory in case of product weights, where is the
number of points in the polynomial lattice point set
A Discrepancy Bound for Deterministic Acceptance-Rejection Samplers Beyond in Dimension 1
In this paper we consider an acceptance-rejection (AR) sampler based on
deterministic driver sequences. We prove that the discrepancy of an element
sample set generated in this way is bounded by ,
provided that the target density is twice continuously differentiable with
non-vanishing curvature and the AR sampler uses the driver sequence
where are real algebraic numbers such that is a
basis of a number field over of degree . For the driver
sequence where is the -th Fibonacci number and
is the fractional part of a non-negative real
number , we can remove the factor to improve the convergence rate to
, where again is the number of samples we accepted.
We also introduce a criterion for measuring the goodness of driver sequences.
The proposed approach is numerically tested by calculating the star-discrepancy
of samples generated for some target densities using and
as driver sequences. These results confirm that achieving a
convergence rate beyond is possible in practice using
and as driver sequences in the
acceptance-rejection sampler
Smoothing the payoff for efficient computation of Basket option prices
We consider the problem of pricing basket options in a multivariate Black
Scholes or Variance Gamma model. From a numerical point of view, pricing such
options corresponds to moderate and high dimensional numerical integration
problems with non-smooth integrands. Due to this lack of regularity, higher
order numerical integration techniques may not be directly available, requiring
the use of methods like Monte Carlo specifically designed to work for
non-regular problems. We propose to use the inherent smoothing property of the
density of the underlying in the above models to mollify the payoff function by
means of an exact conditional expectation. The resulting conditional
expectation is unbiased and yields a smooth integrand, which is amenable to the
efficient use of adaptive sparse grid cubature. Numerical examples indicate
that the high-order method may perform orders of magnitude faster compared to
Monte Carlo or Quasi Monte Carlo in dimensions up to 35
Quasi-Monte Carlo numerical integration on : digital nets and worst-case error
Quasi-Monte Carlo rules are equal weight quadrature rules defined over the
domain . Here we introduce quasi-Monte Carlo type rules for numerical
integration of functions defined on . These rules are obtained by
way of some transformation of digital nets such that locally one obtains qMC
rules, but at the same time, globally one also has the required distribution.
We prove that these rules are optimal for numerical integration in fractional
Besov type spaces. The analysis is based on certain tilings of the Walsh phase
plane
Numerical Integration on Graphs: where to sample and how to weigh
Let be a finite, connected graph with weighted edges. We are
interested in the problem of finding a subset of vertices and
weights such that for functions that are `smooth'
with respect to the geometry of the graph. The main application are problems
where is known to somehow depend on the underlying graph but is expensive
to evaluate on even a single vertex. We prove an inequality showing that the
integration problem can be rewritten as a geometric problem (`the optimal
packing of heat balls'). We discuss how one would construct approximate
solutions of the heat ball packing problem; numerical examples demonstrate the
efficiency of the method
Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order
We define a Walsh space which contains all functions whose partial mixed
derivatives up to order exist and have finite variation. In
particular, for a suitable choice of parameters, this implies that certain
Sobolev spaces are contained in these Walsh spaces. For this Walsh space we
then show that quasi-Monte Carlo rules based on digital
-sequences achieve the optimal rate of convergence of the
worst-case error for numerical integration. This rate of convergence is also
optimal for the subspace of smooth functions. Explicit constructions of digital
-sequences are given hence providing explicit quasi-Monte Carlo
rules which achieve the optimal rate of convergence of the integration error
for arbitrarily smooth functions
Proof Techniques in Quasi-Monte Carlo Theory
In this survey paper we discuss some tools and methods which are of use in
quasi-Monte Carlo (QMC) theory. We group them in chapters on Numerical
Analysis, Harmonic Analysis, Algebra and Number Theory, and Probability Theory.
We do not provide a comprehensive survey of all tools, but focus on a few of
them, including reproducing and covariance kernels, Littlewood-Paley theory,
Riesz products, Minkowski's fundamental theorem, exponential sums, diophantine
approximation, Hoeffding's inequality and empirical processes, as well as other
tools. We illustrate the use of these methods in QMC using examples.Comment: Revised versio
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