180 research outputs found
On the separability of multivariate functions
Separability of multivariate functions alleviates the difficulty in finding a
minimum or maximum value of a function such that an optimal solution can be
searched by solving several disjoint problems with lower dimensionalities. In
most of practical problems, however, a function to be optimized is black-box
and we hardly grasp its separability. In this study, we first describe a
general separability condition which a function defined over an arbitrary
domain satisfies if and only if the function is separable with respect to given
disjoint subsets of variables. By introducing an alternative separability
condition, we propose a Monte Carlo-based algorithm to estimate the
separability of a function defined over unit cube with respect to given
disjoint subsets of variables. Moreover, we extend our algorithm to estimate
the number of disjoint subsets and the disjoint subsets such that a function is
separable with respect to them. Computational complexity of our extended
algorithm is function-dependent and varies from linear to exponential in the
dimension
Construction of scrambled polynomial lattice rules over with small mean square weighted discrepancy
The discrepancy is one of several well-known quantitative
measures for the equidistribution properties of point sets in the
high-dimensional unit cube. The concept of weights was introduced by Sloan and
Wo\'{z}niakowski to take into account the relative importance of the
discrepancy of lower dimensional projections. As known under the name of
quasi-Monte Carlo methods, point sets with small weighted
discrepancy are useful in numerical integration. This study investigates the
component-by-component construction of polynomial lattice rules over the finite
field whose scrambled point sets have small mean square weighted
discrepancy. An upper bound on this discrepancy is proved,
which converges at almost the best possible rate of for all
, where denotes the number of points. Numerical experiments
confirm that the performance of our constructed polynomial lattice point sets
is comparable or even superior to that of Sobol' sequences
Good interlaced polynomial lattice rules for numerical integration in weighted Walsh spaces
Quadrature rules using higher order digital nets and sequences are known to
exploit the smoothness of a function for numerical integration and to achieve
an improved rate of convergence as compared to classical digital nets and
sequences for smooth functions. A construction principle of higher order
digital nets and sequences based on a digit interlacing function was introduced
in [J. Dick, SIAM J. Numer. Anal., 45 (2007) pp.~2141--2176], which interlaces
classical digital nets or sequences whose number of components is a multiple of
the dimension.
In this paper, we study the use of polynomial lattice point sets for
interlaced components. We call quadrature rules using such point sets {\em
interlaced polynomial lattice rules}. We consider weighted Walsh spaces
containing smooth functions and derive two upper bounds on the worst-case error
for interlaced polynomial lattice rules, both of which can be employed as a
quality criterion for the construction of interlaced polynomial lattice rules.
We investigate the component-by-component construction and the Korobov
construction as a means of explicit constructions of good interlaced polynomial
lattice rules that achieve the optimal rate of the worst-case error. Through
this approach we are able to obtain a good dependence of the worst-case error
bounds on the dimension under certain conditions on the weights, while
significantly reducing the construction cost as compared to higher order
polynomial lattice rules
On the discrepancy of two-dimensional folded Hammersley point sets
We give an explicit construction of two-dimensional point sets whose
discrepancy is of best possible order for all . It is
provided by folding Hammersley point sets in base by means of the -adic
baker's transformation which has been introduced by Hickernell (2002) for
and Goda, Suzuki and Yoshiki (2013) for arbitrary , .
We prove that both the minimum Niederreiter-Rosenbloom-Tsfasman weight and the
minimum Dick weight of folded Hammersley point sets are large enough to achieve
the best possible order of discrepancy for all
Quasi-Monte Carlo integration using digital nets with antithetics
Antithetic sampling, which goes back to the classical work by Hammersley and
Morton (1956), is one of the well-known variance reduction techniques for Monte
Carlo integration. In this paper we investigate its application to digital nets
over for quasi-Monte Carlo (QMC) integration, a deterministic
counterpart of Monte Carlo, of functions defined over the -dimensional unit
cube. By looking at antithetic sampling as a geometric technique in a compact
totally disconnected abelian group, we first generalize the notion of
antithetic sampling from base to an arbitrary base . Then we
analyze the QMC integration error of digital nets over with
-adic antithetics. Moreover, for a prime , we prove the existence of good
higher order polynomial lattice point sets with -adic antithetics for QMC
integration of smooth functions in weighted Sobolev spaces. Numerical
experiments based on Sobol' point sets up to show that the rate of
convergence can be improved for smooth integrands by using antithetic sampling
technique, which is quite encouraging beyond the reach of our theoretical
result and motivates future work to address
Fast construction of higher order digital nets for numerical integration in weighted Sobolev spaces
Higher order digital nets are special classes of point sets for quasi-Monte
Carlo rules which achieve the optimal convergence rate for numerical
integration of smooth functions. An explicit construction of higher order
digital nets was proposed by Dick, which is based on digitally interlacing in a
certain way the components of classical digital nets whose number of components
is a multiple of the dimension . In this paper we give a fast computer
search algorithm to find good classical digital nets suitable for interlaced
components by using polynomial lattice point sets.
We consider certain weighted Sobolev spaces of smoothness of arbitrarily high
order, and derive an upper bound on the mean square worst-case error for
digitally shifted higher order digital nets. Employing this upper bound as a
quality criterion, we prove that the component-by-component construction can be
used efficiently to find good polynomial lattice point sets suitable for
interlaced components. Through this approach we are able to get some
tractability results under certain conditions on the weights. Fast construction
using the fast Fourier transform requires the construction cost of operations using memory, where is the number of points and
is the dimension. This implies a significant reduction in the construction cost
as compared to higher order polynomial lattice point sets. Numerical
experiments confirm that the performance of our constructed point sets often
outperforms those of higher order digital nets with Sobol' sequences and
Niederreiter-Xing sequences used for interlaced components, indicating the
usefulness of our algorithm
Constructing good higher order polynomial lattice rules with modulus of reduced degree
In this paper we investigate multivariate integration in weighted unanchored
Sobolev spaces of smoothness of arbitrarily high order. As quadrature points we
employ higher order polynomial lattice point sets over which
are randomly digitally shifted and then folded using the tent transformation.
We first prove the existence of good higher order polynomial lattice rules
which achieve the optimal rate of the mean square worst-case error, while
reducing the required degree of modulus by half as compared to higher order
polynomial lattice rules whose quadrature points are randomly digitally shifted
but not folded using the tent transformation. Thus we are able to restrict the
search space of generating vectors significantly. We then study the
component-by-component construction as an explicit means of obtaining good
higher order polynomial lattice rules. In a way analogous to [J. Baldeaux, J.
Dick, G. Leobacher, D. Nuyens, F. Pillichshammer, Numer. Algorithms, 59 (2012)
403--431], we show how to calculate the quality criterion efficiently and how
to obtain the fast component-by-component construction using the fast Fourier
transform. Our result generalizes the previous result shown by [L.L. Cristea,
J. Dick, G. Leobacher, F. Pillichshammer, Numer. Math., 105 (2007) 413--455],
in which the degree of smoothness is fixed at 2 and classical polynomial
lattice rules are considered
Computing the variance of a conditional expectation via non-nested Monte Carlo
Computing the variance of a conditional expectation has often been of
importance in uncertainty quantification. Sun et al. has introduced an unbiased
nested Monte Carlo estimator, which they call -level simulation
since the optimal inner-level sample size is bounded as the computational
budget increases. In this letter we construct unbiased non-nested Monte Carlo
estimators based on the so-called pick-freeze scheme due to Sobol'. An
extension of our approach to compute higher order moments of a conditional
expectation is also discussed
Stability of lattice rules and polynomial lattice rules constructed by the component-by-component algorithm
We study quasi-Monte Carlo (QMC) methods for numerical integration of
multivariate functions defined over the high-dimensional unit cube. Lattice
rules and polynomial lattice rules, which are special classes of QMC methods,
have been intensively studied and the so-called component-by-component (CBC)
algorithm has been well-established to construct rules which achieve the almost
optimal rate of convergence with good tractability properties for given
smoothness and set of weights. Since the CBC algorithm constructs rules for
given smoothness and weights, not much is known when such rules are used for
function classes with different smoothness and/or weights.
In this paper we prove that a lattice rule constructed by the CBC algorithm
for the weighted Korobov space with given smoothness and weights achieves the
almost optimal rate of convergence with good tractability properties for
general classes of smoothness and weights which satisfy some summability
conditions. Such a stability result also can be shown for polynomial lattice
rules in weighted Walsh spaces. We further give bounds on the weighted star
discrepancy and discuss the tractability properties for these QMC rules. The
results are comparable to those obtained for Halton, Sobol and Niederreiter
sequences
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
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