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An intractability result for multiple integration
Our aim is to show that in the worst case setting the integration problem is intractable. The implications of the intractability result for lattice methods are considered briefly in Section 3
Construction of lattice rules for multiple integration based on a weighted discrepancy
High-dimensional integrals arise in a variety of areas, including quantum physics, the physics and chemistry of molecules, statistical mechanics and more recently, in financial applications.
In order to approximate multidimensional integrals, one may use Monte Carlo methods in which the quadrature points are generated randomly or quasi-Monte Carlo methods, in which points are generated deterministically. One particular class of quasi-Monte Carlo methods for multivariate integration is represented by lattice rules. Lattice rules constructed throughout this thesis allow good approximations to integrals of functions belonging to certain
weighted function spaces. These function spaces were proposed as an explanation as to why integrals in many variables appear to be successfully approximated although the standard theory indicates that the number of quadrature
points required for reasonable accuracy would be astronomical because of the large number of variables.
The purpose of this thesis is to contribute to theoretical results regarding the construction of lattice rules for multiple integration. We consider both lattice rules for integrals over the unit cube and lattice rules suitable for integrals over Euclidean space. The research reported throughout the thesis is devoted to finding the generating vector required to produce lattice rules that have what is termed a low weighted discrepancy . In simple terms, the discrepancy is a measure of the uniformity of the distribution of the quadrature points or in other settings, a worst-case error. One of the assumptions used in these weighted function spaces is that variables are arranged in the decreasing order of their importance and the assignment of weights in this situation results in so-called product weights . In other applications it is rather the importance of group of variables that matters. This situation is modelled by using function spaces in which the weights are general . In the weighted
settings mentioned above, the quality of the lattice rules is assessed by the weighted discrepancy mentioned earlier. Under appropriate conditions on the weights, the lattice rules constructed here produce a convergence rate of the error that ranges
from O(nā1/2) to the (believed) optimal O(nā1+Ī“) for any Ī“ gt 0, with the involved constant independent of the dimension
Determination of the rank of an integration lattice
The continuing and widespread use of lattice rules for high-dimensional numerical quadrature is driving the development of a rich and detailed theory. Part of this theory is devoted to computer searches for rules, appropriate to particular situations.
In some applications, one is interested in obtaining the (lattice) rank of a lattice rule Q(Ī) directly from the elements of a generator matrix B (possibly in upper triangular lattice form) of the corresponding dual lattice Īā„. We treat this problem in detail, demonstrating the connections between this (lattice) rank and the conventional matrix rank deficiency of modulo p versions of B
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
Quasi-Monte Carlo methods for high-dimensional integration: the standard (weighted Hilbert space) setting and beyond
This paper is a contemporary review of quasi-Monte Carlo (QMC) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube . It first introduces the by-now standard setting of weighted Hilbert spaces of functions with square-integrable mixed first derivatives, and then indicates alternative settings, such as non-Hilbert spaces, that can sometimes be more suitable. Original contributions include the extension of the fast component-by-component (CBC) construction of lattice rules that achieve the optimal convergence order (a rate of almost , where is the number of points, independently of dimension) to so-called Ć¢ā¬Åproduct and order dependentĆ¢ā¬ (POD) weights, as seen in some recent applications. Although the paper has a strong focus on lattice rules, the function space settings are applicable to all QMC methods. Furthermore, the error analysis and construction of lattice rules can be adapted to polynomial lattice rules from the family of digital nets.
doi:10.1017/S144618111200007
Constructive approaches to quasi-Monte Carlo methods for multiple integration
Recently, quasi-Monte Carlo methods have been successfully used for approximating multiple integrals in hundreds of dimensions in mathematical finance, and were significantly more efficient than Monte Carlo methods.
To understand the apparent success of quasi-Monte Carlo methods for multiple integration, one popular approach is to study worst-case error bounds in weighted function spaces in which the importance of the variables is moderated by some sequences of weights. Ideally, a family of quasi-Monte Carlo methods in some weighted function space should be strongly tractable. Strong tractability means that the minimal number of quadrature points n needed to reduce the initial error by a factor of Īµ is bounded by a polynomial in Īµā»Ā¹ independently of the dimension d. Several recent publications show the existence of lattice rules that satisfy the strong tractability error bounds in weighted Korobov spaces of periodic integrands and weighted Sobolev spaces of non-periodic integrands. However, those results were non-constructive and thus give no clues as to how to actually construct these lattice rules.
In this thesis, we focus on the construction of quasi-Monte Carlo methods that are strongly tractable. We develop and justify algorithms for the construction of lattice rules that achieve strong tractability error bounds in weighted Korobov and Sobolev spaces. The parameters characterizing these lattice rules are found ācomponent-by-componentā: the (d + 1)-th components are obtained by successive 1-dimensional searches, with the previous d components kept unchanged. The cost of these algorithms vary from O(ndĀ²) to O(nĀ³dĀ²) operations. With currently available technology, they allow construction of rules easily with values of n up to several million and dimensions d up to several hundred
Component-by-component construction of good intermediate-rank lattice rules
It is known that the generating vector of a rank-1 lattice rule can be constructed component-by-component to achieve strong tractability error bounds in both weighted Korobov spaces and weighted Sobolev spaces. Since the weights for these spaces are nonincreasing, the first few variables are in a sense more important than the rest. We thus propose to copy the points of a rank-1 lattice rule a number of times in the first few dimensions to yield an intermediate-rank lattice rule. We show that the generating vector (and in weighted Sobolev spaces, the shift also) of an intermediate-rank lattice rule can also be constructed component-by-component to achieve strong tractability error bounds. In certain circumstances, these bounds are better than the corresponding bounds for rank-1 lattice rules
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