4. Recent advances in higher order quasi-Monte Carlo methods

Abstract

In this article we review some of recent results on higher order quasi-Monte Carlo (HoQMC) methods. After a seminal work by Dick (2007Dick ( , 2008 who originally introduced the concept of HoQMC, there have been significant theoretical progresses on HoQMC in terms of discrepancy as well as multivariate numerical integration. Moreover, several successful and promising applications of HoQMC to partial differential equations with random coefficients and Bayesian estimation/inversion problems have been reported recently. In this article we start with standard quasi-Monte Carlo methods based on digital nets and sequences in the sense of Niederreiter, and then move onto their higher order version due to Dick. The Walsh analysis of smooth functions plays a crucial role in developing the theory of HoQMC, and the aim of this article is to provide a unified picture on how the Walsh analysis enables recent developments of HoQMC both for discrepancy and numerical integration. P ⊂ [0, 1] s be a finite multiset, that is, if an element occurs multiple times, it is counted according to its multiplicity. Then I(f ) is approximated bywhere |P | denotes the cardinality of P . It is obvious that this algorithm is exact for any choice of P if f is a constant function, but except for such a trivial case, a careful design of P and the accompanying theoretical analysis are required to show that the algorithm works well for various functions f . One fairly easy but sensible approach is to choose each point x independently and uniformly from [0, 1] s . This is widely known under the name of Monte Carlo methods [39]. Many fundamental results in probability theory, including the law of large numbers and the central limit theorem, apply to this approach. Looking at I(f ; P ) as a random variable (with P being the underlying stochastic variable), we have E[I(f ; P )] = I(f ) and V[I(f ; P )] = V[f ] |P | ,

A Tool for Custom Construction of QMC and RQMC Point Sets

“…This greedy approach can reduce by a huge factor (exponential in the dimension) the total number of cases that are examined in comparison with the exhaustive search. What is very interesting is that for most types of QMC constructions and FOMs implemented in LatNet Builder, the convergence rate for the worst-case error or variance obtained with this restricted approach is provably the same as for the exhaustive search [8,15].…”

“…Another important choice for H is a Sobolev space of functions whose mixed partial derivatives of order up to α are square-integrable. It is known that for this space, one can construct point sets whose discrepancy converges as O((log n) (s−1)/2 n −α ), and that this is the best possible rate [4,8,15,16,17,18]. The main classes of QMC point sets are lattice points and digital nets.…”

A Tool for Custom Construction of QMC and RQMC Point Sets

“…This greedy approach can reduce by a huge factor (exponential in the dimension) the total number of cases that are examined in comparison with the exhaustive search. What is very interesting is that for most types of QMC constructions and FOMs implemented in LatNet Builder, the convergence rate for the worst-case error or variance obtained with this restricted approach is provably the same as for the exhaustive search [8,14].…”

“…Another important choice for H is a Sobolev space of functions whose mixed partial derivatives of order up to α are square-integrable. It is known that for this space, one can construct point sets whose discrepancy converges as O((log n) (s−1)/2 n −α ), and that this is the best possible rate [4,8,14,15,16,17].…”

m-CUBES An efficient and portable implementation of multi-dimensional integration for gpus

“…The first challenge, "curse of dimensionality", can be addressed to some extent by using a Monte Carlo based algorithm for multi-dimensional integration, as the convergence rate of such methods is independent of the dimension d. The convergence rate can sometimes be further improved by utilizing low-discrepancy sequences (Quasi-Monte Carlo) instead of pseudo-random samples [1] [6]. When utilizing Monte Carlo based approaches, the second challenge of consolidating sampling efforts on the "ill-behaved' areas of the integration space, is addressed through "stratified" and/or "importance" sampling, which aim to reduce the variance of the random samples.…”