A gentle introduction to approximating high-dimensional integrals with lattice rules and a maybe-not-so-gentle introduction to the relevant function spaces for QMC integration
These lecture notes give both a practical and theoretical introduction to high-dimensional integration with lattice rules. We introduce lattice points and rules in combination with accompanying Julia code. Next we introduce the typical function spaces used in the literature. We compare classical Sobolev spaces with Sobolev spaces of dominating mixed smoothness. We derive lower bounds and show that the curse of dimensionality is unavoidable for these unweighted function spaces. Next we switch to weighted periodic spaces and derive a constructive upper bound using the component-by-component construction for the generating vector of a lattice rule. We pick periodic spaces because they are a perfect match with lattice rules. The upper bound shows that under suitable conditions on the weights we obtain dimension-independent worst-case error bounds. We then use embedding constants to transfer these bounds to some common norms for these periodic spaces.
These lecture notes give both a practical and theoretical introduction to high-dimensional integration with lattice rules. We introduce lattice points and rules in combination with accompanying Julia code. Next we introduce the typical function spaces used in the literature. We compare classical Sobolev spaces with Sobolev spaces of dominating mixed smoothness. We derive lower bounds and show that the curse of dimensionality is unavoidable for these unweighted function spaces. Next we switch to weighted periodic spaces and derive a constructive upper bound using the component-by-component construction for the generating vector of a lattice rule. We pick periodic spaces because they are a perfect match with lattice rules. The upper bound shows that under suitable conditions on the weights we obtain dimension-independent worst-case error bounds. We then use embedding constants to transfer these bounds to some common norms for these periodic spaces.
In this talk I will show the beautiful simplicity of approximating high-dimensional integrals using lattice rules. In current engineering applications it is relatively easy to come up with very high and ultra high dimensional integrals, which might be expectations, that depend on hundreds, thousands, or even an infinite number of variables. As the minimal non-trivial tensor product rule, constructed from a 2-point rule, in 100 dimensions already uses 2^{100} = 1267650600228229401496703205376 ≈ 10^{30} function evaluations—which would take 10^{24} seconds at 1 function evaluation per microsecond and we remember the age of the universe to be only approximately 10^{17} seconds—we all know why to use Monte Carlo or quasi-Monte Carlo methods where we can freely choose the number of function evaluations independent of the number of dimensions. While plain vanilla Monte Carlo methods are very robust and easy to apply, in the case of quasi-Monte Carlo methods a lot more details can pop up. I will first introduce the most elegant—in my biased opinion, which I will defend—of all quasi-Monte Carlo methods: the rank-1 lattice rule. This method is defined by one single integer generating vector together with the total number of points in the rule. Approximating a multivariate integral by this method can be done in one line of Matlab code! As one is often interested in successive approximations I will next introduce a rank-1 lattice sequence. Maybe surprisingly, also here, one simple line of Matlab code is sufficient to calculate a segment of a lattice sequence (with a special property). While for Monte Carlo integration the usual assumption is L2 integrability, leading to an error bound of O(N^{−1/2}). For quasi-Monte Carlo methods we generally work with more involved function space settings, leading to error bounds of the form O(N^{−α}) with α ≥ 1/2. I will show the standard function space settings and dive into the weighted function spaces which are being used in current theory to be able to make the error bounds independent of the number of dimensions. We will gloss over the Korobov space of periodic smooth functions, the unanchored Sobolev space with first mixed derivatives, and the cosine space which is a special space of smooth non-periodic functions. All of which are spaces of so-called mixed dominating smoothness, which in terms of mixed derivatives would mean ∂^{k_1+···+k_s} /(∂x_1^{k_1} · · · ∂x_s^{k_s}) exists for all (k_1, . . . , k_s) ≤ (α, . . . , α). For non-periodic function spaces lattice rules and sequences can only reach O(N^{−1}) convergence. To arrive at higher-order convergence of O(N^{−α}) for non-periodic functions we dive into higher-order Sobolev spaces (of mixed dominating smoothness) in combination with digital nets, but here disguised as a lattice rule, namely polynomial lattice rules. Here, all arithmetic is over a finite field (we will choose the one of order 2) and our lattice rules now take the form of a lattice of polynomials which we map back to points in the unit cube. Higher order can now be obtained by means of the interlacing method from Josef Dick. Although more complicated, again, given the generating matrices of the digital net (as a bunch of integers), and two lines for a magical de Bruijn sequence, we can approximate an integral by such a digital sequence with a single line of (complicated) Matlab code. While Matlab/Octave one-liners might be fun, and somehow instructive, I will also point you to my online resources where code is available to (a) construct good lattice rules, lattice sequences and polynomial lattice rules (for product, order-dependent, product-and-order-dependent aka POD, and smoothness-driven product-and-order dependent aka SPOD) weighted spaces, and (b) generate points given the generating vector or matrices. Armed with this software you will be able to attack your own problems in no time. This talk is based upon the results of collaborations with many colleagues.
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 ε is bounded by a polynomial in ε⁻¹ 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 ‘component-by-component’: 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(nd²) to O(n³d²) 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.
We study multivariate numerical integration of smooth functions in weighted Sobolev spaces with dominating mixed smoothness α≥2 defined over the s-dimensional unit cube. We propose a new quasi-Monte Carlo (QMC)-based quadrature rule, named the extrapolated polynomial lattice rule, which achieves the almost optimal rate of convergence. Extrapolated polynomial lattice rules are constructed in two steps: (i) construction of classical polynomial lattice rules over Fb with α consecutive sizes of nodes, bm−α+1,…,bm, and (ii) recursive application of Richardson extrapolation to a chain of α approximate values of the integral obtained by consecutive polynomial lattice rules. We prove the existence of good extrapolated polynomial lattice rules achieving the almost optimal order of convergence of the worst-case error in Sobolev spaces with general weights. Then, by restricting to product weights, we show that such good extrapolated polynomial lattice rules can be constructed by the fast component-by-component algorithm under a computable quality criterion. The required total construction cost is of order (s+α)NlogN, which improves the currently known result for the interlaced polynomial lattice rule, which is of order sαNlogN. We also study the dependence of the worst-case error bound on the dimension. A big advantage of our method compared to interlaced polynomial lattice rules is that the fast QMC matrix vector method can be used in this setting, while still achieving the same rate of convergence. Such a method was previously not known. Numerical experiments for test integrands support our theoretical result.
This paper is a contemporary review of QMC (‘quasi-Monte Carlo’) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0,1] s , where s may be large, or even infinite. After a general introduction, the paper surveys recent developments in lattice methods, digital nets, and related themes. Among those recent developments are methods of construction of both lattices and digital nets, to yield QMC rules that have a prescribed rate of convergence for sufficiently smooth functions, and ideally also guaranteed slow growth (or no growth) of the worst-case error as s increases. A crucial role is played by parameters called ‘weights’, since a careful use of the weight parameters is needed to ensure that the worst-case errors in an appropriately weighted function space are bounded, or grow only slowly, as the dimension s increases. Important tools for the analysis are weighted function spaces, reproducing kernel Hilbert spaces, and discrepancy, all of which are discussed with an appropriate level of detail.
We introduce a new method to approximate integrals ∫ R d f ( x ) d x \int _{\mathbb {R}^d} f(\boldsymbol {x}) \,\mathrm {d}\boldsymbol {x} which simply scales lattice rules from the unit cube [ 0 , 1 ] d [0,1]^d to properly sized boxes on R d \mathbb {R}^d , hereby achieving higher-order convergence that matches the smoothness of the integrand function f f in a certain Sobolev space of dominating mixed smoothness. Our method only assumes that we can evaluate the integrand function f f and does not assume a particular density nor the ability to sample from it. In particular, for the theoretical analysis we show a new result that the method of adding Bernoulli polynomials to a function to make it “periodic” on a box without changing its integral value over the box is equivalent to an orthogonal projection from a well chosen Sobolev space of dominating mixed smoothness to an associated periodic Sobolev space of the same dominating mixed smoothness, which we call a Korobov space. We note that the Bernoulli polynomial method is often not used because of its excessive computational complexity and also here we only make use of it in our theoretical analysis. We show that our new method of applying scaled lattice rules to increasing boxes can be interpreted as orthogonal projections with decreasing projection error. Such a method would not work on the unit cube since then the committed error caused by non-periodicity of the integrand would be constant, but for integration on the Euclidean space we can use the certain decay towards zero when the boxes grow. Hence we can bound the truncation error as well as the projection error and show higher-order convergence in applying scaled lattice rules for integration on Euclidean space. We illustrate our theoretical analysis by numerical experiments which confirm our findings.
We develop and justify an algorithm for the construction of quasi–Monte Carlo (QMC) rules for integration in weighted Sobolev spaces; the rules so constructed are shifted rank-1 lattice rules. The parameters characterising the shifted lattice rule are found “component-by-component”: the ( d + 1 d+1 )-th component of the generator vector and the shift are obtained by successive 1 1 -dimensional searches, with the previous d d components kept unchanged. The rules constructed in this way are shown to achieve a strong tractability error bound in weighted Sobolev spaces. A search for n n -point rules with n n prime and all dimensions 1 to d d requires a total cost of O ( n 3 d 2 ) O(n^3d^2) operations. This may be reduced to O ( n 3 d ) O(n^3d) operations at the expense of O ( n 2 ) O(n^2) storage. Numerical values of parameters and worst-case errors are given for dimensions up to 40 and n n up to a few thousand. The worst-case errors for these rules are found to be much smaller than the theoretical bounds.