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 $n$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-$1$ 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 $O(n^{-1/2})$. 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-$1$ 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 $O(n^{-\lambda/2})$ for all $1 \leq \lambda < 2 \alpha$, where $\alpha$ 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

Construction of quasi-Monte Carlo rules for multivariate integration in spaces of permutation-invariant functions

We study multivariate integration of functions that are invariant under the permutation (of a subset) of their arguments. Recently, in Nuyens, Suryanarayana, and Weimar (Adv. Comput. Math. (2016), 42(1):55--84), the authors derived an upper estimate for the $n$th minimal worst case error for such problems, and showed that under certain conditions this upper bound only weakly depends on the dimension. We extend these results by proposing two (semi-) explicit construction schemes. We develop a component-by-component algorithm to find the generating vector for a shifted rank-$1$ lattice rule that obtains a rate of convergence arbitrarily close to $\mathcal{O}(n^{-\alpha})$, where $\alpha>1/2$ denotes the smoothness of our function space and $n$ is the number of cubature nodes. Further, we develop a semi-constructive algorithm that builds on point sets which can be used to approximate the integrands of interest with a small error; the cubature error is then bounded by the error of approximation. Here the same rate of convergence is achieved while the dependence of the error bounds on the dimension $d$ is significantly improved

Multilevel Monte Carlo simulation for Levy processes based on the Wiener-Hopf factorisation

In Kuznetsov et al. (2011) a new Monte Carlo simulation technique was introduced for a large family of Levy processes that is based on the Wiener-Hopf decomposition. We pursue this idea further by combining their technique with the recently introduced multilevel Monte Carlo methodology. Moreover, we provide here for the first time a theoretical analysis of the new Monte Carlo simulation technique in Kuznetsov et al. (2011) and of its multilevel variant for computing expectations of functions depending on the historical trajectory of a Levy process. We derive rates of convergence for both methods and show that they are uniform with respect to the "jump activity" (e.g. characterised by the Blumenthal-Getoor index). We also present a modified version of the algorithm in Kuznetsov et al. (2011) which combined with the multilevel methodology obtains the optimal rate of convergence for general Levy processes and Lipschitz functionals. This final result is only a theoretical one at present, since it requires independent sampling from a triple of distributions which is currently only possible for a limited number of processes

Strang splitting in combination with rank-11 and rank-rr lattices for the time-dependent Schr\"odinger equation

We approximate the solution for the time dependent Schr\"odinger equation (TDSE) in two steps. We first use a pseudo-spectral collocation method that uses samples of functions on rank-1 or rank-r lattice points with unitary Fourier transforms. We then get a system of ordinary differential equations in time, which we solve approximately by stepping in time using the Strang splitting method. We prove that the numerical scheme proposed converges quadratically with respect to the time step size, given that the potential is in a Korobov space with the smoothness parameter greater than $9/2$. Particularly, we prove that the required degree of smoothness is independent of the dimension of the problem. We demonstrate our new method by comparing with results using sparse grids from [12], with several numerical examples showing large advantage for our new method and pushing the examples to higher dimensionality. The proposed method has two distinctive features from a numerical perspective: (i) numerical results show the error convergence of time discretization is consistent even for higher-dimensional problems; (ii) by using the rank-$1$ lattice points, the solution can be efficiently computed (and further time stepped) using only $1$-dimensional Fast Fourier Transforms.Comment: Modified. 40pages, 5 figures. The proof of Lemma 1 is updated after the paper is publishe

Advanced Quasi-Monte Carlo Algorithms for High-dimensional Integration and Approximation

The aim of this research is to develop algorithms to approximate the solutions of problems defined on spaces of d-variate functions, where d can be arbitrarily large. We study quasi-Monte Carlo methods, which in contrast to Monte Carlo methods, use deterministic point sets. These methods converge significantly faster with well chosen points that minimize deterministic error bounds. Additionally, we require that the computational hardness of these methods has only a moderate or no dependence on d. We study this using the information complexity n(d, epsilon), the minimal number of information operations required to achieve an approximation with an error of epsilon or less. If n(d, epsilon) grows exponentially in d, the problem is said to have the 'curse of dimension' and the goal is to vanquish this curse. Our research resulted in several contributions. We studied multivariate integration for a Hilbert space of functions that are invariant under permutations of the variables; this is inspired from the symmetry constraints induced on solutions of the Schrödinger equation for indistinguishable particles. We showed that there exist quasi-Monte Carlo methods that achieve a convergence rate of O(n^−1/2) with the implied constant depending only polynomially on d and under some conditions independent of d (and that shifted rank-1 lattice rules can attain the same). We then provided a construction algorithm for a shifted rank-1 lattice that obtains the (almost) optimal rate of convergence. Next, we extended the use of rank-1 lattice points, traditionally used for the cubature and approximation of periodic functions, to the non-periodic setting. We did this in the cosine space setting, where we extensively studied the use of tent-transformed lattice points for cubature and approximation of nonperiodic functions. We were able to reuse algorithms from the periodic setting by establishing relations between the cosine space and the Korobov space of periodic functions. Finally we studied some special cases of approximation in two dimensions. We derived explicit expressions for various degrees of exactness for the Fibonacci lattice points. We end the thesis with an insight on the Lebesgue constant of trigonometric interpolation using lattice points as interpolation nodes.status: publishe

Integration and Approximation with Fibonacci lattice points

We study the properties of a special rank-$1$ point set in $2$ dimensions --- Fibonacci lattice points. We present the analysis of these point sets for cubature and approximation of bivariate periodic functions with decaying spectral coefficients. We are interested in truncating the frequency space into index sets based on different degrees of exactness. The numerical results support that the Lebesgue constant of these point sets grows like the conjectured optimal rate $\ln^2(N)$, where $N$ is the number of sample points.status: publishe

Collocation for non-periodic functions with lattice points

Spectral collocation and reconstruction methods have been widely studied for periodic functions using Fourier expansions. We investigate the use of modified Fourier series for the approximation and collocation of $d$-variate non-periodic functions with frequency support on a hyperbolic cross. We show that rank-$1$ lattice points can be used as collocation points in the approximation of non-periodic functions and these lattice points can be constructed by a component-by-component algorithm.status: publishe

Quasi Monte-Carlo methods for the TDSE (Time dependent Schrödinger equation)

We approximate the solution for the time dependent Schrödinger equation (TDSE) in two steps. We first use a pseudo-spectral collocation method that uses samples of the functions on rank-1 lattice points. We then get a system of ordinary differential equations in time, which we solve approximately by stepping in time using the Strang splitting method.status: publishe

Rank-1 lattices and Strang splitting for the time-dependent Schrödinger equation

http://www.hda2017.unsw.edu.au/program/#A5status: publishe

Reconstruction and collocation of a class of non-periodic functions by sampling along tent-transformed rank-1 lattices

Spectral collocation and reconstruction methods have been widely studied for periodic functions using Fourier expansions.We investigate the use of cosine series for the approximation and collocation of multivariate non-periodic functions with frequency support mainly determined by hyperbolic crosses.We seek methods that work for an arbitrary number of dimensions.We show that applying the tent-transformation on rank-1 lattice points renders them suitable to be collocation/sampling points for the approximation of non-periodic functions with perfect numerical stability. Moreover, we show that the approximation degree—in the sense of approximating inner products of basis functions up to a certain degree exactly—of the tent-transformed lattice point set with respect to cosine series, is the same as the approximation degree of the original lattice point set with respect to Fourier series, although the error can still be reduced in the case of cosine series. A component-by-component algorithm is studied to construct such a point set. We are then able to reconstruct a non-periodic function from its samples and approximate the solutions to certain PDEs subject to Neumann and Dirichlet boundary conditions. Finally, we present some numerical results.status: publishe