Fast QMC matrix-vector multiplication

Quasi-Monte Carlo (QMC) rules $1/N \sum_{n=0}^{N-1} f(\boldsymbol{y}_n A)$ can be used to approximate integrals of the form $\int_{[0,1]^s} f(\boldsymbol{y} A) \,\mathrm{d} \boldsymbol{y}$, where $A$ is a matrix and $\boldsymbol{y}$ is row vector. This type of integral arises for example from the simulation of a normal distribution with a general covariance matrix, from the approximation of the expectation value of solutions of PDEs with random coefficients, or from applications from statistics. In this paper we design QMC quadrature points $\boldsymbol{y}_0, ..., \boldsymbol{y}_{N-1} \in [0,1]^s$ such that for the matrix $Y = (\boldsymbol{y}_{0}^\top, ..., \boldsymbol{y}_{N-1}^\top)^\top$ whose rows are the quadrature points, one can use the fast Fourier transform to compute the matrix-vector product $Y \boldsymbol{a}^\top$, $\boldsymbol{a} \in \mathbb{R}^s$, in $\mathcal{O}(N \log N)$ operations and at most $s-1$ extra additions. The proposed method can be applied to lattice rules, polynomial lattice rules and a certain type of Korobov $p$-set. The approach is illustrated computationally by three numerical experiments. The first test considers the generation of points with normal distribution and general covariance matrix, the second test applies QMC to high-dimensional, affine-parametric, elliptic partial differential equations with uniformly distributed random coefficients, and the third test addresses Finite-Element discretizations of elliptic partial differential equations with high-dimensional, log-normal random input data. All numerical tests show a significant speed-up of the computation times of the fast QMC matrix method compared to a conventional implementation as the dimension becomes large

Kuramoto model with coupling through an external medium

Synchronization of coupled oscillators is often described using the Kuramoto model. Here we study a generalization of the Kuramoto model where oscillators communicate with each other through an external medium. This generalized model exhibits interesting new phenomena such as bistability between synchronization and incoherence and a qualitatively new form of synchronization where the external medium exhibits small-amplitude oscillations. We conclude by discussing the relationship of the model to other variations of the Kuramoto model including the Kuramoto model with a bimodal frequency distribution and the Millennium Bridge problem.Comment: 9 pages, 3 figure

Multilevel Quasi-Monte Carlo Methods for Lognormal Diffusion Problems

In this paper we present a rigorous cost and error analysis of a multilevel estimator based on randomly shifted Quasi-Monte Carlo (QMC) lattice rules for lognormal diffusion problems. These problems are motivated by uncertainty quantification problems in subsurface flow. We extend the convergence analysis in [Graham et al., Numer. Math. 2014] to multilevel Quasi-Monte Carlo finite element discretizations and give a constructive proof of the dimension-independent convergence of the QMC rules. More precisely, we provide suitable parameters for the construction of such rules that yield the required variance reduction for the multilevel scheme to achieve an $\varepsilon$-error with a cost of $\mathcal{O}(\varepsilon^{-\theta})$ with $\theta < 2$, and in practice even $\theta \approx 1$, for sufficiently fast decaying covariance kernels of the underlying Gaussian random field inputs. This confirms that the computational gains due to the application of multilevel sampling methods and the gains due to the application of QMC methods, both demonstrated in earlier works for the same model problem, are complementary. A series of numerical experiments confirms these gains. The results show that in practice the multilevel QMC method consistently outperforms both the multilevel MC method and the single-level variants even for non-smooth problems.Comment: 32 page

Electronic Aharonov-Bohm Effect Induced by Quantum Vibrations

Mechanical displacements of a nanoelectromechanical system (NEMS) shift the electron trajectories and hence perturb phase coherent charge transport through the device. We show theoretically that in the presence of a magnetic feld such quantum-coherent displacements may give rise to an Aharonov-Bohm-type of effect. In particular, we demonstrate that quantum vibrations of a suspended carbon nanotube result in a positive nanotube magnetoresistance, which decreases slowly with the increase of temperature. This effect may enable one to detect quantum displacement fluctuations of a nanomechanical device.Comment: 4 pages, 3 figure

CORRECTION TO "QUASI-MONTE CARLO METHODS FOR HIGH-DIMENSIONAL INTEGRATION: THE STANDARD (WEIGHTED HILBERT SPACE) SETTING AND BEYOND”

ISSN:1446-1811ISSN:1446-873

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 $[0,1]^s$. 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 $1/N$, where $N$ is the number of points, independently of dimension) to so-called “product and order dependent†(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