Randomizing the trapezoidal rule gives the optimal RMSE rate in Gaussian Sobolev spaces

Randomized quadratures for integrating functions in Sobolev spaces of order $\alpha \ge 1$, where the integrability condition is with respect to the Gaussian measure, are considered. In this function space, the optimal rate for the worst-case root-mean-squared error (RMSE) is established. Here, optimality is for a general class of quadratures, in which adaptive non-linear algorithms with a possibly varying number of function evaluations are also allowed. The optimal rate is given by showing matching bounds. First, a lower bound on the worst-case RMSE of $O(n^{-\alpha-1/2})$ is proven, where $n$ denotes an upper bound on the expected number of function evaluations. It turns out that a suitably randomized trapezoidal rule attains this rate, up to a logarithmic factor. A practical error estimator for this trapezoidal rule is also presented. Numerical results support our theory.Comment: revision, 21 page

Dynamically Orthogonal Approximation for Stochastic Differential Equations

In this paper, we set the mathematical foundations of the Dynamical Low Rank Approximation (DLRA) method for high-dimensional stochastic differential equations. DLRA aims at approximating the solution as a linear combination of a small number of basis vectors with random coefficients (low rank format) with the peculiarity that both the basis vectors and the random coefficients vary in time. While the formulation and properties of DLRA are now well understood for random/parametric equations, the same cannot be said for SDEs and this work aims to fill this gap. We start by rigorously formulating a Dynamically Orthogonal (DO) approximation (an instance of DLRA successfully used in applications) for SDEs, which we then generalize to define a parametrization independent DLRA for SDEs. We show local well-posedness of the DO equations and their equivalence with the DLRA formulation. We also characterize the explosion time of the DO solution by a loss of linear independence of the random coefficients defining the solution expansion and give sufficient conditions for global existence.Comment: 32 page

Stability properties of a projector-splitting scheme for dynamical low rank approximation of random parabolic equations

We consider the Dynamical Low Rank (DLR) approximation of random parabolic equations and propose a class of fully discrete numerical schemes. Similarly to the continuous DLR approximation, our schemes are shown to satisfy a discrete variational formulation. By exploiting this property, we establish stability of our schemes: we show that our explicit and semi-implicit versions are conditionally stable under a parabolic type CFL condition which does not depend on the smallest singular value of the DLR solution; whereas our implicit scheme is unconditionally stable. Moreover, we show that, in certain cases, the semi-implicit scheme can be unconditionally stable if the randomness in the system is sufficiently small. Furthermore, we show that these schemes can be interpreted as projector-splitting integrators and are strongly related to the scheme proposed by Lubich et al. [BIT Num. Math., 54:171-188, 2014; SIAM J. on Num. Anal., 53:917-941, 2015], to which our stability analysis applies as well. The analysis is supported by numerical results showing the sharpness of the obtained stability conditions.Comment: 48 pages, 14 figure

Derandomised lattice rules for high dimensional integration

We seek shifted lattice rules that are good for high dimensional integration over the unit cube in the setting of an unanchored weighted Sobolev space of functions with square-integrable mixed first derivatives. Many existing studies rely on random shifting of the lattice, whereas here we work with lattice rules with a deterministic shift. Specifically, we consider "half-shifted" rules, in which each component of the shift is an odd multiple of $1/(2N)$, where $N$ is the number of points in the lattice. We show, by applying the principle that \emph{there is always at least one choice as good as the average}, that for a given generating vector there exists a half-shifted rule whose squared worst-case error differs from the shift-averaged squared worst-case error by a term of order only ${1/N^2}$. Numerical experiments, in which the generating vector is chosen component-by-component (CBC) as for randomly shifted lattices and then the shift by a new "CBC for shift" algorithm, yield encouraging results

Suboptimality of Gauss–Hermite quadrature and optimality of the trapezoidal rule for functions with finite smoothness

The suboptimality of Gauss–Hermite quadrature and the optimality of the trapezoidal rule are proved in the weighted Sobolev spaces of square integrable functions of order α, where the optimality is in the sense of worst-case error. For Gauss–Hermite quadrature, we obtain matching lower and upper bounds, which turn out to be merely of the order n−α/2 with n function evaluations, although the optimal rate for the best possible linear quadrature is known to be n−α. Our proof of the lower bound exploits the structure of the Gauss–Hermite nodes; the bound is independent of the quadrature weights, and changing the Gauss–Hermite weights cannot improve the rate n−α/2. In contrast, we show that a suitably truncated trapezoidal rule achieves the optimal rate up to a logarithmic factor

Fast approximation by periodic kernel-based lattice-point interpolation with application in uncertainty quantification

This paper deals with the kernel-based approximation of a multivariate periodic function by interpolation at the points of an integration lattice -- a setting that, as pointed out by Zeng, Leung, Hickernell (MCQMC2004, 2006) and Zeng, Kritzer, Hickernell (Constr. Approx., 2009), allows fast evaluation by fast Fourier transform, so avoiding the need for a linear solver. The main contribution of the paper is the application to the approximation problem for uncertainty quantification of elliptic partial differential equations, with the diffusion coefficient given by a random field that is periodic in the stochastic variables, in the model proposed recently by Kaarnioja, Kuo, Sloan (SIAM J. Numer. Anal., 2020). The paper gives a full error analysis, and full details of the construction of lattices needed to ensure a good (but inevitably not optimal) rate of convergence and an error bound independent of dimension. Numerical experiments support the theory.Comment: 37 pages, 5 figure