Quasi-Monte Carlo (QMC) integration of output functionals of solutions of the diffusion problem with a log-normal random coefficient is considered. The random coefficient is assumed to be given by an exponential of a Gaussian random field that is represented by a series expansion of some system of functions. Graham et al. [16] developed a latticebased QMC theory for this problem and established a quadrature error decay rate ≈ 1 with respect to the number of quadrature points. The key assumption there was a suitable summability condition on the aforementioned system of functions. As a consequence, product-order-dependent (POD) weights were used to construct the lattice rule. In this paper, a different assumption on the system is considered. This assumption, originally considered by Bachmayr et al. [3] to utilise the locality of support of basis functions in the context of polynomial approximations applied to the same type of the diffusion problem, is shown to work well in the same lattice-based QMC method considered by Graham et al.: the assumption leads us to product weights, which enables the construction of the QMC method with a smaller computational cost than Graham et al. A quadrature error decay rate ≈ 1 is established, and the theory developed here is applied to a wavelet stochastic model. By a characterisation of the Besov smoothness, it is shown that a wide class of path smoothness can be treated with this framework.
An existence result is presented for the dynamical low rank (DLR) approximation for random semi-linear evolutionary equations. The DLR solution approximates the true solution at each time instant by a linear combination of products of deterministic and stochastic basis functions, both of which evolve over time. A key to our proof is to find a suitable equivalent formulation of the original problem. The so-called Dual Dynamically Orthogonal formulation turns out to be convenient. Based on this formulation, the DLR approximation is recast to an abstract Cauchy problem in a suitable linear space, for which existence and uniqueness of the solution in the maximal interval are established.
A fully discretised polynomial approximation on spherical shells is considered. The radial direction and the angular direction of the spherical shells are treated differently: for the radial direction, polynomial interpolation is employed, whereas for the angular direction, hyperinterpolation is employed. The error in terms of a weighted L 2 -norm is bounded by the sum of two terms, one for the radial, one for the angular direction. Numerical results support our result.
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 et al. (Monte Carlo and Quasi-Monte Carlo Methods 2004, Springer, New York, 2006) and Zeng et al. (Constr. Approx. 30: 529–555, 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 et al. (SIAM J Numer Anal 58(2): 1068–1091, 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.
A sub-optimality of Gauss-Hermite quadrature and an 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 the matching lower and upper bounds, which turns 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 on 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.
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