Smolyak Quadrature

This thesis is an introduction to the theoretical foundation and practical usage of the Smolyak quadrature rule, which is used to evaluate high-dimensional integrals over regions of Euclidean spaces. Given a sequence of univariate quadrature rules, the Smolyak construction is defined in terms of tensor products taken over the univariate rules' consecutive differences. The evaluation points of the resulting multivariate quadrature rule are distributed more sparsely than those of e.g. tensor product quadrature. It can be shown that a multivariate quadrature rule formulated in this way inherits many useful properties of the underlying sequence of univariate quadrature rules, such as the polynomial exactness. The original formulation of the Smolyak rule is prone to a copious amount of cancellation of terms in practice. This issue can be circumvented by isolating the occurrence of duplicates to a separate term, which can be computed a priori. The resulting combination method forms the basis for a numerical implementation of the Smolyak quadrature rule, which we have provided using the MATLAB scripting language. Our tests suggest that the Smolyak rule provides a competitive alternative in the realm of multidimensional integration routines saturated by the stochastic Monte Carlo method and the deterministic Quasi-Monte Carlo method. This statement is especially valid in the case of smooth integrands and it is backed by the error analysis developed in the second chapter of this thesis. The classical convergence rate is also derived for integrands of sufficient smoothness in the case of a bounded integration region. The third chapter serves as a qualitative approach to generalized sparse grid quadrature. Especially of interest is the dimension-adaptive construction. While it lacks the theoretical foundation of the Smolyak quadrature rule, it has the added benefit of adapting to the spatial structure of the integrand. A MATLAB implementation of this routine is presented vis-à-vis the Smolyak quadrature rule

Lattice-based kernel approximation and serendipitous weights for parametric PDEs in very high dimensions

We describe a fast method for solving elliptic partial differential equations (PDEs) with uncertain coefficients using kernel interpolation at a lattice point set. By representing the input random field of the system using the model proposed by Kaarnioja, Kuo, and Sloan (SIAM J.~Numer.~Anal.~2020), in which a countable number of independent random variables enter the random field as periodic functions, it was shown by Kaarnioja, Kazashi, Kuo, Nobile, and Sloan (Numer.~Math.~2022) that the lattice-based kernel interpolant can be constructed for the PDE solution as a function of the stochastic variables in a highly efficient manner using fast Fourier transform (FFT). In this work, we discuss the connection between our model and the popular ``affine and uniform model'' studied widely in the literature of uncertainty quantification for PDEs with uncertain coefficients. We also propose a new class of weights entering the construction of the kernel interpolant -- \emph{serendipitous weights} -- which dramatically improve the computational performance of the kernel interpolant for PDE problems with uncertain coefficients, and allow us to tackle function approximation problems up to very high dimensionalities. Numerical experiments are presented to showcase the performance of the serendipitous weights

Uncertainty quantification for random domains using periodic random variables

We consider uncertainty quantification for the Poisson problem subject to domain uncertainty. For the stochastic parameterization of the random domain, we use the model recently introduced by Kaarnioja, Kuo, and Sloan (SIAM J. Numer. Anal., 2020) in which a countably infinite number of independent random variables enter the random field as periodic functions. We develop lattice quasi-Monte Carlo (QMC) cubature rules for computing the expected value of the solution to the Poisson problem subject to domain uncertainty. These QMC rules can be shown to exhibit higher order cubature convergence rates permitted by the periodic setting independently of the stochastic dimension of the problem. In addition, we present a complete error analysis for the problem by taking into account the approximation errors incurred by truncating the input random field to a finite number of terms and discretizing the spatial domain using finite elements. The paper concludes with numerical experiments demonstrating the theoretical error estimates.Comment: 38 pages, 3 figure

Uncertainty quantification for random domains using periodic random variables

We consider uncertainty quantification for the Poisson problem subject to domain uncertainty. For the stochastic parameterization of the random domain, we use the model recently introduced by Kaarnioja et al. (SIAM J. Numer. Anal., 2020) in which a countably infinite number of independent random variables enter the random field as periodic functions. We develop lattice quasi-Monte Carlo (QMC) cubature rules for computing the expected value of the solution to the Poisson problem subject to domain uncertainty. These QMC rules can be shown to exhibit higher order cubature convergence rates permitted by the periodic setting independently of the stochastic dimension of the problem. In addition, we present a complete error analysis for the problem by taking into account the approximation errors incurred by truncating the input random field to a finite number of terms and discretizing the spatial domain using finite elements. The paper concludes with numerical experiments demonstrating the theoretical error estimates