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

The construction of good lattice rules and polynomial lattice rules

A comprehensive overview of lattice rules and polynomial lattice rules is given for function spaces based on $\ell_p$ semi-norms. Good lattice rules and polynomial lattice rules are defined as those obtaining worst-case errors bounded by the optimal rate of convergence for the function space. The focus is on algebraic rates of convergence $O(N^{-\alpha+\epsilon})$ for $\alpha \ge 1$ and any $\epsilon > 0$, where $\alpha$ is the decay of a series representation of the integrand function. The dependence of the implied constant on the dimension can be controlled by weights which determine the influence of the different dimensions. Different types of weights are discussed. The construction of good lattice rules, and polynomial lattice rules, can be done using the same method for all $1 < p \le \infty$; but the case $p=1$ is special from the construction point of view. For $1 < p \le \infty$ the component-by-component construction and its fast algorithm for different weighted function spaces is then discussed

Hot new directions for quasi-Monte Carlo research in step with applications

This article provides an overview of some interfaces between the theory of quasi-Monte Carlo (QMC) methods and applications. We summarize three QMC theoretical settings: first order QMC methods in the unit cube $[0,1]^s$ and in $\mathbb{R}^s$, and higher order QMC methods in the unit cube. One important feature is that their error bounds can be independent of the dimension $s$ under appropriate conditions on the function spaces. Another important feature is that good parameters for these QMC methods can be obtained by fast efficient algorithms even when $s$ is large. We outline three different applications and explain how they can tap into the different QMC theory. We also discuss three cost saving strategies that can be combined with QMC in these applications. Many of these recent QMC theory and methods are developed not in isolation, but in close connection with applications

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

FAA Center of Excellence for Alternative Jet Fuels & Environment: Annual Technical Report 2021: For the Period October 1, 2020 - September 30, 2021: Volume 2

FAA Award Number 13-C.This report covers the period October 1, 2020, through September 30, 2021. The Center was established by the authority of FAA solicitation 13-C-AJFE-Solicitation. During that time the ASCENT team launched a new website, which can be viewed at ascent.aero. The next meeting will be held April 5-7, 2022, in Alexandria, VA

11th International Coral Reef Symposium Proceedings

A defining theme of the 11th International Coral Reef Symposium was that the news for coral reef ecosystems are far from encouraging. Climate change happens now much faster than in an ice-age transition, and coral reefs continue to suffer fever-high temperatures as well as sour ocean conditions. Corals may be falling behind, and there appears to be no special silver bullet remedy. Nevertheless, there are hopeful signs that we should not despair. Reef ecosystems respond vigorously to protective measures and alleviation of stress. For concerned scientists, managers, conservationists, stakeholders, students, and citizens, there is a great role to play in continuing to report on the extreme threat that climate change represents to earth’s natural systems. Urgent action is needed to reduce CO2 emissions. In the interim, we can and must buy time for coral reefs through increased protection from sewage, sediment, pollutants, overfishing, development, and other stressors, all of which we know can damage coral health. The time to act is now. The canary in the coral-coal mine is dead, but we still have time to save the miners. We need effective management rooted in solid interdisciplinary science and coupled with stakeholder buy in, working at local, regional, and international scales alongside global efforts to give reefs a chance.https://nsuworks.nova.edu/occ_icrs/1000/thumbnail.jp