256 research outputs found
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 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 for
and any , where 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 ; but the case is special
from the construction point of view. For the
component-by-component construction and its fast algorithm for different
weighted function spaces is then discussed
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Optimal and Near Optimal Configurations on Lattices and Manifolds
Optimal configurations of points arise in many contexts, for example classical ground states for interacting particle systems, Euclidean packings of convex bodies, as well as minimal discrete and continuous energy problems for general kernels. Relevant questions in this area include the understanding of asymptotic optimal configurations, of lattice and periodic configurations, the development of algorithmic constructions of near optimal configurations, and the application of methods in convex optimization such as linear and semidefinite programming
Higher order quasi-Monte Carlo for Bayesian shape inversion
In this article, we consider a Bayesian approach towards data assimilation and uncertainty quantification in diffusion problems on random domains. We provide a rigorous analysis of parametric regularity of the posterior distribution given that the data exhibit only limited smoothness. Moreover, we present a dimension truncation analysis for the forward problem, which is formulated in terms of the domain mapping method. Having these novel results at hand, we shall consider as a practical example Electrical Impedance Tomography in the regime of constant conductivities. We are interested in computing moments, in particular expectation and variance, of the contour of an unknown inclusion, given perturbed surface measurements. By casting the forward problem into the framework of elliptic diffusion problems on random domains, we can directly apply the presented analysis. This straightforwardly yields parametric regularity results for the system response and for the posterior measure, facilitating the application of higher order quadrature methods for the approximation of moments of quantities of interest. As an example of such a quadrature method, we consider here recently developed higher order quasi-Monte Carlo methods. To solve the forward problem numerically, we employ a fast boundary integral solver. Numerical examples are provided to illustrate the presented approach and validate the theoretical findings
Tractability of the Quasi-Monte Carlo quadrature with Halton points for elliptic PDEs with random diffusion
This article is dedicated to the computation of the moments of the solution to stochastic partial differential equations with log-normal distributed diffusion coefficient by the Quasi-Monte Carlo method. Our main result is the polynomial tractability for the Quasi-Monte Carlo method based on the Halton sequence. As a by-product, we obtain also the strong tractability of stochastic partial differential equations with uniformly elliptic diffusion coefficient by the Quasi-Monte Carlo method. Numerical experiments are given to validate the theoretical findings
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