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Quasi-Monte Carlo methods for high dimensional integration - the standard (weighted Hilbert space) setting and beyond
(2012)SAM Research ReportThis paper is a contemporary review of QMC ("quasi-Monte Carlo") methods, i.e., equal-weight rules for the approximate evaluation of high dimensional integrals over the unit cube $[0; 1]^s$. It first introduces the by-now standard setting of weighted Hilbert spaces of functions with square-integrable mixed first derivatives, and then indicates alternative settings, such as non-Hilbert spaces, that can sometimes be more suitable. Original ...Report -
Multilevel Monte-Carlo front-tracking for random scalar conservation laws
(2012)SAM Research ReportWe consider random scalar hyperbolic conservation laws (RSCLs) in spatial dimension $d\ge 1$ with bounded random flux functions which are $\mathbb{P}$-a.s. Lipschitz continuous with respect to the state variable, for which there exists a unique random entropy solution (i.e., a measurable mapping from the probability space into $C(0,T;L^1(\mathbb{R}^d))$ with finite second moments). We present a convergence analysis of a Multi-Level ...Report -
Approximation by harmonic polynomials in star-shaped domains and exponential convergence of Trefftz hp-DGFEM
(2012)SAM Research ReportWe study the approximation of harmonic functions by means of harmonic polynomials in twodimensional, bounded, star-shaped domains. Assuming that the functions possess analytic extensions to a $\delta$-neighbourhood of the domain, we prove exponential convergence of the approximation error with respect to the degree of the approximating harmonic polynomial. All the constants appearing in the bounds are explicit and depend only on the ...Report