Abstract.A number of approaches for discretizing partial differential equations with random data are based on generalized polynomial chaos expansions of random variables. These constitute generalizations of the polynomial chaos expansions introduced by Norbert Wiener to expansions in polynomials orthogonal with respect to non-Gaussian probability measures. We present conditions on such measures which imply mean-square convergence of generalized polynomial chaos expansions to the correct limit and complement these with illustrative examples.Mathematics Subject Classification. 33C45, 35R60, 40A30, 41A10, 60H35, 65N30.
We consider the application of multilevel Monte Carlo methods to elliptic PDEs with random coefficients. We focus on models of the random coefficient that lack uniform ellipticity and boundedness with respect to the random parameter, and that only have limited spatial regularity. We extend the finite element error analysis for this type of equation, carried out in [6], to more difficult problems, posed on non-smooth domains and with discontinuities in the coefficient. For this wider class of model problem, we prove convergence of the multilevel Monte Carlo algorithm for estimating any bounded, linear functional and any continuously Fréchet differentiable non-linear functional of the solution. We further improve the performance of the multilevel estimator by introducing level dependent truncations of the Karhunen-Loève expansion of the random coefficient. Numerical results complete the paper.
AimsEpidemiological studies indicate that traffic noise increases the incidence of coronary artery disease, hypertension and stroke. The underlying mechanisms remain largely unknown. Field studies with nighttime noise exposure demonstrate that aircraft noise leads to vascular dysfunction, which is markedly improved by vitamin C, suggesting a key role of oxidative stress in causing this phenomenon.Methods and resultsWe developed a novel animal model to study the vascular consequences of aircraft noise exposure. Peak sound levels of 85 and mean sound level of 72 dBA applied by loudspeakers for 4 days caused an increase in systolic blood pressure, plasma noradrenaline and angiotensin II levels and induced endothelial dysfunction. Noise increased eNOS expression but reduced vascular NO levels because of eNOS uncoupling. Noise increased circulating levels of nitrotyrosine, interleukine-6 and vascular expression of the NADPH oxidase subunit Nox2, nitrotyrosine-positive proteins and of endothelin-1. FACS analysis demonstrated an increase in infiltrated natural killer-cells and neutrophils into the vasculature. Equal mean sound pressure levels of white noise for 4 days did not induce these changes. Comparative Illumina sequencing of transcriptomes of aortic tissues from aircraft noise-treated animals displayed significant changes of genes in part responsible for the regulation of vascular function, vascular remodelling, and cell death.ConclusionWe established a novel and unique aircraft noise stress model with increased blood pressure and vascular dysfunction associated with oxidative stress. This animal model enables future studies of molecular mechanisms, mitigation strategies, and pharmacological interventions to protect from noise-induced vascular damage.
We present a Multi-Index Quasi-Monte Carlo method for the solution of elliptic partial differential equations with random coefficients. By combining the multi-index sampling idea with randomly shifted rank-1 lattice rules, the algorithm constructs an estimator for the expected value of some functional of the solution. The efficiency of this new method is illustrated on a three-dimensional subsurface flow problem with log-normal diffusion coefficient with underlying Matérn covariance function. This example is particularly challenging because of the small correlation length considered, and thus the large number of uncertainties that must be included. We show numerical evidence that it is possible to achieve a cost inversely proportional to the requested tolerance on the root-mean-square error, for problems with a smoothly varying random field.
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