Efficient calculation of the worst-case error and (fast) component-by-component construction of higher order polynomial lattice rules

We show how to obtain a fast component-by-component construction algorithm for higher order polynomial lattice rules. Such rules are useful for multivariate quadrature of high-dimensional smooth functions over the unit cube as they achieve the near optimal order of convergence. The main problem addressed in this paper is to find an efficient way of computing the worst-case error. A general algorithm is presented and explicit expressions for base~2 are given. To obtain an efficient component-by-component construction algorithm we exploit the structure of the underlying cyclic group. We compare our new higher order multivariate quadrature rules to existing quadrature rules based on higher order digital nets by computing their worst-case error. These numerical results show that the higher order polynomial lattice rules improve upon the known constructions of quasi-Monte Carlo rules based on higher order digital nets

Landscape influence on small-scale water temperature variations in a moorland catchment

Acknowledgements Iain Malcolm and staff at Marine Scotland (Pitlochry) are thanked for the provision of data from the AWS. Finally, the two anonymous reviewers are greatly acknowledged for their constructive comments.Peer reviewedPostprin

Modelling landscape controls on dissolved organic carbon sources and fluxes to streams

Acknowledgments We thank the Natural Environment Research Council NERC (project NE/K000268/1) for funding. Iain Malcolm and staff at Marine Scotland (Pitlochry) are also thanked for the provision of data from the AWS as are the Scottish Environmental Protection Agency and British Atmospheric Data Centre for the provision of meteorological data.Peer reviewedPublisher PD

New ion trap for atomic frequency standard applications

A novel linear ion trap that permits storage of a large number of ions with reduced susceptibility to the second-order Doppler effect caused by the radio frequency (RF) confining fields has been designed and built. This new trap should store about 20 times the number of ions a conventional RF trap stores with no corresponding increase in second-order Doppler shift from the confining field. In addition, the sensitivity of this shift to trapping parameters, i.e., RF voltage, RF frequency, and trap size, is greatly reduced

Atomic frequency standards for ultra-high-frequency stability

The general features of the Hg-199(+) trapped-ion frequency standard are outlined and compared to other atomic frequency standards, especially the hydrogen maser. The points discussed are those which make the trapped Hg-199(+) standard attractive: high line Q, reduced sensitivity to external magnetic fields, and simplicity of state selection, among others

The JPL trapped mercury ion frequency standard

In order to provide frequency standards for the Deep Space Network (DSN) which are more stable than present-day hydrogen masers, a research task was established under the Advanced Systems Program of the TDA to develop a Hg-199(+) trapped ion frequency standard. The first closed-loop operation of this kind is described. Mercury-199 ions are confined in an RF trap and are state-selected through the use of optical pumping with 194 nm UV light from a Hg-202 discharge lamp. Absorption of microwave radiation at the hyperfine frequency (40.5 GHz) is signaled by atomic fluorescence of the UV light. The frequency of a 40.5 GHz oscillator is locked to a 1.6 Hz wide atomic absorption line of the trapped ions. The measured Allan variance of this locked oscillator is currently gamma sub y (pi) = 4.4 x 10 to the minus 12th/square root of pi for 20 is less than pi is less than 320 seconds, which is better stability than the best commercial cesium standards by almost a factor of 2. This initial result was achieved without magnetic shielding and without regulation of ion number

Stream water age distributions controlled by storage dynamics and nonlinear hydrologic connectivity : Modeling with high-resolution isotope data

Peer reviewedPublisher PD

Operational parameters for the superconducting cavity maser

Tests of the superconducting cavity maser (SCM) ultra-stable frequency source have been made for the first time using a hydrogen maser for a frequency reference. In addition to characterizing the frequency stability, the sensitivity of the output frequency to several crucial parameters was determined for various operating conditions. Based on this determination, the refrigeration and thermal control systems of the SCM were modified. Subsequent tests showed substantially improved performance, especially at the longest averaging times

Successive Coordinate Search and Component-by-Component Construction of Rank-1 Lattice Rules

The (fast) component-by-component (CBC) algorithm is an efficient tool for the construction of generating vectors for quasi-Monte Carlo rank-1 lattice rules in weighted reproducing kernel Hilbert spaces. We consider product weights, which assigns a weight to each dimension. These weights encode the effect a certain variable (or a group of variables by the product of the individual weights) has. Smaller weights indicate less importance. Kuo (2003) proved that the CBC algorithm achieves the optimal rate of convergence in the respective function spaces, but this does not imply the algorithm will find the generating vector with the smallest worst-case error. In fact it does not. We investigate a generalization of the component-by-component construction that allows for a general successive coordinate search (SCS), based on an initial generating vector, and with the aim of getting closer to the smallest worst-case error. The proposed method admits the same type of worst-case error bounds as the CBC algorithm, independent of the choice of the initial vector. Under the same summability conditions on the weights as in [Kuo,2003] the error bound of the algorithm can be made independent of the dimension $d$ and we achieve the same optimal order of convergence for the function spaces from [Kuo,2003]. Moreover, a fast version of our method, based on the fast CBC algorithm by Nuyens and Cools, is available, reducing the computational cost of the algorithm to $O(d \, n \log(n))$ operations, where $n$ denotes the number of function evaluations. Numerical experiments seeded by a Korobov-type generating vector show that the new SCS algorithm will find better choices than the CBC algorithm and the effect is better when the weights decay slower.Comment: 13 pages, 1 figure, MCQMC2016 conference (Stanford

Using geophysical surveys to test tracer-based storage estimates in headwater catchments

Acknowledgements The authors are grateful to Stian Bradford, Chris Gabrielli, and Julie Timms for practical and logistical assistance. The provision of transport by Iain Malcolm and Ross Glover of Marine Scotland Science was greatly appreciated. We also thank the European Research Council ERC (project GA 335910 VEWA) for funding through the VeWa project and the Leverhulme Trust for funding through PLATO (RPG-2014-016).Peer reviewedPostprin