Fast integral equation methods for the modified Helmholtz equation

We present a collection of integral equation methods for the solution to the two-dimensional, modified Helmholtz equation, u(\x) - \alpha^2 \Delta u(\x) = 0, in bounded or unbounded multiply-connected domains. We consider both Dirichlet and Neumann problems. We derive well-conditioned Fredholm integral equations of the second kind, which are discretized using high-order, hybrid Gauss-trapezoid rules. Our fast multipole-based iterative solution procedure requires only O(N) or $O(N\log N)$ operations, where N is the number of nodes in the discretization of the boundary. We demonstrate the performance of the methods on several numerical examples.Comment: Published in Computers & Mathematics with Application

The Lung Screen Uptake Trial (LSUT): protocol for a randomised controlled demonstration lung cancer screening pilot testing a targeted invitation strategy for high risk and ‘hard-to-reach’ patients

Background Participation in low-dose CT (LDCT) lung cancer screening offered in the trial context has been poor, especially among smokers from socioeconomically deprived backgrounds; a group for whom the risk-benefit ratio is improved due to their high risk of lung cancer. Attracting high risk participants is essential to the success and equity of any future screening programme. This study will investigate whether the observed low and biased uptake of screening can be improved using a targeted invitation strategy. Methods/design A randomised controlled trial design will be used to test whether targeted invitation materials are effective at improving engagement with an offer of lung cancer screening for high risk candidates. Two thousand patients aged 60–75 and recorded as a smoker within the last five years by their GP, will be identified from primary care records and individually randomised to receive either intervention invitation materials (which take a targeted, stepped and low burden approach to information provision prior to the appointment) or control invitation materials. The primary outcome is uptake of a nurse-led ‘lung health check’ hospital appointment, during which patients will be offered a spirometry test, an exhaled carbon monoxide (CO) reading, and an LDCT if eligible. Initial data on demographics (i.e. age, sex, ethnicity, deprivation score) and smoking status will be collected in primary care and analysed to explore differences between attenders and non-attenders with respect to invitation group. Those who attend the lung health check will have further data on smoking collected during their appointment (including pack-year history, nicotine dependence and confidence to quit). Secondary outcomes will include willingness to be screened, uptake of LDCT and measures of informed decision-making to ensure the latter is not compromised by either invitation strategy. Discussion If effective at improving informed uptake of screening and reducing bias in participation, this invitation strategy could be adopted by local screening pilots or a national programme. Trial registration This study was registered with the ISRCTN (International Standard Registered Clinical/soCial sTudy Number : ISRCTN21774741) on the 23rd September 2015 and the NIH ClinicalTrials.gov database (NCT0255810) on the 22nd September 2015

A new family of regularized kernels for the harmonic oscillator

In this paper, a new two-parameter family of regularized kernels is introduced, suitable for applying high-order time stepping to N-body systems. These high-order kernels are derived by truncating a Taylor expansion of the non-regularized kernel about (r2+ϵ2), generating a sequence of increasingly more accurate kernels. This paper proves the validity of this two-parameter family of regularized kernels, constructs error estimates, and illustrates the benefits of using high-order kernels through numerical experiments