Quadrature by Expansion: A New Method for the Evaluation of Layer Potentials

Integral equation methods for the solution of partial differential equations, when coupled with suitable fast algorithms, yield geometrically flexible, asymptotically optimal and well-conditioned schemes in either interior or exterior domains. The practical application of these methods, however, requires the accurate evaluation of boundary integrals with singular, weakly singular or nearly singular kernels. Historically, these issues have been handled either by low-order product integration rules (computed semi-analytically), by singularity subtraction/cancellation, by kernel regularization and asymptotic analysis, or by the construction of special purpose "generalized Gaussian quadrature" rules. In this paper, we present a systematic, high-order approach that works for any singularity (including hypersingular kernels), based only on the assumption that the field induced by the integral operator is locally smooth when restricted to either the interior or the exterior. Discontinuities in the field across the boundary are permitted. The scheme, denoted QBX (quadrature by expansion), is easy to implement and compatible with fast hierarchical algorithms such as the fast multipole method. We include accuracy tests for a variety of integral operators in two dimensions on smooth and corner domains

Stimulus-Induced Rhythmic, Periodic, or Ictal Discharges in Coma-Incidence and Interrater Reliability of Continuous EEG After a Standard Stimulation Protocol: A Prospective Study.

PURPOSE: Stimulus-induced rhythmic, periodic, or ictal discharges (SIRPIDs) are often seen during continuous electroencephalographic (cEEG) monitoring in coma. Given their uncertain clinical significance, our prospective study evaluated incidence of SIRPIDs in comatose patients in the neuroscience intensive care unit (NSICU) who underwent a standard stimulation protocol and defined interreader reliability for cEEG. METHODS: Of 146 patients prospectively screened who underwent cEEG during a 6-month period, 53 patients were included and 93 patients were excluded. Our protocol used a sequence of auditory, mild tactile, and painful stimuli tested in a quiet room. Continuous electroencephalogram were then reviewed offline by blinded experts, with interrater agreement assessed by kappa statistic. By Pearson χ and Wilcoxon rank-sum tests, we then compared binary and numerical clinical features between those with and without SIRPIDs. RESULTS: Of 53 patients who underwent our protocol, one patient with a corrupt cEEG file was excluded. Traumatic brain injury was the most common diagnosis. Moderate interrater agreement was observed for 66 total stimulations: 20 patients (38.5%) had possible or definite SIRPIDs by minimum one reviewer. For 19 stimulations reviewed by a third reviewer, consensus was reached in 10 cases making the incidence of SIRPIDs 19.3% in our cohort. There was a moderate interrate agreement with kappa of 0.5 (95% confidence interval: 0.1, 0.7). Median intensive care unit stay was 15 days in patients with SIRPIDs versus 6.5 days in those without (P = 0.021). CONCLUSIONS: Our prospective study of SIRPIDs in the neuroscience intensive care unit found a 19% incidence by cEEG using a standard stimulation protocol, most often rhythmic delta activity, and showed a moderate interrater agreement