Markov Chain Monte Carlo (MCMC) proves to be powerful for Bayesian inference and in particular for exoplanet radial velocity fitting because MCMC provides more statistical information and makes better use of data than common approaches like chisquare fitting. However, the non-linear density functions encountered in these problems can make MCMC time-consuming. In this paper, we apply an ensemble sampler respecting affine invariance to orbital parameter extraction from radial velocity data. This new sampler has only one free parameter, and it does not require much tuning for good performance, which is important for automatization. The autocorrelation time of this sampler is approximately the same for all parameters and far smaller than Metropolis-Hastings, which means it requires many fewer function calls to produce the same number of independent samples. The affine-invariant sampler speeds up MCMC by hundreds of times compared with Metropolis-Hastings in the same computing situation. This novel sampler would be ideal for projects involving large datasets such as statistical investigations of planet distribution. The biggest obstacle to ensemble samplers is the existence of multiple local optima; we present a clustering technique to deal with local optima by clustering based on the likelihood of the walkers in the ensemble. We demonstrate the effectiveness of the sampler on real radial velocity data.

This paper is devoted to the analysis of convergence of Schwarz Waveform Relaxation (SWR) domain decomposition methods (DDM) for solving the stationary linear and nonlinear Schrödinger equations by the imaginary-time method. Although SWR are extensively used for numerically solving high-dimensional quantum and classical wave equations, the analysis of convergence and of the rate of convergence is still largely open for variable coefficients linear and nonlinear equations. The aim of this paper is to tackle this problem for both the linear and nonlinear Schrödinger equations in the two-dimensional setting. By extending ideas and concepts presented earlier [12] and by using pseudodifferential calculus, we prove the convergence and determine some approximate rates of convergence of the two-dimensional Classical SWR method for two subdomains with smooth boundary. Some numerical experiments are also proposed to validate the analysis.

We report the detection of three new exoplanets from Keck Observatory. HD 163607 is a metal-rich G5IV star with two planets. The inner planet has an observed orbital period of 75.29 ± 0.02 days, a semi-amplitude of 51.1 ± 1.4 m s −1 , an eccentricity of 0.73 ± 0.02 and a derived minimum mass of M P sin i= 0.77 ± 0.02 M Jup . This is the largest eccentricity of any known planet in a multi-planet system. The argument of periastron passage is 78.7 ± 2.0 • ; consequently, the planet's closest approach to its parent star is very near the line of sight, leading to a relatively high transit probability of 8%. The outer planet has an orbital period of 3.60 ± 0.02 years, an orbital eccentricity of 0.12 ± 0.06 and a semi-amplitude of 40.4 ± 1.3 m s −1 . The minimum mass is M P sin i= 2.29 ± 0.16 M Jup . HD 164509 is a metal-rich G5V star with a planet in an orbital period of 282.4 ± 3.8 days and an eccentricity of 0.26 ± 0.14. The semi-amplitude of 14.2 ± 2.7 m s −1 implies a minimum mass of 0.48 ± 0.09 M Jup . The radial velocities of HD 164509 also exhibit a residual linear trend of -5.1 ± 0.7 m s −1 per year, indicating the presence of an additional longer period companion in the system. Photometric observations demonstrate that HD 163607 and HD 164509 are constant in brightness to sub-millimag levels on their radial velocity periods. This provides strong support for planetary reflex motion as the cause of the radial velocity variations.