Importance sampling is modified via {\it homotopy continuation} to improve the efficiency and success of the sampler. The homotopy will use a known distribution as a starting empirical importance sampling distribution and generate a continuous schedule which culminates with the target distribution. The focus is the estimation of the normalization constant of the target distribution. The homotopy method is extended to a Bayesian setting, for stationary and time dependent posterior distributions. The numerical implementation uses a combination of sample averages, with sampling parameter N, and homotopy stages M, where M is typically a small number. The algorithm replaces homotopy stages for sampling steps, potentially resulting in a better or more efficient importance sampler. Numerical experiments suggest this is the case. The results also suggest that the method may improve the efficiency of the sampler by concentrating the samples in regions of greater impact.

Wave breaking transfers momentum between waves to currents. In doing so it dissipates wave energy and affects long-term dispersion and transport of tracers, Wave breaking also affects the transfer of wind stresses to the ocean and is a major player in ocean acoustics, the dynamics of near-surface gas exchanges and the dynamics of phytoplankton and buoyant pollutants. Progressive waves were generated numerically using Gerris [6]. The velocity field was then used to estimate Lagrangian parcel dynamics in the neighborhood of the breaking event. The trans- port generated by the breaking events is shown to be significant, when compared to the wave-generated residual flow, as captured by the Stokes drift. We formulate a parametrization of the break- ing velocity based upon an analysis of the Largrangian dynamics of fluid parcels. Further, we describe how the breaking velocity is incorporated in the vortex force formulation of wave current interactions [1, 2].

We present the mathematical analysis of the Isolation Random Forest Method (IRF Method) for anomaly detection, introduced in {\sc F.~T. Liu, K.~M. Ting, Z.-H. Zhou:}, {\it Isolation-based anomaly detection}, TKDD 6 (2012) 3:1–3:39. We prove that the IRF space can be endowed with a probability induced by the Isolation Tree algorithm (iTree). In this setting, the convergence of the IRF method is proved, using the Law of Large Numbers. A couple of counterexamples are presented to show that the method is inconclusive and no certificate of quality can be given, when using it as a means to detect anomalies. Hence, an alternative version of the method is proposed whose mathematical foundation is fully justified. Furthermore, a criterion for choosing the number of sampled trees needed to guarantee confidence intervals of the numerical results is presented. Finally, numerical experiments are presented to compare the performance of the classic method with the proposed one.