Importance sampling the union of rare events with an application to power systems analysis

We consider importance sampling to estimate the probability $\mu$ of a union of $J$ rare events $H_j$ defined by a random variable $\boldsymbol{x}$. The sampler we study has been used in spatial statistics, genomics and combinatorics going back at least to Karp and Luby (1983). It works by sampling one event at random, then sampling $\boldsymbol{x}$ conditionally on that event happening and it constructs an unbiased estimate of $\mu$ by multiplying an inverse moment of the number of occuring events by the union bound. We prove some variance bounds for this sampler. For a sample size of $n$, it has a variance no larger than $\mu(\bar\mu-\mu)/n$ where $\bar\mu$ is the union bound. It also has a coefficient of variation no larger than $\sqrt{(J+J^{-1}-2)/(4n)}$ regardless of the overlap pattern among the $J$ events. Our motivating problem comes from power system reliability, where the phase differences between connected nodes have a joint Gaussian distribution and the $J$ rare events arise from unacceptably large phase differences. In the grid reliability problems even some events defined by $5772$ constraints in $326$ dimensions, with probability below $10^{-22}$, are estimated with a coefficient of variation of about $0.0024$ with only $n=10{,}000$ sample values

Effect of Pore Geometry on the Compressibility of a Confined Simple Fluid

Fluids confined in nanopores exhibit properties different from the properties of the same fluids in bulk, among these properties are the isothermal compressibility or elastic modulus. The modulus of a fluid in nanopores can be extracted from ultrasonic experiments or calculated from molecular simulations. Using Monte Carlo simulations in the grand canonical ensemble, we calculated the modulus for liquid argon at its normal boiling point (87.3~K) adsorbed in model silica pores of two different morphologies and various sizes. For spherical pores, for all the pore sizes (diameters) exceeding 2~nm, we obtained a logarithmic dependence of fluid modulus on the vapor pressure. Calculation of the modulus at saturation showed that the modulus of the fluid in spherical pores is a linear function of the reciprocal pore size. The calculation of the modulus of the fluid in cylindrical pores appeared too scattered to make quantitative conclusions. We performed additional simulations at higher temperature (119.6~K), at which Monte Carlo insertions and removals become more efficient. The results of the simulations at higher temperature confirmed both regularities for cylindrical pores and showed quantitative difference between the fluid moduli in pores of different geometries. Both of the observed regularities for the modulus stem from the Tait-Murnaghan equation applied to the confined fluid. Our results, along with the development of the effective medium theories for nanoporous media, set the groundwork for analysis of the experimentally-measured elastic properties of fluid-saturated nanoporous materials

EFFECTIVENESS OF DIFFERENT METHODS FOR FATIGUE LIFE ENHANCEMENT OF FASTENER HOLES IN D16AT ALUMINUM ALLOY

An evaluation of the effectiveness of three different methods for enhancement of fatigue life of fastener holes in D16AT aircraft Al-alloy has been made. Objects of comparative analysis are the friction stir hole expansion (FSHE), solid mandrel cold working and symmetric cold expansion (SCE) methods. The results are generalized on the basis of fatigue tests, S-N curves obtained, X-ray diffraction and micro-structural analyses. Under the high-cycle fatigue performance, the SCE provides more than 66 times longer fatigue life as compared to solid mandrel method and more than 82 times greater fatigue life in comparison with FSHE method. Through X-ray diffraction analysis it has been found out that the higher efficiency of the SCE method is due to the symmetric distribution (with respect to the plate middle plane) of the introduced residual hoop stresses around the hole. On the other hand, the solid mandrel cold working method causes a significant gradient of the residual stress distribution in the thickness plate direction, which is a precondition for nucleation and propagation of corner fatigue cracks. It has been established that the FSHE method efficiency depends primarily on the heat generated and the equivalent plastic strain size. The combination of these factors determines the beneficial micro-effect of the microstructure modifying immediately around the hole and the useful macro-effect due to the introduced compressive residual stresses. It has been concluded that SCE method should be used for pre-stressing of fastener holes in the most loaded components in the D16AT aircraft structures - wings and fuselage, while FSHE method can be applied for processing of fastener holes in less loaded aircraft components