It is well known that Markov chain Monte Carlo (MCMC) methods scale poorly with dataset size. A popular class of methods for solving this issue is stochastic gradient MCMC (SGMCMC). These methods use a noisy estimate of the gradient of the log-posterior, which reduces the per iteration computational cost of the algorithm. Despite this, there are a number of results suggesting that stochastic gradient Langevin dynamics (SGLD), probably the most popular of these methods, still has computational cost proportional to the dataset size. We suggest an alternative log-posterior gradient estimate for stochastic gradient MCMC which uses control variates to reduce the variance. We analyse SGLD using this gradient estimate, and show that, under log-concavity assumptions on the target distribution, the computational cost required for a given level of accuracy is independent of the dataset size. Next, we show that a different control-variate technique, known as zero variance control variates, can be applied to SGMCMC algorithms for free. This postprocessing step improves the inference of the algorithm by reducing the variance of the MCMC output. Zero variance control variates rely on the gradient of the log-posterior; we explore how the variance reduction is affected by replacing this with the noisy gradient estimate calculated by SGMCMC.

Seismic hazard and risk analyses underpin the loadings prescribed by engineering design codes, the decisions by asset owners to retrofit structures, the pricing of insurance policies, and many other activities. This is a comprehensive overview of the principles and procedures behind seismic hazard and risk analysis. It enables readers to understand best practises and future research directions. Early chapters cover the essential elements and concepts of seismic hazard and risk analysis, while later chapters shift focus to more advanced topics. Each chapter includes worked examples and problem sets for which full solutions are provided online. Appendices provide relevant background in probability and statistics. Computer codes are also available online to help replicate specific calculations and demonstrate the implementation of various methods. This is a valuable reference for upper level students and practitioners in civil engineering, and earth scientists interested in engineering seismology.

For performance-based design, non-linear dynamic structural analysis for various types of input ground motions is required. Stochastic (simulated) ground motions are sometimes useful as input motions, because unlike recorded motions they are not limited in number and because their properties can be varied systematically to understand the impact of ground motion properties on structural response. Here a stochastic ground motion model with time and frequency nonstationarity is developed using wavelet packets. Wavelet transform is a tool for analyzing time-series data with time and frequency nonstationarity, as well as simulating such data. Wavelet packet transform is an operation that decomposes time-series data into wavelet packets in the time and frequency domain, and its inverse transform reconstructs a time-series from wavelet packets. The characteristics of a nonstationary ground motion therefore can be modeled intuitively by specifying the amplitudes of wavelet packets at each time and frequency. In the proposed model, 13 parameters are sufficient to completely describe the time and frequency characteristics of a ground motion. These parameters can be computed from a specific target ground motion recording or by regression analysis based on a large database of recordings. The simulated ground motions produced by the proposed model reasonably match the target ground motion recordings in several respects including the spectral acceleration, inelastic response spectra, duration, bandwidth, and time and frequency nonstationarity. In addition, the median and logarithmic standard deviation of the spectral acceleration of the simulated ground motions match those of the published empirical ground motion prediction. These results suggest that the synthetic ground motions generated by the proposed model can be used for the non-linear dynamic structural analysis as the input ground motions.

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%'hen care is taken to prevent contamination, the superconducting transition temperatures of thin Al, Sn, Tl, and amorphous-Bi films are depressed by overlayers of Ne or Ar. For a given film, the ratio of the shift in T, produced by Ar to that produced by Ne is observed to be 1.8. The results can be explained in terms of a modification of the phonon spectrum by the presence of the noble gas. A simple model predicts the observed ratio of the effect of Ar to that of Ne and the relative magnitudes of the shifts for the four different metals.