Evaluating crystallographic likelihood functions using numerical quadratures

Intensity-based likelihood functions in crystallographic applications have the potential to enhance the quality of structures derived from marginal diffraction data. Their usage, however, is complicated by the ability to efficiently compute these target functions. Here, a numerical quadrature is developed that allows the rapid evaluation of intensity-based likelihood functions in crystallographic applications. By using a sequence of change-of-variable transformations, including a nonlinear domain-compression operation, an accurate, robust and efficient quadrature is constructed. The approach is flexible and can incorporate different noise models with relative ease.

Evaluating crystallographic likelihood functions using numerical quadratures

Intensity-based likelihood functions in crystallographic applications have the potential to enhance the quality of structures derived from marginal diffraction data. Their usage however is complicated by the ability to efficiently compute these targets functions. Here a numerical quadrature is developed that allows for the rapid evaluation of intensity-based likelihood functions in crystallographic applications. By using a sequence of change of variable transformations, including a non-linear domain compression operation, an accurate, robust, and efficient quadrature is constructed. The approach is flexible and can incorporate different noise models with relative ease.

Asymptotically Compatible Schemes for Peridynamics Based on Numerical Quadratures

We present some studies of numerical schemes for nonlocal peridynamic and nonlocal diffusion models. We describe asymptotically compatible (AC) schemes recently developed for robust discretizations of nonlocal models. The AC schemes for peridynamic models provide convergent approximations to nonlocal models associated with fixed horizon parameter as well as their limiting local models. We illustrate what quadrature based discretizations can be AC schemes and what may fail to be AC.