Loop integration results have been obtained using numerical integration and extrapolation. An extrapolation to the limit is performed with respect to a parameter in the integrand which tends to zero. Results are given for a non-scalar four-point diagram. Extensions to accommodate loop integration by existing integration packages are also discussed. These include: using previously generated partitions of the domain and roundoff error guards.

Abstract. Viewing a large graph in limited display space has traditionally been accomplished using either reduced scale rendering of the graph or by attaching scrollbars to a view window which shows only a small portion of the entire graph. Recent work, however, has concentrated on integrating a locally detailed view with a globally scaled view. We present an algorithm for constructing a view which smoothly integrates local detail and global context in a single view window and describe user interaction with such a display.

A back propagation artificial neural network approach is applied to three common challenges in engineering geology: (1) characterization of subsurface geometry/position of the slip (or failure surface) of active landslides, (2) assessment of slope displacements based on ground water elevation and climate, and (3) assessment of groundwater elevations based on climate data. Series of neural network models are trained, validated, and applied to a landslide study along Lake Michigan and cases from the literature. The subsurface characterization results are also compared to a limit equilibrium circular failure surface search with specific adopted boundary conditions. It is determined that the neural network models predict slip surfaces better than the limit equilibrium slip surface search using the most conservative criteria. Displacements and groundwater elevations are also predicted fairly well, in real time. The models' ability to predict displacements and groundwater elevations provides a foundational framework for building future warning systems with additional inputs.

We consider multivariate integrals which can be expressed as iterated integrals over product regions. The iteration over the dimensions is applied recursively for a numerical evaluation. We evaluate a scheme for setting the tolerated error in the interface between the integration levels and address the efficiency of the resulting method with respect to time and space requirements.

We revisit the iterated numerical integration method and show that it is extremely efficient in solving certain classes of problems. A multidimensional integral can be approximated by a combination of lower-dimensional or one-dimensional adaptive methods iteratively. When an integrand contains sharp ridges which are not parallel with any axis, iterated methods often outperform adaptive cubature methods in low dimensions. We use examples to support our analysis.