Algorithmic

Hydraulic fracture modeling relies on a non-linear, non-local moving boundary problem with multiple nested length scales possesses a harsh numerical stability criterion for explicit time-stepping methods and a demonstrable tendency for algorithmic details to have a first order impact on predictions. Bringing together predictions from a variety of types of hydraulic fracture simulators with data from both the field and laboratory, this talk will highlight some lessons that have been learned over the past decade of research.

The M-ellipsoid of a convex body K is a fundamental object introduced by V. Milman. It has the same volume as K and can cover K with the union a small number of translations. There are several proofs of the existence of an M-ellipsoid. We present algorithms for constructing M-ellipsoids based on two of these proofs. The first is randomized and is via a proof by Klartag giving optimal covering bounds. The second, based on a proof by Pisier, achieves slightly weaker bounds but leads to a deterministic algorithm.