Numerical linear algebra underlies much of the modern world. It is essential to a wide variety situations, and as such, it is of great interest to analyze and optimize the underlying algorithms. In recent years, there has been an increased interest in optimizing algorithms to reduce communication, which is frequently many orders of magnitude slower than performing calculations. The Hölder-Brascamp-Lieb inequality and the Red-Blue Pebbling Game are two powerful approaches to theoretical analysis of communication, and both are used to derive communication optimal lower bounds. Once these bounds have been derived, the next challenge is to implement an algorithm that attains these minimums, and thus, is communication minimizing. Many currently implemented algorithms do not meet these... Read More