SCAIM Seminar: Uri Ascher

Computer simulation is widely recognized nowadays as a major tool for scientific discovery as well as engineering testing. Correspondingly, typical challenges in today's scientific computing can be profoundly different from most numerical analysis considerations of only a few decades ago. The luxury of proceeding to compute only with a full theoretical backing is becoming rarer, to the chagrin of both mathematicians and their admirers, and various types of attempts in the community to bridge the perceived gap may be observed. In a particularly unfortunate scenario, the natural thirst for solid theory may in fact lead to the highlighting of inferior methods or irrelevant computational task in preference over better but more complicated or less mathematically appealing options. In this pizza-time talk I will discuss and highlight various aspects of this situation. Case studies will be described to illustrate the concepts. These include preserving energy in Hamiltonian systems, proving that faster gradient descent methods are faster, proving bounds on Monte Carlo methods for trace estimation, justifying level set methods for computational inverse problems, and employing variational PDE methods in image processing.