# Courant Lectures with Andrew Stuart

## Time and Location:

March 30, 2023 at 11AM; Warren Weaver Hall, Room 109## Notes:

**Andrew Stuart**, *Bren Professor of Computing and Mathematical Sciences* at Caltech, will present two lectures at Courant at the end of March. The first—"The Legacy of Rudolph Kalman: Applications, Algorithms and Analysis"— is designed for a general audience and will take place on March 30th, with a reception to follow in the lounge. A second, more technical lecture—"The Mean-Field Ensemble Kalman Filter"—will be held on March 31st.

*You can find the RSVP form here.*

**"The Legacy of Rudolph Kalman: Applications, Algorithms and Analysis"**

**March 30th at 11AM, WWH 109**

In 1960 Rudolph Kalman published what is arguably the first paper to develop a systematic, principled approach to the use of data to improve the predictive capability of mathematical models. As our ability to gather data grows at an enormous rate, the importance of this work continues to grow too. The lecture will describe Kalman's paper and how it leads to other algorithms, including ensemble Kalman filtering. These methods have revolutionized applications such as navigation, weather prediction, oceanography, oil recovery and medical imaging; and they shows promise to impact other areas such as machine learning. Various applications will be highlighted in the talk.

Algorithms will be introduced, and interpreted, through a simple iterative optimization framework. Despite their widespread use in applications, ensemble Kalman methods have attracted limited analysis. The talk will conclude by explaining a framework for their analysis, based on mean-field interacting particle systems, and recent theoretical results stemming from this framework.

**"The Mean-Field Ensemble Kalman Filter"**

**March 31st at 4PM, WWH 1302**

Ensemble Kalman filters constitute a methodology for incorporating noisy data into complex dynamical models to enhance predictive capability. They are widely adopted in the geophysical sciences, underpinning weather forecasting for example, and are starting to be used throughout the sciences and engineering; furthermore, they have been adapted to function as a general-purpose tool for parametric inference. The strength of these methods stems from their ability to operate using complex models as a black box, together with their natural adaptation to high performance computers. In this talk we introduce theory which, for the firsttime, elucidates conditions under which this widely adopted methodology provides accurate model predictions and uncertainties for discrete time filtering. The theory rests on a mean-field formulation of the methodology and an error analysis controlling differences between probability measure propagation under the mean-field model and under the true filtering distribution.

The mean-field formulation is based on joint work with Edoardo Calvello (Caltech) and Sebastian Reich (Potsdam).

The error analysis is based on joint work with Jose Carrillo (Oxford), Franca Hoffmann (Caltech) and Urbain Vaes (Paris).

**Andrew Stuart** obtained his undergraduate degree in Mathematics, from Bristol University in 1983, his PhD from the Oxford University Computing Laboratory in 1987 and was then a postdoc at MIT in the period 1987--1989. Before joining Caltech he held permanent positions at Bath University (1989--1992), Stanford University (1992--1999) and Warwick University (1999--2016). His research interests focus on computational applied mathematics; and in particular challenges presented by the age of information, such as the integration of data with mathematical models and the mathematics of machine learning. He has been awarded the Leslie Fox Prize for Numerical Analysis in 1989, the IPST Monroe Martin Prize in 1995, the Whitehead Prize from the London Mathematical Society in 2000, and the James Wilkinson Prize in Numerical Analysis and Scientific Computing (1997), the Germund Dahlquist Prize (1997) and the J.D. Crawford Prize (2007), all from SIAM. He was elected an inaugural SIAM Fellow in 2009. He delivered invited lectures at the International Congress of Industrial and Applied Mathematics (ICIAM) in 2007 and 2023, at the European Congress of Mathematicians (ECM) in 2012 and at the International Congress of Mathematicians (ICM) in 2014. He was elected as a Fellow of The Royal Society in 2023.