The conference was a NSF/CBMS Regional Conference in the Mathematical Sciences, and took place at the Department of Applied Mathematics of the Illinois Institute of Technology (IIT), August 9--13, 2010, Chicago. It was organised by Jinqiao Duan ([email protected]), Igor Cialenco, and Fred J. Hickernell.

This conference was about new methods for numerical simulations of stochastic partial differential equations and some applications. The numerical solution of stochastic partial differential equations (SPDEs) is at a stage of development similar to that of stochastic ordinary differential equations (SODEs) in the 1970s. Past developments have been ad hoc and fragmentary and it was not known how to derive higher order schemes. The breakthrough for SODEs came via stochastic Taylor expansions which are based on iterated applications of the Ito formula, the stochastic chain rule. This theory was expounded in the influential monograph of Kloeden & Platen which first appeared in 1992. Unfortunately there is no general Ito formula for SPDEs. Moreover, many additional difficulties arise with SPDEs, in particular the infinite dimensionality of the driving noise processes. Very recently, Jentzen & Kloeden have shown how stochastic Taylor expansions for the solution for SPDEs have been established on the basis of the integral representation of mild solutions of SPDEs and new higher order numerical schemes for SPDEs are now being derived.

The lectures by Peter E. Kloeden aimed to crystallize the foundational developments in the field of numerical SPDEs and set the stage for an explosion of new results that will bring the field to maturity in the coming decade. The above issues are closely related and are handled in the lectures.

Supplemental presentations by a few participants, such as P. Sounganidis, Arnulf Jentzen, Anthony Roberts, Salah Mohemmed, Richard Sowers, Hassan Allouba, and Klaus Ritter, added relevant topics to the conference, especially making connections with quantifying model uncertainty, complexity, moving boundary problems, parameter estimates, and numerical implementation.

Eight graduate students, junior faculty members and postdocs presented their current research in the poster session, which was on display for a whole day. The posters attracted many local students as well as participants.
Two break down sessions were conducted to discuss current research topics, developments and importance of numerical solutions for stochastic processes (Moderator Klaus Ritter for session 1 and Moderator Richard Sowers for session 2). Participants raised many questions and some were answered by the audience.

One panel session was conducted to discuss issues related to the lectures. The moderator was Anthony Roberts and the panel consisted of Hassan Allouba, Salah Mohemmed, Klaus Ritter, Panagiotis Souganidis and Richard Sowers.
One break down session was conducted to discuss Future Research Directions in Numerical Approximation of Stochastic Processes (hosted by Peter E. Kloeden and Jinqiao Duan). Various topics were proposed by the participants concerning what they think are the important issues which lie ahead for numerical solutions of stochastic partial differential equations.

Three co-organizers have randomly talked with many participants about their comments on this conference. They all liked the location (Chicago- easy to come and go by driving, flying or train). The graduate students enjoyed the poster session and they liked the presentation on the first day about Matlab simulation of stochastic differential equations (by a IIT graduate student, Miss Xingye Kan). Several senior participants, for example P. Souganidis of the University of Chicago, thought it was great to have this event in Chicago to bring related people together for learning and research discussions.