Computing with Uncertainty: Mathematical Modeling, Numerical Approximation and Large Scale Optimization of Complex Systems with Uncertainty

A Saturday-Sunday tutorial on Computing with Uncertainty precedes this workshop.

Mathematical and computational models for an increasing number of complex systems in basic sciences, engineering and, increasingly, also in life sciences and socioeconomic modeling, involve uncertainty: the input data could be random parameters expressing information that may only be revealed in the future, or simply reflect measurement error or inherent variability.

Uncertainty can also arise on a more primitive level due to insufficient knowledge about particular components of the system under consideration. We then not only confront descriptive issues but also have to question the validity of the prescriptive implications we might derive from the solutions of such mathematical models in the presence of uncertainty. With the prodigious increase in computational capabilities, already seen and expected to continue, and accompanied by a corresponding increase in data, new strategies in mathematical and computational modeling as well as system optimization will be required.

Nevertheless, there will inevitably remain a gap, never to be fully bridged, between the mathematical formulation of problem classes with uncertain inputs and incomplete information, the available data for these models, and implementable solution and optimization algorithms predicting complex system behavior under such circumstances.

This means that, in addition to the design of efficient numerical procedures which can operate on mathematical models with uncertain data on a very large scale, some pivotal questions must be answered:

• How can we determine if a mathematical model is a satisfactory approximation of a system under consideration for given, limited experimental data?
• What are efficient forward simulations for partial differential equation models of complex systems with random inputs?
• How can efficient deterministic forward simulations be coupled with data assimilation and online experimentation?
• What are implications of data and solution uncertainty on numerical optimization of complex systems on high dimensional parameter spaces?
• How can the quality of approximation be validated when a problem, after being formulated by a mathematical model, must be replaced by a reduced mathematical model computationally more tractable?

Description

The mathematical description and the design of efficient, deterministic forward simulations and optimizations of complex systems with uncertainty poses challenges to mathematics, statistics and scientific computing alike.

There is a strong need to understand the behavior of novel mathematical models of complex systems with uncertainties on high dimensional parameter spaces, their efficient numerical solution and numerical optimization with particular attention to stability, consistency, and convergence rates, as well as scalability of their computer implementations. This workshop addresses Uncertainty Quantification and Stochastic Optimization, with focus on mathematical modeling, numerical analysis and large scale scientific computing. It will have a strongly interdisciplinary nature, bringing together applied mathematicians, computational scientists and researchers working on a wide range of applications, from engineering and the sciences.