Mathematical models intended for computational simulation of complex real-world processes are a crucial ingredient in virtually every field of science, engineering, medicine, and business. Two related but independent phenomena have led to the near-ubiquity of models: the remarkable growth in computing power, and the matching gains in algorithmic speed and accuracy. Together, these factors have vastly increased the applicability and reliability of simulation, not only by drastically reducing simulation time, thus permitting solution of larger and larger problems, but also by allowing simulation of previously intractable problems.
Statistical models similarly play a crucial role in modeling real-world processes in virtually every field of science, engineering, medicine, and business. The recent explosion in the availability of data, and the computing power to deal with it, have only increased their centrality. Statistical models tend to be used for modeling processes for which there is abundant data, while math models tend to be developed for data-poor environments, although there are many exceptions.
It is rarely the case that either a mathematical model or a statistical model of a process can be constructed, with assurance from its construction that the model accurately represents or predicts the complete process. Thus, in the crucial final step of utilizing the model for prediction or understanding of the real-world process, it is of central importance to understand the uncertainties inherent in using the model. In mathematics and engineering, this is called Uncertainty Quantification (UQ) and has become a major part of applied mathematics. In statistics this is called Model Uncertainty (MU), and has long been one of the most prominent fields of statistics, and includes hypothesis testing, model selection, model averaging, model criticism, and many other specialty areas.
In large part because of efforts of SAMSI over the last 12 years, UQ has seen considerable involvement of both applied mathematicians and statisticians. Indeed there is a SIAM activity group on UQ, an ASA activity group on UQ, a joint ASA/SIAM Journal on UQ, and regular conferences that are amongst the largest of specialized SIAM or ASA conferences. As an example, the recent joint conference in Lausanne had 850 participants.
On the other hand, UQ and MU have developed mostly independently and with almost completely separate communities, because of their different origins and the very different nature of the models being considered. But the goals of the two communities are often the same, and include:
- Quantifying the uncertainty in the model itself in a way that allows quantification of the uncertainty in predictions or inferences when using the model, as well as quantifying the uncertainty in model inputs.
- Determining unknown parameters or initial conditions of the model (‘inverse problems’ in UQ and ‘parameter estimation’ in MU).
- Reducing the complexity of the model.
- Determining the best among competing models, or, alternatively, optimally utilizing all available models.
- Finding the most influential components of the model.
- Developing methods for model criticism, often with the aim of improving the model.
- Dealing with the uncertainty in linking different models, often owing to different physical processes and/or different length and time scales, and often linking mathematical and statistical models.
Bringing together researchers from the two communities to attack these common goals is the primary goal of this SAMSI program.
Although we have been focusing on applied mathematics and statistics in this discussion, the intellectual content in dealing with uncertainty of models comes from a variety of disciplines, including engineering, probability, operations research, and machine learning, and the application areas are remarkably diverse. The MUMS program will seek to engage all these disciplines and a wide variety of applications
WG1: UQ in Materials, Wei Chen, Northwestern University / Ralph Smith, NC State University
WG2: Reduce Order Models (ROMs) Theory and Application, Elaine Spiller, Marquette University; Ralph Smith, NC State University; Youssef Marzouk, MIT; and Matthew Plumlee, Northwestern University
WG3: Prediction Uncertainty and Extrapolation, David Higdon, Virginia Tech / Pierre Barbillon, Agro Paris tech
WG4: Data Fusion, Dongchu Sun , University of Missouri; Chong He, University of Missouri; and Jim Berger, Duke University
WG5: Foundations of Model Uncertainty, Rui Paulo, Universidade de Lisboa / Jan Hannig, University of North Carolina – Chapel Hill
WG6: Storm Surge Hazard & Risk, Taylor Asher , University of North Carolina – Chapel Hill / Whitney Huang, University of Victoria (CAN)
Planning for this program is ongoing. As more information becomes available, it will be provided.
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