, Purdue University

Pages: pp. 8-9

Simulations of physical systems are often compromised by the lack of accurate and sufficient data characterization and the inability to quantify the predictive uncertainty introduced by incomplete model parameterizations. Even essentially deterministic systems must be treated stochastically when their parameters are underspecified by data. To complicate matters further, it isn't uncommon to encounter several competing conceptual models that purport to describe a given phenomenon. These parametric and structural (model) uncertainties limit our ability to accurately simulate and predict the manufacturing of composite materials, earthquake and seismic monitoring, reservoir exploitation, carbon sequestration, environmental remediation, nuclear waste disposal, and other complex systems. Consequently, uncertainty quantification (UQ) has emerged as an essential part of science-based predictions.

The increasing complexity of modern simulations demands a method for quantifying uncertainty in system-state estimates. We can divide existing approaches to UQ into two classes: statistical and stochastic. Statistical approaches include brute-force Monte Carlo simulations (MCS), accelerated MCS (such as quasi-Monte Carlo and Markov chain Monte Carlo methods), importance-sampling techniques, variance-reduction schemes, and response-surface methods. Stochastic approaches consist of direct methods (such as interval analysis, operator-based approaches, and polynomial chaos expansions) and indirect methods (such as Fokker-Plank and moment equations). This special issue of *Computing in Science & Engineering* magazine is devoted to recent advances in stochastic methods.

Stochastic frameworks for UQ treat uncertain system parameters as random fields and model the corresponding physical phenomena with stochastic partial differential equations (SPDEs). The theory of SPDEs gives a rigorous probabilistic framework for propagating uncertainty in system parameters to estimates of state variables. Researchers have traditionally used SPDEs to model classical systems exposed to random influences or operating in random environments. Lately, though, SPDEs have also arisen as scaling limits of interacting particle models in statistical physics, as limits of models for the distribution of gene types in population genetics, and as hydrodynamic limits of physical systems' microscopic descriptions. A probabilistic representation of solutions lets us exploit the power of stochastic calculus and probabilistic limit theory when analyzing deterministic problems; it also offers new perspectives on the phenomena for modeling purposes. Such approaches can be quite effective in sorting out multiple-scale structures and in developing Monte Carlo-type numerical methods.

This special issue of *CiSE* highlights various aspects of and outstanding problems in stochastic modeling of complex systems. The techniques covered include stochastic spectral methods, the information-theoretic approach, and sensitivity analysis for uncertainty quantification. Two application articlesâ€”one in the field of ecology and the other in the field of manufacturing designâ€”emphasize the diversity of research disciplines in which the ability to handle uncertainty is crucial to obtaining reliable science-based predictions.

"Being Sensitive to Uncertainty" demonstrates how sensitivity and uncertainty analyses can help identify the relationships between input and output variations, leading to the construction of more accurate models.

"Stochastic Computational Fluid Mechanics" presents examples of stochastic simulations of compressible and incompressible flows and provides analytical solutions for verifying these newly emerging methods for stochastic modeling.

"An Information-Theoretic Approach to Stochastic Materials Modeling" introduces a new method that relies on information-theoretic principles to build stochastic microstructural models and generate comprehensive databases of stochastic material models.

"Explaining Noise as Environmental Variations in Population Dynamics" shows how to use models of contaminant/stressor fate and transport with biological organism movement to capture the impact of spatial and temporal heterogeneity on population dynamics.

Finally, "Integrated Stochastic Supply-Chain Design Models" demonstrates how uncertainty within a supply chain can be quantified and integrated into supply-chain design models to provide optimal performance.

These few articles only begin to sample some of the methods in development and use today. This field is still rather new, and in the near future, we expect to see many other novel approaches to solving problems in this exciting and important area.

Daniel M. Tartakovsky is an associate professor in the Department of Mechanical and Aerospace Engineering at the University of California, San Diego. His technical interests include SPDEs, uncertainty quantification, inverse modeling, and hybrid methods. Tartakovsky has a PhD in hydrology from the University of Arizona. He is a member of SIAM, the American Geophysical Union, and the International Association of Hydrological Sciences. Contact him at dmt@ucsd.edu.

Dongbin Xiu is an assistant professor in the Department of Mathematics at Purdue University. His technical interests include stochastic modeling in engineering applications, high-order and spectral methods for PDEs, multiscale simulations, and large-scale simulations of complex systems. Xiu has a PhD in applied mathematics from Brown University. He is a member of SIAM. Contact him at dxiu@math.purdue.edu.

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