Multi-Scale Fusion of Information for Uncertainty Quantification and Management in Large-Scale Simulations
Technical Report,01 Sep 2009,31 Aug 2015
Brown University, Providence Providence
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We developed an integrated methodology for uncertainty quantification UQ that proceedsfrom initial problem definition to engineering applications. Towards this goal, we worked onfive research areas 1 Mathematical analysis of SPDEs and multiscale formulation 2 Nu-merical solution of SPDEs 3 Reduced-Order modeling 4 EstimationInverse problems and5 Robust optimization and control. This work set the mathematical foundations of Uncer-tainty Qantification methods used by many diverse communities in computational mechanics,fluid dynamics, plasma dynamics, and materials science. We have pioneered methods for efficienthigh-dimensional representations of stochastic processes, established Wick-Malliavin approxima-tion for nonlinear SPDEs, theoretical error estimates for multiscale parametric and stochasticPDEs, a new approach to design of experiment and UQ on parametric manifolds, multi-fidelityoptimization-under-uncertainty, a data-driven Bayasian framework and probabilistic graphicalmodels for UQ, and information-based coarse graining methods. We have also demonstratedan integration of our UQ methodology and all five areas for a benchmark problem. We havepublished more than 150 papers in top mathematical journals, obtained one patent MIT,and have established one software company MIT.
- Numerical Mathematics
- Information Science
- Statistics and Probability