报告人：Peng Chen (Oden Institute, UT Austin)
地点：Room 1418, Sciences Building No. 1
Abstract: Recent advances in scientific computing have enabled large-scale modeling and simulation for complex systems in many science and engineering fields, while big volume of data become increasingly available to enhance system prediction, control, and design by integration with models. The integration of data and models, subject to various inevitable uncertainties, can be generally formulated as optimization problems under uncertainty, which include data-driven model reduction, Bayesian inversion, stochastic optimization, optimal experimental design, data assimilation, etc. One critical computational challenge to solve such problems is the high dimensionality of the uncertain parameters and/or optimization variables, which leads to intractable complexity growth with respect to the dimension by traditional methods. To tackle this challenge, several intrinsic properties have been investigated, such as the intrinsic low dimensionality, low rankness, sparsity, high regularity, anisotropy. In this talk, I will focus on the intrinsic low dimensionality informed by the Hessian of a given quantity of interest (QoI), which is a square matrix (symmetric operator) of the second-order partial derivatives of a function (functional) with respect to its variables and measures the local curvature or geometric property of the function. Efficient Hessian action and decomposition methods will be presented in exploiting the low dimensional structure of the QoI, which can be applied to a large range of optimization problems. I will demonstrate this computational strategy by a few exemplary problems, including model reduction, stochastic optimization, and Bayesian inversion, for which the reduced computational complexity that only depends on the intrinsic dimension is achieved.