Charles Talbot (UNC-CH), GMA & Visions Seminar

Title: A Kernel PCA-Based Adjoint Method and Its Goal-Oriented Extensions

Abstract: The adjoint state method is a staple technique for efficiently computing gradients involved in numerical optimization, and continues to be applied to inverse problems in a broad array of fields ranging from geophysics to finance.  Forproblems involving large, high dimensional datasets, however, the high computational cost of conventional uncertainty quantification methods warrants the development of new techniques utilizing ideas from machine learning for efficient data assimilation.  To this end, we demonstrate how the adjoint method, coupled with optimization based on methods of machine learning, can facilitate the minimization of an objective function defined on a space of significantly reduced dimension.  In particular, we consider inverse problems involving elastic channelized media, and the determination of its spatial distribution of material parameters given a measurement of deformation.  By viewing the spatial distribution of uncertain parameters as constituting a random field, the Karhunen-Loeve expansion and its nonlinear extensions applied to candidate ‘snapshot’ data furnish an optimal basis with respect to which optimization using L-BFGS can be carried out. Since certain subsets of the original data are characterized by different features, the convergence rate of the method depends in part on, and may be limited by, the observations used to furnish the kernel principal component basis. By attributing a significance ‘weight’ to each realization of the random field, then, one might hope to accelerate the convergence rate of the hybrid method.  With this aim in mind, we present a formulation of Weighted Kernel PCA, and how it leads to a goal-oriented hybrid method. We compare the convergence of the weighted and unweighted methods, and discuss further extensions of the kernel PCA-based adjoint method.