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1. Choi, Minseok. Time-dependent Karhunen-Loeve type decomposition methods for SPDEs.

Degree: PhD, Applied Mathematics, 2014, Brown University

A new hybrid methodology for the stochastic partial differential equations (SPDEs) is developed based on the dynamically-orthogonal (DO) and bi-orthogonal (BO) expansions; both approaches are an extension of the Karhunen-Loeve (KL) expansion. The original KL expansion provides a low-dimensional representation for square integrable random processes since it is optimal in the mean square sense. The solution to SPDEs is represented in a way that it follows the characteristics of KL expansion on-the-fly at any given time. To this end, both the spatial and stochastic basis in the representation are time-dependent unlike the traditional methods such as polynomial chaos (PC), where only one of them is time-dependent. In order to overcome the redundancy the DO imposes the dynamical constraints on the spatial basis while the BO imposes the static constraints on the spatial and stochastic basis. We examine the relation of the BO and DO and prove theoretically and illustrate numerically their equivalence, in the sense that one method is an exact reformulation of the other by deriving an invertible and linear transformation matrix governed by a matrix differential equation that connects the BO and the DO. We also examine the pathology of the BO that occurs when there is an eigenvalue crossing leading to the numerical instability. On the other hand we observe that the DO suffers numerically when there is a high condition number of the covariance matrix for the stochastic basis. To this end, we propose a unified hybrid framework of the two methods by utilizing an invertible and linear transformation between them. We also present an adaptive algorithm to add or remove modes to better capture the transient behavior. Several numerical examples, linear and nonlinear, are presented to illustrate the DO and BO methods, their equivalence, and adaptive strategies. It is also shown numerically that two methods converge exponentially fast with respect to the number of modes giving the same levels of accuracy, which is comparable with the PC method but with substantially smaller computational cost compared to stochastic collocation, especially when the involved parametric space is high-dimensional. Advisors/Committee Members: Karniadakis, George (Director), Rozovsky, Boris (Reader), Themistoklis, Sapsis (Reader).

Subjects/Keywords: uncertainty quantification

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APA (6th Edition):

Choi, M. (2014). Time-dependent Karhunen-Loeve type decomposition methods for SPDEs. (Doctoral Dissertation). Brown University. Retrieved from

Chicago Manual of Style (16th Edition):

Choi, Minseok. “Time-dependent Karhunen-Loeve type decomposition methods for SPDEs.” 2014. Doctoral Dissertation, Brown University. Accessed January 16, 2021.

MLA Handbook (7th Edition):

Choi, Minseok. “Time-dependent Karhunen-Loeve type decomposition methods for SPDEs.” 2014. Web. 16 Jan 2021.


Choi M. Time-dependent Karhunen-Loeve type decomposition methods for SPDEs. [Internet] [Doctoral dissertation]. Brown University; 2014. [cited 2021 Jan 16]. Available from:

Council of Science Editors:

Choi M. Time-dependent Karhunen-Loeve type decomposition methods for SPDEs. [Doctoral Dissertation]. Brown University; 2014. Available from: