Commun. Comput. Phys., 16 (2014), pp. 75-95.

Approximation of Spatio-Temporal Random Processes Using Tensor Decomposition

Debraj Ghosh 1*, Anup Suryawanshi 1

1 Department of Civil Engineering, Indian Institute of Science, Bangalore 560012, India.

Received 20 November 2012; Accepted (in revised version) 19 November 2013
Available online 31 March 2014


A new representation of spatio-temporal random processes is proposed in this work. In practical applications, such processes are used to model velocity fields, temperature distributions, response of vibrating systems, to name a few. Finding an efficient representation for any random process leads to encapsulation of information which makes it more convenient for a practical implementations, for instance, in a computational mechanics problem. For a single-parameter process such as spatial or temporal process, the eigenvalue decomposition of the covariance matrix leads to the well-known Karhunen-Loeve (KL) decomposition. However, for multiparameter processes such as a spatio-temporal process, the covariance function itself can be defined in multiple ways. Here the process is assumed to be measured at a finite set of spatial locations and a finite number of time instants. Then the spatial covariance matrix at different time instants are considered to define the covariance of the process. This set of square, symmetric, positive semi-definite matrices is then represented as a third-order tensor. A suitable decomposition of this tensor can identify the dominant components of the process, and these components are then used to define a closed-form representation of the process. The procedure is analogous to the KL decomposition for a single-parameter process, however, the decompositions and interpretations vary significantly. The tensor decompositions are successfully applied on (i) a heat conduction problem, (ii) a vibration problem, and (iii) a covariance function taken from the literature that was fitted to model a measured wind velocity data. It is observed that the proposed representation provides an efficient approximation to some processes. Furthermore, a comparison with KL decomposition showed that the proposed method is computationally cheaper than the KL, both in terms of computer memory and execution time.

AMS subject classifications: 60G12, 65F99, 65D15

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Key words: Random process, spatio-temporal process, tensor decomposition, uncertainty quantification, probabilistic mechanics.

*Corresponding author.
Email:, (D. Ghosh), (A. Suryawanshi)

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