Approximation of Spatio-Temporal Random Processes Using Tensor Decomposition

Ghosh, Debraj and Suryawanshi, Anup (2014) Approximation of Spatio-Temporal Random Processes Using Tensor Decomposition. In: COMMUNICATIONS IN COMPUTATIONAL PHYSICS, 16 (1). pp. 75-95.

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Abstract

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.

Item Type:	Journal Article
Publication:	COMMUNICATIONS IN COMPUTATIONAL PHYSICS
Publisher:	GLOBAL SCIENCE PRESS
Additional Information:	Copy right for this article belongs to the GLOBAL SCIENCE PRESS, ROOM 3208, CENTRAL PLAZA, 18 HARBOUR RD, WANCHAI, HONG KONG 00000, PEOPLES R CHINA
Keywords:	Random process; spatio-temporal process; tensor decomposition; uncertainty quantification; probabilistic mechanics
Department/Centre:	Division of Mechanical Sciences > Civil Engineering
Date Deposited:	25 Sep 2014 06:02
Last Modified:	25 Sep 2014 06:02
URI:	http://eprints.iisc.ac.in/id/eprint/49947

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