An Iterative Proper Orthogonal Decomposition (I-POD) Technique with Application to the Filtering of Partial Differential
- PDF / 7,497,153 Bytes
- 26 Pages / 439.642 x 666.49 pts Page_size
- 28 Downloads / 206 Views
An Iterative Proper Orthogonal Decomposition (I-POD) Technique with Application to the Filtering of Partial Differential Equations Dan Yu1 · Suman Chakravorty2
Published online: 10 November 2015 © American Astronautical Society 2015
Abstract In this paper, we consider the filtering of systems governed by partial differential equations (PDE). We adopt a reduced order model (ROM) based strategy to solve the problem. We propose an iterative version of the snapshot proper orthogonal decomposition (POD) technique, termed I-POD, to sequentially construct a single ROM for PDEs that is capable of capturing their behavior over the entire state space of the system, and not just around the snapshot trajectory. Further, the technique is entirely data based, and is applicable to forced as well as unforced systems. The IPOD is compared to two other ROM techniques: the Balanced POD ( BPOD) and the dynamic mode decomposition (DMD). We apply the ROM generated using the I-POD technique to construct reduced order Kalman filters to solve the filtering problem. The methodology is tested on several 1-dimensional PDEs of interest including the heat equation, the wave equation and 2-dimensional pollutant transport equation. Keywords Model reduction · Proper orthogonal decomposition · Kalman filtering
Introduction In this paper, we are interested in the filtering/ data assimilation in systems that are governed by partial differential equations (PDE). We take a reduced order model
Suman Chakravorty
[email protected] 1
Department of Aerospace Engineering, Texas A&M University, 301A RDMC, TAMU, College Station, TX 77840, USA
2
Department of Aerospace Engineering, Texas A&M University, 741A HRBB, College Station, TX, 77840, USA
J of Astronaut Sci (2013) 60:468–493
469
(ROM) based approach to the problem. We propose an iterative version of the snapshot proper orthogonal decomposition (POD) technique that allows us to form an ROM of the PDE of interest in terms of the eigenfunctions of the PDE operator. The I-POD is compared to two other ROM techniques: the Balanced POD (BPOD) and the dynamic mode decomposition (DMD) technique. We then apply this ROM, along with the Kalman filtering technique, to form a reduced order filter for the PDE. The filter is constructed in an offline-online fashion where the expensive computations for the ROM construction is accomplished offline, while the online part consists of the reduced order Kalman filter which is much more computational tractable than the full problem. The technique is applied to several one and two dimensional partial differential equations. We take a ROM based approach to solving the problem of filtering in PDEs. In particular, we rely on the so-called proper orthogonal decomposition (POD), more precisely, the snapshot POD technique, to construct ROMs for the PDE of interest. The POD technique has been used extensively in the Fluids community to produce ROMs of fluid physics phenomenon such as turbulence and fluid structure interaction [1–4]. There has also been work recently to pro
Data Loading...
