Calibration of reduced-order model for a coupled Burgers equations based on PC-EnKF

¹ School of Mathematics and Statistics, Nanjing University of Information Science and Technology (NUIST), Nanjing 210044, P.R. China
² Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA
³ School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USA

^* Corresponding author: guanglin@purdue.edu

Received: 17 October 2017
Accepted: 18 October 2017

Abstract

The proper orthogonal decomposition (POD) and the discrete empirical interpolation method (DEIM) are applied to coupled Burgers equations to develop its reduced-order model (ROM) by the Galerkin projection. A calibrated POD ROM is developed in the current study through adding and multiplying a set of time-dependent random parameters to recover the loss of accuracy due to the truncation of the POD modes. Calibrating the ROM becomes essentially a high-dimensional statistical inverse inference problem. To reduce the computational effort, the polynomial chaos based ensemble Kalman filter (PC-EnKF) is adopted in this work. By using a sparse optimization algorithm, a sparse PC expansion is obtained to facilitate further calculation of statistical moments used in ensemble Kalman filter. We apply the well-defined calibrated POD ROM for the coupled Burgers equations with the Reynolds number Re = 10 000. The numerical results show that the PC-EnKF method is efficient in reducing the uncertainty included in the initial guess of input parameters and feasible in correcting the behavior of the POD ROM. The study suggests that the PC-EnKF is quite general as a calibration tool for calibrating the POD ROM.