Calibration Of Reduced-order Model For A Coupled Burgers Equations Based On PC-EnKF

The model reduction procedure,developed for optimization design, optimal control and inverse problem research that require computational efficiency and physical accuracy, intends to construct a reduce-order model which can present a powerful representation of the dynamics of large-scale systems with a smaller number of degrees of freedom, and therefore results in a significant time reduction.The proper orthogonal decomposition (POD) has become one of the most important reduced-order model methods due to its ability to rebuild numerical signals. The POD reduced-order model (ROM) of coupled Burgers equations is first constructed by combining POD with Galerkin projection. In order to further reduce the complex computation, the nonlinear reduced-order terms of the POD ROM are further dealt with by the discrete empirical interpolation method (DEIM). The loss of accuracy of solutions is then inevitably incurred due to the truncation of the POD modes and the numerical discretization algorithm itself. The higher POD modes indeed sometimes play an important role in keeping the stability and accuracy of ROM. However, this will increase the computational burden. A calibrated POD ROM is developed in the current study through adding and multiplying a set of time-dependent random parameters. So calibrating the ROM becomes essentially a statistical inverse inference for these parameters in high-dimensional random space. The computational effort involved is thus challenging due to the ill-posed feature. To address these issues the polynomial chaos-based ensemble Kalman filter (PC-EnKF) is adopted.And the introduction of a sparse algorithm for the purpose of determining model input parameters and output state variable of interest in the form of PC expansion helps to detect the unnecessary bases in the PC expansion and obtain sparse coefficients that facilitate further calculation of statistical moments used in ensemble Kalman filter. With the retrieved input parameters a well-defined calibrated POD ROM is ultimately obtained for the coupled Burgers equations with the Reynolds number Re = 10000. 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 solution of the dynamical system. The study suggests that the PC-EnKF is quite general as a calibration tool and very promising to efficiently extend its application to a higher-dimensional calibration problem.