Retrieval Of Initial Condition For Burgers Equations Via EnKF Method Based On POD Sparse Sensing

Uncertainty quantification(UQ)appearing in the disaster assessment,as well as climate change,has received more and more attention.To reduce the uncertainty of model output,it can be attained through data assimilation method.Data assimilation is to optimize the input of the problem for determining solution through combining the numerical prediction model(i.e.,numerical discrete scheme of partial differential equation,PDE)with the observation data for the purpose of achieving the best fitting(or forecast)to the observation.Currently,the reducedorder model has played an important role in reducing computational costs when replacing the full-order model to implement data assimilation in a low-dimensional space.However,it is still necessary to explore the determination of the optimal observation position in the original space for further reducing the calculation and application cost.This is an active research area.The basic idea of compressed sensing is introduced in the framework of ensemble Kalman filtering(En KF)to explore the optimal observation(position),and establish an adaptive ensemble Kalman filter algorithm based on the reduced-order model when optimizing the input field(such as the initial conditions of the PDE)of the full-order model to realize the reconstruction of the forecast field.The detailed process includes two aspects: first,sensing measurement that intends to obtain the best observation by calculating the observation matrix C derived from the POD reduced order basis via the QD algorithm;second,performing the reconstruction of the coefficients of reduced-order Kalman filter,and then returning it to the original space for accomplishing the update of snapshots.In order to demonstrate the feasibility and effectiveness of this algorithm,we applied it to retrieve the initial condition of Burgers equation.The main research results include:(1)under the cases of Reynolds number 250 Re = and 1000 Re =,the retrieval of initial condition and thus the solution reconstruction are successfully realized;(2)For the construction of the reduced-order model,its degree of freedom(Do F)determined by the termination condition in the algorithm is less than the one obtained through the singular value truncation via the traditional energy law;(3)The implementation process of the algorithm depends completely on the reduced-order model;(4)The data obtained at the optimal observation position is effective for the retrieval of the initial conditions.These results above will lay a foundation for further application to the research of high-dimensional problems.