Generalization Of Empirical Interpolation Method (EIM) Within Evolutionary Context With Application To Prediction

This paper focuses on how to apply the empirical interpolation method(EIM)to evolution context and to the prediction.We are the first to combine EIM with other reduced-order methods(ROM)or data assimilation to predict quickly.The key point of EIM prediction is to limit the time steps used for prediction in a specified time range.As long as all the interpolation time steps are within the known time range,we can extrapolate the data outside the known time.Besides,when we only predict the coefficients obtained by ROM or data assimilation,the calculation time is greatly reduced.We have considered four types of problems.The first is to use the EIM approximation and EIM prediction for one-dimensional function in the form of f =u(x,t;?).The second and third problems are,in the two-dimensional heat transfer problem,predicting the coefficients of EIM approximation and parameterizedbackground data-weak approach(PBDW)approximation respectively.The last problem is the time-dependant parametrized problem and we propose a model to simulate the variation of the size of two interacting bacterias.Our results prove that even though the prediction accuracy is limited by the approximation accuracy,which is the price of the prediction based on the approximation coefficients,the prediction results are satisfactory.The results of two interacting bacterias show that the application of EIM prediction is not limited to fast prediction,but can also be used to optimize the model and maintain a good fit longer.