Title: Efficient Covariance Estimation for Sparsified Functional Data. Abstract: To avoid prohibitive computation cost of sending entire data, we propose four sparsification schemes Random-knots, Random-knots-Spatial, B-spline, Bspline-Spatial, and present corresponding nonparametric estimation of the covariance function. The covariance estimators are asymptotically equivalent to the sample covariance computed directly from the original data. And the estimated functional principal components effectively approximate the infeasible principal components under regularity conditions. The convergence rate reflects that leveraging spatial correlation and B-spline interpolation helps to reduce information loss. Data-driven selection method is further applied to determine the number of eigenfunctions in the model. Extensive numerical experiments are conducted to illustrate the theoretical results. 