  Massive data bring the big challenges of memory and computation to
researchers, which can be tackled to some extent by taking subsamples from the
full data as a surrogate. For functional data, it is common to collect
measurements intensively over their domains, which require more memory and
computation time when the sample size is large. The situation would be much
worse when the statistical inference is made through bootstrap samples. To the
best of our knowledge, there is no work to study the subsampling for the
functional linear regression or its generation systematically. In this article,
based on the functional L-optimality criterion we propose an optimal
subsampling method for the functional linear model. When the response is a
discrete or categorical variable, we further extend this subsampling method to
the functional generalized linear model. We establish the asymptotic properties
of the resultant estimators by the subsampling methods. The finite sample
performance of our proposed subsampling methods is investigated by extensive
simulation studies. We also apply our proposed subsampling methods to analyze
the global climate data and the kidney transplant data. The results from the
analysis of these data show that the optimal subsampling methods motivated by
the functional L-optimality criterion are much better than the uniform
subsampling method and can well approximate the results based on full data.
