  Functional principal component analysis is one of the most commonly employed
approaches in functional and longitudinal data analysis and we extend it to
analyze functional/longitudinal data observed on a general $d$-dimensional
domain. The computational issues emerging in the extension are fully addressed
with our proposed solutions. The local linear smoothing technique is employed
to perform estimation because of its capabilities of performing large-scale
smoothing and of handling data with different sampling schemes (possibly on
irregular domain) in addition to its nice theoretical properties. Besides
taking the fast Fourier transform strategy in smoothing, the modern GPGPU
(general-purpose computing on graphics processing units) architecture is
applied to perform parallel computation to save computation time. To resolve
the out-of-memory issue due to large-scale data, the random projection
procedure is applied in the eigendecomposition step. We show that the proposed
estimators can achieve the classical nonparametric rates for longitudinal data
and the optimal convergence rates for functional data if the number of
observations per sample is of the order $(n/ \log n)^{d/4}$. Finally, the
performance of our approach is demonstrated with simulation studies and the
fine particulate matter (PM 2.5) data measured in Taiwan.
