  Proper Orthogonal Decomposition (POD) is a widely used technique for the
construction of low-dimensional approximation spaces from high-dimensional
input data. For large-scale applications and an increasing amount of input data
vectors, however, computing the POD often becomes prohibitively expensive. This
work presents a generic, easy-to-implement approach to compute an approximate
POD based on arbitrary tree hierarchies of worker nodes, where each worker
computes a POD of only a small amount of input vectors. The tree hierarchy can
be freely adapted to optimally suit the available computational resources. In
particular, this hierarchical approximate POD (HAPOD) allows for both simple
parallelization with low communication overhead, as well as live sequential POD
computation under restricted memory capacities. Rigorous error estimates ensure
the reliability of our approach, and extensive numerical examples underline its
performance.
