Abstract

Proper Orthogonal Decomposition (POD) is a widely used technique for constructing low-order approximation spaces from high-dimensional input data. Apart from numerous applications in the data sciences, POD is also a fundamental tool for the basis generation in projection-based reduced order modelling methods.

For large-scale applications with an increasing amount of input data vectors, however, computing the POD quickly becomes prohibitively expensive, in particular when the generated data is so large that it cannot be stored entirely in memory.

In this contribution we introduce 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 the given 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. We present rigorous error estimates and numerical examples which underline the performance and reliability of our approach.