Hierarchical Approximate Proper Orthogonal Decomposition

TL;DR

This paper introduces a hierarchical approximate POD method that efficiently computes low-dimensional representations from large datasets by leveraging tree hierarchies, enabling scalable and memory-efficient computations with reliable error control.

Contribution

It proposes a flexible, tree-based hierarchical approach to approximate POD, improving scalability and efficiency for large-scale data analysis.

Findings

Enables parallel and sequential POD computations with low communication overhead.

Provides rigorous error estimates for the approximate POD method.

Demonstrates superior performance through extensive numerical examples.

Abstract

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.