Decomposition of flow data via gradient-based transport optimization

TL;DR

This paper introduces a gradient-based optimization method for decomposing flow data into transported modes, enhancing data compression in flow applications by allowing transformations.

Contribution

It extends proper orthogonal decomposition by incorporating transformed modes and proves the existence of solutions for the resulting optimization problem.

Findings

Proved existence of solutions for the infinite-dimensional problem.

Developed a gradient computation and discretization scheme.

Validated approach with three challenging numerical examples.

Abstract

We study an optimization problem related to the approximation of given data by a linear combination of transformed modes. In the simplest case, the optimization problem reduces to a minimization problem well-studied in the context of proper orthogonal decomposition. Allowing transformed modes in the approximation renders this approach particularly useful to compress data with transported quantities, which are prevalent in many flow applications. We prove the existence of a solution to the infinite-dimensional optimization problem. Towards a numerical implementation, we compute the gradient of the cost functional and derive a suitable discretization in time and space. We demonstrate the theoretical findings with three challenging numerical examples.