Approximate Low Dimensional Models Based on Proper Orthogonal Decomposition for Black-Box Applications

Abstract

Many industrial applications in engineering and science are solved using commercial engineering solvers that function as black-box simulation tools. Besides being time consuming and computationally expensive, the actual mathematical models and underlying structure of these problems are for most part unknown. This paper presents a method for constructing approximate low-dimensional models for such problems using the proper orthogonal decomposition (POD) technique. We consider a heat diffusion problem and a contamination transport problem, where the actual mathematical models are assumed to be unknown but numerical data in time are available so as to enable the formation of an ensemble of snapshots The POD technique is then used to produce a set of basis functions that spans the snapshot collection corresponding to each problem. The key idea is then to assume some functional form of the low-order model, and use the available snapshot data to perform a least squares fit in the POD basis coordinate system to determine the unknown model coefficients. Initial results based on this approximation method seem to hold some promise in creating a predictive low-order model, that is, one able to predict solutions not included in the original snapshot set. Some issues arising from this approximation method are also discussed in this paper.