Model Reduction with MapReduce-enabled Tall-and-Skinny Singular Value Decomposition

TL;DR

This paper introduces a scalable MapReduce-based method for model reduction of parameterized PDEs using singular value decomposition, interpolation, and a novel subset selection technique that provides confidence measures.

Contribution

It presents a new approach combining MapReduce, SVD, and gradient-based subset selection for efficient, accurate reduced-order modeling of large-scale PDE data.

Findings

Effective reduction of 4TB data using MapReduce and SVD.

Improved accuracy over scalar response surfaces in capturing local features.

Confidence measures enable assessment of model predictions.

Abstract

We present a method for computing reduced-order models of parameterized partial differential equation solutions. The key analytical tool is the singular value expansion of the parameterized solution, which we approximate with a singular value decomposition of a parameter snapshot matrix. To evaluate the reduced-order model at a new parameter, we interpolate a subset of the right singular vectors to generate the reduced-order model's coefficients. We employ a novel method to select this subset that uses the parameter gradient of the right singular vectors to split the terms in the expansion yielding a mean prediction and a prediction covariance---similar to a Gaussian process approximation. The covariance serves as a confidence measure for the reduce order model. We demonstrate the efficacy of the reduced-order model using a parameter study of heat transfer in random media. The high-fidelity simulations produce more than 4TB of data; we compute the singular value decomposition and evaluate the reduced-order model using scalable MapReduce/Hadoop implementations. We compare the accuracy of our method with a scalar response surface on a set of temperature profile measurements and find that our model better captures sharp, local features in the parameter space.