Polynomial Chaos Expansions on Principal Geodesic Grassmannian
  Submanifolds for Surrogate Modeling and Uncertainty Quantification
Dimitris G. Giovanis Dimitrios Loukrezis Ioannis G. Kevrekidis Michael D. Shields
In this work we introduce a manifold learning-based surrogate modeling
framework for uncertainty quantification in high-dimensional stochastic
systems. Our first goal is to perform data mining on the available simulation
data to identify a set of low-dimensional (latent) descriptors that efficiently
parameterize the response of the high-dimensional computational model. To this
end, we employ Principal Geodesic Analysis on the Grassmann manifold of the
response to identify a set of disjoint principal geodesic submanifolds, of
possibly different dimension, that captures the variation in the data. Since
operations on the Grassmann require the data to be concentrated, we propose an
adaptive algorithm based on Riemanniann K-means and the minimization of the
sample Frechet variance on the Grassmann manifold to identify "local" principal
geodesic submanifolds that represent different system behavior across the
parameter space. Polynomial chaos expansion is then used to construct a mapping
between the random input parameters and the projection of the response on these
local principal geodesic submanifolds. The method is demonstrated on four test
cases, a toy-example that involves points on a hypersphere, a Lotka-Volterra
dynamical system, a continuous-flow stirred-tank chemical reactor system, and a
two-dimensional Rayleigh-Benard convection problem