Math @ Duke
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Publications [#361540] of Ruda Zhang
Papers Published
- Zhang, R; Mak, S; Dunson, D, Gaussian Process Subspace Regression for Model Reduction
(July, 2021)
(last updated on 2022/08/06)
Abstract: Subspace-valued functions arise in a wide range of problems, including
parametric reduced order modeling (PROM). In PROM, each parameter point can be
associated with a subspace, which is used for Petrov-Galerkin projections of
large system matrices. Previous efforts to approximate such functions use
interpolations on manifolds, which can be inaccurate and slow. To tackle this,
we propose a novel Bayesian nonparametric model for subspace prediction: the
Gaussian Process Subspace regression (GPS) model. This method is extrinsic and
intrinsic at the same time: with multivariate Gaussian distributions on the
Euclidean space, it induces a joint probability model on the Grassmann
manifold, the set of fixed-dimensional subspaces. The GPS adopts a simple yet
general correlation structure, and a principled approach for model selection.
Its predictive distribution admits an analytical form, which allows for
efficient subspace prediction over the parameter space. For PROM, the GPS
provides a probabilistic prediction at a new parameter point that retains the
accuracy of local reduced models, at a computational complexity that does not
depend on system dimension, and thus is suitable for online computation. We
give four numerical examples to compare our method to subspace interpolation,
as well as two methods that interpolate local reduced models. Overall, GPS is
the most data efficient, more computationally efficient than subspace
interpolation, and gives smooth predictions with uncertainty quantification.
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