A Distributed Active Subspace Method for Scalable Surrogate Modeling of Function Valued Outputs
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A Distributed Active Subspace Method for Scalable Surrogate Modeling of Function Valued Outputs Hayley Guy1 · Alen Alexanderian1 · Meilin Yu2 Received: 7 August 2019 / Revised: 8 October 2020 / Accepted: 12 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We present a distributed active subspace method for training surrogate models of complex physical processes with high-dimensional inputs and function valued outputs. Specifically, we represent the model output with a truncated Karhunen–Loève (KL) expansion, screen the structure of the input space with respect to each KL mode via the active subspace method, and finally form an overall surrogate model of the output by combining surrogates of individual output KL modes. To ensure scalable computation of the gradients of the output KL modes, needed in active subspace discovery, we rely on adjoint-based gradient computation. The proposed method combines benefits of active subspace methods for input dimension reduction and KL expansions used for spectral representation of the output field. We provide a mathematical framework for the proposed method and conduct an error analysis of the mixed KL active subspace approach. Specifically, we provide an error estimate that quantifies errors due to active subspace projection and truncated KL expansion of the output. We demonstrate the numerical performance of the surrogate modeling approach with an application example from biotransport. Keywords Distributed active subspace · Karhunen–Loève expansion · Dimension reduction · Function valued outputs · Porous medium flow · Biotransport
1 Introduction Models with uncertain input parameters are common in modeling of complex systems. Computational studies such as forward uncertainty propagation, optimization, or parame-
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Meilin Yu [email protected] Hayley Guy [email protected] Alen Alexanderian [email protected]
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Department of Mathematics, North Carolina State University, Raleigh, USA
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Department of Mechanical Engineering, The University of Maryland, Baltimore County, Baltimore, USA 0123456789().: V,-vol
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Journal of Scientific Computing
(2020) 85:36
ter estimation require repeated evaluation of the model. These tasks become challenging for expensive-to-evaluate complex models. To address this challenge, surrogate models are often used. By approximating the mapping from the uncertain input parameters to output quantities of interest (QoIs), using a surrogate model, one can replace expensive model evaluations by inexpensive surrogate model evaluations. Examples of surrogate modeling tools include polynomial chaos expansion [24,33], multivariate adaptive regression splines [21], and Gaussian processes [42]. In the present work, we consider surrogate construction for models of the form y = f (x, ξ ),
(1)
where x belongs to a spatial domain and ξ ∈ R N p is a vector of uncertain parameters. In our target applications f is defined in terms of the solution of a partial differential equation (PDE) that is parameteriz
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