A Distributed Active Subspace Method for Scalable Surrogate Modeling of Function Valued Outputs

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.

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Proofs

Proofs

Proof of Theorem 1

Note that,

$$\begin{aligned} e(\varvec{x},\varvec{\xi }) =\bigg | \sum _{k=1}^N (F_k(\varvec{\xi }) -{G}_k(\varvec{W}_{k,1}^T\varvec{\xi }))\phi _k(\varvec{x}) \bigg | \le \sum _{k=1}^N |F_k(\varvec{\xi }) -{G}_k(\varvec{W}_{k,1}^T\varvec{\xi })||\phi _k(\varvec{x})|. \end{aligned}$$

Then,

$$\begin{aligned} {\mathbb {E}}\{{\bar{e}}\}&= {\mathbb {E}} \bigg \{\int _{\mathcal {X}}\sum _{k=1}^N |F_k(\varvec{\xi }) -{G}_k(\varvec{W}_{k,1}^T\varvec{\xi })||\phi _k(\varvec{x})| d\varvec{x} \bigg \}\\&= \sum _{k=1}^N \bigg (\int _{\mathcal {X}}|\phi _k(\varvec{x})| d\varvec{x} \bigg ) {\mathbb {E}}\{|F_k(\varvec{\xi }) -{G}_k(\varvec{W}_{k,1}^T\varvec{\xi })| \} \\&\le \displaystyle \sum _{k=1}^N \left[ \int _{\mathcal {X}}|\phi _k(\varvec{x})|^2 d\varvec{x}\right] ^\frac{1}{2}\left[ \int _{\mathcal {X}}1^2 d\varvec{x}\right] ^\frac{1}{2} {\mathbb {E}}\{|F_k(\varvec{\xi }) -{G}_k(\varvec{W}_{k,1}^T\varvec{\xi })|^2\}^{1/2} \le |{\mathcal {X}}|^\frac{1}{2} \displaystyle \sum _{k=1}^N \delta _k \end{aligned}$$

where we have used Cauchy–Schwarz inequlity and Lemma 2(a). \(\square \)

Proof of Theorem 2

We have

$$\begin{aligned} {\bar{E}}(\varvec{\xi }) = \int _{\mathcal {X}}E(\varvec{x},\varvec{\xi }) d\varvec{x}&= \int _{\mathcal {X}}|f(\varvec{x},\varvec{\xi })-f_N(\varvec{x},\varvec{\xi })+ f_N(\varvec{x},\varvec{\xi })-{\hat{f}}_N(\varvec{x},\varvec{\xi })| \\&\le \int _{\mathcal {X}}|f(\varvec{x},\varvec{\xi })-f_N(\varvec{x},\varvec{\xi })|d\varvec{x} + \int _{\mathcal {X}}|f_N(\varvec{x},\varvec{\xi })-{\hat{f}}_N(\varvec{x},\varvec{\xi })|d\varvec{x} \\&\le \bigg \{\int _{\mathcal {X}}|f(\varvec{x},\varvec{\xi })-f_N(\varvec{x},\varvec{\xi })|^2 d\varvec{x}\bigg \}^{1/2}|{\mathcal {X}}|^\frac{1}{2} + {\bar{e}}(\varvec{\xi }) \\&= \bigg \{ \sum _{k=N+1}^\infty F_k(\varvec{\xi })^2 \bigg \}^{1/2}|{\mathcal {X}}|^\frac{1}{2} + {\bar{e}}(\varvec{\xi }). \end{aligned}$$

Now we consider

$$\begin{aligned} {\mathbb {E}}\{{\bar{E}}(\varvec{\xi })\}&\le {\mathbb {E}}\Bigg (\bigg \{ \sum _{k=N+1}^\infty F_k(\varvec{\xi })^2 \bigg \}^{1/2}\Bigg )|{\mathcal {X}}|^\frac{1}{2} + {\mathbb {E}}\{{\bar{e}}(\varvec{\xi })\} \\&\le |{\mathcal {X}}|^\frac{1}{2} \Bigg [{\mathbb {E}}\Bigg (\bigg \{\sum _{k=N+1}^\infty F_k(\varvec{\xi })^2\bigg \}^{1/2}\Bigg ) +\sum _{k=1}^N \delta _k \Bigg ] \le |{\mathcal {X}}|^\frac{1}{2}\Bigg [{\mathbb {E}}\Bigg (\sum _{k=N+1}^\infty F_k(\varvec{\xi })^2\Bigg )^{1/2}+\sum _{k=1}^N \delta _k \Bigg ]. \end{aligned}$$

Note that \({\mathbb {E}}\Bigg (\sum _{k=N+1}^\infty F_k(\varvec{\xi })^2\Bigg ) = \sum _{k=N+1}^\infty {\mathbb {E}}\big (F_k(\varvec{\xi })^2\big ) = \sum _{k=N+1}^\infty \lambda _k(C_f)\). Therefore, we have our desired result

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\{{\bar{E}}(\varvec{\xi })\}&\le |{\mathcal {X}}|^\frac{1}{2}\Bigg [\Bigg (\sum _{k=N+1}^\infty \lambda _k(C_f)\Bigg )^{1/2} + \sum _{k=1}^N \delta _k\Bigg ]\\&= |{\mathcal {X}}|^\frac{1}{2}\Bigg [\Bigg (\sum _{k=N+1}^\infty \lambda _k(C_f)\Bigg )^{1/2} + \sum _{k=1}^N \left( \sum _{j = r_k+1}^{N_p}\lambda _j(\mathbf {{S}}_k)\right) ^{1/2}\Bigg ]. \end{aligned}. \end{aligned}$$

\(\square \)

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Guy, H., Alexanderian, A. & Yu, M. A Distributed Active Subspace Method for Scalable Surrogate Modeling of Function Valued Outputs. J Sci Comput 85, 36 (2020). https://doi.org/10.1007/s10915-020-01346-2

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Keywords

  • Distributed active subspace
  • Karhunen–Loève expansion
  • Dimension reduction
  • Function valued outputs
  • Porous medium flow
  • Biotransport