Publication: Using automatic differentiation for compressive sensing in uncertainty quantification
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Using automatic differentiation for compressive sensing in uncertainty quantification

- Article in a journal -
 

Author(s)
Mu Wang , Guang Lin , Alex Pothen

Published in
Special issue of Optimization Methods & Software: Advances in Algorithmic Differentiation Optimization Methods & Software

Editor(s)
Bruce Christianson, Shaun A. Forth, Andreas Griewank

Year
2018

Publisher
Taylor & Francis

Abstract
This paper employs automatic differentiation (ad) in the compressive sensing-based generalized polynomial chaos (gPC) expansion, which computes a sparse approximation of the Quantity of Interest (QoI) using orthogonal polynomials as basis functions. An earlier approach without ad relies on an iterative procedure to refine the solution by approximating the gradient of the QoI. With ad, the gradient can be accurately evaluated, and a set of basis functions of the gPC expansion associated with new random variables can be efficiently identified. The computational complexity of the algorithm using ad is independent of the number of basis functions, whereas an earlier algorithm had complexity proportional to the square of this number. Our test problems include synthetic problems and a high-dimensional stochastic partial differential equation. With the new basis, the coefficient vector in the gPC expansion is sparser than the original basis. We demonstrate that introducing ad can greatly improve the performance by computing solutions 2 to 10 times faster than an earlier approach. The accuracy of the gPC expansion is also improved; sparse gpC expansions are obtained without iterative refinement, even for high dimensions when an earlier approach fails.

Cross-References
Christianson2018Sio

BibTeX
@ARTICLE{
         Wang2018Uad,
       crossref = "Christianson2018Sio",
       author = "Mu Wang and Guang Lin and Alex Pothen",
       title = "Using automatic differentiation for compressive sensing in uncertainty
         quantification",
       journal = "Optimization Methods \& Software",
       volume = "33",
       number = "4--6",
       pages = "799--812",
       year = "2018",
       publisher = "Taylor \& Francis",
       doi = "10.1080/10556788.2017.1359267",
       url = "https://doi.org/10.1080/10556788.2017.1359267",
       eprint = "https://doi.org/10.1080/10556788.2017.1359267",
       abstract = "This paper employs automatic differentiation (AD) in the compressive sensing-based
         generalized polynomial chaos (gPC) expansion, which computes a sparse approximation of the Quantity
         of Interest (QoI) using orthogonal polynomials as basis functions. An earlier approach without AD
         relies on an iterative procedure to refine the solution by approximating the gradient of the QoI.
         With AD, the gradient can be accurately evaluated, and a set of basis functions of the gPC expansion
         associated with new random variables can be efficiently identified. The computational complexity of
         the algorithm using AD is independent of the number of basis functions, whereas an earlier algorithm
         had complexity proportional to the square of this number. Our test problems include synthetic
         problems and a high-dimensional stochastic partial differential equation. With the new basis, the
         coefficient vector in the gPC expansion is sparser than the original basis. We demonstrate that
         introducing AD can greatly improve the performance by computing solutions 2 to 10 times faster than
         an earlier approach. The accuracy of the gPC expansion is also improved; sparse gpC expansions are
         obtained without iterative refinement, even for high dimensions when an earlier approach fails.",
       booktitle = "Special issue of Optimization Methods \& Software: Advances in
         Algorithmic Differentiation",
       editor = "Bruce Christianson and Shaun A. Forth and Andreas Griewank"
}


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