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Iterative Sampled Methods for Massive and Separable Nonlinear Inverse Problems

  • Julianne ChungEmail author
  • Matthias Chung
  • J. Tanner Slagel
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11603)

Abstract

In this paper, we consider iterative methods based on sampling for computing solutions to separable nonlinear inverse problems where the entire dataset cannot be accessed or is not available all-at-once. In such scenarios (e.g., when massive amounts of data exceed memory capabilities or when data is being streamed), solving inverse problems, especially nonlinear ones, can be very challenging. We focus on separable nonlinear problems, where the objective function is nonlinear in one (typically small) set of parameters and linear in another (larger) set of parameters. For the linear problem, we describe a limited-memory sampled Tikhonov method, and for the nonlinear problem, we describe an approach to integrate the limited-memory sampled Tikhonov method within a nonlinear optimization framework. The proposed method is computationally efficient in that it only uses available data at any iteration to update both sets of parameters. Numerical experiments applied to massive super-resolution image reconstruction problems show the power of these methods.

Keywords

Tikhonov regularization Sampled methods Variable projection Kaczmarz methods Super-resolution Medical imaging and other applications 

Notes

Acknowledgements

We gratefully acknowledge support by the National Science Foundation under grants NSF DMS 1723005 and NSF DMS 1654175.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Julianne Chung
    • 1
    Email author
  • Matthias Chung
    • 1
  • J. Tanner Slagel
    • 1
  1. 1.Department of MathematicsVirginia TechBlacksburgUSA

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