A nonlinear pde-based method for sparse deconvolution

Yu Mao, Bin Dong, Stanley Osher

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


In this paper, we introduce a new nonlinear evolution partial differential equation (PDE) for sparse deconvolution problems. The proposed PDE has the form of a continuity equation that arises in various research areas, e.g., fluid dynamics and optimal transportation, and thus has some interesting physical and geometric interpretations. The underlying optimization model that we consider is the standard ℓ1 minimization with linear equality constraints, i.e., minu{∥u ∥1 : Au = f}, with A being an undersampled convolution operator. We show that our PDE preserves the ℓ1 norm while lowering the residual ∥Au-f ∥2. More importantly the solution of the PDE becomes sparser asymptotically, which is illustrated numerically. Therefore, it can be treated as a natural and helpful plug-in to some algorithms for ℓ1 minimization problems, e.g., Bregman iterative methods introduced for sparse reconstruction problems in [W. Yin, S. Osher, D. Goldfarb, and J. Darbon, SIAM J. Imaging Sci., 1 (2008), pp. 143-168]. Numerical experiments show great improvements in terms of both convergence speed and reconstruction quality. copyright

Original languageEnglish (US)
Pages (from-to)965-976
Number of pages12
JournalMultiscale Modeling and Simulation
Issue number3
StatePublished - 2010


  • Iterative regularization
  • Sparse deconvolution
  • Transport equation
  • ℓ1 minimization

ASJC Scopus subject areas

  • Chemistry(all)
  • Modeling and Simulation
  • Ecological Modeling
  • Physics and Astronomy(all)
  • Computer Science Applications


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