Using ℓ1 Regularization to Improve Numerical Partial Differential Equation Solvers

Theresa Scarnati, Anne Gelb, Rodrigo Platte

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

Abstract

Sparse regularization plays a central role in many recent developments in imaging and other related fields. However, it is still of limited use in numerical solvers for partial differential equations (PDEs). In this paper we investigate the use of ℓ1 regularization to promote sparsity in the shock locations of hyperbolic PDEs. We develop an algorithm that uses a high order sparsifying transform which enables us to effectively resolve shocks while still maintaining stability. Our method does not require a shock tracking procedure nor any prior information about the number of shock locations. It is efficiently implemented using the alternating direction method of multipliers. We present our results on one and two dimensional examples using both finite difference and spectral methods as underlying PDE solvers.

Original languageEnglish (US)
Pages (from-to)225-252
Number of pages28
JournalJournal of Scientific Computing
Volume75
Issue number1
DOIs
StatePublished - Apr 1 2018

Keywords

  • Alternating direction method of multipliers
  • Image denoising
  • Numerical conservation laws
  • ℓ regularization

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science
  • Numerical Analysis
  • Engineering(all)
  • Computational Theory and Mathematics
  • Computational Mathematics
  • Applied Mathematics

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