Efficient multiscale methods for optimal in situ bioremediation design

Yong Liu, Barbara S. Minsker

Research output: Contribution to journalArticlepeer-review


This paper presents a multiscale derivative method for solving a successive approximation linear quadratic regulator model for optimal in situ bioremediation design. An efficient one-sided forward divided difference numerical derivatives calculation was implemented as the first stage of the method, which only required assembling the right-hand-side vector of the linear systems of equations of the simulation model and performing backward substitution. The derivative calculation was reduced from O(N3) to nearly O(N2), where N is the number of non-Dirichlet state variables. A V-cycle multiscale derivatives approximation was implemented as the second stage, which used coarser mesh derivatives to interpolate finer mesh derivatives. Implementing the numerical derivatives method in a case study with over 1,600 state variables caused a reduction of more than two-thirds in computing time over the previous analytical derivatives method without loss of accuracy. Using the V-cycle multiscale derivatives approximation further reduced computing time by 29%, resulting in an overall 77% reduction compared to the previous analytical derivatives method. The reduction will be even greater for applications with more state variables, enabling the solution of much larger-scale problems than was previously possible.

Original languageEnglish (US)
Pages (from-to)227-236
Number of pages10
JournalJournal of Water Resources Planning and Management
Issue number3
StatePublished - May 2002
Externally publishedYes


  • Biological treatment
  • Groundwater management
  • Optimization
  • Remedial action

ASJC Scopus subject areas

  • Civil and Structural Engineering
  • Geography, Planning and Development
  • Water Science and Technology
  • Management, Monitoring, Policy and Law


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