Certified Rapid Solution of Partial Differential Equations for Real-Time Parameter Estimation and Optimization

Martin Grepl, Ngoc C. Nguyen, Karen Veroy, Anthony T. Patera, Gui R. Liu

Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademicpeer-review


Engineering analysis requires the prediction of selected “outputs” s relevant to ultimate component and system performance; typical outputs include critical stresses or strains, flow rates or pressure drops, and various measures of temperature and heat flux. These outputs are functions of “inputs” μ that serve to identify a particular configuration of the component or system; typical inputs reflect geometry, properties, and boundary conditions and loads.

In many cases, the input-output function is best articulated as a (say) linear functional ℓ of a field variable u(μ) that is the solution to an input-parameterized partial differential equation (PDE); typical field variables and associated PDEs include temperature and steady/unsteady conduction, displacement and equilibrium elasticity/Helmholtz, and velocity and steady incompressible Navier—Stokes. System behavior is thus described by an input-output relation s(μ) = ℓ(u(μ)), the evaluation of which requires solution of the underlying PDE.

Our focus is on “deployed” systems—components or processes in operation in the field—and associated “Assess-Act” scenarios. In the Assess stage we pursue robust parameter estimation (inverse) procedures that map measured-observable outputs to (all) possible system-characteristic and environment-state inputs. In the subsequent Act stage we then pursue adaptive design (optimization) procedures that map mission-objective outputs to best control-variable inputs.
Original languageEnglish
Title of host publicationReal-Time PDE-Constrained Optimization
EditorsLorentz T. Biegler , Omar Ghattas, Matthias Heinkenschloss, David Keyes, Bart van Bloemen Waanders
Place of PublicationPhiladelphia
PublisherSociety for Industrial and Applied Mathematics (SIAM)
Number of pages18
ISBN (Electronic)9780898718935
ISBN (Print)9780898716214
Publication statusPublished - 2007
Externally publishedYes

Publication series

NameComputational Science & Engineering
PublisherSociety for Industrial and Applied Mathematics


  • optimal control
  • parameter estimation
  • real-time optimization
  • partial differential equations
  • science and engineering applications

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