Optimization of PDEs with Uncertain Inputs

  • Drew P. KouriEmail author
  • Alexander Shapiro
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 163)


Uncertainty pervades nearly all science and engineering applications including the optimal control and design of systems governed by partial differential equations (PDEs). In many applications, it is critical to determine optimal solutions that are resilient to the inherent uncertainty in unknown boundary conditions, inaccurate coefficients, and unverifiable modeling assumptions. In this tutorial, we develop a general theory for PDE-constrained optimization problems in which inputs or coefficients of the PDE are uncertain. We discuss numerous approaches for incorporating risk preference and conservativeness into the optimization problem formulation, motivated by concrete engineering applications. We conclude with a discussion of nonintrusive solution methods and numerical examples.



This work was supported by DARPA EQUiPS grant SNL 014150709.

Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia LLC, a wholly owned subsidiary of Honeywell International Inc. for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government.


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© National Technology & Engineering Solutions of Sandia, LLC. Under the terms of Contract DE-NA0003525, there is a non-exclusive license for use of this work by or on behalf of the U.S. Government 2018

Authors and Affiliations

  1. 1.Center for Computing ResearchSandia National LaboratoriesAlbuquerqueUSA
  2. 2.School of Industrial and Systems EngineeringGeorgia Institute of TechnologyAtlantaUSA

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