Library Subscription: Guest
Begell Digital Portal Begell Digital Library eBooks Journals References & Proceedings Research Collections
International Journal for Uncertainty Quantification
IF: 4.911 5-Year IF: 3.179 SJR: 1.008 SNIP: 0.983 CiteScore™: 5.2

ISSN Print: 2152-5080
ISSN Online: 2152-5099

Open Access

International Journal for Uncertainty Quantification

DOI: 10.1615/Int.J.UncertaintyQuantification.2020032480
pages 225-248


Joseph Hart
Optimization and Uncertainty Quantification, Sandia National Laboratories, P.O. Box 5800, Albuquerque, New Mexico 87123-1320, USA
Bart van Bloemen Waanders
Optimization and Uncertainty Quantification, Sandia National Laboratories, P.O. Box 5800, Albuquerque, New Mexico 87123-1320, USA
Roland Herzog
Technical University Chemnitz, Faculty of Mathematics, 09107 Chemnitz, Germany


Many problems in engineering and sciences require the solution of large scale optimization constrained by partial differential equations (PDEs). Though PDE-constrained optimization is itself challenging, most applications pose additional complexity, namely, uncertain parameters in the PDEs. Uncertainty quantification (UQ) is necessary to characterize, prioritize, and study the influence of these uncertain parameters. Sensitivity analysis, a classical tool in UQ, is frequently used to study the sensitivity of a model to uncertain parameters. In this article, we introduce "hyperdifferential sensitivity analysis" which considers the sensitivity of the solution of a PDE-constrained optimization problem to uncertain parameters. Our approach is a goal-oriented analysis which may be viewed as a tool to complement other UQ methods in the service of decision making and robust design. We formally define hyperdifferential sensitivity indices and highlight their relationship to the existing optimization and sensitivity analysis literatures. Assuming the presence of low rank structure in the parameter space, computational efficiency is achieved by leveraging a generalized singular value decomposition in conjunction with a randomized solver which converts the computational bottleneck of the algorithm into an embarrassingly parallel loop. Two multiphysics examples, consisting of nonlinear steady state control and transient linear inversion, demonstrate efficient identification of the uncertain parameters which have the greatest influence on the optimal solution.


  1. Vogel, C.R., Sparse Matrix Computations Arising in Distributed Parameter Identification, SIAM J. Matrix Anal. Appl., 20:1027-1037, 1999.

  2. Ascher, U.M. and Haber, E., Grid Refinement and Scaling for Distributed Parameter Estimation Problems, Inverse Probl., 17:571-590,2001.

  3. Haber, E. and Ascher, U.M., Preconditioned All-At-Once Methods for Large, Sparse Parameter Estimation Problems, Inverse Probl, 17:1847-1864, 2001.

  4. Vogel, C.R., Computational Methods for Inverse Problems, SIAM Frontiers in Applied Mathematics Series, Philadelphia: SIAM, 2002.

  5. Biegler, L.T., Ghattas, O., Heinkenschloss, M., and van Bloemen Waanders, B., Large-Scale PDE-Constrained Optimization, Vol. 30, Berlin: Springer, 2003.

  6. Biros, G. and Ghattas, O., Parallel Lagrange-Newton-Krylov-Schur Methods for PDE-Constrained Optimization. Parts I-II, SIAM J. Sci. Comput., 27:687-738,2005.

  7. Laird, C.D., Biegler, L.T., van Bloemen Waanders, B., and Bartlett, R.A., Time Dependent Contaminant Source Determination for Municipal Water Networks Using Large Scale Optimization, ASCE J. Water Resour. Mgt. Plan., 131:125-134,2005.

  8. Hintermuller, M. and Vicente, L.N., Space Mapping for Optimal Control of Partial Differential Equations, SIAM J. Opt:., 15:1002-1025,2005.

  9. Hazra, S.B. and Schulz, V., Simultaneous Pseudo-Timestepping for Aerodynamic Shape Optimization Problems with State Constraints, SIAM J. Sci. Comput., 28:1078-1099, 2006.

  10. Biegler, L.T., Ghattas, O., Heinkenschloss, M., Keyes, D., and van Bloemen Waanders, B., Eds., Real-Time PDE-Constrained Optimization, Vol. 3, Philadelphia: SIAM, 2007.

  11. Borzi, A., High-Order Discretization and Multigrid Solution of Elliptic Nonlinear Constrained Optimal Control Problems, J. Comput. Appl. Math, 200:67-85,2007.

  12. Hinze, M., Pinnau, R., Ulbrich, M., and Ulbrich, S., Optimization with PDE Constraints, Berlin: Springer, 2009.

  13. Biegler, L., Biros, G., Ghattas, O., Heinkenschloss, M., Keyes, D., Mallick, B., Marzouk, Y., Tenorio, L., van Bloemen Waanders, B., and Willcox, K., Eds., Large-Scale Inverse Problems and Quantification of Uncertainty, New York: Wiley, 2011.

  14. Iooss, B. and Saltelli, A., Introduction to Sensitivity Analysis, Berlin: Springer, 2016.

  15. Saltelli, A., Ratto, M., Andres, T., Campolongo, F., Cariboni, J., Gatelli, D., Saisana, M., and Tarantola, S., Global Sensitivity Analysis: The Primer, New York: Wiley, 2008.

  16. Iooss, B. andLemaitre, P., A Review on Global Analysis Methods, Berlin: Springer, 2015.

  17. Borgonovo, E., A New Uncertainty Importance Measure, Reliab. Eng. Syst. Saf., 92:771-784, 2007.

  18. Kucherenko, S. and Iooss, B., Derivative-Based Global Sensitivity Measures, in Handbook of Uncertainty Quantification, R. Ghanem, D. Higdon, and H. Owhadi, Eds., Cham, Switzerland: Springer, 2016.

  19. Prieur, C. and Tarantola, S., Variance-Based Sensitivity Analysis: Theory and Estimation Algorithms, in Handbook of Uncertainty Quantification, R. Ghanem, D. Higdon, and H. Owhadi, Eds., Cham, Switzerland: Springer, 2016.

  20. Song, E., Nelson, B.L., and Staum, J., Shapley Effects for Global Sensitivity Analysis: Theory and Computation, SIAM/ASA J. Uncertain. Quantif., 4:1060-1083,2016.

  21. Brandes, K. and Griesse, R., Quantitative Stability Analysis of Optimal Solutions in PDE-Constrained Optimization, J. Com put. Appl. Math, 206:908-926, 2007.

  22. Biiskens, C. and Griesse, R., Parametric Sensitivity Analysis of Perturbed PDE Optimal Control Problems with State and Control Constraints, J. Optim. Theory Appl, 131(1):17-35, 2006.

  23. Griesse, R., Parametric Sensitivity Analysis in Optimal Control of a Reaction-Diffusion System-Part II: Practical Methods and Examples, Optim.. Methods Software, 19(2):217-242,2004.

  24. Griesse, R., Parametric Sensitivity Analysis in Optimal Control of a Reaction Diffusion System. I. Solution Differentiability, Numer. Funct. Anal. Optim, 25(1-2):93-117,2004.

  25. Griesse, R., Stability and Sensitivity Analysis in Optimal Control of Partial Differential Equations, Habilitation Thesis, Faculty of Natural Sciences, Karl-Franzens University, 2007.

  26. Griesse, R. and Vexler, B., Numerical Sensitivity Analysis for the Quantity of Interest in PDE-Constrained Optimization, SIAMJ. Sci. Comput., 29(1):22-48, 2007.

  27. Griesse, R. and Volkwein, S., Parametric Sensitivity Analysis for Optimal Boundary Control of a 3D Reaction-Difusion System, in Large-Scale Nonlinear Optimization, G.D. Pillo and M. Roma, Eds., Berlin: Springer, 2006.

  28. Griesse, R. and Walther, A., Parametric Sensitivities for Optimal Control Problems Using Automatic Differentiation, Opt. Control Appl. Methods, 24:297-314, 2003.

  29. Saibaba, A.K., Lee, J., and Kitanidis, P.K., Randomized Algorithms for Generalized Hermitian Eigenvalue Problems with Application to Computing Karhunen-Loeve Expansion, Numer. Linear Algebra Appl., 23:314-339,2016.

  30. Murthy, P.R., Operations Research, 2nd ed., New Delhi: New Age International Publishers, 2007.

  31. Bonnans, J.F. and Shapiro, A., Optimization Problems with Perturbations: A Guided Tour, SIAMRev., 40(2):228-264, 1998.

  32. Rakovec, O., Hill, M.C., Clark, M.P., Weerts, A.H., Teuling, A.J., and Uijlenhoet, R., Distributed Evaluation of Local Sensitivity Analysis (DELSA), with Application to Hydrologic Models, Water Resour. Res., 50:409-426, 2014.

  33. Sobol', I. and Kucherenko, S., Derivative based Global Sensitivity Measures and the Link with Global Sensitivity Indices, Math. Comput. Simul., 79:3009-3017,2009.

  34. Sobol', I. and Kucherenko, S., A New Derivative based Importance Criterion for Groups of Variables and Its Link with the Global Sensitivity Indices, Comput. Phys. Commun., 181:1212-1217, 2010.

  35. Morris, M., Factorial Sampling Plans for Preliminary Computational Experiments, Technometrics, 33:161-174, 1991.

  36. Constantine, P.G., Active Subspaces: Emerging Ideas for Dimension Reduction in Parameter Studies, Philadelphia: SIAM, 2015.

  37. Constantine, P.G. and Diaz, P., Global Sensitivity Metrics from Active Subspaces, Reliab. Eng. Syst. Saf., 162:1-13, 2017.

  38. Fassbender, H. and Kressner, D., Structured Eigenvalue Problems, GAMM-Mitteilungen, 29(2):297-318, 2006.

  39. Halko, N., Martinsson, P.G., and Tropp, J.A., Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions, SIAMRev., 53(2):217-288, 2011.

  40. Ito, K. and Ravindran, S.S., Optimal Control of Thermally Convected Fluid Flows, SIAMJ. Sci. Comput., 19:1847-1869, 1998.

  41. Kouri, D.P. and Ridzal, D., Inexact Trust-Region Methods for PDE-Constrained Optimization, New York: Springer, pp. 83-121,2018.

  42. Kouri, D.P., von Winckel, G., and Ridzal, D., Rapid Optimization Library, Albuquerque, NM: Sandia National Lab (SNL-NM), Rep. No. SAND2017-12025PE, 2017.

  43. Heroux, M., Bartlett, R., Howle, V., Hoekstra, R., Hu, J., Kolda, T., Lehoucq, R., Long, K., Pawlowski, R., Phipps, E., Salinger, A., Thornquist, H., Tuminaro, R., Willenbring, J., and Williams, A., An Overview of Trilinos, Albuquerque, NM: Sandia National Laboratories, Tech. Rep. SAND2003-2927, 2003.

Articles with similar content:

Journal of Machine Learning for Modeling and Computing, Vol.1, 2020, issue 1
Maha Youssef, Roland Pulch
Approach to the Study of Global Asymptotic Stability of Lattice Differential Equations with Delay for Modeling of Immunosensors
Journal of Automation and Information Sciences, Vol.51, 2019, issue 2
Andrey S. Sverstiuk , Igor Ye. Andrushchak, Vasiliy P. Martsenyuk
International Heat Transfer Conference 11, Vol.16, 1998, issue
P. Floquet, S. Domenech, L. Pibouleau , G. Athier
Information Technologies of Visual Control. Part 2. Mathematical Models. Synthesis of Algorithms
Journal of Automation and Information Sciences, Vol.35, 2003, issue 2
Mikhail V. Artyushenko, Dmitriy V. Lebedev, Oleg V. Nikitenko
Laurent David, Romain Leroux, Ludovic Chatellier