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.


  1. 1.
    R. A. Adams. Sobolev Spaces. Academic Press, New York, 1975.zbMATHGoogle Scholar
  2. 2.
    E. Andreassen, B. S. Lazarov, and O. Sigmund. Design of manufacturable 3d extremal elastic microstructure. Mechanics of Materials, 69(1):1–10, 2014.Google Scholar
  3. 3.
    V. Artus, J. L. Durlofsky, J. Onwunalu, and K. Aziz. Optimization of nonconventional wells under uncertainty using statistical proxies. Computational Geosciences, 10(4):389–404, 2006.zbMATHGoogle Scholar
  4. 4.
    Ph. Artzner, F. Delbaen, J.-M. Eber, and D. Heath. Coherent measures of risk. Math. Finance, 9(3):203–228, 1999.MathSciNetzbMATHGoogle Scholar
  5. 5.
    A. Asadpoure, M. Tootkaboni, and J. K. Guest. Robust topology optimization of structures with uncertainties in stiffness – applications to trust structures. Computers & Structures, 89(11–12):1131–1141, 2011.Google Scholar
  6. 6.
    H. Attouch, G. Buttazzo, and G. Michaille. Variational analysis in Sobolev and BV spaces, volume 6 of MPS/SIAM Series on Optimization. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2006.Google Scholar
  7. 7.
    I. Babuška, F. Nobile, and R. Tempone. A stochastic collocation method for elliptic partial differential equations with random input data. SIAM Rev., 52(2):317–355, 2010.MathSciNetzbMATHGoogle Scholar
  8. 8.
    I. Babuška, R. Tempone, and G. E. Zouraris. Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal., 42(2):800–825 (electronic), 2004.MathSciNetzbMATHGoogle Scholar
  9. 9.
    I. Babuška, R. Tempone, and G. E. Zouraris. Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation. Comput. Methods Appl. Mech. Engrg., 194(12–16):1251–1294, 2005.MathSciNetzbMATHGoogle Scholar
  10. 10.
    W. Bangerth, H. Klie, M. F. Wheeler, P. L. Stoffa, and M. K. Sen. On optimization algorithms for the reservoir oil well placement problem. Computational Geosciences, 10(3):303–319, 2006.zbMATHGoogle Scholar
  11. 11.
    K. Barty, J.-S. Roy, and C. Strugarek. Hilbert-valued perturbed subgradient algorithms. Mathematics of Operations Research, 32(3):551–562, 2007.MathSciNetzbMATHGoogle Scholar
  12. 12.
    H. H. Bauschke and P. L. Combettes. Convex Analysis and Montone Operator Theory in Hilbert Space. CMS Books in Mathematics. Springer New York, 2011.zbMATHGoogle Scholar
  13. 13.
    R. E. Bellman. Dynamic Programming. Princeton University Press, Princeton, NJ, 1957.zbMATHGoogle Scholar
  14. 14.
    A. Ben-Tal, L. E. Ghaoui, and A. Nemirovski. Robust Optimization. Princeton Series in Applied Mathematics. Princeton University Press, 2009.zbMATHGoogle Scholar
  15. 15.
    A. Ben-Tal, D. Den Hertog, A. De Waegenaere, B. Melenberg, and G. Rennen. Robust solutions of optimization problems affected by uncertain probabilities. Management Science, 59(2):341–357, 2013.Google Scholar
  16. 16.
    A. Ben-Tal and M. Teboulle. Penalty functions and duality in stochastic programming via phi-divergence functionals. Mathematics of Operations Research, 12:224–240, 1987.MathSciNetzbMATHGoogle Scholar
  17. 17.
    A. Ben-Tal and M. Teboulle. An old-new concept of convex risk measures: The optimized certainty equivalent. Mathematical Finance, 17(3):449–476, 2007.MathSciNetzbMATHGoogle Scholar
  18. 18.
    J. O. Berger. The robust Bayesian viewpoint (with discussion). Robustness of Bayesian Analysis, pages 63–124, 1985.Google Scholar
  19. 19.
    J. O. Berger. Statistical Decision Theory and Bayesian Analysis. Springer Series in Statistics. Springer, 1985.zbMATHGoogle Scholar
  20. 20.
    J. O. Berger. An overview of robust Bayesian analysis. Test, 3(1):5–124, 1994.MathSciNetzbMATHGoogle Scholar
  21. 21.
    J. G. Berryman and G. W. Milton. Microgeometry of random composites and porous media. Journal of Physics D: Applied Physics, 21(1):87, 1988.Google Scholar
  22. 22.
    D. P. Bertsekas. Constrained Optimization and Lagrange Multiplier Methods. Academic Press, New York,. London, Paris, San Diego, San Francisco, 1982.Google Scholar
  23. 23.
    D. Bertsimas, D. B. Brown, and C. Caramanis. Theory and applications of robust optimization. SIAM Review, 53(3):464–501, 2011.MathSciNetzbMATHGoogle Scholar
  24. 24.
    D. Bertsimas and J. Sethuraman. Moment problems and semidefinite optimization. In H. Wolkowicz, R. Saigal, and L. Vandenberghe, editors, Handbook of Semidefinite Programming, pages 469–510. Kluwer Academic Publishers, 2000.Google Scholar
  25. 25.
    J. R. Birge and F. Louveaux. Introduction to stochastic programming. Springer-Verlag, New York, 1997.zbMATHGoogle Scholar
  26. 26.
    J. F. Bonnans and A. Shapiro. Perturbation Analysis of Optimization Problems. Springer Verlag, Berlin, Heidelberg, New York, 2000.zbMATHGoogle Scholar
  27. 27.
    A. Borzì and G. von Winckel. A POD framework to determine robust controls in PDE optimization. Comput. Vis. Sci., 14:91–103, 2011.MathSciNetzbMATHGoogle Scholar
  28. 28.
    S. C. Brenner and L. R. Scott. The Mathematical Theory of Finite Element Methods. Springer Verlag, Berlin, Heidelberg, New York, second edition, 2002.zbMATHGoogle Scholar
  29. 29.
    P. Cheridito and T. Li. Risk measures on Orlicz hearts. Mathematical Finance, 19(2):189–214, 2009.MathSciNetzbMATHGoogle Scholar
  30. 30.
    F. H. Clarke. Nonsmooth Analysis and Control Theory. Graduate Texts in Mathematics. Springer, 1998.zbMATHGoogle Scholar
  31. 31.
    A. Cohen, R. DeVore, and C. Schwab. Convergence rates of best n-term Galerkin approximations for a class of elliptic sPDEs. Foundations of Computational Mathematics, 10(6):615–646, 2010.MathSciNetzbMATHGoogle Scholar
  32. 32.
    A. R. Conn, N. I. M. Gould, and Ph. L. Toint. Trust–Region Methods. SIAM, Philadelphia, 2000.zbMATHGoogle Scholar
  33. 33.
    J. B. Conway. A Course in Functional Analysis. Graduate Texts in Mathematics. Springer New York, 1985.zbMATHGoogle Scholar
  34. 34.
    I. Csiszár. Eine informationstheoretische ungleichung und ihre anwendung auf den beweis der ergodizitat von markoffschen ketten. Magyar. Tud. Akad. Mat. Kutato Int. Kozls, 8, 1063.Google Scholar
  35. 35.
    A. Defant and K. Floret. Tensor Norms and Operator Ideals. North-Holland Mathematics Studies. Elsevier Science, 1993.zbMATHGoogle Scholar
  36. 36.
    E. Delage and Y. Ye. Distributionally robust optimization under moment uncertainty with application to data-driven problems. Operations Research, 58:595–6127, 2010.MathSciNetzbMATHGoogle Scholar
  37. 37.
    D. Dentcheva, S. Penev, and A. Ruszczyński. Kusuoka representation of higher order dual risk measures. Annals of Operations Research, 181(1):325–335, 2010.MathSciNetzbMATHGoogle Scholar
  38. 38.
    D. Dentcheva and A. Ruszczyński. Optimization with stochastic dominance constraints. SIAM Journal on Optimization, 14(2):548–566, 2003.MathSciNetzbMATHGoogle Scholar
  39. 39.
    I. T. Dimov. Monte Carlo methods for applied scientists. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008.zbMATHGoogle Scholar
  40. 40.
    O. Dorn and R. Villegas. History matching of petroleum reservoirs using a level set technique. Inverse Problems, 24(3):035015, 2008.MathSciNetzbMATHGoogle Scholar
  41. 41.
    J. Dupačová. Uncertainties in minimax stochastic programs. Optimization, 60(10–11):1235–1250, 2011.MathSciNetzbMATHGoogle Scholar
  42. 42.
    J. Eckstein and D. P. Bertsekas. On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators. Mathematical Programming, 55(1):293–318, Apr 1992.MathSciNetzbMATHGoogle Scholar
  43. 43.
    Y. M. Ermoliev and A. A. Gaivoronski. Stochastic methods for solving minimax problems. Cybernetics, 19(4):550–559, 1983.MathSciNetGoogle Scholar
  44. 44.
    Y. M. Ermoliev, A. A. Gaivoronski, and C. Nedeva. Stochastic optimization problems with incomplete information on distribution functions. SIAM Journal on Control and Optimization, 23(5):697–716, 1985.MathSciNetzbMATHGoogle Scholar
  45. 45.
    G. B. Folland. Real analysis. Modern techniques and their applications. Pure and Applied Mathematics (New York). John Wiley & Sons Inc., New York, second edition, 1999.Google Scholar
  46. 46.
    A. A. Gaivoronski. A numerical method for solving stochastic programming problems with moment constraints on a distribution function. Annals of Operations Research, 31(1):347–369, 1991.MathSciNetGoogle Scholar
  47. 47.
    S. Garreis and M. Ulbrich. Constrained optimization with low-rank tensors and applications to parametric problems with PDEs. SIAM Journal on Scientific Computing, 39(1):A25–A54, 2017.MathSciNetzbMATHGoogle Scholar
  48. 48.
    T. Gerstner and M. Griebel. Numerical integration using sparse grids. Numer. Algorithms, 18(3–4):209–232, 1998.MathSciNetzbMATHGoogle Scholar
  49. 49.
    T. Gerstner and M. Griebel. Dimension-adaptive tensor-product quadrature. Computing, 71(1):65–87, 2003.MathSciNetzbMATHGoogle Scholar
  50. 50.
    M. Grigoriu. Reduced order models for random functions. application to stochastic problems. Applied Mathematical Modelling, 33(1):161–175, 2009.MathSciNetzbMATHGoogle Scholar
  51. 51.
    M. Grigoriu. A method for solving stochastic equations by reduced order models and local approximations. Journal of Computational Physics, 231(19):6495–6513, 2012.MathSciNetzbMATHGoogle Scholar
  52. 52.
    V. Hauk. Structural and Residual Stress Analysis by Nondestructive Methods: Evaluation - Application - Assessment. Elsevier Science, 1997.zbMATHGoogle Scholar
  53. 53.
    E. Hille and R. S. Phillips. Functional analysis and semi-groups. American Mathematical Society Colloquium Publications, vol. 31. American Mathematical Society, Providence, R. I., 1957. rev. ed.Google Scholar
  54. 54.
    K. Ito and K. Kunisch. Lagrange Multiplier Approach to Variational Problems and Applications. Society for Industrial and Applied Mathematics, 2008.Google Scholar
  55. 55.
    P. Kall and S. W. Wallace. Stochastic Programming. Wiley, Chichester etc., 1994.zbMATHGoogle Scholar
  56. 56.
    S. Kalpakjian and S. R. Schmid. Manufacturing Engineering and Technology. Prentice Hall, 2010.Google Scholar
  57. 57.
    K. Karhunen. Über lineare Methoden in der Wahrscheinlichkeitsrechnung. Ann. Acad. Sci. Fennicae. Ser. A. I. Math.-Phys., 1947(37):79, 1947.Google Scholar
  58. 58.
    G. E. Karniadakis, C.-H. Su, D. Xiu, D. Lucor, C. Schwab, and R. A. Todor. Generalized polynomial chaos solution for differential equations with random inputs. Technical Report 2005–01, Seminar for Applied Mathematics, ETH Zurich, Zurich, Switzerland, 2005.Google Scholar
  59. 59.
    B. Khoromskij and C. Schwab. Tensor-structured Galerkin approximation of parametric and stochastic elliptic PDEs. SIAM J. Sci. Comput., 33(1):364–385, 2011.MathSciNetzbMATHGoogle Scholar
  60. 60.
    D. P. Kouri. A multilevel stochastic collocation algorithm for optimization of PDEs with uncertain coefficients. SIAM/ASA Journal on Uncertainty Quantification, 2(1):55–81, 2014.MathSciNetzbMATHGoogle Scholar
  61. 61.
    D. P. Kouri, M. Heinkenschloss, D. Ridzal, and B. G. van Bloemen Waanders. A trust-region algorithm with adaptive stochastic collocation for PDE optimization under uncertainty. SIAM Journal on Scientific Computing, 35(4):A1847–A1879, 2013.MathSciNetzbMATHGoogle Scholar
  62. 62.
    D. P. Kouri, M. Heinkenschloss, D. Ridzal, and B. G. van Bloemen Waanders. Inexact objective function evaluations in a trust-region algorithm for PDE-constrained optimization under uncertainty. SIAM Journal on Scientific Computing, 36(6):A3011–A3029, 2014.MathSciNetzbMATHGoogle Scholar
  63. 63.
    D. P. Kouri and T. M. Surowiec. Existence and optimality conditions for risk-averse PDE-constrained optimization. SIAM/ASA Journal on Uncertainty Quantification, 6(2):787–815, 2018.MathSciNetzbMATHGoogle Scholar
  64. 64.
    D. P. Kouri and T. M. Surowiec. Risk-averse PDE-constrained optimization using the conditional value-at-risk. SIAM Journal on Optimization, 26(1):365–396, 2016.MathSciNetzbMATHGoogle Scholar
  65. 65.
    J. R. Krebs, J. E. Anderson, D. Hinkley, R. Neelamani, S. Lee, A. Baumstein, and M. D. Lacasse. Fast full-waveform seismic inversion using encoded sources. Geophysics, 74(6):177–188, 2009.Google Scholar
  66. 66.
    P. A. Krokhmal. Higher moment coherent risk measures. Quantitative Finance, 7(4):373–387, 2007.MathSciNetzbMATHGoogle Scholar
  67. 67.
    B. Lazarov, M. Schevenels, and O. Sigmund. Topology optimization considering material and geometric uncertainties using stochastic collocation methods. Structural and Multidisciplinary Optimization, pages 1–16, 2012. online first.Google Scholar
  68. 68.
    O. P. Le Maitre and O. M. Knio. Spectral Methods for Uncertainty Quantification With Applications to Computational Fluid Dynamics. Scientific Computation. Springer-Verlag, Berlin, 2010.Google Scholar
  69. 69.
    M. Loève. Probability theory. II. Graduate Texts in Mathematics, Vol. 46. Springer-Verlag, New York, fourth edition, 1978.Google Scholar
  70. 70.
    D. Love and G. Bayraksan. Phi-divergence constrained ambiguous stochastic programs. Technical report, Technical report, Program in Applied Mathematics, University of Arizona, 2013.Google Scholar
  71. 71.
    A. Mafusalov and S. Uryasev. Buffered probability of exceedance: mathematical properties and optimization. SIAM Journal on Optimization, 28(2):1077–1103, 2018.MathSciNetzbMATHGoogle Scholar
  72. 72.
    M. M. Mäkelä and N. Neittaanmäki. Nonsmooth Optimization: Analysis And Algorithms With Applications To Optimal Control. World Scientific Publishing Company, 1992.zbMATHGoogle Scholar
  73. 73.
    E. M. Makhlouf, W. H. Chen, M. L. Wasserman, and J. H. Seinfeld. A general history matching algorithm for three-phase, three-dimensional petroleum reserviors. Society of Petroleum Engineers, 1(2), 1993.Google Scholar
  74. 74.
    H. Markowitz. Portfolio selection. The Journal of Finance, 7(1):pp. 77–91, 1952.Google Scholar
  75. 75.
    K. Marti, editor. Stochastic Optimization. Numerical Methods and Technical Applications. Springer, Berlin, 1992. LN in Economics and Math. Systems 379.Google Scholar
  76. 76.
    K. Marti. Differentiation formulas for probability functions: The transformation method. Mathematical Programming, 75:201–220, 1996.MathSciNetzbMATHGoogle Scholar
  77. 77.
    K. Maute. Topology Optimization under Uncertainty, pages 457–471. Springer Vienna, Vienna, 2014.Google Scholar
  78. 78.
    K. Maute and D. M. Frangopol. Reliability-based design of mems mechanisms by topology optimization. Computers & Structures, 81(8–11):813–824, 2003.Google Scholar
  79. 79.
    T. Morimoto. Markov processes and the h-theorem. J. Phys. Soc. Jap., 18(3):328–333, 1963.zbMATHGoogle Scholar
  80. 80.
    A. Nemirovski, A. Juditsky, G. Lan, and A. Shapiro. Robust stochastic approximation approach to stochastic programming. SIAM Journal on Optimization, 19(4):1574–1609, 2009.MathSciNetzbMATHGoogle Scholar
  81. 81.
    A. Nemirovski and A. Shapiro. Convex approximations of chance constrained programs. SIAM Journal on Optimization, 17(4):969–996, 2007.MathSciNetzbMATHGoogle Scholar
  82. 82.
    A. Nemirovski and D. Yudin. On Cezari’s convergence of the steepest descent method for approximating saddle point of convex-concave functions. Soviet Math. Dokl., 239:1056–1059, 1978.Google Scholar
  83. 83.
    F. Nobile, R. Tempone, and C. G. Webster. An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal., 46(5):2411–2442, 2008.MathSciNetzbMATHGoogle Scholar
  84. 84.
    F. Nobile, R. Tempone, and C. G. Webster. A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM Journal on Numerical Analysis, 46(5):2309–2345, 2008.MathSciNetzbMATHGoogle Scholar
  85. 85.
    E. Novak and K. Ritter. High-dimensional integration of smooth functions over cubes. Numer. Math., 75(1):79–97, 1996.MathSciNetzbMATHGoogle Scholar
  86. 86.
    E. Novak and K. Ritter. Simple cubature formulas with high polynomial exactness. Constr. Approx., 15(4):499–522, 1999.MathSciNetzbMATHGoogle Scholar
  87. 87.
    B.K. Pagnoncelli, S. Ahmed, and A. Shapiro. Sample average approximation method for chance constrained programming: theory and applications. J. Optim. Theory Appl., 142(2):399–416, 2009.MathSciNetzbMATHGoogle Scholar
  88. 88.
    J. S. Pang and S. Leyffer. On the global minimization of the value-at-risk. Optimization Methods and Software, 19(5):611–631, 2004.MathSciNetzbMATHGoogle Scholar
  89. 89.
    B.T. Polyak. New method of stochastic approximation type. Automat. Remote Control, 51:937–946, 1990.MathSciNetzbMATHGoogle Scholar
  90. 90.
    A. Prékopa. Probabilistic programming. In Stochastic programming, volume 10 of Handbooks Oper. Res. Management Sci., pages 267–351. Elsevier Sci. B. V., Amsterdam, 2003.Google Scholar
  91. 91.
    H. Robbins and S. Monro. A stochastic approximation method. Ann. Math. Statist., 22(3):400–407, 9 1951.MathSciNetzbMATHGoogle Scholar
  92. 92.
    R. T. Rockafellar. Monotone operators and the proximal point algorithm. SIAM Journal on Control and Optimization, 14(5):877–898, 1976.MathSciNetzbMATHGoogle Scholar
  93. 93.
    R. T. Rockafellar and J. O. Royset. On buffered failure probability in design and optimization of structures. Reliability Engineering & System Safety, 95(5):499–510, 2010.Google Scholar
  94. 94.
    R. T. Rockafellar and S. Uryasev. Conditional value-at-risk for general loss distributions. Journal of Banking & Finance, 26(7):1443–1471, 2002.Google Scholar
  95. 95.
    R. T. Rockafellar and S. Uryasev. The fundamental risk quadrangle in risk management, optimization and statistical estimation. Surveys in Operations Research and Management Science, 18(1–2):33–53, 2013.MathSciNetGoogle Scholar
  96. 96.
    R. T. Rockafellar and Roger J.-B. Wets. Scenarios and policy aggregation in optimization under uncertainty. Math. Oper. Res., 16(1):119–147, 1991.MathSciNetzbMATHGoogle Scholar
  97. 97.
    W. W. Rogosinski. Moments of non-negative mass. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 245(1240):1–27, 1958.MathSciNetzbMATHGoogle Scholar
  98. 98.
    J. O. Royset and E. Polak. Extensions of stochastic optimization results to problems with system failure probability functions. Journal of Optimization Theory and Applications, 133(1):1–18, 2007.MathSciNetzbMATHGoogle Scholar
  99. 99.
    A. Ruszczyński and A. Shapiro. Optimization of risk measures. In G. Calafiore and F. Dabbene, editors, Probabilistic and Randomized Methods for Design Under Uncertainty, pages 119–157, London, 2006. Springer Verlag.Google Scholar
  100. 100.
    R. A. Ryan. Introduction to tensor products of Banach spaces. Springer Monographs in Mathematics. Springer-Verlag London Ltd., London, 2002.zbMATHGoogle Scholar
  101. 101.
    F. Santosa and W. W. Symes. Linear inversion of band-limited reflection seismograms. SIAM Journal on Scientific and Statistical Computing, 7(4):1307–1330, 1986.MathSciNetzbMATHGoogle Scholar
  102. 102.
    P. Sarma, L. J. Durlofsky, K. Aziz, and W. H. Chen. Efficient real-time reservoir management using adjoint-based optimal control and model updating. Computational Geosciences, 10(1):3–36, 2006.MathSciNetzbMATHGoogle Scholar
  103. 103.
    H. Scarf. A min-max solution of an inventory problem. In Studies in the Mathematical Theory of Inventory and Production, pages 201–209. Stanford University Press, 1958.Google Scholar
  104. 104.
    C. Schwab and C. J. Gittelson. Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs. Acta Numer., 2011:291–467, 2011.MathSciNetzbMATHGoogle Scholar
  105. 105.
    A. Shapiro. On concepts of directional differentiability. J. Optim. Theory Appl., 66(3):477–487, 1990.MathSciNetzbMATHGoogle Scholar
  106. 106.
    A. Shapiro. Monte Carlo sampling methods. In A. Ruszczynski and A. Shapiro, editors, Stochastic Programming, Handbooks in Operations Research and Management Science, Vol. 10, pages 353–425. Elsevier, 2003.Google Scholar
  107. 107.
    A. Shapiro. Distributionally robust stochastic programming. SIAM J. Optimization, 27(4):2258–2275, 2017.MathSciNetzbMATHGoogle Scholar
  108. 108.
    A. Shapiro, D. Dentcheva, and A. Ruszczynski. Lectures on Stochastic Programming: Modeling and Theory, Second Edition. MOS-SIAM Series on Optimization. Society for Industrial and Applied Mathematics, Philadelphia, 2014.zbMATHGoogle Scholar
  109. 109.
    O. Sigmund. Manufacturing tolerant topology optimization. Acta Mechanica Sinica, 25(2):227–239, 2009.zbMATHGoogle Scholar
  110. 110.
    S. A. Smoljak. Quadrature and interpolation formulae on tensor products of certain function classes. Soviet Math. Dokl., 4:240–243, 1963.Google Scholar
  111. 111.
    W. W. Symes and J. J. Carazzone. Velocity inversion by differential semblance optimization. Geophysics, 56(5):654–663, 1991.Google Scholar
  112. 112.
    H. Tiesler, R. M. Kirby, D. Xiu, and T. Preusser. Stochastic collocation for optimal control problems with stochastic PDE constraints. SIAM Journal on Control and Optimization, 50(5):2659–2682, 2012.MathSciNetzbMATHGoogle Scholar
  113. 113.
    S. Uryasev. Derivatives of probability functions and integrals over sets given by inequalities. J. Comput. Appl. Math., 56(1–2):197–223, 1994. Stochastic programming: stability, numerical methods and applications (Gosen, 1992).Google Scholar
  114. 114.
    S. Uryasev. Derivatives of probability functions and some applications. Ann. Oper. Res., 56:287–311, 1995. Stochastic programming (Udine, 1992).MathSciNetzbMATHGoogle Scholar
  115. 115.
    S. Uryasev and R. T. Rockafellar. Conditional value-at-risk: Optimization approach. In S. Uryasev and P. M. Pardalos, editors, Stochastic optimization: algorithms and applications. Papers from the conference held at the University of Florida, Gainesville, FL, February 20–22, 2000, volume 54 of Appl. Optim., pages 411–435. Kluwer Acad. Publ., Dordrecht, 2001.Google Scholar
  116. 116.
    M. M. Vainberg. Variational methods for the study of nonlinear operators. Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1964. With a chapter on Newton’s method by L. V. Kantorovich and G. P. Akilov. Translated and supplemented by Amiel Feinstein.Google Scholar
  117. 117.
    W. van Ackooij and R. Henrion. (Sub-)gradient formulae for probability functions of random inequality systems under Gaussian distribution. SIAM/ASA Journal on Uncertainty Quantification, 5(1):63–87, 2017.MathSciNetzbMATHGoogle Scholar
  118. 118.
    B. van den Bosch and J. H. Seinfeld. History matching in two-phase petroleum reserviors: Incompressible flow. Society of Petroleum Engineers, 17(6), 1977.Google Scholar
  119. 119.
    G. van Essen, M. Zandvliet, P. van den Hof, O. Bosgra, and J. D. Jansen. Robust waterflooding optimization of multiple geological scenarios. Society of Petroleum Engineers, 14(1), 2009.Google Scholar
  120. 120.
    J. E. Warner, M. D. Grigoriu, and W. Aquino. Stochastic reduced order models for random vectors: Application to random eigenvalue problems. Probabilistic Engineering Mechanics, 31:1–11, 2013.Google Scholar
  121. 121.
    W. Wiesemann, D. Kuhn, and M. Sim. Distributionally robust convex optimization. Operations Research, 62(6):1358–1376, 2014.MathSciNetzbMATHGoogle Scholar
  122. 122.
    D. Xiu and G. E. Karniadakis. Modeling uncertainty in flow simulations via generalized polynomial chaos. J. Comput. Phys., 187(1):137–167, 2003.MathSciNetzbMATHGoogle Scholar

Copyright information

© 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

Personalised recommendations