Hostname: page-component-6898cfcb55-th4rj Total loading time: 0 Render date: 2026-11-11T13:39:25.277Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimization problems governed by systems of PDEs with uncertainties

Part of:Optimality conditionsExistence theoriesPartial differential equations, boundary value problems

Published online by Cambridge University Press:  01 July 2025

Matthias HeinkenschlossOpen the ORCID record for Matthias Heinkenschloss[Opens in a new window] and
Drew P. Kouri
Matthias Heinkenschloss
Affiliation:
Department of Computational Applied Mathematics and Operations Research, Rice University, 6100 Main St, Houston, TX 77005, USA E-mail: heinken@rice.edu
Drew P. Kouri
Affiliation:
Optimization and Uncertainty Quantification, Sandia National Laboratories, PO Box 5800, Albuquerque, NM 87125, USA E-mail: dpkouri@sandia.gov
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

This paper reviews current theoretical and numerical approaches to optimization problems governed by partial differential equations (PDEs) that depend on random variables or random fields. Such problems arise in many engineering, science, economics and societal decision-making tasks. This paper focuses on problems in which the governing PDEs are parametrized by the random variables/fields, and the decisions are made at the beginning and are not revised once uncertainty is revealed. Examples of such problems are presented to motivate the topic of this paper, and to illustrate the impact of different ways to model uncertainty in the formulations of the optimization problem and their impact on the solution. A linear–quadratic elliptic optimal control problem is used to provide a detailed discussion of the set-up for the risk-neutral optimization problem formulation, study the existence and characterization of its solution, and survey numerical methods for computing it. Different ways to model uncertainty in the PDE-constrained optimization problem are surveyed in an abstract setting, including risk measures, distributionally robust optimization formulations, probabilistic functions and chance constraints, and stochastic orders. Furthermore, approximation-based optimization approaches and stochastic methods for the solution of the large-scale PDE-constrained optimization problems under uncertainty are described. Some possible future research directions are outlined.

MSC classification

Primary:49K20: Problems involving partial differential equations49K45: Problems involving randomness90C15: Stochastic programming
Secondary:49J50: Fréchet and Gateaux differentiability65K05: Mathematical programming methods65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods

Information

Type
Research Article
Information
Acta Numerica , Volume 34 , July 2025, pp. 491 - 577
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2025. Published by Cambridge University Press

References

Adams, R. A. and Fournier, J. J. F. (2003), Sobolev Spaces, second edition, Elsevier Science.Google Scholar
Ali, A. Ahmad, Ullmann, E. and Hinze, M. (2017), Multilevel Monte Carlo analysis for optimal control of elliptic PDEs with random coefficients, SIAM/ASA J. Uncertain. Quantif. 5, 466–492.10.1137/16M109870XCrossRefGoogle Scholar
Aliprantis, C. D. and Border, K. C. (2006), Infinite Dimensional Analysis: A Hitchhiker’s Guide, third edition, Springer.Google Scholar
Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D. (1999), Coherent measures of risk, Math. Finance 9, 203–228.10.1111/1467-9965.00068CrossRefGoogle Scholar
Babuška, I., Nobile, F. and Tempone, R. (2007), A stochastic collocation method for elliptic partial differential equations with random input data, SIAM J. Numer. Anal. 45, 1005–1034.10.1137/050645142CrossRefGoogle Scholar
Bandeira, A. S., Scheinberg, K. and Vicente, L. N. (2014), Convergence of trust-region methods based on probabilistic models, SIAM J. Optim. 24, 1238–1264.10.1137/130915984CrossRefGoogle Scholar
Bangerth, W., Klie, H., Wheeler, M., Stoffa, P. and Sen, M. (2006), On optimization algorithms for the reservoir oil well placement problem, Comput. Geosci. 10, 303–319.10.1007/s10596-006-9025-7CrossRefGoogle Scholar
Baraldi, R. J. and Kouri, D. P. (2023), A proximal trust-region method for nonsmooth optimization with inexact function and gradient evaluations, Math. Program. 201, 559–598.10.1007/s10107-022-01915-3CrossRefGoogle Scholar
Baraldi, R. J. and Kouri, D. P. (2024), Local convergence analysis of an inexact trust-region method for nonsmooth optimization, Optim. Lett. 18, 663–680.10.1007/s11590-023-02092-8CrossRefGoogle Scholar
Barty, K., Roy, J.-S. and Strugarek, C. (2007), Hilbert-valued perturbed subgradient algorithms, Math. Oper. Res. 32, 551–562.10.1287/moor.1070.0253CrossRefGoogle Scholar
Bastin, F., Cirillo, C. and Toint, P. L. (2006), An adaptive Monte Carlo algorithm for computing mixed logit estimators, Comput. Manag. Sci. 3, 55–79.10.1007/s10287-005-0044-yCrossRefGoogle Scholar
Bauschke, H. H. and Combettes, P. L. (2017), Convex Analysis and Monotone Operator Theory in Hilbert Spaces , CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, second edition, Springer.Google Scholar
Becker, R., Braack, M., Meidner, D., Rannacher, R. and Vexler, B. (2007), Adaptive finite element methods for PDE-constrained optimal control problems, in Reactive Flows, Diffusion and Transport (Jäger, W., Rannacher, R. and Warnatz, J., eds), Springer, pp. 177–205.10.1007/978-3-540-28396-6_8CrossRefGoogle Scholar
Becker, R., Kapp, H. and Rannacher, R. (2000), Adaptive finite element methods for optimal control of partial differential equations: Basic concepts, SIAM J. Control Optim. 39, 113–132.10.1137/S0363012999351097CrossRefGoogle Scholar
Ben-Tal, A. and Teboulle, M. (2007), An old-new concept of convex risk measures: The optimized certainty equivalent, Math. Finance 17, 449–476.10.1111/j.1467-9965.2007.00311.xCrossRefGoogle Scholar
Bendsøe, M. P. and Sigmund, O. (2003), Topology Optimization: Theory, Methods and Applications, Springer.Google Scholar
Benner, P., Onwunta, A. and Stoll, M. (2016), Block-diagonal preconditioning for optimal control problems constrained by PDEs with uncertain inputs, SIAM J. Matrix Anal. Appl. 37, 491–518.10.1137/15M1018502CrossRefGoogle Scholar
Benzi, M., Golub, G. H. and Liesen, J. (2005), Numerical solution of saddle point problems, Acta Numer. 14, 1–137.10.1017/S0962492904000212CrossRefGoogle Scholar
Berahas, A. S., Cao, L. and Scheinberg, K. (2021), Global convergence rate analysis of a generic line search algorithm with noise, SIAM J. Optim. 31, 1489–1518.10.1137/19M1291832CrossRefGoogle Scholar
Berger, J. O. (1994), An overview of robust Bayesian analysis, Test 3, 5–124.10.1007/BF02562676CrossRefGoogle Scholar
Bertsekas, D. P. (2019), Reinforcement Learning and Optimal Control, Athena Scientific Optimization and Computation Series, Athena Scientific.Google Scholar
Bertsekas, D. P. (2024), Model predictive control and reinforcement learning: A unified framework based on dynamic programming, IFAC-PapersOnLine 58, 363–383.10.1016/j.ifacol.2024.09.056CrossRefGoogle Scholar
Birge, J. R. and Louveaux, F. (2011), Introduction to Stochastic Programming, Springer Series in Operations Research and Financial Engineering, second edition, Springer.10.1007/978-1-4614-0237-4CrossRefGoogle Scholar
Borzì, A. (2010), Multigrid and sparse-grid schemes for elliptic control problems with random coefficients, Comput. Vis. Sci. 13, 153–160.10.1007/s00791-010-0134-4CrossRefGoogle Scholar
Borzì, A. and Schulz, V. (2009), Multigrid methods for PDE optimization, SIAM Rev. 51, 361–395.10.1137/060671590CrossRefGoogle Scholar
Borzì, A. and Schulz, V. (2012), Computational Optimization of Systems Governed by Partial Differential Equations, Vol. 8 of Computational Science & Engineering, SIAM.Google Scholar
Brenner, S. C. and Scott, L. R. (2008), The Mathematical Theory of Finite Element Methods, Vol. 15 of Texts in Applied Mathematics, third edition, Springer.10.1007/978-0-387-75934-0CrossRefGoogle Scholar
Brouwer, D. R. and Jansen, J. D. (2004), Dynamic optimization of waterflooding with smart wells using optimal control theory, SPE J. 9, 391–402.10.2118/78278-PACrossRefGoogle Scholar
Bungartz, H.-J. and Griebel, M. (2004), Sparse grids, Acta Numer. 13, 147–269.10.1017/S0962492904000182CrossRefGoogle Scholar
Carter, R. G. (1991), On the global convergence of trust region algorithms using inexact gradient information, SIAM J. Numer. Anal. 28, 251–265.10.1137/0728014CrossRefGoogle Scholar
Carter, R. G. (1993), Numerical experience with a class of algorithms for nonlinear optimization using inexact function and gradient information, SIAM J. Sci. Comput. 14, 368–388.10.1137/0914023CrossRefGoogle Scholar
Carter, R. G. and Rachford, H. H. Jr (2003), Optimizing line-pack management to hedge against future load uncertainty, in PSIG Annual Meeting, PSIG, art. PSIG-0306.Google Scholar
Charrier, J., Scheichl, R. and Teckentrup, A. L. (2013), Finite element error analysis of elliptic PDEs with random coefficients and its application to multilevel Monte Carlo methods, SIAM J. Numer. Anal. 51, 322–352.10.1137/110853054CrossRefGoogle Scholar
Chaudhuri, A., Kramer, B., Norton, M., Royset, J. O. and Willcox, K. E. (2022), Certifiable risk-based engineering design optimization, AIAA J. 60, 551–565.10.2514/1.J060539CrossRefGoogle Scholar
Chen, P. and Quarteroni, A. (2014), Weighted reduced basis method for stochastic optimal control problems with elliptic PDE constraint, SIAM/ASA J. Uncertain. Quantif. 2, 364–396.10.1137/130940517CrossRefGoogle Scholar
Chen, P., Quarteroni, A. and Rozza, G. (2016), Multilevel and weighted reduced basis method for stochastic optimal control problems constrained by Stokes equations, Numer. Math. 133, 67–102.10.1007/s00211-015-0743-4CrossRefGoogle Scholar
Cheridito, P. and Li, T. (2009), Risk measures on Orlicz hearts, Math. Finance 19, 189–214.10.1111/j.1467-9965.2009.00364.xCrossRefGoogle Scholar
Ciaramella, G., Nobile, F. and Vanzan, T. (2024), A multigrid solver for PDE-constrained optimization with uncertain inputs, J. Sci. Comput. 101, art. 13.10.1007/s10915-024-02646-7CrossRefGoogle Scholar
Cohen, A. and DeVore, R. (2015), Approximation of high-dimensional parametric PDEs, Acta Numer. 24, 1–159.10.1017/S0962492915000033CrossRefGoogle Scholar
Cohen, A., DeVore, R. and Schwab, C. (2010), Convergence rates of best N-term Galerkin approximations for a class of elliptic PDEs, Found. Comput. Math. 10, 615–646.Google Scholar
Collis, S. S. and Heinkenschloss, M. (2002), Analysis of the streamline upwind/Petrov Galerkin method applied to the solution of optimal control problems. Technical report TR02–01, Department of Computational and Applied Mathematics, Rice University, Houston, TX. Also available at arXiv:2411.09828.Google Scholar
Conn, A. R., Gould, N. I. M. and Toint, P. L. (2000), Trust-Region Methods, MPS/SIAM Series on Optimization, SIAM.10.1137/1.9780898719857CrossRefGoogle Scholar
Conti, S., Held, H., Pach, M., Rumpf, M. and Schultz, R. (2011), Risk averse shape optimization, SIAM J. Control Optim. 49, 927–947.10.1137/090754315CrossRefGoogle Scholar
Conti, S., Rumpf, M., Schultz, R. and Tölkes, S. (2018), Stochastic dominance constraints in elastic shape optimization, SIAM J. Control Optim. 56, 3021–3034.10.1137/16M108313XCrossRefGoogle Scholar
Curtis, F. E., Scheinberg, K. and Shi, R. (2019), A stochastic trust region algorithm based on careful step normalization, INFORMS J. Optim. 1, 200–220.10.1287/ijoo.2018.0010CrossRefGoogle Scholar
Dambrine, M., Dapogny, C. and Harbrecht, H. (2015), Shape optimization for quadratic functionals and states with random right-hand sides, SIAM J. Control Optim. 53, 3081–3103.10.1137/15M1017041CrossRefGoogle Scholar
Dedé, L. and Quarteroni, A. (2005), Optimal control and numerical adaptivity for advection-diffusion equations, M2AN Math. Model. Numer. Anal. 39, 1019–1040.10.1051/m2an:2005044CrossRefGoogle Scholar
Dennis, J. E. Jr and Schnabel, R. B. (1996), Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Vol. 16 of Classics in Applied Mathematics, SIAM.10.1137/1.9781611971200CrossRefGoogle Scholar
Dentcheva, D. and Ruszczyński, A. (2003), Optimization with stochastic dominance constraints, SIAM J. Optim. 14, 548–566.10.1137/S1052623402420528CrossRefGoogle Scholar
Dick, J., Kuo, F. Y. and Sloan, I. H. (2013), High-dimensional integration: The quasi-Monte Carlo way, Acta Numer. 22, 133–288.10.1017/S0962492913000044CrossRefGoogle Scholar
Eckstein, J. and Bertsekas, D. P. (1992), On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators, Math. Program. 55, 293–318.10.1007/BF01581204CrossRefGoogle Scholar
Ekeland, I. and Temam, R. (1999), Convex Analysis and Variational Problems, Vol. 28 of Classics in Applied Mathematics, SIAM.10.1137/1.9781611971088CrossRefGoogle Scholar
Eldred, M. S. (2011), Design under uncertainty employing stochastic expansion methods, Int. J. Uncertain. Quantif. 1, 119–146.10.1615/Int.J.UncertaintyQuantification.v1.i2.20CrossRefGoogle Scholar
Fabbri, G., Gozzi, F. and Świech, A. (2017), Stochastic Optimal Control in Infinite Dimension: Dynamic Programming and HJB Equations, Vol. 82 of Probability Theory and Stochastic Modelling, Springer.10.1007/978-3-319-53067-3CrossRefGoogle Scholar
Folland, G. B. (1999), Real Analysis: Modern Techniques and Their Applications , Pure and Applied Mathematics, second edition, Wiley.Google Scholar
Föllmer, H. and Schied, A. (2002), Convex measures of risk and trading constraints, Finance Stoch. 6, 429–447.10.1007/s007800200072CrossRefGoogle Scholar
Föllmer, H. and Schied, A. (2010), Convex and coherent risk measures, in Encyclopedia of Quantitative Finance, Wiley, pp. 355–363.Google Scholar
Garcke, J. and Griebel, M., eds (2013), Sparse Grids and Applications, Vol. 88 of Lecture Notes in Computational Science and Engineering, Springer.10.1007/978-3-642-31703-3CrossRefGoogle Scholar
Garreis, S. and Ulbrich, M. (2017), Constrained optimization with low-rank tensors and applications to parametric problems with PDEs, SIAM J. Sci. Comput. 39, A25–A54.10.1137/16M1057607CrossRefGoogle Scholar
Garreis, S., Surowiec, T. and Ulbrich, M. (2021), An interior-point approach for solving risk-averse PDE-constrained optimization problems with coherent risk measures, SIAM J. Optim. 31, 1–29.10.1137/19M125039XCrossRefGoogle Scholar
Geiersbach, C. and Henrion, R. (2024), Optimality conditions in control problems with random state constraints in probabilistic or almost-sure form, Math. Oper. Res. Available at doi:10.1287/moor.2023.0177.CrossRefGoogle Scholar
Geiersbach, C. and Hintermüller, M. (2022), Optimality conditions and Moreau–Yosida regularization for almost sure state constraints, ESAIM Control Optim. Calc. Var. 28, art. 80.10.1051/cocv/2022070CrossRefGoogle Scholar
Geiersbach, C. and Pflug, G. C. (2019), Projected stochastic gradients for convex constrained problems in Hilbert spaces, SIAM J. Optim. 29, 2079–2099.10.1137/18M1200208CrossRefGoogle Scholar
Geiersbach, C. and Scarinci, T. (2023), A stochastic gradient method for a class of nonlinear PDE-constrained optimal control problems under uncertainty, J. Differential Equations 364, 635–666.10.1016/j.jde.2023.04.034CrossRefGoogle Scholar
Geiersbach, C. and Wollner, W. (2020), A stochastic gradient method with mesh refinement for PDE-constrained optimization under uncertainty, SIAM J. Sci. Comput. 42, A2750–A2772.10.1137/19M1263297CrossRefGoogle Scholar
Geiersbach, C., Henrion, R. and Pérez-Aros, P. (2025), Numerical solution of an optimal control problem with probabilistic and almost sure state constraints, J. Optim. Theory Appl. 204, art. 7.10.1007/s10957-024-02578-0CrossRefGoogle Scholar
Geihe, B., Lenz, M., Rumpf, M. and Schultz, R. (2013), Risk averse elastic shape optimization with parametrized fine scale geometry, Math. Program. 141, 383–403.10.1007/s10107-012-0531-1CrossRefGoogle Scholar
Gerstner, T. and Griebel, M. (1998), Numerical integration using sparse grids, Numer. Algorithms 18, 209–232.10.1023/A:1019129717644CrossRefGoogle Scholar
Gerstner, T. and Griebel, M. (2003), Dimension-adaptive tensor-product quadrature, Computing 71, 65–87.10.1007/s00607-003-0015-5CrossRefGoogle Scholar
Giles, M. B. (2015), Multilevel Monte Carlo methods, Acta Numer. 24, 259–328.10.1017/S096249291500001XCrossRefGoogle Scholar
Giles, M. B. and Haji-Ali, A.-L. (2019), Multilevel nested simulation for efficient risk estimation, SIAM/ASA J. Uncertain. Quantif. 7, 497–525.10.1137/18M1173186CrossRefGoogle Scholar
Gratton, S., Sartenaer, A. and Toint, P. L. (2008), Recursive trust-region methods for multiscale nonlinear optimization, SIAM J. Optim. 19, 414–444.10.1137/050623012CrossRefGoogle Scholar
Griebel, M. (2006), Sparse grids and related approximation schemes for higher dimensional problems, in Foundations of Computational Mathematics, Santander 2005 (Pardo, L. M. et al., eds), Vol. 331 of London Mathematical Society Lecture Note Series, Cambridge University Press, pp. 106–161.10.1017/CBO9780511721571.004CrossRefGoogle Scholar
Grundvig, D. S. and Heinkenschloss, M. (2024), Line-search based optimization using function approximations with tunable accuracy, Optim. Methods Softw. Available at doi:10.1080/10556788.2024.2436192.CrossRefGoogle Scholar
Grüne, L. and Pannek, J. (2011), Nonlinear Model Predictive Control: Theory and Algorithms , Communications and Control Engineering Series, Springer.Google Scholar
Gunzburger, M. D. (2003), Perspectives in Flow Control and Optimization, SIAM.Google Scholar
Gunzburger, M. D., Webster, C. G. and Zhang, G. (2014), Stochastic finite element methods for partial differential equations with random input data, Acta Numer. 23, 521–650.10.1017/S0962492914000075CrossRefGoogle Scholar
Guth, P. A., Kaarnioja, V., Kuo, F. Y., Schillings, C. and Sloan, I. H. (2021), A quasi-Monte Carlo method for optimal control under uncertainty, SIAM/ASA J. Uncertain. Quantif. 9, 354–383.10.1137/19M1294952CrossRefGoogle Scholar
Heinkenschloss, M. and Leykekhman, D. (2010), Local error estimates for SUPG solutions of advection-dominated elliptic linear–quadratic optimal control problems, SIAM J. Numer. Anal. 47, 4607–4638.10.1137/090759902CrossRefGoogle Scholar
Heinkenschloss, M. and Vicente, L. N. (2002), Analysis of inexact trust-region SQP algorithms, SIAM J. Optim. 12, 283–302.10.1137/S1052623499361543CrossRefGoogle Scholar
Heinkenschloss, M., Kramer, B. and Takhtaganov, T. (2020), Adaptive reduced-order model construction for conditional value-at-risk estimation, SIAM/ASA J. Uncertain. Quantif. 8, 668–692.10.1137/19M1257433CrossRefGoogle Scholar
Heinkenschloss, M., Kramer, B., Takhtaganov, T. and Willcox, K. (2018), Conditional-value-at-risk estimation via reduced-order models, SIAM/ASA J. Uncertain. Quantif. 6, 1395–1423.10.1137/17M1160069CrossRefGoogle Scholar
Hintermüller, M. and Hoppe, R. H. W. (2008), Goal-oriented adaptivity in control constrained optimal control of partial differential equations, SIAM J. Control Optim. 47, 1721–1743.10.1137/070683891CrossRefGoogle Scholar
Hinze, M. (2005), A variational discretization concept in control constrained optimization: The linear–quadratic case, Comput. Optim. Appl. 30, 45–63.10.1007/s10589-005-4559-5CrossRefGoogle Scholar
Hinze, M., Pinnau, R., Ulbrich, M. and Ulbrich, S. (2009), Optimization with PDE Constraints, Vol. 23 of Mathematical Modelling, Theory and Applications, Springer.10.1007/978-1-4020-8839-1_3CrossRefGoogle Scholar
Ito, K. and Ravindran, S. S. (1998), Optimal control of thermally convected fluid flows, SIAM J. Sci. Comput. 19, 1847–1869.10.1137/S1064827596299731CrossRefGoogle Scholar
Jahn, J. (2007), Introduction to the Theory of Nonlinear Optimization, third edition, Springer.Google Scholar
Jakeman, J. D., Kouri, D. P. and Huerta, J. G. (2022), Surrogate modeling for efficiently, accurately and conservatively estimating measures of risk, Reliab. Engrg Syst. Saf. 221, art. 108280.10.1016/j.ress.2021.108280CrossRefGoogle Scholar
Kolvenbach, P., Lass, O. and Ulbrich, S. (2018), An approach for robust PDE-constrained optimization with application to shape optimization of electrical engines and of dynamic elastic structures under uncertainty, Optim. Eng. 19, 697–731.10.1007/s11081-018-9388-3CrossRefGoogle Scholar
Kouri, D. P. (2012), An approach for the adaptive solution of optimization problems governed by partial differential equations with uncertain coefficients. PhD thesis, Department of Computational and Applied Mathematics, Rice University, Houston, TX.Google Scholar
Kouri, D. P. (2014), A multilevel stochastic collocation algorithm for optimization of PDEs with uncertain coefficients, SIAM/ASA J. Uncertain. Quantif. 2, 55–81.10.1137/130915960CrossRefGoogle Scholar
Kouri, D. P. (2017), A measure approximation for distributionally robust PDE-constrained optimization problems, SIAM J. Numer. Anal. 55, 3147–3172.10.1137/15M1036944CrossRefGoogle Scholar
Kouri, D. P. (2019), Higher-moment buffered probability, Optim. Lett. 13, 1223–1237.10.1007/s11590-018-1359-2CrossRefGoogle Scholar
Kouri, D. P. and Ridzal, D. (2018), Inexact trust-region methods for PDE-constrained optimization, in Frontiers in PDE-Constrained Optimization (Antil, H. et al., eds), Vol. 163 of IMA Volumes in Mathematics and its Applications, Springer, pp. 83–121.10.1007/978-1-4939-8636-1_3CrossRefGoogle Scholar
Kouri, D. P. and Shapiro, A. (2018), Optimization of PDEs with uncertain inputs, in Frontiers in PDE-Constrained Optimization (Antil, H. et al., eds), Vol. 163 of IMA Volumes in Mathematics and its Applications, Springer, pp. 41–81.10.1007/978-1-4939-8636-1_2CrossRefGoogle Scholar
Kouri, D. P. and Surowiec, T. M. (2016), Risk-averse PDE-constrained optimization using the conditional value-at-risk, SIAM J. Optim. 26, 365–396.10.1137/140954556CrossRefGoogle Scholar
Kouri, D. P. and Surowiec, T. M. (2018), Existence and optimality conditions for risk-averse PDE-constrained optimization, SIAM/ASA J. Uncertain. Quantif. 6, 787–815.10.1137/16M1086613CrossRefGoogle Scholar
Kouri, D. P. and Surowiec, T. M. (2020a), Epi-regularization of risk measures, Math. Oper. Res. 45, 774–795.10.1287/moor.2019.1013CrossRefGoogle Scholar
Kouri, D. P. and Surowiec, T. M. (2020b), Risk-averse optimal control of semilinear elliptic PDEs, ESAIM Control Optim. Calc. Var. 26, art. 53.10.1051/cocv/2019061CrossRefGoogle Scholar
Kouri, D. P. and Surowiec, T. M. (2022), A primal–dual algorithm for risk minimization, Math. Program. 193, 337–363.10.1007/s10107-020-01608-9CrossRefGoogle Scholar
Kouri, D. P., Heinkenschloss, M., Ridzal, D. andvan Bloemen Waanders, B. G. (2013), A trust-region algorithm with adaptive stochastic collocation for PDE optimization under uncertainty, SIAM J. Sci. Comput. 35, A1847–A1879.10.1137/120892362CrossRefGoogle Scholar
Kouri, D. P., Heinkenschloss, M., Ridzal, D. and van Bloemen Waanders, B. G. (2014), Inexact objective function evaluations in a trust-region algorithm for PDE-constrained optimization under uncertainty, SIAM J. Sci. Comput. 36, A3011–A3029.10.1137/140955665CrossRefGoogle Scholar
Kouri, D. P., Staudigl, M. and Surowiec, T. M. (2023), A relaxation-based probabilistic approach for PDE-constrained optimization under uncertainty with pointwise state constraints, Comput. Optim. Appl. 85, 441–478.10.1007/s10589-023-00461-8CrossRefGoogle Scholar
Krokhmal, P. A. (2007), Higher moment coherent risk measures, Quant. Finance 7, 373–387.10.1080/14697680701458307CrossRefGoogle Scholar
Krumscheid, S. and Nobile, F. (2018), Multilevel Monte Carlo approximation of functions, SIAM/ASA J. Uncertain. Quantif. 6, 1256–1293.10.1137/17M1135566CrossRefGoogle Scholar
Kuo, F. Y. and Nuyens, D. (2016), Application of quasi-Monte Carlo methods to elliptic PDEs with random diffusion coefficients: A survey of analysis and implementation, Found. Comput. Math. 16, 1631–1696.10.1007/s10208-016-9329-5CrossRefGoogle Scholar
Kuo, F. Y., Scheichl, R., Schwab, C., Sloan, I. H. and Ullmann, E. (2017), Multilevel quasi-Monte Carlo methods for lognormal diffusion problems, Math. Comp. 86, 2827–2860.10.1090/mcom/3207CrossRefGoogle Scholar
Kushner, H. J. and Dupuis, P. (2001), Numerical Methods for Stochastic Control Problems in Continuous Time , Vol. 24 of Stochastic Modelling and Applied Probability, second edition, Springer.Google Scholar
Lazarov, B. S. and Sigmund, O. (2011), Filters in topology optimization based on Helmholtz-type differential equations, Int. J. Numer. Methods Engrg 86, 765–781.10.1002/nme.3072CrossRefGoogle Scholar
Lazarov, B. S., Schevenels, M. and Sigmund, O. (2012a), Topology optimization considering material and geometric uncertainties using stochastic collocation methods, Struct. Multidiscip. Optim. 46, 597–612.10.1007/s00158-012-0791-7CrossRefGoogle Scholar
Lazarov, B. S., Schevenels, M. and Sigmund, O. (2012b), Topology optimization with geometric uncertainties by perturbation techniques, Int. J. Numer. Methods Engrg 90, 1321–1336.10.1002/nme.3361CrossRefGoogle Scholar
Levy, H. (1992), Stochastic dominance and expected utility: Survey and analysis, Manag. Sci. 38, 555–593.10.1287/mnsc.38.4.555CrossRefGoogle Scholar
Lewis, R. M. and Nash, S. G. (2005), Model problems for the multigrid optimization of systems governed by differential equations, SIAM J. Sci. Comput. 26, 1811–1837.10.1137/S1064827502407792CrossRefGoogle Scholar
Leykekhman, D. (2012), Investigation of commutative properties of discontinuous Galerkin methods in PDE constrained optimal control problems, J. Sci. Comput. 53, 483–511.10.1007/s10915-012-9582-yCrossRefGoogle Scholar
Leykekhman, D. and Heinkenschloss, M. (2012), Local error analysis of discontinuous Galerkin methods for advection-dominated elliptic linear–quadratic optimal control problems, SIAM J. Numer. Anal. 50, 2012–2038.10.1137/110826953CrossRefGoogle Scholar
Lions, J.-L. (1971), Optimal Control of Systems Governed by Partial Differential Equations, Vol. 170 of Grundlehren der mathematischen Wissenschaften, Springer.10.1007/978-3-642-65024-6CrossRefGoogle Scholar
Litvinov, W. G. (2000), Optimization in Elliptic Problems with Applications to Mechanics of Deformable Bodies and Fluid Mechanics , Vol. 119 of Operator Theory: Advances and Applications, Birkhäuser.Google Scholar
Lord, G. J., Powell, C. E. and Shardlow, T. (2014), An Introduction to Computational Stochastic PDEs, Cambridge Texts in Applied Mathematics, Cambridge University Press.10.1017/CBO9781139017329CrossRefGoogle Scholar
Mafusalov, A. and Uryasev, S. (2018), Buffered probability of exceedance: Mathematical properties and optimization, SIAM J. Optim. 28, 1077–1103.10.1137/15M1042644CrossRefGoogle Scholar
Markowitz, H. (1952), Portfolio selection, J. Finance 7, 77–91.Google Scholar
Markowski, M. (2022), Efficient solution of smoothed risk-averse PDE-constrained optimization problems. PhD thesis, Department of Computational and Applied Mathematics, Rice University, Houston, TX.Google Scholar
Martin, M. and Nobile, F. (2021), PDE-constrained optimal control problems with uncertain parameters using SAGA, SIAM/ASA J. Uncertain. Quantif. 9, 979–1012.10.1137/18M1224076CrossRefGoogle Scholar
Martin, M., Krumscheid, S. and Nobile, F. (2021), Complexity analysis of stochastic gradient methods for PDE-constrained optimal control problems with uncertain parameters, ESAIM Math. Model. Numer. Anal. 55, 1599–1633.10.1051/m2an/2021025CrossRefGoogle Scholar
Meyn, S. (2022), Control Systems and Reinforcement Learning, Cambridge University Press.10.1017/9781009051873CrossRefGoogle Scholar
Milz, J. (2023a), Consistency of Monte Carlo estimators for risk-neutral PDE-constrained optimization, Appl. Math. Optim. 87, art. 57.10.1007/s00245-023-09967-3CrossRefGoogle Scholar
Milz, J. (2023b), Reliable error estimates for optimal control of linear elliptic PDEs with random inputs, SIAM/ASA J. Uncertain. Quantif. 11, 1139–1163.10.1137/22M1503889CrossRefGoogle Scholar
Milz, J. (2023c), Sample average approximations of strongly convex stochastic programs in Hilbert spaces, Optim. Lett. 17, 471–492.10.1007/s11590-022-01888-4CrossRefGoogle Scholar
Milz, J. and Ulbrich, M. (2024), Sample size estimates for risk-neutral semilinear PDE-constrained optimization, SIAM J. Optim. 34, 844–869.10.1137/22M1512636CrossRefGoogle Scholar
Mohammadi, B. and Pironneau, O. (2004), Shape optimization in fluid mechanics, Annu. Rev. Fluid Mech. 36, 255–279.10.1146/annurev.fluid.36.050802.121926CrossRefGoogle Scholar
Nash, S. G. (2000), A multigrid approach to discretized optimization problems, Optim. Methods Softw. 14, 99–116.10.1080/10556780008805795CrossRefGoogle Scholar
Nemirovski, A. and Yudin, D. (1978), On Cezari’s convergence of the steepest descent method for approximating saddle point of convex–concave functions, Soviet Math. Dokl. 19, 258–269.Google Scholar
Nemirovski, A., Juditsky, A., Lan, G. and Shapiro, A. (2009), Robust stochastic approximation approach to stochastic programming, SIAM J. Optim. 19, 1574–1609.10.1137/070704277CrossRefGoogle Scholar
Nobile, F. and Vanzan, T. (2023), Preconditioners for robust optimal control problems under uncertainty, Numer. Linear Algebra Appl. 30, art. e2472.10.1002/nla.2472CrossRefGoogle Scholar
Nobile, F. and Vanzan, T. (2024), A combination technique for optimal control problems constrained by random PDEs, SIAM/ASA J. Uncertain. Quantif. 12, 693–721.10.1137/22M1532263CrossRefGoogle Scholar
Nocedal, J. and Wright, S. J. (2006), Numerical Optimization, second edition, Springer.Google Scholar
Pearson, J. W. and Pestana, J. (2020), Preconditioners for Krylov subspace methods: an overview, GAMM-Mitt. 43, art. e202000015.10.1002/gamm.202000015CrossRefGoogle Scholar
Peherstorfer, B. (2019), Multifidelity Monte Carlo estimation with adaptive low-fidelity models, SIAM/ASA J. Uncertain. Quantif. 7, 579–603.10.1137/17M1159208CrossRefGoogle Scholar
Peherstorfer, B., Willcox, K. and Gunzburger, M. D. (2016), Optimal model management for multifidelity Monte Carlo estimation, SIAM J. Sci. Comput. 38, A3163–A3194.10.1137/15M1046472CrossRefGoogle Scholar
Peherstorfer, B., Willcox, K. and Gunzburger, M. D. (2018), Survey of multifidelity methods in uncertainty propagation, inference, and optimization, SIAM Rev. 60, 550–591.10.1137/16M1082469CrossRefGoogle Scholar
Pflug, G. C. and Pichler, A. (2014), Multistage Stochastic Optimization, Springer Series in Operations Research and Financial Engineering, Springer.10.1007/978-3-319-08843-3CrossRefGoogle Scholar
Phelps, C., Royset, J. O. and Gong, Q. (2016), Optimal control of uncertain systems using sample average approximations, SIAM J. Control Optim. 54, 1–29.10.1137/140983161CrossRefGoogle Scholar
Quarteroni, A. (2009), Numerical Models for Differential Problems , Vol. 2 of MS&A: Modeling, Simulation and Applications, Springer.Google Scholar
Rawlings, J. B., Mayne, D. Q. and Diehl, M. M. (2024), Model Predictive Control: Theory, Computation, and Design, second edition, Nob Hill Publishing.Google Scholar
Recht, B. (2019), A tour of reinforcement learning: The view from continuous control, Annu. Rev. 2, 253–279.Google Scholar
Robbins, H. and Monro, S. (1951), A stochastic approximation method, Ann. Math. Statist. 22, 400–407.10.1214/aoms/1177729586CrossRefGoogle Scholar
Rockafellar, R. T. (1971), Convex integral functionals and duality, in Contributions to Nonlinear Functional Analysis (Zarantonello, E. H., ed.), Elsevier, pp. 215–236.10.1016/B978-0-12-775850-3.50012-1CrossRefGoogle Scholar
Rockafellar, R. T. (1973), Penalty methods and augmented Lagrangians in nonlinear programming, in Fifth Conference on Optimization Techniques, Part I (Conti, R. and Ruberti, A., eds), Vol. 3 of Lecture Notes in Computer Science, Springer, pp. 418–425.10.1007/3-540-06583-0_41CrossRefGoogle Scholar
Rockafellar, R. T. (2018), Solving stochastic programming problems with risk measures by progressive hedging, Set-Valued Var. Anal. 26, 759–768.10.1007/s11228-017-0437-4CrossRefGoogle Scholar
Rockafellar, R. T. and Royset, J. O. (2010), On buffered failure probability in design and optimization of structures, Reliab. Engrg Syst. Saf. 95, 499–510.10.1016/j.ress.2010.01.001CrossRefGoogle Scholar
Rockafellar, R. T. and Uryasev, S. (2002), Conditional value-at-risk for general loss distributions, J. Banking Finance 26, 1443–1471.10.1016/S0378-4266(02)00271-6CrossRefGoogle Scholar
Rockafellar, R. T. and Uryasev, S. (2020), Minimizing buffered probability of exceedance by progressive hedging, Math. Program. 181, 453–472.10.1007/s10107-019-01462-4CrossRefGoogle Scholar
Rockafellar, R. T. and Wets, R. J.-B. (1991), Scenarios and policy aggregation in optimization under uncertainty, Math. Oper. Res. 16, 119–147.10.1287/moor.16.1.119CrossRefGoogle Scholar
Rogosinski, W. W. (1958), Moments of non-negative mass, Proc. R. Soc. A 245(1240), 1–27.Google Scholar
Römisch, W. and Surowiec, T. M. (2024), Asymptotic properties of Monte Carlo methods in elliptic PDE-constrained optimization under uncertainty, Numer. Math. 156, 1887–1914.10.1007/s00211-024-01436-5CrossRefGoogle Scholar
Rosseel, E. and Wells, G. N. (2012), Optimal control with stochastic PDE constraints and uncertain controls, Comput. Methods Appl. Mech. Engrg 213–216, 152–167.10.1016/j.cma.2011.11.026Google Scholar
Royset, J. O., Bonfiglio, L., Vernengo, G. and Brizzolara, S. (2017), Risk-adaptive set-based design and applications to shaping a hydrofoil, J. Mech. Design 139, art. 101403.10.1115/1.4037623CrossRefGoogle Scholar
Ruszczynski, A. and Shapiro, A., eds (2003), Stochastic Programming , Vol. 10 of Handbooks in Operations Research and Management Science, Elsevier.Google Scholar
Scarf, H. (1958), A min-max solution of an inventory problem, in Studies in The Mathematical Theory of Inventory and Production, Stanford University Press, pp. 201–209.Google Scholar
Schulz, V. and Schillings, C. (2013), Optimal aerodynamic design under uncertainty, in Management and Minimisation of Uncertainties and Errors in Numerical Aerodynamics: Results of the German collaborative project MUNA (Eisfeld, B. et al., eds), Springer, pp. 297–338.10.1007/978-3-642-36185-2_13CrossRefGoogle Scholar
Shapiro, A. (2017), Distributionally robust stochastic programming, SIAM J. Optim. 27, 2258–2275.10.1137/16M1058297CrossRefGoogle Scholar
Shapiro, A., Dentcheva, D. and Ruszczyński, A. (2014), Lectures on Stochastic Programming: Modeling and Theory, Vol. 9 of MPS/SIAM Series on Optimization, second edition, SIAM.10.1137/1.9781611973433CrossRefGoogle Scholar
Smith, R. C. (2014), Uncertainty Quantification: Theory, Implementation, and Applications, Vol. 12 of Computational Science & Engineering, SIAM.Google Scholar
Smolyak, S. A. (1963), Quadrature and interpolation formulas for tensor products of certain classes of functions, Dokl. Akad. Nauk SSSR 148, 1042–1045.Google Scholar
Spingarn, J. E. (1983), Partial inverse of a monotone operator, Appl. Math. Optim. 10, 247–265.10.1007/BF01448388CrossRefGoogle Scholar
Spingarn, J. E. (1985), Applications of the method of partial inverses to convex programming: Decomposition, Math. Program. 32, 199–223.10.1007/BF01586091CrossRefGoogle Scholar
Teckentrup, A. L., Scheichl, R., Giles, M. B. and Ullmann, E. (2013), Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients, Numer. Math. 125, 569–600.10.1007/s00211-013-0546-4CrossRefGoogle Scholar
Tiesler, H., Kirby, R. M., Xiu, D. and Preusser, T. (2012), Stochastic collocation for optimal control problems with stochastic PDE constraints, SIAM J. Control Optim. 50, 2659–2682.10.1137/110835438CrossRefGoogle Scholar
Toint, P. L. (1988), Global convergence of a class of trust-region methods for nonconvex minimization in Hilbert space, IMA J. Numer. Anal. 8, 231–252.10.1093/imanum/8.2.231CrossRefGoogle Scholar
Tröltzsch, F. (2010), Optimal Control of Partial Differential Equations: Theory, Methods and Applications, Vol. 112 of Graduate Studies in Mathematics, American Mathematical Society.Google Scholar
Uryasev, S. (1994), Derivatives of probability functions and integrals over sets given by inequalities, J. Comput. Appl. Math. 56, 197–223.10.1016/0377-0427(94)90388-3CrossRefGoogle Scholar
Uryasev, S. (1995), Derivatives of probability functions and some applications, Ann. Oper. Res. 56, 287–311.10.1007/BF02031712CrossRefGoogle Scholar
Uryasev, S., ed. (2000), Probabilistic Constrained Optimization: Methodology and Applications, Vol. 49 of Nonconvex Optimization and its Applications, Kluwer Academic.10.1007/978-1-4757-3150-7_1CrossRefGoogle Scholar
Uryasev, S. and Rockafellar, R. T. (2001), Conditional value-at-risk: Optimization approach, in Stochastic Optimization: Algorithms and Applications (Uryasev, S. and Pardalos, P. M., eds), Vol. 54 of Applied Optimization, Kluwer, pp. 411–435.10.1007/978-1-4757-6594-6_17CrossRefGoogle Scholar
Van Barel, A. and Vandewalle, S. (2019), Robust optimization of PDEs with random coefficients using a multilevel Monte Carlo method, SIAM/ASA J. Uncertain. Quantif. 7, 174–202.10.1137/17M1155892CrossRefGoogle Scholar
Van Barel, A. and Vandewalle, S. (2021), MG/OPT and multilevel Monte Carlo for robust optimization of PDEs, SIAM J. Optim. 31, 1850–1876.10.1137/20M1347164CrossRefGoogle Scholar
von Neumann, J. and Morgenstern, O. (2007), Theory of Games and Economic Behavior, sixtieth-anniversary edition, Princeton University Press.Google Scholar
Wathen, A. J. (2015), Preconditioning, Acta Numer. 24, 329–376.10.1017/S0962492915000021CrossRefGoogle Scholar
Wen, T. and Zahr, M. J. (2023), A globally convergent method to accelerate large-scale optimization using on-the-fly model hyperreduction: Application to shape optimization, J. Comput. Phys. 484, art. 112082.10.1016/j.jcp.2023.112082CrossRefGoogle Scholar
Xiu, D. (2010), Numerical Methods for Stochastic Computations: A Spectral Method Approach, Princeton University Press.Google Scholar
Zahr, M. J. and Farhat, C. (2015), Progressive construction of a parametric reduced-order model for PDE-constrained optimization, Int. J. Numer. Methods Engrg 102, 1111–1135.10.1002/nme.4770CrossRefGoogle Scholar
Zahr, M. J., Carlberg, K. T. and Kouri, D. P. (2019), An efficient, globally convergent method for optimization under uncertainty using adaptive model reduction and sparse grids, SIAM/ASA J. Uncertain. Quantif. 7, 877–912.10.1137/18M1220996CrossRefGoogle Scholar
Zandvliet, M., Van Essen, G. and Brouwer, D. (2008), Adjoint-based well-placement optimization under production constraints, SPE J. 13, 392–399.10.2118/105797-PACrossRefGoogle Scholar
Zhou, M., Lazarov, B. S. and Sigmund, O. (2014), Topology optimization for optical projection lithography with manufacturing uncertainties, Appl. Optics 53, 2720–2729.10.1364/AO.53.002720CrossRefGoogle ScholarPubMed
Ziems, J. C. (2013), Adaptive multilevel inexact SQP-methods for PDE-constrained optimization with control constraints, SIAM J. Optim. 23, 1257–1283.10.1137/110848645CrossRefGoogle Scholar
Ziems, J. C. and Ulbrich, S. (2011), Adaptive multilevel inexact SQP methods for PDE-constrained optimization, SIAM J. Optim. 21, 1–40.10.1137/080743160CrossRefGoogle Scholar
Zou, Z., Kouri, D. P. and Aquino, W. (2017), An adaptive sampling approach for solving PDEs with uncertain inputs and evaluating risk, in 19th AIAA Non-Deterministic Approaches Conference, AIAA SciTech Forum, art. AIAA 2017-1325.Google Scholar
Zou, Z., Kouri, D. P. and Aquino, W. (2018), A locally adapted reduced basis method for solving risk-averse PDE-constrained optimization problems, in 2018 AIAA Non-Deterministic Approaches Conference, AIAA SciTech Forum, art. AIAA 2018-2174.Google Scholar
Zou, Z., Kouri, D. P. and Aquino, W. (2019), An adaptive local reduced basis method for solving PDEs with uncertain inputs and evaluating risk, Comput. Methods Appl. Mech. Engrg 345, 302–322.10.1016/j.cma.2018.10.028CrossRefGoogle Scholar
Zou, Z., Kouri, D. P. and Aquino, W. (2022), A locally adapted reduced-basis method for solving risk-averse PDE-constrained optimization problems, SIAM/ASA J. Uncertain. Quantif. 10, 1629–1651.10.1137/21M1411342CrossRefGoogle Scholar