Skip to main content
SpringerLink
Log in

Control of Random PDEs: An Overview

  • Chapter
  • First Online:
Book cover Recent Advances in PDEs: Analysis, Numerics and Control

Part of the book series: SEMA SIMAI Springer Series ((SEMA SIMAI,volume 17))

Abstract

This work reviews theoretical and numerical concepts in the emergent field of optimal control of partial differential equations under uncertainty. The following topics are considered: uncertainty modelling in control problems using probabilistic tools, variational formulation of partial differential equations with random inputs, robust and risk averse formulations of optimal control problems, and numerical resolution methods. The exposition is focused on running the path starting from uncertainty modelling and ending in the practical implementation of numerical schemes for the numerical approximation of the considered problems. To this end, a selected number of illustrative examples is analysed.

Dedicated to Prof. Enrique Fernández-Cara on the occasion of his 60th birthday.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Babuška, I., Nobile, F., Tempone, R.: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM Rev. 52(2), 317–355 (2010)

    Article  MathSciNet  Google Scholar 

  2. Chen, P., Quarteroni, A.: A new algorithm for high-dimensional uncertainty quantification based on dimension-adaptive sparse grid approximation and reduced basis methods. J. Comput. Phys. 298, 176–193 (2015)

    Article  MathSciNet  Google Scholar 

  3. Chen, P., Quarteroni, A., Rozza, G.: Stochastic optimal Robin boundary control problems of advection-dominated elliptic equations. SIAM J. Numer. Anal. 51(5), 2700–2722 (2013)

    Article  MathSciNet  Google Scholar 

  4. Chen, P., Quarteroni, A., Rozza, G.: Comparison between reduced basis and stochastic collocation methods for elliptic problems. J. Sci. Comput. 59(1), 187–216 (2014)

    Article  MathSciNet  Google Scholar 

  5. Chiba, R.: Stochastic analysis of heat conduction and thermal stresses in solids: a review. In: Kazi, S.N. (ed.) Heat Transfer Phenomena and Applications. InTech, London (2012)

    Google Scholar 

  6. Cohen, A., DeVore, R.: Approximation of high-dimensional parametric PDEs. Acta Numer. 24, 1–159 (2015)

    Article  MathSciNet  Google Scholar 

  7. Ghanem, R.G., Spanos, P.D.: Stochastic Finite Elements. A Spectral Approach. Springer, Berlin (1981)

    Google Scholar 

  8. Gittelson, C.J., Andreev, R., Schwab, C.: Optimality of adaptive Galerkin methods for random parabolic partial differential equations. J. Comput. Appl. Math. 263, 189–201 (2014)

    Article  MathSciNet  Google Scholar 

  9. Gunzburger, M.D., Lee, H.-C., Lee, J.: Error estimates of stochastic optimal Neumann boundary control problems. SIAM J. Numer. Anal. 49(4), 1532–1552 (2011)

    Article  MathSciNet  Google Scholar 

  10. Halmos, P.: Measure Theory. Graduate Texts in Mathematics, vol. 18. Springer, New York (1970)

    Google Scholar 

  11. Hesthaven, J.S., Rozza, G., Stamm, B.: Certified Reduced Basis Methods for Parametrized Partial Differential Equations. Springer Briefs in Mathematics. BCAM, Bizkaia (2016)

    Book  Google Scholar 

  12. Hou, L.S., Lee, J., Manouzi, H.: Finite element approximations of stochastic optimal control problems constrained by stochastic elliptic PDEs. J. Math. Anal. Appl. 384(1), 87–103 (2011)

    Article  MathSciNet  Google Scholar 

  13. Keshavarzzadeh, V., Meidani, H., Tortorelli, A.: Gradient based design optimization under uncertainty via stochastic expansion methods. Comput. Methods Appl. Mech. Eng. 3016, 47–76 (2016)

    Article  MathSciNet  Google Scholar 

  14. Labovsky, A., Gunzburger, M.: An efficient and accurate method for the identification of the most influential random parameters appearing in the input data for PDEs. SIAM/ASA J. Uncertain. Quantif. 2(1), 82–105 (2014)

    Article  MathSciNet  Google Scholar 

  15. Light, W.A., Cheney, E.W.: Approximation Theory in Tensor Product Spaces. Lecture Notes in Mathematics, vol. 1169. Springer, New York (1985)

    Chapter  Google Scholar 

  16. Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)

    Book  Google Scholar 

  17. Lions, J.L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications, vol. I. Springer, New York (1972)

    MATH  Google Scholar 

  18. Lord, G.J., Powell, C.E., Shardlow, T.: An Introduction to Computational Stochastic PDEs. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2014)

    Book  Google Scholar 

  19. Marín, F.J., Martínez-Frutos, J., Periago, F.: Robust averaged control of vibrations for the Bernoulli-Euler beam equation. J. Optim. Theory Appl. 174(2), 428–454 (2017)

    Article  MathSciNet  Google Scholar 

  20. Marn, F.J., Martínez-Frutos, J., Periago, F.: A polynomial chaos-based approach to risk-averse piezoelectric control of random vibrations of beams. Int. J. Numer. Methods Eng. 1–18 (2018). https://doi.org/10.1002/nme.5823

    Article  MathSciNet  Google Scholar 

  21. Martínez-Frutos, J., Kessler, M., Periago, F.: Robust optimal shape design for an elliptic PDE with uncertainty in its input data. ESAIM Control Optim. Calc. Var. 21(4), 901–923 (2015)

    Article  MathSciNet  Google Scholar 

  22. Martínez-Frutos, J., Kessler, M., Münch, A., Periago, F.: Robust optimal Robin boundary control for the transient heat equation with random input data. Int. J. Numer. Methods Eng. 108(2), 116–135 (2016)

    Article  MathSciNet  Google Scholar 

  23. Martínez-Frutos, J., Herrero-Pérez, D., Kessler, M., Periago, F.: Robust shape optimization of continuous structures via the level set method. Comput. Methods Appl. Mech. Eng. 305, 271–291 (2016)

    Article  MathSciNet  Google Scholar 

  24. Nobile, F., Tempone, R., Webster, C.G.: An anisotropic sparse grid stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 46(5), 2411–2442 (2008)

    Article  MathSciNet  Google Scholar 

  25. Nouy, A.: Recent developments in spectral stochastic methods for the numerical solution of stochastic partial differential equations. Arch. Comput. Meth. Eng. 16(3), 251–285 (2009)

    Article  MathSciNet  Google Scholar 

  26. Rosseel, E., Wells, G.N.: Optimal control with stochastic PDE constrains and uncertain controls. Comput. Methods Appl. Mech. Eng. 213/216, 152–167 (2012)

    Article  MathSciNet  Google Scholar 

  27. Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods and Applications. Graduate Studies in Mathematics, vol. 112. AMS, Providence (2010)

    Google Scholar 

  28. Wloka, J.: Partial Differential Equations. Cambridge University Press, Cambridge (1987)

    Book  Google Scholar 

  29. Zhang, D.: Stochastic Methods for Flow in Porous Media: Coping with Uncertainties. Academic Press, San Diego (2002)

    Google Scholar 

Download references

Acknowledgements

Work supported by projects MTM2013-47053 from Ministerio de Economía y Competitividad (Spain), the AEI/FEDER and UE under the contract DPI2016-77538-R, and 19274/PI/14 from Fundación Séneca (Agencia de Ciencia y Tecnología de la Región de Murcia, Spain). The first author was also supported by a Ph.D. grant from Fundación Séneca.

Author information

Authors and Affiliations

  1. Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, Cartagena, Spain

    Francisco J. Marín & Francisco Periago

  2. Departamento de Estructuras y Construcción, Universidad Politécnica de Cartagena, Cartagena, Spain

    Jesús Martínez-Frutos

Authors
  1. Francisco J. Marín
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jesús Martínez-Frutos
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Francisco Periago
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Francisco Periago .

Editor information

Editors and Affiliations

  1. Facultad de Matemáticas, Universidad de Sevilla, Sevilla, Spain

    Anna Doubova

  2. Facultad de Matemáticas, Universidad de Sevilla, Sevilla, Spain

    Manuel González-Burgos

  3. Facultad de Matemáticas, Universidad de Sevilla, Sevilla, Spain

    Francisco Guillén-González

  4. Departamento de Informática y Análisis Numérico, Universidad de Córdoba, Córdoba, Spain

    Mercedes Marín Beltrán

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Marín, F.J., Martínez-Frutos, J., Periago, F. (2018). Control of Random PDEs: An Overview. In: Doubova, A., González-Burgos, M., Guillén-González, F., Marín Beltrán, M. (eds) Recent Advances in PDEs: Analysis, Numerics and Control. SEMA SIMAI Springer Series, vol 17. Springer, Cham. https://doi.org/10.1007/978-3-319-97613-6_10

Download citation

Publish with us

Policies and ethics

Search

Navigation