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Multilevel and weighted reduced basis method for stochastic optimal control problems constrained by Stokes equations

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Abstract

In this paper we develop and analyze a multilevel weighted reduced basis method for solving stochastic optimal control problems constrained by Stokes equations. We prove the analytic regularity of the optimal solution in the probability space under certain assumptions on the random input data. The finite element method and the stochastic collocation method are employed for the numerical approximation of the problem in the deterministic space and the probability space, respectively, resulting in many large-scale optimality systems to solve. In order to reduce the unaffordable computational effort, we propose a reduced basis method using a multilevel greedy algorithm in combination with isotropic and anisotropic sparse-grid techniques. A weighted a posteriori error bound highlights the contribution stemming from each method. Numerical tests on stochastic dimensions ranging from 10 to 100 demonstrate that our method is very efficient, especially for solving high-dimensional and large-scale optimization problems.

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References

  1. Arroyo, M., Heltai, L., Millán, D., DeSimone, A.: Reverse engineering the euglenoid movement. Proc Nat Acad Sci 109(44), 17874–17879 (2012)

    Article  Google Scholar 

  2. Babuška, I., Nobile, F., Tempone, R.: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J Numer Anal 45(3), 1005–1034 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Babuška, I., Tempone, R., Zouraris, G.E.: Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J Numer Anal 42(2), 800–825 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Barrault, M., Maday, Y., Nguyen, N.C., Patera, A.T.: An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations. Comptes Rendus Mathematique Anal Numérique 339(9), 667–672 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Binev, P., Cohen, A., Dahmen, W., DeVore, R., Petrova, G., Wojtaszczyk, P.: Convergence rates for greedy algorithms in reduced basis methods. SIAM J Math Anal 43(3), 1457–1472 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bochev, P.B., Gunzburger, M.D.: Least-squares finite-element methods for optimization and control problems for the Stokes equations. Comput Math Appl 48(7), 1035–1057 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bochev, P.B., Gunzburger, M.D.: Least-squares finite element methods, vol. 166. Springer (2009)

  8. Boyaval, S., Le Bris, C., Lelièvre, T., Maday, Y., Nguyen, N.C., Patera, A.T.: Reduced basis techniques for stochastic problems. Arch Comput Methods Eng 17, 435–454 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral methods: fundamentals in single domains. Springer (2006)

  10. Chen, P., Quarteroni, A.: Weighted reduced basis method for stochastic optimal control problems with elliptic PDE constraints. SIAM/ASA J Uncertain Quantif 2(1), 364–396 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen, P., Quarteroni, A., Rozza, G.: A weighted reduced basis method for elliptic partial differential equations with random input data. SIAM J Numer Anal 51(6), 3163–3185 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Chen, P., Quarteroni, A., Rozza, G.: A weighted empirical interpolation method: a priori convergence analysis and applications. ESAIM Math Modell Numer Anal 48(7):943–953 (2014)

  15. Durrett, R.: Probability: theory and examples. Cambridge University Press (2010)

  16. Elman, H., Liao, Q.: Reduced basis collocation methods for partial differential equations with random coefficients. SIAM/ASA J Uncertain Quantif 1(1), 192–217 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Evans, L.C.: Partial differential equations, graduate studies in mathematics, vol. 19, American Mathematical Society (2009)

  18. Formaggia, L., Quarteroni, A., Veneziani, A.: Cardiovascular mathematics: modeling and simulation of the circulatory system, vol. 1. Springer, MS&A (2009)

    Book  MATH  Google Scholar 

  19. Gerner, A.L., Veroy, K.: Certified reduced basis methods for parametrized saddle point problems. SIAM J Sci Comput 34(5), A2812–A2836 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gerstner, T., Griebel, M.: Dimension-adaptive tensor-product quadrature. Computing 71(1), 65–87 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Glowinski, R., Lions, J.L.: Exact and approximate controllability for distributed parameter systems. Cambridge University Press (1996)

  22. Gunzburger, M.D.: Perspectives in flow control and optimization, vol. 5. SIAM (2003)

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Gunzburger, M.D., Manservisi, S.: Analysis and approximation of the velocity tracking problem for Navier-Stokes flows with distributed control. SIAM J Numer Anal 37(5), 1481–1512 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  25. Haasdonk, B., Urban, K., Wieland, B.: Reduced basis methods for parameterized partial differential equations with stochastic influences using the Karhunen-Love expansion. SIAM/ASA J Uncertain Quantif 1(1), 79–105 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. 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  MATH  Google Scholar 

  27. Huynh, D.B.P., Knezevic, D.J., Chen, Y., Hesthaven, J.S., Patera, A.T.: A natural-norm successive constraint method for inf-sup lower bounds. Comput Method Appl Mech Eng 199(29), 1963–1975 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. Junseok, K.: Phase-field models for multi-component fluid flows. Commun Comput Phy 12(3), 613–661 (2012)

    MathSciNet  Google Scholar 

  29. Kärcher M., Grepl, M.: A certified reduced basis method for parametrized elliptic optimal control problems. Accepted in ESAIM: Control, Optimisation and Calculus of Variations(2013)

  30. Kärcher, M., Grepl, M., Veroy, K.: Certified reduced basis methods for parametrized distributed optimal control problems. Manuscript (2014)

  31. Kouri, D.P., Heinkenschloos, D., Ridzal, M., Van Bloemen Waanders, B.G.: A trust-region algorithm with adaptive stochastic collocation for PDE optimization under uncertainty. Preprint ANL/MCS-P3035-0912 (2012)

  32. Kunisch, K., Volkwein, S.: Proper orthogonal decomposition for optimality systems. ESAIM Math Modell Numer Anal 42(1), 1 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  33. Lions, J.L.: Optimal control of systems governed by partial differential equations. Springer (1971)

  34. Manzoni, A.: Reduced models for optimal control, shape optimization and inverse problems in haemodynamics. PhD thesis, EPFL (2012)

  35. Negri, F.: Reduced basis method for parametrized optimal control problems governed by PDEs, Master Thesis, Politecnico di Milano (2011)

  36. Negri, F., Manzoni, A., Rozza, G.: Reduced basis approximation of parametrized optimal flow control problems for the Stokes equations. EPFL, MATHICSE Report 02, submitted (2014)

  37. Negri, F., Rozza, G., Manzoni, A., Quarteroni, A.: Reduced basis method for parametrized elliptic optimal control problems. SIAM J Sci Comput 35(5), A2316–A2340 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  39. Nobile, F., Tempone, R., Webster, C.G.: A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J Numer Anal 46(5), 2309–2345 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  40. Prudhomme, C., Rovas, D.V., Veroy, K., Machiels, L., Maday, Y., Patera, A.T., Turinici, G.: Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods. J Fluids Eng 124(1), 70–80 (2002)

    Article  Google Scholar 

  41. Quarteroni, A.: Numerical models for differential problems. Springer, MS & A 8 (2013)

  42. Rees, T., Wathen, A.J.: Preconditioning iterative methods for the optimal control of the Stokes equations. SIAM J Sci Comput 33(5), 2903–2926 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  43. Rosseel, E., Wells, G.N.: Optimal control with stochastic PDE constraints and uncertain controls. Comput Methods Appl Mech Eng 213C216(0):152–167 (2012)

  44. Rovas, D.V.: Reduced-basis output bound methods for parametrized partial differential equations. PhD thesis, Massachusetts Institute of Technology (2003)

  45. Rozza, G., Huynh, D.B.P., Patera, A.T.: Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch Comput Methods Eng 15(3), 229–275 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  46. Rozza, G., Manzoni, A., Negri, F.: Reduction strategies for PDE-constrained oprimization problems in Haemodynamics. In: Proceedings of ECCOMAS CFD (2012)

  47. Rozza, G., Veroy, K.: On the stability of the reduced basis method for Stokes equations in parametrized domains. Comput Methods Appl Mech Eng 196(7), 1244–1260 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  48. Schöberl, J., Zulehner, W.: Symmetric indefinite preconditioners for saddle point problems with applications to PDE-constrained optimization problems. SIAM J Matrix Anal Appl 29(3), 752–773 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  49. Schwab, C., Todor, R.A.: Karhunen-Loève approximation of random fields by generalized fast multipole methods. J Comput Phy 217(1), 100–122 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  50. Tiesler, H., Kirby, R.M., Xiu, D., Preusser, T.: Stochastic collocation for optimal control problems with stochastic PDE constraints. SIAM J Control Optim 50(5), 2659–2682 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  51. Tröltzsch, F.: Optimal control of partial differential equations: theory, methods, and applications, vol. 112. American Mathematical Society (2010)

  52. Veroy, K., Prudhomme, C., Rovas, D.V., Patera, A.T.: A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations. In: Proceedings of the 16th AIAA computational fluid dynamics conference, vol. 3847 (2003)

  53. Xiu, D., Hesthaven, J.S.: High-order collocation methods for differential equations with random inputs. SIAM J Sci Comput 27(3), 1118–1139 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  54. Xu, J., Zikatanov, L.: Some observations on Babuška and Brezzi theories. Numerische Mathematik 94(1), 195–202 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

We would like to acknowledge the reviewers for many helpful comments and suggestions. The Matlab packages MLife previously developed by Prof. Fausto Saleri from MOX, Politecnico di Milano and rbMIT developed by Prof. Anthony Patera and his coworkers from Massachusetts Institute of Technology are acknowledged. This work is partially supported by FNS \(200021\_141034\). G. Rozza acknowledges NOFYSAS excellence grant of SISSA.

Author information

Authors and Affiliations

  1. Seminar for Applied Mathematics ETH Zurich, Rämistrasse 101, 8092, Zurich, Switzerland

    Peng Chen

  2. Chair of Modelling and Scientific Computing, Mathematics Institute of Computational Science and Engineering, EPFL, Station 8, 1015, Lausanne, Switzerland

    Alfio Quarteroni

  3. SISSA MathLab, International School for Advanced Studies, via Bonomea 265, 34136, Trieste, Italy

    Gianluigi Rozza

Authors
  1. Peng Chen
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  2. Alfio Quarteroni
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  3. Gianluigi Rozza
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Corresponding author

Correspondence to Peng Chen.

Appendix

Appendix

We follow the analysis of a general saddle point problem in Ch. 16.3.2 of [41] to prove a variation of the Brezzi theorem that has been used to prove (3.9) and (3.14). Let us introduce the affine manifold

$$\begin{aligned} X^{\sigma } = \{ v\in X: \mathcal {B}(v,\mu ) = (\sigma , \mu ) \quad \forall \mu \in M\}, \end{aligned}$$
(8.1)

and the kernal of \(\mathcal {B}\) as

$$\begin{aligned} X^0 = \{v\in X: \mathcal {B}(v,\mu ) = 0 \quad \forall \mu \in M\}, \end{aligned}$$
(8.2)

where we specify \(X = V\times Q\times G\) with element \(v = \{\mathbf {v},q,\mathbf {g}\}\), and \(M = V\times Q\) with element \(\mu = \{\mathbf {v},q\}\) in our particular case. Moreover, we specify \(\sigma \) and l such that \(<\sigma , \mu > = r.h.s. (3.8)_2\) and \(<l,v> = r.h.s. (3.8)_1\). We can therefore associate (3.8) with the following reduced problem

$$\begin{aligned} \text {find } u \in X^{\sigma } \text { such that } \mathcal {A}(u,v) =\; <l,v> \quad \forall v \in X^0. \end{aligned}$$
(8.3)

Thanks to the inf-sup condition of \(\mathcal {B}\), we can infer that there exists a unique function \(u^{\sigma } \in (X^0)^{\bot }\) such that \(\mathcal {B}(u^{\sigma }, \mu ) = <\sigma , \mu > \, \forall \mu \in M\). Moreover, since

$$\begin{aligned} |<\sigma , \mu >| = |(\mathbf {h}_{\tau }, \mathbf {v})_{\partial D_N}- (\nu _{\tau }(y)\nabla \mathbf {u}, \nabla \mathbf {v})| \le \left( ||\mathbf {h}_{\tau }||_H + \frac{|\nu _{\tau }(y)|}{\nu (\bar{y})}||\mathbf {u}||_V\right) ||\mathbf {v}||_{V}, \end{aligned}$$
(8.4)

we obtain (by denoting the inf-sup constant of \(\mathcal {B}\) as \(\beta ^*\))

$$\begin{aligned} ||u^{\sigma }||_X \le \frac{1}{\beta ^*} \left( ||\mathbf {h}_{\tau }||_H + \frac{\nu _{\tau }(y)}{\nu (\bar{y})}||\mathbf {u}||_V\right) . \end{aligned}$$
(8.5)

The reduced problem (8.3) can therefore be restated as

$$\begin{aligned} \text {find } \tilde{u} \in X^0 \text { such that } \mathcal {A}(\tilde{u},v) = \,<l,v> - \mathcal {A}(u^{\sigma },v) \quad \forall v \in X^0. \end{aligned}$$
(8.6)

Thanks to the coercivity and continuity of \(\mathcal {A}\), existence and uniqueness of the solution \(\tilde{u}\) follow by the Lax-Milgram theorem. Moreover, since

$$\begin{aligned} |<l,v>| = |- (\nu _{\tau }(y)\nabla \mathbf {u}^a,\nabla \mathbf {v}^a)| \le \frac{|\nu _{\tau }(y)|}{\nu (\bar{y})} ||\mathbf {u}^a||_V ||\mathbf {v}^a||_V, \end{aligned}$$
(8.7)

we obtain (by denoting the coercivity and continuity constants of \(\mathcal {A}\) as \(\alpha \) and \(\gamma \))

$$\begin{aligned} ||\tilde{u}||_X \le \frac{1}{\alpha }\left( \frac{|\nu _{\tau }(y)|}{\nu (\bar{y})} ||\mathbf {u}^a||_V + \gamma ||u^{\sigma }||_X\right) \end{aligned}$$
(8.8)

Therefore, the solution u can be bounded by

$$\begin{aligned} ||u||_X \le ||u^{\sigma }||_X + ||\tilde{u}||_X \le \frac{1}{\alpha } \frac{|\nu _{\tau }(y)|}{\nu (\bar{y})} ||\mathbf {u}^a||_V + \frac{\alpha +\gamma }{\alpha \beta ^*} \left( ||\mathbf {h}_{\tau }||_H + \frac{\nu _{\tau }(y)}{\nu (\bar{y})}||\mathbf {u}||_V\right) . \end{aligned}$$
(8.9)

Let A be such that \(<Au, v> = \mathcal {A}(u,v) \, \forall u \in X^{\sigma }, v\in X^0\); we can restate (8.3) as \(<Au - l, v> = 0 \, \forall v \in X^0\). It follows that \((Au-l) \in X_{polar}^0\), where \(X_{polar}^0\) is the polar set of \(X^0\) defined as \( X_{polar}^0 = \{g\in X': <g,v> = 0 \quad \forall v \in X^0\}. \) Therefore, there exists a unique \(\eta \in M\) such that \(-\mathcal {B}(v,\eta ) = <Au-l,v> \, \forall v \in X\). Moreover,

$$\begin{aligned} ||\eta ||_M&\le \frac{1}{\beta ^*} \left( \gamma ||u||_X + \frac{|\nu _{\tau }(y)|}{\nu (\bar{y})} ||\mathbf {u}^a||_V \right) \nonumber \\&\le \frac{\alpha + \gamma }{\alpha \beta ^*} \frac{|\nu _{\tau }(y)|}{\nu (\bar{y})} ||\mathbf {u}^a||_V + \frac{\gamma (\alpha +\gamma )}{\alpha (\beta ^*)^2}\left( ||\mathbf {h}_{\tau }||_H + \frac{\nu _{\tau }(y)}{\nu (\bar{y})}||\mathbf {u}||_V\right) . \end{aligned}$$
(8.10)

We can identify the constants in (3.1) as

$$\begin{aligned} \alpha _1 = \frac{1}{\alpha }; \quad \alpha _2 = \frac{\alpha + \gamma }{\alpha \beta ^*}; \quad \beta _1 = \frac{\alpha +\gamma }{\alpha \beta ^*}; \quad \beta _2 = \frac{\gamma (\alpha +\gamma )}{\alpha (\beta ^*)^2}. \end{aligned}$$
(8.11)

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Chen, P., Quarteroni, A. & Rozza, G. Multilevel and weighted reduced basis method for stochastic optimal control problems constrained by Stokes equations. Numer. Math. 133, 67–102 (2016). https://doi.org/10.1007/s00211-015-0743-4

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