Abstract
In this paper, we investigate a stochastic Galerkin approximation scheme for an optimal control problem governed by an elliptic PDE with random field in its coefficients. The objective is to minimize the expectation of a cost functional with the deterministic constrained control. We represent the random elliptic PDE in term of the generalized polynomial chaos expansion and obtain the deterministic optimal problem. By applying the well-known Lions’ Lemma to the reduced optimal problem, we obtain the necessary and sufficient optimality conditions. We establish a scheme to approximate the optimality system through the discretization with respect to both the spatial space and the probability space by Stochastic Galerkin method. Then a priori error estimates are derived for the state, the co-state and the control variables. Numerical examples are presented to illustrate our theoretical results.
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References
Babuska, I., Chatzipantelidis, P.: On solving elliptic stochastic partial differential equations. Comput. Methods Appl. Mech. Eng. 191, 4093–4122 (2002)
Babuska, I., Liu, K., Tempone, R.: Solving stochastic partial differential equations based on the experimental data. Math. Models Methods Appl. Sci. 13(3), 415–444 (2003)
Deb, M.K., Babuska, I., Oden, J.T.: Solution of stochastic partial differential equations using Galerkin finite element techniques. Comput. Methods Appl. Mech. Eng. 190, 6359–6372 (2001)
Schwab, C., Todor, R.A.: Sparse finite elements for elliptic problems with stochastic loading. Numer. Math. 95, 707–734 (2003)
Papadrakakis, M., Papadopoulos, V.: Robust and efficient methods for stochastic finite element analysis using Monte Carlo simulation. Comput. Methods Appl. Mech. Eng. 134, 325–340 (1996)
Doltsinis, I.: Inelastic deformation processes with random parameters methods of analysis and design. Comput. Methods Appl. Mech. Eng. 192, 2405–2423 (2003)
Fishman, G.: Monte Carlo, Concepts, Algorithms, and Applications. Springer, New York (1996)
Kleiber, M., Hien, T.D.: The Stochastic Finite Element Method. Wiley, Chichester (1992)
Babuska, 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)
Xiu, D., Hesthaven, J.S.: High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27(3), 1118–1139 (2005)
Foo, J., Wan, X., Karniadakis, G.E.: The multi-element probabilistic collocation method (ME-PCM): error analysis and applications. J. Comput. Phys. 227(22), 9572–9595 (2008)
Ma, X., Zabaras, N.: An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations. J. Comput. Phys. 228(8), 3084–3113 (2009)
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)
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)
Beck, J., Nobile, F., Tamellini, L., Tempone, R.: Stochastic spectral Galerkin and collocation methods for PDEs with random coefficients: a numerical comparison. In: Spectral and High Order Methods for Partial Differential Equations, vol. 76. Springer, Berlin, pp. 43–62 (2011)
Ghanem, R., Spanos, P.D.: Stochastic Finite Elements: A Spectral Approach. Springer, Berlin (1991)
Babuska, I., Tempone, R., Zouraris, G.E.: Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42(2), 800–825 (2004)
Ghanem, R.: Ingredients for a general purpose stochastic finite elements implementation. Comput. Methods Appl. Mech. Eng. 168, 19–34 (1999)
Keese, A., Matthies, H.G.: Parallel solution of stochastic PDEs. Proc. Appl. Math. Mech. 2, 485–486 (2003)
Xiu, D., Karniadakis, G.E.: The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24, 619–644 (2002)
Xiu, D., Lucor, D., Su, C.H., Karniadakis, G.E.: Stochastic modeling of flow-structure interactions using generalized polynomial chaos. ASME J. Fluid Eng. 124, 51–69 (2002)
Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)
Glowinski, R., Lions, J.L.: Exact and Approximate Controllability for Distributed Parameter Systems. Cambridge University Press, Cambridge (1996)
Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods, and Applications, vol. 112. American Mathematical Society, Providence (2010)
Liu, W.B., Tiba, D.: Error estimates for the finite element approximation of a class of nonlinear optimal control problems. J. Numer. Funct. Optim. 22, 953–972 (2001)
Liu, W.B., Yan, N.N.: A posteriori error estimates for convex boundary control problems. SIAM Numer. Anal. 39, 73–99 (2001)
Liu, W.B., Yan, N.N.: Adaptive Finite Element Methods for Optimal Control Governed by PDEs. Series in Information and Computational Science, vol. 41. Science Press, Beijing (2008)
Sun, T.J.: Discontinuous Galerkin finite element method with interior penalties for convection diffusion optimal control problem. Int. J. Numer. Anal. Model 7(1), 87–107 (2010)
Sun, T.J., Ge, L., Liu, W.B.: Equivalent a posteriori error estimates for a constrained optimal control problem governed by parabolic equations. Int. J. Numer. Anal. Model 10(1), 1–23 (2013)
Liu, W.B., Yang, D.P., Yuan, L., Ma, C.Q.: Finite elemnet approximation of an optimal control problem with integral state constraint. SIAM J. Numer. Anal. 48(3), 1163–1185 (2010)
Du, N., Shi, J.T., Liu, W.B.: An effective gradient projection method for stochastic optimal control. Int. J. Numer. Anal. Model. 10(4), 757–774 (2013)
Shen, W.F., Ge, L., Yang, D.P.: Finite element methods for optimal control problems governed by linear quasi-parabolic integer-differential equations. Int. J. Numer. Anal. Model 10(3), 536–550 (2013)
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)
Rosseel, E., Wells, G.N.: Optimal control with stochastic PDE constrains and uncertain controls. Comput. Methods Appl. Mech. Eng. 213–216, 152–167 (2012)
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, 87–103 (2011)
Lee, H.C., Lee, J.: A stochastic Galerkin method for stochastic control problems. Commun. Comput. Phys. 14(1), 77–106 (2013)
Adams, R.: Sobolev Spaces. Academic, New York (1975)
Wiener, N.: The homogeneous chaos. Am. J. Math. 60, 897–936 (1938)
Malliavin, P.: Stochastic Analysis. Springer, Berlin (1997)
Keese, A.: Numerical Solution of Systems with Stochastic Uncertainties: A General Purpose Framework for Stochastic Finite Elements. Ph.D. Thesis, Technical University Braunschweig, Braunschweig, Germany (2004)
Evans, L.: Partial differential equations. Grad. Stud. Math., vol. 19. AMS, Providence (1998)
Øksendal, B.: Stochastic Differential Equations: An Introduction with Application, 5th edn. Spring, Berlin (1998)
Fursikov, A.V.: Optimal Control of Distributed Systems, Theory and Applications. American Mathematical Society, Providence (2000)
Becker, R., Vexler, B.: Optimal control of the convection-diffusion equation using stabilized finite element methods. Numer. Math. 106, 349–367 (2007)
Ciarlet, P.G.: The finite element method for elliptic problems. In: Classics Appl. Math., vol. 40. SIAM, Philadelphia (2002)
Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 2nd edn. Springer, Berlin (2002)
Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Longman Higher Education, London (1986)
Todor, R.A., Schwab, C.: Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients. IMA J. Numer. Anal. 27, 232–261 (2007)
Cohen, A., DeVore, R., Schwab, C.: Convergence rates of best N-term Galerkin approximations for a class of elliptic sPDEs. Found. Comput. Math. 10, 615–646 (2010)
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This work is supported by the NSF of China (Nos. 11271231, 11301300 and 11501326), the NSF of Shandong Province (No. ZR2010AQ019).
Appendix 1: Convergence Analysis
Appendix 1: Convergence Analysis
In this appendix, we present some remarks about the convergence of the truncated problem to the original stochastic problem. It is a complicated theoretical problem which can not be discussed overall clearly here. We consider the general expansion of the stochastic diffusion coefficient \(a(x,\omega )\). The readers also can see some relevant discussions in [17, 48, 49].
1.1 Proof of Convergence
It is natural to see that some assumptions on the expansion of \(a(x,\omega )\) are needed to guarantee this convergence. Generally, we assume that the deterministic and stochastic variables in \(a(x,\omega )\) can be separated (cf. [34, 48]).
Assumption 7.1
The random field \(a\in L^\infty (\varOmega \times D)\) in (2.2) can be represented as
with known deterministic \(\bar{a},\,\psi _m \in \, L^\infty (D)\) and stochastic \(X_m \in \, L^\infty (\varOmega )\). Here, \(\bar{a}(x)\) is not necessarily equal to the mean field \(\mathbb {E}_a\). Without loss of generality, we also assume that \(\psi _m,\,X_m\, \ne 0\) for all \(m\in \mathbb {N_+}\).
Since computations can handle only finite data sets, we truncate the fluctuation expansion (7.1) and introduce, for \(M\in \mathbb {N}\), the truncated stochastic function as
In order to allow control of the error in the solution to (2.2) after truncation of r, we also require uniform convergence of the fluctuation r.
Assumption 7.2
The fluctuation r in the representation of (7.1) satisfies
This assumption is reasonable. Here, we recall similar assumptions were made in the references as follows.
Remark 7.1
In [17], the truncated K–L expansion of the stochastic process a is
where \(\{\lambda _n,\,b_n(x)\}_{n=1}^\infty \) are eigenvalues and corresponding orthogonal eigenfunctions of a compact self-adjoint operator, the real random variables \(\{Y_n(\omega )\}_{n=1}^\infty \) are mutually uncorrelated and have mean zero and unit variance. By using Mercers theorem, the convergence analysis of the truncated problem was proved based on the following result
Remark 7.2
Furthermore, In [49], for general expansion (7.1) in an orthogonal basis, algebraic convergence rates independent of N were derived for the spectral approach under rather mild assumptions on the smoothness of a. The key feature lies in the choice of a particular sparse tensor product polynomial space which can be interpreted as a form of non-linear best N-term approximation.
Due to Assumption 7.2, the following pointwise error estimate holds.
Proposition 7.3
If Assumption 7.2 holds, then there exists a truncation order \(M_r\in \mathbb {N}\) of the expansion (7.1) such that
The new elliptic problem with truncated diffusion coefficient \(a_M\) is therefore well-posed for M large enough (depending on a). This follows immediately from Strang’s lemma, which allows also explicit control of the error in the solution y to (2.2).
Corollary 7.4
If the stochastic diffusion coefficient a satisfy Assumptions 7.1 and 7.2, then there exists a truncation order \(M_r\in \mathbb {N}\) of the expansion (7.1) such that (7.8) below is well-posed in \(L^\infty (\varOmega ; H_0^{1}(D))\) for any \(M\ge M_r\). Moreover, if y and \(y_M\) are the unique solutions in \(L^2(\varOmega ; H_0^1(D))\) of
and
respectively, then
for all \(M\ge M_r\). Here, \(C>0\) is the Poincaré constant for the domain D.
Proof
Since y solves (7.7) and \(y_M\) solves (7.8), using \(v\in L^2(\varOmega ; H_0^1(D))\) leads to the following weak variational formulations
and
respectively.
Subtracting equation (7.11) from equation (7.10), taking \(v=y-y_M\), we have
Using the diffusion coefficient condition (2.3) in (7.12) allows us to write
Using Poincaré inequality and Proposition 7.3, dividing the both-hand terms of(7.13) by \(\Vert y-y_M\Vert _{L^2(\varOmega ; H_0^1(D))}\), which leads to (7.9). \(\square \)
From Corollary 7.4, we can prove the convergence of the truncated problem to the original stochastic problem.
Corollary 7.5
If the stochastic diffusion coefficient a satisfy Assumptions 7.1 and 7.2, then there exists a truncation order \(M_r\in \mathbb {N}\) of the expansion (7.1) such that the truncated problem (7.8) converges to the original stochastic problem (2.2) for \(M\ge M_r\) and \(M\rightarrow +\infty \).
1.2 Convergence Rate
If the convergence rate of the truncated problem to the original stochastic problem can be estimated, it will contribute much to many practical applications. Of course, to derive the convergence rate, some more precise assumptions on the expansion of \(a(x,\omega )\) are needed. In [48], exponential convergence rates independent of the truncation order N of the expansion a were proved under the following assumption.
Assumption 7.3
Assume that the terms in the expansion (7.1) decay exponentially to 0 in the \(L^\infty \) norm, i.e. there exist constants \(c_{r},\,c_{1,r},\,k>0\) such that
Several choices satisfying the above assumption are possible, such as the Legendre and KL expansions, which are discussed in [48]. And it was shown that the assumption (7.14) holds with \(k=1/d\) if the fluctuation r is piecewise analytic in the physical variable \(x\in \,D \in \mathbb {R}^d\).
By Assumption 7.3, the following pointwise error estimate holds.
Proposition 7.4
If Assumption 7.3 holds, then
Similarly to the proof of Corollary 7.4, we can obtain:
Corollary 7.6
If the stochastic diffusion coefficient a satisfy Assumptions 7.1 and 7.3, then there exists a truncation order \(M_r\in \mathbb {N}\) of the expansion (7.1) such that
for all \(M\ge M_r\). Here, \(C>0\) is the Poincaré constant for the domain D.
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Sun, T., Shen, W., Gong, B. et al. A Priori Error Estimate of Stochastic Galerkin Method for Optimal Control Problem Governed by Stochastic Elliptic PDE with Constrained Control. J Sci Comput 67, 405–431 (2016). https://doi.org/10.1007/s10915-015-0091-7
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DOI: https://doi.org/10.1007/s10915-015-0091-7