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A Priori Error Estimate of Stochastic Galerkin Method for Optimal Control Problem Governed by Stochastic Elliptic PDE with Constrained Control

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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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Correspondence to Wenbin Liu.

Additional information

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

$$\begin{aligned} a(x,\omega )=\bar{a}(x)+r(x,\omega ), \quad \text{ where } \quad r(x,\omega )=\sum \limits _{m=1}^{\infty }\psi _m(x)\otimes X_m(\omega ), \quad \forall \,\, (x, \omega )\in D\times {\varOmega }, \end{aligned}$$
(7.1)

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

$$\begin{aligned} a_M(x,\omega )=\bar{a}(x)+\sum \limits _{m=1}^{M}\psi _m(x)\otimes X_m(\omega ), \quad \forall \,\, (x, \omega )\in D\times {\varOmega }. \end{aligned}$$
(7.2)

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

$$\begin{aligned} \Vert \psi _m\otimes X_m\Vert _{L^\infty (D\times \varOmega )}\rightarrow 0, \quad \text{ as }\quad m\rightarrow +\infty . \end{aligned}$$
(7.3)

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

$$\begin{aligned} a_N(x,\omega )=\mathbb {E}[a](x)+\sum \limits _{n=1}^{N}\sqrt{\lambda _n} b_n(x)Y_n(\omega ), \end{aligned}$$
(7.4)

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

$$\begin{aligned} \sup \limits _{x\in D}\mathbb {E}[(a-a_N)^2](x)=\sup \limits _{x\in D}\sum \limits _{n=N+1}^{+\infty }\lambda _n b_n^2(x)\rightarrow 0 \quad \text{ as }\quad N\rightarrow +\infty . \end{aligned}$$
(7.5)

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

$$\begin{aligned} \Vert a-a_M\Vert _{L^\infty (D\times \varOmega )}\rightarrow 0, \quad \text{ as }\quad M\ge M_r \,\, \,\text{ and } \,\, \,M\rightarrow +\infty . \end{aligned}$$
(7.6)

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

$$\begin{aligned} -\nabla \cdot [ a(x,\omega )\nabla y(x,\omega )]=u(x), \quad \,\,\, x\in D,\,\, \omega \in {\varOmega }, \end{aligned}$$
(7.7)

and

$$\begin{aligned} -\nabla \cdot [ a_M(x,\omega )\nabla y_M(x,\omega )]=u(x), \quad \,\,\, x\in D,\,\, \omega \in {\varOmega }, \end{aligned}$$
(7.8)

respectively, then

$$\begin{aligned} \Vert y-y_M\Vert _{L^2(\varOmega ; H_0^1(D))}\le \frac{C}{a_{min}} \Vert a-a_M\Vert _{L^\infty (\varOmega \times D)} \Vert y\Vert _{L^2(\varOmega ; H_0^1(D))}, \end{aligned}$$
(7.9)

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

$$\begin{aligned} ( a\nabla y, \nabla v)=(u,v), \quad \forall \,\, v\in L^2(\varOmega ; H_0^1(D)), \end{aligned}$$
(7.10)

and

$$\begin{aligned} ( a_M\nabla y_M, \nabla v)=(u, v), \quad \forall \,\, v\in L^2(\varOmega ; H_0^1(D)), \end{aligned}$$
(7.11)

respectively.

Subtracting equation (7.11) from equation (7.10), taking \(v=y-y_M\), we have

$$\begin{aligned} \bigl ( a_M\nabla (y-y_M), \nabla (y-y_M) \bigr )=-\bigl ((a-a_M)\nabla y,\nabla (y-y_M)\bigr ). \end{aligned}$$
(7.12)

Using the diffusion coefficient condition (2.3) in (7.12) allows us to write

$$\begin{aligned} a_{min}\Vert y-y_M\Vert _{L^2(\varOmega ; H_0^1(D))}^2\le & {} \bigl ( a_M\nabla (y-y_M), \nabla (y-y_M) \bigr ) \nonumber \\= & {} -\bigl ((a-a_M)\nabla y,\nabla (y-y_M)\bigr )\nonumber \\\le & {} \Vert a-a_M\Vert _{L^\infty (\varOmega \times D)}\Vert y\Vert _{L^2(\varOmega ; H_0^1(D))}\Vert y-y_M\Vert _{L^2(\varOmega ; H_0^1(D))}.\nonumber \\ \end{aligned}$$
(7.13)

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

$$\begin{aligned} \Vert \psi _m\otimes X_m\Vert _{L^\infty (D\times \varOmega )}\le c_{r}exp(-c_{1,r}m^k), \quad \forall \,\, m\in \mathbb {N}. \end{aligned}$$
(7.14)

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

$$\begin{aligned} \Vert a-a_M\Vert _{L^\infty (D\times \varOmega )}\le c_{r}exp(-c_{1,r}M^k), \quad \forall \,\, M\in \mathbb {N}. \end{aligned}$$
(7.15)

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

$$\begin{aligned} \Vert y-y_M\Vert _{L^2(\varOmega ; H_0^1(D))}\le \frac{c_{r}C}{a_{min}}exp(-c_{1,r}M^k) \Vert y\Vert _{L^2(\varOmega ; H_0^1(D))}, \end{aligned}$$
(7.16)

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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