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

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.

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)