  In this paper, a local-global model reduction method is presented to solve
stochastic optimal control problems governed by partial differential equations
(PDEs). If the optimal control problems involve uncertainty, we need to use a
few random variables to parameterize the uncertainty. The stochastic optimal
control problems require solving coupled optimality system for a large number
of samples in the stochastic space to quantify the statistics of the system
response and explore the uncertainty quantification. Thus the computation is
prohibitively expensive. To overcome the difficulty, model reduction is
necessary to significantly reduce the computation complexity. We exploit the
advantages from both reduced basis method and Generalized Multiscale Finite
Element Method (GMsFEM) and develop the local-global model reduction method for
stochastic optimal control problems with PDE constraints. This local-global
model reduction can achieve much more computation efficiency than using only
local model reduction approach and only global model reduction approach. We
recast the stochastic optimal problems in the framework of saddle-point
problems and analyze the existence and uniqueness of the optimal solutions of
the reduced model. In the local-global approach, most of computation steps are
independent of each other. This is very desirable for scientific computation.
Moreover, the online computation for each random sample is very fast via the
proposed model reduction method. This allows us to compute the optimality
system for a large number of samples. To demonstrate the performance of the
local-global model reduction method, a few numerical examples are provided for
different stochastic optimal control problems.
