Preconditioners for robust optimal control problems under uncertainty

by   Fabio Nobile, et al.

The discretization of robust quadratic optimal control problems under uncertainty using the finite element method and the stochastic collocation method leads to large saddle-point systems, which are fully coupled across the random realizations. Despite its relevance for numerous engineering problems, the solution of such systems is notoriusly challenging. In this manuscript, we study efficient preconditioners for all-at-once approaches using both an algebraic and an operator preconditioning framework. We show in particular that for values of the regularization parameter not too small, the saddle-point system can be efficiently solved by preconditioning in parallel all the state and adjoint equations. For small values of the regularization parameter, robustness can be recovered by the additional solution of a small linear system, which however couples all realizations. A mean approximation and a Chebyshev semi-iterative method are investigated to solve this reduced system. Our analysis considers a random elliptic partial differential equation whose diffusion coefficient κ(x,ω) is modeled as an almost surely continuous and positive random field, though not necessarily uniformly bounded and coercive. We further provide estimates on the dependence of the preconditioned system on the variance of the random field. Such estimates involve either the first or second moment of the random variables 1/min_x∈Dκ(x,ω) and max_x∈Dκ(x,ω), where D is the spatial domain. The theoretical results are confirmed by numerical experiments, and implementation details are further addressed.


page 1

page 2

page 3

page 4


Robust discretization and solvers for elliptic optimal control problems with energy regularization

We consider the finite element discretization and the iterative solution...

Robust finite element discretization and solvers for distributed elliptic optimal control problems

We consider standard tracking-type, distributed elliptic optimal control...

Reliable Error Estimates for Optimal Control of Linear Elliptic PDEs with Random Inputs

We discretize a risk-neutral optimal control problem governed by a linea...

Certainty Equivalent Quadratic Control for Markov Jump Systems

Real-world control applications often involve complex dynamics subject t...

Subordinated Gaussian Random Fields in Elliptic Partial Differential Equations

To model subsurface flow in uncertain heterogeneous fractured media an e...

The Helmholtz equation with uncertainties in the wavenumber

We investigate the Helmholtz equation with suitable boundary conditions ...

A Streamline upwind Petrov-Galerkin Reduced Order Method for Advection-Dominated Partial Differential Equations under Optimal Control

In this paper we will consider distributed Linear-Quadratic Optimal Cont...

Please sign up or login with your details

Forgot password? Click here to reset