Multilevel quadrature formulae for the optimal control of random PDEs
<|reference_start|>Multilevel quadrature formulae for the optimal control of random PDEs: This manuscript presents a framework for using multilevel quadrature formulae to compute the solution of optimal control problems constrained by random partial differential equations. Our approach consists in solving a sequence of optimal control problems discretized with different levels of accuracy of the physical and probability discretizations. The final approximation of the control is then obtained in a postprocessing step, by suitably combining the adjoint variables computed on the different levels. We present a convergence analysis for an unconstrained linear quadratic problem, and detail our framework for the specific case of a Multilevel Monte Carlo quadrature formula. Numerical experiments confirm the better computational complexity of our MLMC approach compared to a standard Monte Carlo sample average approximation, even beyond the theoretical assumptions.<|reference_end|>
@article{nobile2024multilevel,
    title={Multilevel quadrature formulae for the optimal control of random PDEs},
    author={Fabio Nobile and Tommaso Vanzan},
    journal={arXiv preprint arXiv:2407.06678},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.06678},
    primaryClass={math.NA cs.NA math.OC}
}