Adaptive stochastic gradient descent for optimal control of parabolic equations with random parameters. (English) Zbl 1533.65156
Summary: In this paper we extend the adaptive gradient descent (AdaGrad) algorithm to the optimal distributed control of parabolic partial differential equations with uncertain parameters. This stochastic optimization method achieves an improved convergence rate through adaptive scaling of the gradient step size. We prove the convergence of the algorithm for this infinite dimensional problem under suitable regularity, convexity, and finite variance conditions, and relate these to verifiable properties of the underlying system parameters. Finally, we apply our algorithm to the optimal thermal regulation of lithium battery systems under uncertain loads.
{© 2022 Wiley Periodicals LLC.}
{© 2022 Wiley Periodicals LLC.}
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 93E03 | Stochastic systems in control theory (general) |
| 49M41 | PDE constrained optimization (numerical aspects) |
| 62L20 | Stochastic approximation |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 78A57 | Electrochemistry |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
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