An iteratively regularized stochastic gradient method for estimating a random parameter in a stochastic PDE. A variational inequality approach

doi:10.23952/jnva.5.2021.6.02

Cited by 1 publication

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“…By incorporating regularization, we are able to circumvent this difficulty and prove strong convergence of the proposed scheme. Despite the rich history of stochastic approaches, their use for stochastic optimal control and inverse problems is quite recent; see [12,13] for control problems and [14][15][16] for inverse problems. However, all the methods used in these works are first-order methods, and the present study is the first to employ stochastic second-order methods in a Hilbert space setting.…”

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confidence: 99%

A stochastic regularized second-order iterative scheme for optimal control and inverse problems in stochastic partial differential equations

Dambrine

Khan

Sama

2022

Phil. Trans. R. Soc. A.

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Numerous applied models used in the study of optimal control problems, inverse problems, shape optimization, machine learning, fractional programming, neural networks, image registration and so on lead to stochastic optimization problems in Hilbert spaces. Under a suitable convexity assumption on the objective function, a necessary and sufficient optimality condition for stochastic optimization problems is a stochastic variational inequality. This article presents a new stochastic regularized second-order iterative scheme for solving a variational inequality in a stochastic environment where the primary operator is accessed by employing sampling techniques. The proposed iterative scheme, which fits within the general framework of the stochastic approximation approach, has its almost-sure convergence analysis given in a Hilbert space. We test the feasibility and the efficacy of the proposed stochastic approximation approach for a stochastic optimal control problem and a stochastic inverse problem, both associated with a second-order stochastic partial differential equation. This article is part of the theme issue ‘Non-smooth variational problems and applications’.

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