David Schneiderhan: Multilevel Stochastic Gradient Descent for Risk-Averse PDE Constraint Optimizat… скачать в хорошем качестве

David Schneiderhan: Multilevel Stochastic Gradient Descent for Risk-Averse PDE Constraint Optimizat…

Full title: Multilevel Stochastic Gradient Descent for Risk-Averse PDE Constraint Optimization We introduce a multilevel stochastic gradient descent framework for the optimization of partial differential equation–constrained systems with uncertain input data, including recent extensions to risk‑averse objectives. The method employs a parallel multilevel Monte Carlo estimator to approximate stochastic gradients, providing explicit control over the bias, introduced by numerical discretization and the sampling error arising from incomplete gradient information, while optimally managing computational resources. We show that the method exhibits linear convergence in the number of optimization steps while avoiding the cost of sample average approximation to compute the full gradient. Numerical experiments demonstrate substantial gains in convergence speed and accuracy compared to standard (mini‑batch) stochastic gradient descent with respect to computational resources. The approach is especially well‑suited for high‑dimensional control problems, leveraging parallel computing architectures and a distributed multilevel data structure.