In this work, we tackle the problem of minimising the Conditional-Value-at-Risk (CVaR) of output quantities of complex differential models with random input data, using gradient-based approaches in combination with the Multi-Level Monte Carlo (MLMC) method. In particular, we consider the framework of multi-level Monte Carlo for parametric expectations and propose modifications of the MLMC estimator, error estimation procedure, and adaptive MLMC parameter selection to ensure the estimation of the CVaR and sensitivities for a given design with a prescribed accuracy. We then propose combining the MLMC framework with an alternating inexact minimisation-gradient descent algorithm, for which we prove exponential convergence in the optimisation iterations under the assumptions of strong convexity and Lipschitz continuity of the gradient of the objective function. We demonstrate the performance of our approach on two numerical examples of practical relevance, which evidence the same optimal asymptotic cost-tolerance behaviour as standard MLMC methods for fixed design computations of output expectations.

翻译：在这项工作中,我们利用基于梯度的办法,结合多层次蒙特卡洛(MLMC)方法,处理将复杂差异模型的产出数量与随机输入数据最小化的问题,特别是考虑多层次蒙特卡洛(MLMC)框架的参数参数参数参数参数参数参数参数参数参数参数参数参数参数参数参数参数,以确保对CVAR的估计和特定设计的敏感度具有规定的准确性。然后,我们提议将MLMC框架与不精确的递减梯度梯度递减算法相结合,为此,我们证明在目标功能梯度的强烈交融和利普西茨连续性假设下,优化比重指数一致。我们用两个具有实际相关性的数字实例展示了我们的做法,这些实例证明了与固定设计产出预期值计算标准MLMC方法一样最优的刺激成本容忍行为。