TTRISK: Tensor train decomposition algorithm for risk averse optimization

This article develops a new algorithm named TTRISK to solve high-dimensional risk-averse optimization problems governed by differential equations (ODEs and/or partial [PDEs]) under uncertainty. As an example, we focus on the so-called Conditional Value at Risk (CVaR), but approach is equally applicable other coherent risk measures. Both full and reduced space formulations are considered. The based low rank tensor approximations of random fields discretized using stochastic collocation. To avoid nonsmoothness objective function underpinning CVaR, propose adaptive strategy select width parameter smoothed CVaR balance smoothing approximation errors. Moreover, unbiased Monte Carlo estimate can be computed as control variate. accelerate computations, introduce efficient preconditioner for Karush–Kuhn–Tucker (KKT) system in formulation.The numerical experiments demonstrate that proposed method enables accurate constrained large-scale systems. In particular, first example consists elliptic PDE with coefficients constraints. second motivated realistic application devise lockdown plan United Kingdom COVID-19. results indicate framework feasible tens variables.