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This article develops a new algorithm named TTRISK to solve high-dimensionalrisk-averse optimization problems governed by differential equations (ODEsand/or PDEs) under uncertainty. As an example, we focus on the so-calledConditional Value at Risk (CVaR), but the approach is equally applicable toother coherent risk measures. Both the full and reduced space formulations areconsidered. The algorithm is based on low rank tensor approximations of randomfields discretized using stochastic collocation. To avoid non-smoothness of theobjective function underpinning the CVaR, we propose an adaptive strategy toselect the width parameter of the smoothed CVaR to balance the smoothing andtensor approximation errors. Moreover, unbiased Monte Carlo CVaR estimate canbe computed by using the smoothed CVaR as a control variate. To accelerate thecomputations, we introduce an efficient preconditioner for the KKT system inthe full space formulation.The numerical experiments demonstrate that theproposed method enables accurate CVaR optimization constrained by large-scalediscretized systems. In particular, the first example consists of an ellipticPDE with random coefficients as constraints. The second example is motivated bya realistic application to devise a lockdown plan for United Kingdom underCOVID-19. The results indicate that the risk-averse framework is feasible withthe tensor approximations under tens of random variables.

TTRISK: Tensor Train Decomposition Algorithm for Risk Averse Optimization