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Abstract
This article develops a new algorithm named TTRISK to solve highdimensional riskaverse optimization problems governed by differential equations (ODEs and/or partial differential equations [PDEs]) under uncertainty. As an example, we focus on the socalled Conditional Value at Risk (CVaR), but the approach is equally applicable to other coherent risk measures. Both the full and reduced space formulations are considered. The algorithm is based on low rank tensor approximations of random fields discretized using stochastic collocation. To avoid nonsmoothness of the objective function underpinning the CVaR, we propose an adaptive strategy to select the width parameter of the smoothed CVaR to balance the smoothing and tensor approximation errors. Moreover, unbiased Monte Carlo CVaR estimate can be computed by using the smoothed CVaR as a control variate. To accelerate the computations, we introduce an efficient preconditioner for the Karush–Kuhn–Tucker (KKT) system in the full space formulation.The numerical experiments demonstrate that the proposed method enables accurate CVaR optimization constrained by largescale discretized systems. In particular, the first example consists of an elliptic PDE with random coefficients as constraints. The second example is motivated by a realistic application to devise a lockdown plan for United Kingdom under COVID19. The results indicate that the riskaverse framework is feasible with the tensor approximations under tens of random variables.
Original language  English 

Journal  Numerical Linear Algebra with Applications 
Early online date  3 Dec 2022 
DOIs  
Publication status  Epub ahead of print  3 Dec 2022 
Keywords
 CVaR
 full space
 preconditioner
 reduced space
 risk measures
 tensor train
 TTRISK
ASJC Scopus subject areas
 Algebra and Number Theory
 Applied Mathematics
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Dive into the research topics of 'TTRISK: Tensor train decomposition algorithm for risk averse optimization'. Together they form a unique fingerprint.Projects
 2 Active

Overcoming the curse of dimensionality in dynamic programming by tensor decompositions
Engineering and Physical Sciences Research Council
10/05/21 → 9/05/23
Project: Research council

Tensor decomposition sampling algorithms for Bayesian inverse problems
Engineering and Physical Sciences Research Council
1/03/21 → 28/02/24
Project: Research council