TTRISK: Tensor train decomposition algorithm for risk averse optimization

Harbir Antil, Sergey Dolgov, Akwum Onwunta

Research output: Contribution to journalArticlepeer-review

2 Citations (SciVal)


This article develops a new algorithm named TTRISK to solve high-dimensional risk-averse optimization problems governed by differential equations (ODEs and/or partial differential equations [PDEs]) under uncertainty. As an example, we focus on the so-called 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 large-scale 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 COVID-19. The results indicate that the risk-averse framework is feasible with the tensor approximations under tens of random variables.

Original languageEnglish
Article numbere2481
JournalNumerical Linear Algebra with Applications
Issue number3
Early online date3 Dec 2022
Publication statusPublished - 31 May 2023

Bibliographical note

Funding Information:
Harbir Antil and Akwum Onwunta are partially supported by NSF grants DMS‐2110263, DMS‐1913004 and the Air Force Office of Scientific Research under Award number: FA9550‐19‐1‐0036 and FA9550‐22‐1‐0248. Sergey Dolgov is thankful for the support from Engineering and Physical Sciences Research Council (EPSRC) New Investigator Award EP/T031255/1 and New Horizons grant EP/V04771X/1.

Funding Information:
Air Force Office of Scientific Research, Grant/Award Numbers: FA9550‐19‐1‐0036; FA9550‐22‐1‐0248; Engineering and Physical Sciences Research Council, Grant/Award Number: EP/T031255/1; EP/V04771X/1; National Science Foundation, Grant/Award Numbers: DMS‐2110263; DMS‐1913004 Funding information


  • CVaR
  • full space
  • preconditioner
  • reduced space
  • risk measures
  • tensor train

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Applied Mathematics


Dive into the research topics of 'TTRISK: Tensor train decomposition algorithm for risk averse optimization'. Together they form a unique fingerprint.

Cite this