TTRISK: Tensor train decomposition algorithm for risk averse optimization
TTRISK: Tensor train decomposition algorithm for risk averse optimization
复制标题
DOI：
10.1002/nla.2481
复制
发表时间：
2021-11
影响因子：
4.3
通讯作者：
Harbir Antil;S. Dolgov;Akwum Onwunta
中科院分区：
文献类型：
--
作者：
Harbir Antil;S. Dolgov;Akwum Onwunta
本文开发了一种名为 TTRISK 的新算法，用于解决不确定性下由微分方程（ODE 和/或偏微分方程 [PDE]）控制的高维风险规避优化问题。例如，我们关注所谓的条件风险价值（CVaR），但该方法同样适用于其他相关风险衡量标准。完整空间公式和缩小空间公式均被考虑。该算法基于使用随机搭配离散化的随机场的低秩张量近似。为了避免支撑 CVaR 的目标函数不平滑，我们提出了一种自适应策略来选择平滑 CVaR 的宽度参数，以平衡平滑和张量逼近误差。此外，可以通过使用平滑的 CVaR 作为控制变量来计算无偏蒙特卡洛 CVaR 估计。为了加速计算，我们在全空间公式中引入了 Karush-Kuhn-Tucker (KKT) 系统的高效预处理器。数值实验表明，所提出的方法能够在大型离散系统约束下实现精确的 CVaR 优化。特别是，第一个示例由以随机系数作为约束的椭圆偏微分方程组成。第二个例子的动机是为英国制定 COVID-19 封锁计划的实际应用。结果表明，风险规避框架在数十个随机变量下的张量近似下是可行的。
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.