Taylor Approximation for Chance Constrained Optimization Problems Governed by Partial Differential Equations with High-Dimensional Random Parameters
Taylor Approximation for Chance Constrained Optimization Problems Governed by Partial Differential Equations with High-Dimensional Random Parameters
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DOI：
10.1137/20m1381381
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发表时间：
2020-11
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影响因子：
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通讯作者：
Peng Chen;O. Ghattas
中科院分区：
文献类型：
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作者：
Peng Chen;O. Ghattas
提出了一种快速、可扩展的优化方法来求解高维随机参数偏微分方程（PDE）约束的机会或概率约束优化问题。为了解决昂贵的偏微分方程解和高维不确定性的关键计算挑战，我们通过泰勒近似构建约束函数的代理，这依赖于导数的有效计算，Hessian的低秩近似和特征值分解的随机算法。为了解决不等式机会约束不可微的困难，我们利用机会约束中不连续指示函数的光滑逼近，并应用罚函数法将不等式约束优化问题转化为无约束优化问题.此外，我们设计了一个基于梯度的优化方案，逐渐增加平滑和惩罚参数，以实现收敛，为此，我们提出了一个有效的计算近似成本函数的梯度的泰勒近似。基于一个问题的地下水优化管理的数值实验，我们证明了泰勒近似的准确性，它的能力，大大加快约束评估，连续优化方案的收敛性，以及所提出的方法的可扩展性的PDE解决的数量增加随机参数的尺寸从一千到几十万。
We propose a fast and scalable optimization method to solve chance or probabilistic constrained optimization problems governed by partial differential equations (PDEs) with high-dimensional random parameters. To address the critical computational challenges of expensive PDE solution and high-dimensional uncertainty, we construct surrogates of the constraint function by Taylor approximation, which relies on efficient computation of the derivatives, low rank approximation of the Hessian, and a randomized algorithm for eigenvalue decomposition. To tackle the difficulty of the non-differentiability of the inequality chance constraint, we use a smooth approximation of the discontinuous indicator function involved in the chance constraint, and apply a penalty method to transform the inequality constrained optimization problem to an unconstrained one. Moreover, we design a gradient-based optimization scheme that gradually increases smoothing and penalty parameters to achieve convergence, for which we present an efficient computation of the gradient of the approximate cost functional by the Taylor approximation. Based on numerical experiments for a problem in optimal groundwater management, we demonstrate the accuracy of the Taylor approximation, its ability to greatly accelerate constraint evaluations, the convergence of the continuation optimization scheme, and the scalability of the proposed method in terms of the number of PDE solves with increasing random parameter dimension from one thousand to hundreds of thousands.