Reduced Order Model Hessian Approximations in Newton Methods for Optimal Control
Reduced Order Model Hessian Approximations in Newton Methods for Optimal Control
复制标题
最优控制牛顿法中的降阶模型 Hessian 近似
DOI：
10.1007/978-3-030-95157-3_18
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发表时间：
2022
期刊：
影响因子：
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通讯作者：
Magruder, Caleb
中科院分区：
文献类型：
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作者：
Heinkenschloss, Matthias;Magruder, Caleb
本文介绍了降阶模型（ROM）为基础的Hessian近似用于不精确牛顿法的解决方案的优化问题的隐式约束的大规模系统，通常是一个离散化的偏微分方程（PDE）。直接应用不精确牛顿法求解这个问题需要每次优化迭代求解多个偏微分方程。为了降低计算复杂度，提出了一种ROM Hessian近似。由于只有Hessian近似，但使用原始目标函数及其梯度，因此在适当的假设下，所得的不精确牛顿法保持一阶全局收敛性。因此，甚至可以使用计算上廉价的较低保真度ROM，这不同于ROM方法，ROM方法用ROM优化问题的序列来代替原始优化问题，并且通常需要精确地近似原始问题的函数和梯度信息。在所提出的方法中，ROM Hessian近似的质量决定了收敛速度，但不是方法是否收敛。基于投影的ROM由状态和伴随快照构造，并且计算相对便宜。半线性抛物型最优控制问题的数值例子表明，所提出的方法可以导致大量节省的整体PDE解决方案所需的。
This paper introduces reduced order model (ROM) based Hessian approximations for use in inexact Newton methods for the solution of optimization problems implicitly constrained by a large-scale system, typically a discretization of a partial differential equation (PDE). The direct application of an inexact Newton method to this problem requires the solution of many PDEs per optimization iteration. To reduce the computational complexity, a ROM Hessian approximation is proposed. Since only the Hessian is approximated, but the original objective function and its gradient is used, the resulting inexact Newton method maintains the first-order global convergence property, under suitable assumptions. Thus even computationally inexpensive lower fidelity ROMs can be used, which is different from ROM approaches that replace the original optimization problem by a sequence of ROM optimization problem and typically need to accurately approximate function and gradient information of the original problem. In the proposed approach, the quality of the ROM Hessian approximation determines the rate of convergence, but not whether the method converges. The projection based ROM is constructed from state and adjoint snapshots, and is relatively inexpensive to compute. Numerical examples on semilinear parabolic optimal control problems demonstrate that the proposed approach can lead to substantial savings in terms of overall PDE solves required.