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Randomized Sketching Algorithms for Low-Memory Dynamic Optimization
SIAM Journal on Optimization ( IF 3.1 ) Pub Date : 2021-05-10 , DOI: 10.1137/19m1272561
Ramchandran Muthukumar , Drew P. Kouri , Madeleine Udell

SIAM Journal on Optimization, Volume 31, Issue 2, Page 1242-1275, January 2021.
This paper develops a novel limited-memory method to solve dynamic optimization problems. The memory requirements for such problems often present a major obstacle, particularly for problems with PDE constraints such as optimal flow control, full waveform inversion, and optical tomography. In these problems, PDE constraints uniquely determine the state of a physical system for a given control; the goal is to find the value of the control that minimizes an objective. While the control is often low dimensional, the state is typically more expensive to store. This paper suggests using randomized matrix approximation to compress the state as it is generated and shows how to use the compressed state to reliably solve the original dynamic optimization problem. Concretely, the compressed state is used to compute approximate gradients and to apply the Hessian to vectors. The approximation error in these quantities is controlled by the target rank of the sketch. This approximate first- and second-order information can readily be used in any optimization algorithm. As an example, we develop a sketched trust-region method that adaptively chooses the target rank using a posteriori error information and provably converges to a stationary point of the original problem. Numerical experiments with the sketched trust-region method show promising performance on challenging problems such as the optimal control of an advection-reaction-diffusion equation and the optimal control of fluid flow past a cylinder.


中文翻译:

低内存动态优化的随机草绘算法

SIAM优化杂志,第31卷,第2期,第1242-1275页,2021年1月。
本文提出了一种新的有限内存方法来解决动态优化问题。对于此类问题的存储要求通常会成为主要障碍,特别是对于PDE约束(例如最佳流量控制,全波形反转和光学层析成像)的问题。在这些问题中,PDE约束唯一地确定了给定控件的物理系统状态。目的是找到使目标最小化的控制价值。尽管控件通常是低维的,但是状态通常更昂贵。本文建议使用随机矩阵逼近来压缩状态生成时的状态,并说明如何使用压缩状态可靠地解决原始的动态优化问题。具体来说,压缩状态用于计算近似梯度并将Hessian应用于向量。这些数量的近似误差由草图的目标等级控制。这种近似的一阶和二阶信息可以轻松地用于任何优化算法中。例如,我们开发了一种草绘的信任区域方法,该方法使用后验误差信息自适应地选择目标等级,并证明收敛到原始问题的平稳点。用草绘的信任区方法进行的数值实验显示出在具有挑战性的问题上的有希望的性能,例如对流-反应-扩散方程的最佳控制和流经圆柱体的流体的最佳控制。这种近似的一阶和二阶信息可以轻松地用于任何优化算法中。例如,我们开发了一种草绘的信任区域方法,该方法使用后验误差信息自适应地选择目标等级,并证明收敛到原始问题的平稳点。用草绘的信任区方法进行的数值实验显示出在具有挑战性的问题上的有希望的性能,例如对流-反应-扩散方程的最佳控制和流经圆柱体的流体的最佳控制。这种近似的一阶和二阶信息可以轻松地用于任何优化算法中。例如,我们开发了一种草绘的信任区域方法,该方法使用后验误差信息自适应地选择目标等级,并证明收敛到原始问题的平稳点。用草绘的信任区方法进行的数值实验显示出在具有挑战性的问题上的有希望的性能,例如对流-反应-扩散方程的最佳控制和流经圆柱体的流体的最佳控制。
更新日期:2021-05-20
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