Two of the main challenges in optimal control are solving problems with state-dependent running costs and developing efficient numerical solvers that are computationally tractable in high dimensions.In this paper,we p...Two of the main challenges in optimal control are solving problems with state-dependent running costs and developing efficient numerical solvers that are computationally tractable in high dimensions.In this paper,we provide analytical solutions to certain optimal control problems whose running cost depends on the state variable and with constraints on the control.We also provide Lax-Oleinik-type representation formulas for the corresponding Hamilton-Jacobi partial differential equations with state-dependent Hamiltonians.Additionally,we present an efficient,grid-free numerical solver based on our representation formulas,which is shown to scale linearly with the state dimension,and thus,to overcome the curse of dimensionality.Using existing optimization methods and the min-plus technique,we extend our numerical solvers to address more general classes of convex and nonconvex initial costs.We demonstrate the capabilities of our numerical solvers using implementations on a central processing unit(CPU)and a field-programmable gate array(FPGA).In several cases,our FPGA implementation obtains over a 10 times speedup compared to the CPU,which demonstrates the promising performance boosts FPGAs can achieve.Our numerical results show that our solvers have the potential to serve as a building block for solving broader classes of high-dimensional optimal control problems in real-time.展开更多
为能够在合理计算规模下准确模拟空爆自由场冲击波特征的网格划分方案,获得可靠的计算结果,本文运用验证与确认(verification and validation,V&V)方法,通过开展网格敏感性分析,确认了炸药空爆计算模型的数值解、外推估值、网格收...为能够在合理计算规模下准确模拟空爆自由场冲击波特征的网格划分方案,获得可靠的计算结果,本文运用验证与确认(verification and validation,V&V)方法,通过开展网格敏感性分析,确认了炸药空爆计算模型的数值解、外推估值、网格收敛指标与比例距离的定量关系,给出了满足不同网格收敛指标要求的最大比例网格尺寸随比例距离位置的变化关系。据此,在比例距离Z为0~40 m·kg^(-1/3)范围内,给出了划分渐变网格的优化方案。针对1维、2维、3维计算模型,分别比较了细网格、粗网格、渐变网格方案的计算精度与计算耗时情况,讨论了本文提出网格划分方法的适用性。同时,采用本文建议的网格优化方案,给出了动爆冲击波毁伤飞机数值模拟场景的应用算例。结果表明,本文建议的网格优化方案可在几乎不降低计算精度的前提下显著提升计算效率,可为空爆自由场或者近似计算场景的数值模型网格划分提供参考。展开更多
基金supported by the DOE-MMICS SEA-CROGS DE-SC0023191 and the AFOSR MURI FA9550-20-1-0358supported by the SMART Scholarship,which is funded by the USD/R&E(The Under Secretary of Defense-Research and Engineering),National Defense Education Program(NDEP)/BA-1,Basic Research.
文摘Two of the main challenges in optimal control are solving problems with state-dependent running costs and developing efficient numerical solvers that are computationally tractable in high dimensions.In this paper,we provide analytical solutions to certain optimal control problems whose running cost depends on the state variable and with constraints on the control.We also provide Lax-Oleinik-type representation formulas for the corresponding Hamilton-Jacobi partial differential equations with state-dependent Hamiltonians.Additionally,we present an efficient,grid-free numerical solver based on our representation formulas,which is shown to scale linearly with the state dimension,and thus,to overcome the curse of dimensionality.Using existing optimization methods and the min-plus technique,we extend our numerical solvers to address more general classes of convex and nonconvex initial costs.We demonstrate the capabilities of our numerical solvers using implementations on a central processing unit(CPU)and a field-programmable gate array(FPGA).In several cases,our FPGA implementation obtains over a 10 times speedup compared to the CPU,which demonstrates the promising performance boosts FPGAs can achieve.Our numerical results show that our solvers have the potential to serve as a building block for solving broader classes of high-dimensional optimal control problems in real-time.
文摘为能够在合理计算规模下准确模拟空爆自由场冲击波特征的网格划分方案,获得可靠的计算结果,本文运用验证与确认(verification and validation,V&V)方法,通过开展网格敏感性分析,确认了炸药空爆计算模型的数值解、外推估值、网格收敛指标与比例距离的定量关系,给出了满足不同网格收敛指标要求的最大比例网格尺寸随比例距离位置的变化关系。据此,在比例距离Z为0~40 m·kg^(-1/3)范围内,给出了划分渐变网格的优化方案。针对1维、2维、3维计算模型,分别比较了细网格、粗网格、渐变网格方案的计算精度与计算耗时情况,讨论了本文提出网格划分方法的适用性。同时,采用本文建议的网格优化方案,给出了动爆冲击波毁伤飞机数值模拟场景的应用算例。结果表明,本文建议的网格优化方案可在几乎不降低计算精度的前提下显著提升计算效率,可为空爆自由场或者近似计算场景的数值模型网格划分提供参考。
