不确定欠定系统的可验证可扩展包围域
Certified Scalable Enclosures for Uncertain Underdetermined Systems
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中文总结 AI 辅助
针对不确定欠定系统的解集可验证难题，开发梯形线性化方法结合线性规划实现可扩展包围域，经多类应用验证其通用性并分析紧密度与成本的权衡。
中文摘要 AI 辅助
具有有界不确定性的欠定模型（如非线性设计与估计）的核心挑战是对解集进行可验证。传统求解方法，如基于牛顿法或基于采样的不确定性量化的方法，要么因欠定特性而不适用，要么在收敛的前提下无法保证已找到所有解。我们针对非线性设计中出现的问题解决了该挑战，该问题需给定目标稳态区间并求解参数。我们开发了梯形线性化方法，该方法可严格包围所有解，并将其与易处理的线性规划方法结合以计算分量级边界。我们证明所得线性规划族可迭代收缩初始参数区域，对目标状态和初始参数区域进行细分可提升收缩效果并提供更紧密的包围域。梯形松弛得到的线性规划族还可通过Oettli-Prager特性求解区间线性系统的解。我们通过不同应用展示了该方法的通用性，包括生物分子电路中的非线性设计、灵敏度分析方法及压缩感知场景，并讨论了包围域紧密度与计算成本之间的权衡。
英文摘要
The central challenge in underdetermined models with bounded uncertainty, such as in nonlinear design and estimation, is certifying the solution sets. Conventional solution methodologies, such as those based on Newton's method or on sampling-based uncertainty quantification, are either not applicable due to the underdetermined nature or do not give guarantees that all solutions have been found, assuming they converge. We addressed this issue for a problem that arises in nonlinear design, where a target steady-state box is prescribed and the parameters have to be found. We developed a trapezoidal linearisation method that rigorously encloses all solutions and combined it with a tractable linear programming method to compute the component-wise bounds. We showed that the resulting families of linear programs can iteratively contract an initial parameter region. A subdivision of the target state and the initial parameter region can improve the contraction and provide tighter enclosures. The trapezoidal relaxation gives a family of linear programs that can also be used for finding solutions for interval linear systems via the Oettli-Prager characterisation. We demonstrate the generality of the method through different applications, including nonlinear design in biomolecular circuits, a sensitivity analysis method, and in a compressed sensing context, and discuss the trade-off between enclosure tightness and computational cost.
发表机构
- Indian Institute of Technology Delhi（印度德里理工学院）
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