基于优化的时空超材料波动方程有效系数识别
Optimization-Based Identification of Effective Coefficients for Wave Equations in Spatio-Temporal Metamaterials
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中文总结 AI 辅助
该研究将优化系数识别技术拓展至波动方程，可从波传播测量中识别时空超材料波动方程的有效系数，经理论证明与数值实验验证其有效性。
中文摘要 AI 辅助
我们研究异质介质中波动方程的有效系数识别问题。这类方程用于建模时空超材料，其基础材料属性随空间和时间变化。尽管均匀化理论可在微观尺度趋于零的渐近情形下提供有效模型，但从波传播观测中确定宏观参数仍具挑战性。我们提出一种基于优化的方法，通过最小化代价泛函来识别常数有效系数。该方法适用于基础时空相关系数未知、而解可通过测量获取空间和时间信息的场景。我们将基于优化的系数识别技术从椭圆多尺度问题拓展至波动方程，并证明若存在均匀化极限，当微观尺度趋于零时，识别出的系数会收敛至均匀化系数。此外，我们还证明了对应有效解向异质解的收敛性，包括$L^2$强收敛以及一阶空间和时间导数的弱收敛。数值实验进一步验证了该方法在非周期时空异质介质中的性能，包括那些尚未明确存在均匀化极限的情形。
英文摘要
We study the identification of effective coefficients for wave equations in heterogeneous media. Such equations arise in the modeling of spatio-temporal metamaterials, where the underlying material properties exhibit variations in both space and time. While homogenization provides effective models in the asymptotic regime of vanishing microscopic scales, determining macroscopic parameters from observations of wave propagation remains challenging. We introduce an optimization-based method that identifies a constant effective coefficient by minimizing a cost functional. The approach is designed for situations in which the underlying space-time-dependent coefficient is unknown, while the solution is available in space and time by measurements. We extend optimization-based coefficient identification techniques from elliptic multiscale problems to wave equations and prove that, provided a homogenized limit exists, the identified coefficient converges to the homogenized coefficient as the microscopic scale tends to zero. Furthermore, we establish convergence of the corresponding effective solution towards the heterogeneous solution, including strong convergence in $L^2$ and weak convergence of first-order space and time derivatives. Numerical experiments further demonstrate the performance of the method for non-periodic space-time heterogeneous media, including cases for which a homogenized limit is not known to exist.
