有限测量下非齐次热方程初始条件的重构
Reconstruction of the Initial Condition for a Non-Homogeneous Heat Equation from Finite Measurements
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中文总结 AI 辅助
针对一维非齐次热方程，提出利用有限时间点测量值重构初始温度的显式方案，在固定时间范围内实现高效恢复，并建立代数收敛速率及数值验证。
中文摘要 AI 辅助
我们研究一个逆问题：在Dirichlet边界条件下，从单个空间位置上的有限个时间点测量值重构一维非齐次热方程的初始温度分布。我们提出了一种显式逼近方案，该方案包含源项并利用精细的测量时间序列。与以往通常依赖指数增长观测时间的方法不同，我们选择的时间序列允许在固定时间范围内实现高效恢复。我们在初始数据和强迫项的适当正则性假设下建立了代数收敛速率，并通过不同测量次数的数值实验验证了理论上的最优性。
英文摘要
We address the inverse problem of reconstructing the initial temperature distribution in a one-dimensional non-homogeneous heat equation with Dirichlet boundary conditions from a finite number of pointwise-in-time measurements at a single spatial location. We propose an explicit approximation scheme that incorporates the source term and utilizes a refined sequence of measurement times. Unlike previous approaches that often rely on exponentially growing observation times, our chosen time sequence allows for efficient recovery within a fixed time horizon. We establish algebraic convergence rates under suitable regularity assumptions on the initial data and the forcing, and we validate the theoretical sharpness with numerical experiments for varying numbers of measurements.
发表机构
- Indiana University East（印第安纳大学东校区）
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