AI 中文总结
针对半无限带域泊松方程的分段光滑逆源重建问题，提出结合交错分区、GTSVD正则化等的局域傅里叶延拓方法，在中低噪声下鲁棒性优于对比方法。
AI 中文摘要
我们从半无限带域内沿内部直线测得的带噪解值，重建泊松方程中的分段光滑源项。对观测值应用一维Dirichlet拉普拉斯算子，可将该逆问题简化为正则化二阶微分，再经有界可逆校正处理。微分迹与源项相差一个解析平滑项，因此具有相同的内部奇异支撑和跳跃数据。该结构启发了一种局域傅里叶延拓方法，其结合了两个交错检测分区、经GTSVD正则化的局部导数系数、磨光共轭傅里叶和与结构对齐的数值微分。随后通过指数衰减谱校正恢复源项。对于精确分区，重建继承分段微分速率$\bar{\tau}(\bar{s}-2)/\bar{s}$；局部峰值和分区扰动估计描述断点误差的附加影响。与全网格总变分正则化、截断傅里叶逆变换的对比显示，该方法在高噪声下具有竞争力，低噪声下优势显著；重复高斯噪声测试证实其在中、低噪声区域的鲁棒性。
英文摘要
We reconstruct a piecewise-smooth source in a Poisson equation on a semi-infinite strip from noisy solution values measured along an interior line. Applying the one-dimensional Dirichlet Laplacian to the observation reduces the inverse problem to regularized second-order differentiation followed by a boundedly invertible correction. The differentiated trace and the source differ by an analytic smoothing term and therefore have the same interior singular support and jump data. This structure motivates a localized Fourier extension method that combines two staggered detection partitions, GTSVD-regularized local derivative coefficients, mollified conjugate Fourier sums, and structure-aligned numerical differentiation. The source is then recovered by an exponentially decaying spectral correction. For an exact partition, the reconstruction inherits the piecewise differentiation rate $\mathcal O(δ^{(\bar s-2)/\bar s})$; local peak and partition perturbation estimates describe the additional effect of breakpoint errors. Comparisons with full-grid total-variation regularization and truncated Fourier inversion show competitive high-noise performance and a pronounced low-noise advantage of the localized method. Repeated Gaussian-noise tests confirm robustness in the moderate- and low-noise regimes.
