有限测量下测地线射线变换的反演：稳定性与重构
Inverting the geodesic ray transform with finite measurements: stability and reconstruction
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中文总结 AI 辅助
本文研究有限测量下测地线射线变换的反演，证明Lipschitz稳定性并开发基于最速梯度下降和共轭梯度的重构算法，数值实验验证了方法的有效性。
中文摘要 AI 辅助
我们研究了从数据的有限多个局部平均值中反演测地线射线变换的问题。在适当的几何假设下，当测量划分足够精细时，我们证明了在任意固定的有限维重构空间上的Lipschitz稳定性。我们开发了基于最速梯度下降和共轭梯度的收敛重构算法，并使用分片常数有限元空间实现。二维和三维数值实验说明了几何和测量离散化对重构质量的影响。
英文摘要
We study the inversion of the geodesic ray transform from finitely many local averages of its data. Under suitable geometric assumptions, we prove Lipschitz stability on any fixed finite-dimensional reconstruction space when the measurement partition is sufficiently fine. We develop convergent reconstruction algorithms based on Steepest Gradient Descent and Conjugate Gradients and implement them using piecewise constant finite element spaces. Numerical experiments in 2D and 3D illustrate the influence of geometry and measurement discretization on reconstruction quality.
