随机抛物方程的几何逆源问题
A Geometric Inverse Source Problem for Stochastic Parabolic Equations
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中文总结 AI 辅助
本文针对随机抛物方程，提出形状梯度重建方法，通过部分边界通量测量首次同时恢复漂移项和扩散项中的两个未知源支撑，数值实验验证了其区分源通道的有效性。
中文摘要 AI 辅助
本文研究随机抛物方程中同时恢复两个未知确定性源支撑的几何逆问题，其中一个源出现在漂移项中，另一个出现在扩散项中。我们证明仅通过部分边界通量测量就能唯一确定两个源支撑，还证明了周长正则化目标泛函的极小元存在性；对于光滑界面，推导了形状导数的Hadamard型边界表示，据我们所知，这是SPDE框架下首次同时恢复漂移源支撑和扩散源支撑的此类公式，该导数具有真正的随机双通道结构：伴随状态控制漂移源界面的灵敏度，鞅分量控制扩散源界面的灵敏度。基于此公式，我们提出了形状梯度重建方法，数值实验验证了该方法的有效性，以及其在全边界观测和部分边界观测下区分两个源通道的能力。
英文摘要
This paper addresses the geometric inverse problem of simultaneously recovering two unknown deterministic source supports in a stochastic parabolic equation, where one source appears in the drift term and the other in the diffusion term. We establish that partial boundary flux measurements alone uniquely determine both supports. Moreover, we prove the existence of minimizers for a perimeter-regularized objective functional. For smooth interfaces, we derive a Hadamard-type boundary representation of the shape derivative. To the best of our knowledge, this is the first such formula for the simultaneous recovery of a drift-source support and a diffusion-source support in an SPDE setting. The derivative exhibits a genuinely stochastic two-channel structure: the adjoint state governs the sensitivity of the drift-source interface, while the martingale component controls the sensitivity of the diffusion-source interface. Based on this formula, we develop a shape-gradient reconstruction method. Numerical experiments demonstrate its effectiveness and its capacity to distinguish between the two source channels under both full and partial boundary observations.
