无导数恢复自由边界DCIS模型中的非线性项
Derivative-Free Recovery of a Nonlinearity in a Free-Boundary DCIS Model
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中文总结 AI 辅助
针对导管原位癌（DCIS）自由边界模型的逆系数问题，提出无导数的同伦延拓方法，结合皮卡迭代与斯佩纳搜索，通过数值实验验证了理论结果。
中文摘要 AI 辅助
我们研究了导管原位癌（DCIS）的多维自由边界模型的逆系数问题，该模型中肿瘤界面由营养物浓度、组织压力和曲率的非线性耦合所支配，未知的营养物消耗函数可通过针吸活检获得的营养物浓度时间迹线进行恢复。对于正问题，我们在可容许的消耗函数类上建立了一致局部适定性。该逆问题被重述为一个不动点问题：通过有限维空间近似可容许集可得到离散迭代算子，我们证明了这些离散算子存在不动点，且离散不动点的子序列强收敛到连续算子的一个不动点，该不动点在一致性条件下可求解逆问题。为近似这些不动点，我们开发了一种同伦延拓方法，将线性收敛的皮卡迭代与立方体斯佩纳搜索相结合，无需对目标泛函求导或计算伴随状态。对径向对称和非对称DCIS模型开展的若干数值实验证实了这些理论结果。
英文摘要
We investigate an inverse coefficient problem for a multidimensional free-boundary model of ductal carcinoma in situ (DCIS), in which the tumor interface is governed by the nonlinear coupling of nutrient concentration, tissue pressure and curvature, and the unknown nutrient consumption function is recovered from a temporal trace of the nutrient concentration obtained by needle aspiration biopsy. For the forward problem, we establish uniform local well-posedness over an admissible class of consumption functions. The inverse problem is recast as a fixed-point problem: approximating the admissible set by finite-dimensional spaces yields discrete iteration operators, for which we prove the existence of fixed points, and the strong convergence of a subsequence of discrete fixed points to a fixed point of the continuous operator, which solves the inverse problem under a consistency condition. To approximate these fixed points, we develop a homotopy-continuation method combining a linearly convergent Picard iteration with a cubical Sperner search, without differentiating an objective functional or computing an adjoint state. Several numerical experiments on radially symmetric and non-symmetric DCIS models corroborate the theoretical findings.
