从部分边界数据确定抛物型方程中非线性项的稳定性与重构
Stability and Reconstruction of a Nonlinearity in a Parabolic Equation from Partial Boundary Data
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中文总结 AI 辅助
研究从边界任意子集的单次通量确定非线性抛物型方程半线性项的反问题，建立赫尔德型稳定性估计，提出迭代重构算法并通过数值实验验证其准确性。
中文摘要 AI 辅助
本研究探讨的是，基于在边界任意子集上测得的单次边界通量，确定非线性抛物型方程中半线性项的反问题。更确切地说，我们研究了该反问题的唯一性与稳定性问题，并建立了新的赫尔德型稳定性估计，该赫尔德指数明确依赖于测量构型以及半线性项的正则性性质。分析过程依赖于一种新方法，该方法基于推导伴随方程解的合适积分恒等式，这使得反问题可被重构为带有变号源项的反源问题。主要结果通过结合抛物型方程的基本性质（包括最大值原理和合适的能量估计）得到。最后，我们结合反源问题的思想补充了迭代重构算法，并通过多个数值实验验证了其准确性。
英文摘要
In this work, we investigate the inverse problem of determining a semilinear term in a nonlinear parabolic equation from one single boundary flux measurement taken on an arbitrary subset of the boundary. More precisely, we address both uniqueness and stability issues of the inverse problem and establish new Hölder-type stability estimates. The Hölder exponent depends explicitly on the measurement configuration as well as on regularity properties of the semilinear term. The analysis relies on a novel approach based on the derivation of a suitable integral identity involving solutions of the associated adjoint equation. This allows reformulating the inverse problem as an inverse source problem with a sign-changing source term. The main results are obtained by combining fundamental properties of parabolic equations, including maximum principle and appropriate energy estimates. Finally, we complement the theoretical analysis with an iterative reconstruction algorithm inspired by inverse source problems, and illustrate its accuracy on several numerical experiments.
