AI 中文总结
针对带条件稳定性的抛物型逆源问题，提出无伴随积分反馈方法，通过正向求解实现源项重构，经分析该方法为迭代正则化方案，数值实验验证了其重构的准确性与稳定性。
AI 中文摘要
我们研究带加法源项 $f(t)+\zeta(t,x)$ 的线性非自治抛物型方程的逆源问题，其中未知分量仅依赖于时间，需通过解的积分观测值进行恢复。将该问题归约为等价的线性逆问题后，我们建立了Tikhonov正则化解的存在性与唯一性，并推导了一阶最优性条件。我们证明正算子在时间域内具有Volterra表示，由此得到 $H^{-1}(0,T)$ 空间下的弱范数稳定性估计；在源项满足先验 $H^r(0,T)$ 界的条件下，还得到了 $L^2(0,T)$ 空间下的条件Hölder稳定性估计。受该Volterra结构启发，我们提出一种仅需正向求解的无伴随积分反馈方法来重构源项。我们通过证明该反馈迭代的适定性、不动点性质、精确数据下的收敛性以及带噪数据下的有限迭代稳定性，对其进行了分析。我们进一步表明，采用合适的依赖于噪声的停止规则时，该方法构成一种迭代正则化方案。针对光滑源项和分段常数源项的数值实验，结合时间正则化与自动参数选择，证明了其在噪声存在时可实现准确且稳定的重构。
英文摘要
We study an inverse source problem for a linear non-autonomous parabolic equation with additive source term $f(t)+ζ(t,x)$, where the unknown component depends only on time and is recovered from an integral observation of the solution. After reducing the problem to an equivalent linear inverse problem, we establish existence and uniqueness of the Tikhonov-regularized solution and derive a first-order optimality condition. We show that the forward operator admits a Volterra representation in time, yielding a weak-norm stability estimate in $H^{-1}(0,T)$ and, under an a priori $H^r(0,T)$ bound on the source, a conditional Hölder stability estimate in $L^2(0,T)$. Motivated by this Volterra structure, we introduce an adjoint-free integral feedback method that reconstructs the source using only forward solves. We analyze the feedback iteration by establishing its well-definedness and fixed-point properties, convergence for exact data, and finite-iteration stability with respect to noisy data. We further show that, with an appropriate noise-dependent stopping rule, the method constitutes an iterative regularization scheme. Numerical experiments for smooth and piecewise constant sources, supplemented by temporal regularization and automatic parameter selection, demonstrate accurate and stable reconstructions in the presence of noise.
Comments35 pages, 4 figures, 2 tables
