非线性记忆输运方程逆初值问题的Carleman--Picard方法及时空降维
Carleman--Picard and time-dimensional reduction for inverse initial-data problems in nonlinear transport with memory
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中文总结 AI 辅助
针对带非线性与记忆效应的拟线性输运方程逆初值问题，本文采用Legendre--指数时空降维法结合Carleman权与Tikhonov正则化的Picard方法，实现含噪数据下初态的准确鲁棒重构，收敛迅速。
中文摘要 AI 辅助
我们研究一类带非线性与记忆效应的拟线性输运方程的逆初值问题：给定流入边界数据，从流出边界的时变测量值中重构未知初态。首先，我们应用Legendre--指数时空降维法，将控制方程转化为空间上的有限非线性系统；随后，提出结合Carleman权与Tikhonov正则化的Picard方法，每次迭代中用前一次迭代值计算非线性项，得到具有唯一解的线性极小化问题。利用主输运算子的Carleman估计证明：当Carleman参数足够大时，所得Picard映射在给定容许集上是压缩映射，因此该方法可从该集内任意初始猜测收敛；还建立了针对含噪流出数据的稳定性，上述分析结果针对截断并正则化后的降维问题成立。二维数值实验表明，该方法可准确重构单个及多个包含体，对噪声具有鲁棒性，且Picard迭代收敛迅速。
英文摘要
We study an inverse initial-data problem for a quasilinear transport equation with nonlinear and memory effects. The unknown initial state is reconstructed from time-dependent measurements on the outflow boundary, with prescribed inflow data. We first apply a Legendre--exponential time-dimensional reduction to transform the governing equation into a finite nonlinear system in space. We then develop a Carleman-weighted and Tikhonov-regularized Picard method. At each iteration, the nonlinear terms are evaluated using the previous iterate, leading to a linear minimization problem with a unique solution. A Carleman estimate for the principal transport operator is used to prove that, for a sufficiently large Carleman parameter, the resulting Picard map is contractive on a prescribed admissible set. Consequently, the method converges from an arbitrary initial guess in that set. We also establish stability with respect to noisy outflow data. These analytical results concern the truncated and regularized reduced problem. Two-dimensional numerical experiments demonstrate accurate reconstruction of single and multiple inclusions, robustness with respect to noise, and rapid convergence of the Picard iteration.
