一个终端值准线性抛物问题的准反向方法分析
Analysis of a quasi-reversibility method for a terminal value quasi-linear parabolic problem with measurements
AI总结：
本文提出了一种改进的准反向方法，用于计算具有噪声数据的非局部终端-边界值抛物问题的指数不稳定解，通过数据测量使用滤波正则化算子扰动问题，采用变分框架分析新的近似问题，并讨论了该方法在半线性情况下的分析及一般化应用。
摘要：
本文提出了一种改进的准反向方法，用于计算具有噪声数据的非局部终端-边界值抛物问题的指数不稳定解。基于数据测量，我们通过所谓的滤波正则化算子扰动问题，设计了一个近似问题。不同于最近发展的传统谱方法，我们采用变分框架分析这个新的近似问题，其中可以应用有限元方法。为了看到该方法的整体骨架，我们的主要结果集中在半线性情况的分析，并讨论了该分析可以适应的一些一般化情况。正如在许多物理过程中所普遍存在的，可能有大量模型源于这个更简单的案例，例如脑肿瘤的源定位问题和核科学中的非线性汇热传导问题。针对每种噪声水平，我们利用Faedo-Galerkin方法研究近似问题的弱可解性。依靠能量似分析，我们提供了所提出方法在真解足够光滑时的L^2-H^1收敛率。根据域的维度，我们获得了L^r中的一些误差估计，其中r>2。本文还描绘了准线性系统的反向唯一性的证明。为了证明可接受的正则性假设，讨论了若干物理应用。
英文摘要：
This paper presents a modified quasi-reversibility method for computing the exponentially unstable solution of a nonlocal terminal-boundary value parabolic problem with noisy data. Based on data measurements, we perturb the problem by the so-called filter regularized operator to design an approximate problem. Different from recently developed approaches that consist in the conventional spectral methods, we analyze this new approximation in a variational framework, where the finite element method can be applied. To see the whole skeleton of this method, our main results lie in the analysis of a semi-linear case and we discuss some generalizations where this analysis can be adapted. As is omnipresent in many physical processes, there is likely a myriad of models derived from this simpler case, such as source localization problems for brain tumors and heat conduction problems with nonlinear sinks in nuclear science. With respect to each noise level, we benefit from the Faedo-Galerkin method to study the weak solvability of the approximate problem. Relying on the energy-like analysis, we provide detailed convergence rates in $L^2$-$H^1$ of the proposed method when the true solution is sufficiently smooth. Depending on the dimensions of the domain, we obtain an error estimate in $L^{r}$ for some $r>2$. Proof of the backward uniqueness for the quasi-linear system is also depicted in this work. To prove the regularity assumptions acceptable, several physical applications are discussed.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
