The present paper analyzes a spectral regularization of a time-reversed reaction-diffusion problem with globally and locally Lipschitz nonlinearities. This type of inverse and ill-posed problems arises in a variety of real-world applications concerning heat conduction and tumour source localization. In accordance with the weak solvability result for the forward problem, we focus on the inverse problem with high-order Sobolev-Gevrey smoothness and with Sobolev measurements. As expected from the well-known results for the linear case, we prove that this nonlinear spectral regularization possesses a logarithmic rate of convergence in a high-order Sobolev norm. The proof can be done by the verification of variational source condition; this way validates such a fine strategy in the framework of inverse problems for nonlinear partial differential equations. Ultimately, we study a semi-discrete version of the regularization method for a class of reaction-diffusion problems with non-degenerate nonlinearity. The convergence of this iterative scheme is also investigated.
翻译:本文分析了全球和地方Lipschitz非线性和非线性的时间逆反反扩散问题的光谱正规化问题。 这种反向和错误的问题出现在有关热导和肿瘤源本地化的各种现实应用中。 根据前期问题的软软溶性结果,我们侧重于高阶Sobolev-Gevrey光滑和Sobolev测量的反向问题。正如线性案例众所周知的结果所预期的那样,我们证明这种非线性光谱正规化在高阶Sobolev规范中具有对数趋同率。通过核实变异源条件可以证明这一点;这种方式在非线性部分差异方程式的反向问题框架内验证了这种精细战略。最后,我们研究一种非线性反射方法的半分解版本,以非线性非线性反射问题为一类。这种迭代办法的趋同也得到了调查。