论文标题
基于亚级别的Lavrentiev单调问题的正规化
Subgradient-based Lavrentiev regularisation of monotone ill-posed problems
论文作者
论文摘要
We introduce subgradient-based Lavrentiev regularisation of the form \begin{equation*} \mathcal{A}(u) + α\partial \mathcal{R}(u) \ni f^δ\end{equation*} for linear and nonlinear ill-posed problems with monotone operators $\mathcal{A}$ and general regularisation functionals $ \ MATHCAL {R} $。与Tikhonov正则化相反，此方法将方程式本身呈现，并避免使用$ \ Mathcal {a} $的衍生物的伴随。因此，它特别适用于仅取决于过去信息并允许实时计算正则化解决方案的时间造成问题。我们在BANACH空间中建立了一般体系理论，并证明了具有变异源条件的收敛速率结果。此外，我们证明了它在半连接抛物线PDE中的第一类和参数识别问题的线性伏特拉积分运算符中的总变量降级中的应用。
We introduce subgradient-based Lavrentiev regularisation of the form \begin{equation*} \mathcal{A}(u) + α\partial \mathcal{R}(u) \ni f^δ\end{equation*} for linear and nonlinear ill-posed problems with monotone operators $\mathcal{A}$ and general regularisation functionals $\mathcal{R}$. In contrast to Tikhonov regularisation, this approach perturbs the equation itself and avoids the use of the adjoint of the derivative of $\mathcal{A}$. It is therefore especially suitable for time-causal problems that only depend on information in the past and allows for real-time computation of regularised solutions. We establish a general well-posedness theory in Banach spaces and prove convergence-rate results with variational source conditions. Furthermore, we demonstrate its application in total-variation denoising in linear Volterra integral operators of the first kind and parameter-identification problems in semilinear parabolic PDEs.
