IRGNM Tikhonov和IRGNM Ivanov方法在Banach空间中在切线锥条件下的收敛性与自适应离散化
Convergence and adaptive discretization of the IRGNM Tikhonov and the IRGNM Ivanov method under a tangential cone condition in Banach space
AI总结：
本文研究了在Banach空间中在切线锥条件下，IRGNM Tikhonov和Ivanov方法的收敛性与自适应离散化，通过目标导向加权对偶残差估计器提供误差界。
摘要：
本文考虑了迭代正则化Gauss-Newton方法（IRGNM）的经典Tikhonov版本和Ivanov类型版本，其中正则化通过限制解的范围实现。在一般Banach空间设置和切线锥条件下，收敛性（无需源条件，因此无需速率）此前仅在对算子非线性更强的限制或空间限制下得到证明。此外，我们为离散问题提供了收敛结果，并展示了如何通过目标导向加权对偶残差估计器提供所需的误差界。结果通过非线性椭圆边值问题的反源问题进行了说明，包括测度值和L^∞源的情况。对于后者，我们还提供了Ivanov类型的IRGNM的数值结果。
英文摘要：
In this paper we consider the Iteratively Regularized Gauss-Newton Method (IRGNM) in its classical Tikhonov version and in an Ivanov type version, where regularization is achieved by imposing bounds on the solution. We do so in a general Banach space setting and under a tangential cone condition, while convergence (without source conditions, thus without rates) has so far only been proven under stronger restrictions on the nonlinearity of the operator and/or on the spaces. Moreover, we provide a convergence result for the discretized problem with an appropriate control on the error and show how to provide the required error bounds by goal oriented weighted dual residual estimators. The results are illustrated for an inverse source problem for a nonlinear elliptic boundary value problem, for the cases of a measure valued and of an $L^\infty$ source. For the latter, we also provide numerical results with the Ivanov type IRGNM.
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