In this paper we consider the solution of monotone inverse problems using the particular example of a parameter identification problem for a semilinear parabolic PDE. For the regularized solution of this problem, we introduce a total variation based regularization method requiring the solution of a monotone inclusion problem. We show well-posedness in the sense of inverse problems of the resulting regularization scheme. In addition, we introduce and analyze a numerical algorithm for the solution of this inclusion problem using a nested inertial primal dual method. We demonstrate by means of numerical examples the convergence of both the numerical algorithm and the regularization method.

翻译：本文以半线性抛物型偏微分方程的参数辨识问题为例，探讨单调逆问题的求解方法。针对该问题的正则化解，我们引入了一种基于全变差的正则化方法，该方法要求解一个单调包含问题。我们证明了所得正则化方案在逆问题意义下的适定性。此外，我们提出并分析了一种求解该包含问题的数值算法，该算法采用嵌套惯性原始对偶方法。通过数值算例，我们验证了数值算法与正则化方法的收敛性。