Solving ill-posed control problems by stabilized finite element methods: an alternative to Tikhonov regularization
Solving ill-posed control problems by stabilized finite element methods: an alternative to Tikhonov regularization
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DOI：
10.1088/1361-6420/aaa32b
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发表时间：
2018-03-01
期刊：
影响因子：
2.1
通讯作者：
Larson, Mats G.
中科院分区：
文献类型：
--
作者：
Burman, Erik;Hansbo, Peter;Larson, Mats G.
Tikhonov正则化是不适定问题正则化中最常用的方法之一。在椭圆型偏微分控制问题的有限元解的设置中，吉洪诺夫正则化相当于将控制变量或其导数的适当加权的最小二乘项添加到确定最优性系统的拉格朗日量。在这份说明中，我们表明，在对流占优的对流扩散问题的设置中开发的离散不适定问题的稳定化方法，可以非常适合于稳定最优控制问题，并且Tikhonov正则化将导致不太准确的离散解。我们认为一些反问题的泊松方程作为一个例子，并获得新的误差估计重建的解决方案从测量数据和重建的源项从测量数据。这些估计包括离散化误差和测量误差的影响。
Tikhonov regularization is one of the most commonly used methods for the regularization of ill-posed problems. In the setting of finite element solutions of elliptic partial differential control problems, Tikhonov regularization amounts to adding suitably weighted least squares terms of the control variable, or derivatives thereof, to the Lagrangian determining the optimality system. In this note we show that the stabilization methods for discretely illposed problems developed in the setting of convection-dominated convection-diffusion problems, can be highly suitable for stabilizing optimal control problems, and that Tikhonov regularization will lead to less accurate discrete solutions. We consider some inverse problems for Poisson's equation as an illustration and derive new error estimates both for the reconstruction of the solution from the measured data and reconstruction of the source term from the measured data. These estimates include both the effect of the discretization error and error in the measurements.