论文标题
关于非线性不良问题的梯度类型投影方法的家族
On a family of gradient type projection methods for nonlinear ill-posed problems
论文作者
论文摘要
我们提出和分析了一系列连续的投影方法，其步骤指导与解决方案的非线性不良问题相同，以满足切向锥条件（TCC）。该家族遍布Landweber方法，最小误差法和最陡峭的下降方法； Thush提供了一个统一的框架来分析这些方法。此外，我们在此家族中定义了新方法，这些方法是TCC常数的收敛，范围是Landweber和其他梯度类型方法所需的范围两倍。 TCC广泛用于迭代方法，用于解决非线性不良问题。这项工作的关键思想是使用TCC来构建具有分离属性的特殊凸集，并成功地投射到这些集合上。为非线性2D椭圆参数识别问题提供了数值实验，以验证我们方法的效率。
We propose and analyze a family of successive projection methods whose step direction is the same as Landweber method for solving nonlinear ill-posed problems that satisfy the Tangential Cone Condition (TCC). This family enconpasses Landweber method, the minimal error method, and the steepest descent method; thush providing an unified framework for the analysis of these methods. Moreover, we define in this family new methods which are convergent for the constant of the TCC in a range twice as large as the one required for the Landweber and other gradient type methods. The TCC is widely used in the analysis of iterative methods for solving nonlinear ill-posed problems. The key idea in this work is to use the TCC in order to construct special convex sets possessing a separation property, and to succesively project onto these sets. Numerical experiments are presented for a nonlinear 2D elliptic parameter identification problem, validating the efficiency of our method.
