We investigate the method of nonstationary asymptotical regularization for solving linear ill-posed problems in Hilbert spaces. This method introduces the convex constraints that are proper lower semicontinuous and allowed to be non-smooth, therefore can be used for sparsity and discontinuity reconstruction. Under some suitable conditions , the convergence and regularity of the proposed method are established. Under the discretion of Runge–Kutta method, different iteration modes can be deduced for numerical implementation. The numerical results of iteration modes under one-stage explicit Euler, one-stage implicit Euler and two-stage explicit Runge–Kutta are presented to illustrate the efficiency of the proposed method.

中文翻译：

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线性不适定问题的凸约束非平稳渐近正则化方法

我们研究了解决希尔伯特空间中线性不适定问题的非平稳渐近正则化方法。该方法引入了适当下半连续且允许非光滑的凸约束，因此可用于稀疏性和不连续性重建。在一定条件下，该方法具有收敛性和规律性。在龙格-库塔方法的判断下，可以推导出不同的迭代模式进行数值实现。给出了一阶段显式欧拉、一阶段隐式欧拉和两阶段显式龙格-库塔迭代模式的数值结果，以说明该方法的效率。