This paper considers large-scale linear ill-posed inverse problems whose solutions can be represented as sums of smooth and piecewise constant components. To solve such problems we consider regularizers consisting of two terms that must be balanced. Namely, a Tikhonov term guarantees the smoothness of the smooth solution component, while a total-variation (TV) regularizer promotes blockiness of the non-smooth solution component. A scalar parameter allows to balance between these two terms and, hence, to appropriately separate and regularize the smooth and non-smooth components of the solution. This paper proposes an efficient algorithm to solve this regularization problem by the alternating direction method of multipliers (ADMM). Furthermore, a novel algorithm for automatic choice of the balancing parameter is introduced, using robust statistics. The proposed approach is supported by some theoretical analysis, and numerical experiments concerned with different inverse problems are presented to validate the choice of the balancing parameter.

翻译：本文审议了大规模线性不正确反向问题, 其解决方案可以作为平滑和片段常态组件的总和来代表。 为了解决这些问题, 我们考虑由两个必须平衡的术语组成的正规化者。 也就是说, Tikhonov 术语可以保证平滑解决方案组件的顺利性, 而一个总变量( TV) 常规化器可以促进非单向解决方案组件的阻塞性。 一个标度参数可以平衡这两个术语, 从而适当区分和规范解决方案的平滑和非单向组件。 本文建议了一种高效的算法, 通过乘数交替方向方法( ADMMM)来解决这一正规化问题。 此外, 引入了一种新算法, 用于自动选择平衡参数, 使用可靠的统计数据。 提议的方法得到一些理论分析的支持, 并且提出了与不同反面问题有关的数字实验, 以验证平衡参数的选择 。