We study a linear operator equation that does not satisfy the Hadamard well-posedness conditions. It is assumed that the solution of the equation has different smoothness properties in different regions of its domain. More exactly, the solution is representable as the sum of a smooth and discontinuous components. The Tikhonov regularization method is applied for the construction of a stable approximate solution. In this method, the stabilizer is the sum of the Lebesgue norm and the smoothed \(BV\)-norm. Each of the functionals in the stabilizer depends only on one component and takes into account its properties. Convergence theorems are proved for the regularized solutions and their discrete approximations. It is shown that discrete regularized solutions can be found with the use of the Newton method and nonlinear analogs of \(\alpha\)-processes.

中文翻译：

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包含不连续分量的线性算子方程的正则化算法分析

我们研究了不满足Hadamard适定性条件的线性算子方程。假设方程的解在其域的不同区域具有不同的平滑度属性。更确切地说，该解决方案可表示为平滑分量和不连续分量的总和。Tikhonov正则化方法用于构造稳定的近似解。在这种方法中，稳定器是Lebesgue范数和平滑\（BV \）的和。-规范。稳定剂中的每个功能仅取决于一个组件，并考虑到其性能。证明了正则解及其离散近似的收敛定理。结果表明，使用牛顿法和\（\ alpha \）-过程的非线性类似物可以找到离散的正则解。