ABSTRACT

We study regularization of ill-posed equations involving multiplication operators when the multiplier function is positive almost everywhere and zero is an accumulation point of the range of this function. Such equations naturally arise from equations based on non-compact self-adjoint operators in Hilbert space, after applying unitary transformations arising out of the spectral theorem. For classical regularization theory, when noisy observations are given and the noise is deterministic and bounded, then non-compactness of the ill-posed equations is a minor issue. However, for statistical ill-posed equations with non-compact operators less is known if the data are blurred by white noise. We develop a theory for spectral regularization with emphasis on this case. In this context, we highlight several aspects, in particular, we discuss the intrinsic degree of ill-posedness in terms of rearrangements of the multiplier function. Moreover, we address the required modifications of classical regularization schemes in order to be used for non-compact statistical problems, and we also introduce the concept of the effective ill-posedness of the operator equation under white noise. This study is concluded with prototypical examples for such equations, as these are deconvolution equations and certain final value problems in evolution equations.

摘要

我们研究涉及乘法算子的病态方程的正则化，当乘数函数几乎处处为正且零是该函数范围的累积点时。在应用由谱定理产生的酉变换之后，这些方程自然产生于基于希尔伯特空间中非紧自伴算子的方程。对于经典的正则化理论，当给出噪声观测并且噪声是确定性和有界的时，不适定方程的非紧致性是一个小问题。然而，对于具有非紧致算子的统计不适定方程，如果数据被白噪声模糊，我们对它的了解就更少了。我们开发了一种光谱正则化理论，重点是这种情况。在这种情况下，我们强调几个方面，特别是，我们根据乘数函数的重新排列来讨论内在的不适定程度。此外，我们解决了经典正则化方案所需的修改，以便用于非紧统计问题，并且我们还引入了白噪声下算子方程的有效不适定性的概念。本研究以此类方程的原型示例结束，因为这些是反卷积方程和演化方程中的某些终值问题。