On Ill-Posedness Measures and Space Change in Sobolev Scales

Abstract

The degree of ill-posedness of a linear inverse problem is an important knowledge base to select appropriate regularization methods for the stable approximate solution of such a problem. In this paper, we consider ill-posedness measures for a linear ill-posed operator equation $Ax=y$, where the compact linear operator $A:X\to Y$ maps between infinite dimensional Hilbert spaces. Using the decay rate of singular values of $A$ tending to zero we define an interval of ill-posedness and motivate its meaning by considering lower and upper bounds for the rates of the condition numbers occurring in the numerical solution process of the discretized problem. An equivalent interval information is obtained when compactness measures as $ϵ$-entropy or $ϵ$-capacity are exploited alternatively. For the specific case $X:=L^2(0,1)$, the space change problem of shifting the space $X$ along a Sobolev scale is treated. In detail, we study the change of the interval of ill-posedness if the solutions are restricted to the Sobolev space $W_2^1[0,1)$. The results of these considerations are a warning to characterize the ill-posedness of a problem superficial. Moreover, the interdependences between ill-posedess measures, embedding operators, Hilbert and Sobolev scales are discussed.