On ill-posedness measures and space change in Sobolev scales

作     者：Hofmann, B. Tautenhahn, U. 

作者机构：Technische Universität Chemnitz Fakultät für Mathematik D-09107 Chemnitz Germany Hochschule für Technik Wirtschaft und Sozialwesen Zittau Görlitz (FH) FB Mathematik D-02763 Zittau P.O. Box 261 Germany 

出 版 物：《Zeitschrift für Analysis und ihre Anwendungen》 (Z. Anal. Anwend.)

年 卷 期：1997年第16卷第4期

页      面：979-1000页

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

主　　题：Compact linear operators Degree of ill-posedness Embedding operators Hilbert scale Ill-posed problems Ill-posedness measures Interval of ill-posedness Sobolev scale ε-capacity 

摘      要：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 → Y maps between infinite dimensional Hubert 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 := L2(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 W22[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, Hubert and Sobolev scales are discussed. © Heldermann Verlag.