zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Optimality for ill-posed problems under general source conditions. (English) Zbl 0907.65049

The problem of identifying the unknown x of the ill-posed inverse problem Ax=y is studied, where A(X,Y) is a linear bounded operator between infinite-dimensional Hilbert spaces X and Y with non-closed range R(A) of A and, xM φ,E ={xX; x-x ¯=φ(A * A) 1/2 v, v=E} (x ¯ denotes an initial approximation for the problem Ax=y) with appropriate functions φ. As regards accuracy which can be obtained for identifying x from y δ Y it is proved that under certain conditions

infsupRy δ -x=Eρ -1 (δ 2 /E 2 )

holds with ρ(λ)=λφ -1 (λ), where inf is taken over all methods R:YX and the sup is taken over all xM φ,E and y-y δ δ. In addition, it is proved the optimality of a general class of regularization methods which guarantee this best possible accuracy. In this general class Tikhonov methods and spectral methods are special cases. Different classes of examples are discussed.

65J10Equations with linear operators (numerical methods)
65J20Improperly posed problems; regularization (numerical methods in abstract spaces)
47A50Equations and inequalities involving linear operators, with vector unknowns