Arbitrary divergence speed of the least-squares method in infinite-dimensional inverse ill-posed problems
A standard engineering procedure for approximating the solutions of an infinite-dimensional inverse problem of the form Ax = y, where A is a given compact linear operator on a Hilbert space X and y is the given data, is to find a sequence {XN} of finite-dimensional approximating subspaces of X whose...
| Autores: | , |
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| Formato: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2006 |
| País: | Argentina |
| Recursos: | Consejo Nacional de Investigaciones Científicas y Técnicas |
| Repositorio: | CONICET Digital (CONICET) |
| Idioma: | inglés |
| OAI Identifier: | oai:ri.conicet.gov.ar:11336/84067 |
| Acesso em linha: | http://hdl.handle.net/11336/84067 |
| Access Level: | acceso abierto |
| Palavra-chave: | Arbitrary Divergence Least-Squares Method https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
| Resumo: | A standard engineering procedure for approximating the solutions of an infinite-dimensional inverse problem of the form Ax = y, where A is a given compact linear operator on a Hilbert space X and y is the given data, is to find a sequence {XN} of finite-dimensional approximating subspaces of X whose union is dense in X and to construct the sequence {xN} of least-squares solutions of the problem in X N. In 1980, Seidman showed that if the problem is ill-posed, then, without any additional assumptions on the exact solution or on the sequence of approximating subspaces XN, it cannot be guaranteed that the sequence {xN} will converge to the exact solution. In this paper, this result is extended in the following sense: it is shown that if X is separable, then for any y ∈ X, y ≠ 0 and for any arbitrarily given function there exists an injective, compact linear operator A and an increasing sequence of finite-dimensional subspaces XN ⊂ X such that for all, where xN is the least-squares solution of Ax = y in XN. © 2006 IOP Publishing Ltd. |
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