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Zbl 1063.65040
Janno, Jaan; Tautenhahn, Ulrich
On Lavrentiev regularization for ill-posed problems in Hilbert scales.
(English)
[J] Numer. Funct. Anal. Optimization 24, No. 5-6, 531-555 (2003). ISSN 0163-0563; ISSN 1532-2467/e

Ill-posed problems (1) $Ax = y$ are considered, where $A: X \to X$ is a linear bounded injective operator in a Hilbert space $X$, and $A$ in addition is selfadjoint and nonnegative. The right-hand side $y \in X$ of equation (1) is approximately known, i.e., $y^\delta \in X$ and $\delta > 0$ with $\Vert y^\delta - y \Vert \le \delta$ are given, where $\Vert \cdot \Vert$ denotes the norm of the Hilbert space $X$. \par The main subject of this paper is a Lavrentiev regularization of equation (1) in Hilbert scales, this is, $A x_\alpha^\delta + \alpha B^s x_\alpha^\delta = y^\delta$. Here, $B$ is a linear unbounded selfadjoint strictly positive definite operator in $X$ such that for some $a > 0$, the norm $\Vert B^{-a} x \Vert$ is equivalent to $\Vert Ax \Vert$. In addition, $\alpha > 0$ denotes a regularization parameter and $s$ is a real number. For source representations $x^\dagger \in R(B^{-p})$ with $p > 0$ and real parameters $r$ satisfying some technical conditions, the authors prove estimates of the form $\Vert B^r(x_\alpha^\delta - x^\dagger) \Vert = \cal{O}(\delta^{(p-r)/(p+a)})$ for a priori- as well as a posteriori parameter choices of $\alpha$, respectively. Here, $x^\dagger$ denotes the solution of equation (1). \par The presented results then are extended to a class of regularization methods as well as to infinitely smoothing operators and nonlinear ill-posed problems, respectively. Finally the results are applied to an inverse heat conduction problem and a Volterra integral equation of the first kind, respectively.
[Robert Plato (Berlin)]
MSC 2000:
*65J20 Improperly posed problems (numerical methods in abstract spaces)
47A52 Ill-posed problems etc.
65J15 Equations with nonlinear operators (numerical methods)
47J06 Nonlinear ill-posed problems
65R30 Improperly posed problems (integral equations, numerical methods)
45G10 Nonsingular nonlinear integral equations
35K05 Heat equation
35R30 Inverse problems for PDE
65M32 Inverse problems
65J10 Equations with linear operators (numerical methods)

Keywords: linear ill-posed problems; Lavrentiev regularization; regularization methods; a priori parameter choices; a posteriori parameter choices; nonlinear ill-posed problems; monotone operators; inverse heat conduction problem; Volterra integral equation of the first kind; Hilbert space; Hilbert scales

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