Convergence rates for solutions of inhomogeneous ill-posed problems in Banach space with sufficiently smooth data
type: Item
instance of: scholarly article
title: Convergence rates for solutions of inhomogeneous ill-posed problems in Banach space with sufficiently smooth data
zbMATH Open document ID: 1494.47016
DOI: 10.1007/978-3-030-51945-2_12
author: Matthew A. Fury
publication date: 2021-09-09T00:00:00Z
review text: The subject of this paper is the ill-posed Cauchy problem \[ u^\prime(t) = Au(t) + h(t), \ 0 < t < T, \quad u(0) = \varphi, \tag{1}\] where \( A \) is a closed linear operator on a Banach space \( X \) such that \( -A \) is the infinitesimal generator of a holomorphic semigroup \( e^{-tA} \) of angle \( \theta \in (0,\frac{\pi}{2}) \) on \( X \). In addition, \( \varphi \in X \) as well as the function \( h:[0,T] \to X \) are given, and \( 0 \) belongs to the resolvent set of \( A \). For the regularization of the given ill-posed Cauchy problem, the author considers the regularized Cauchy problem \[v_\beta^\prime(t) = f_\beta(A)v_\beta(t) + h(t), \ 0 < t < T, \quad u(0) = \varphi, \tag{2}\] where, for each parameter \( 0 < \beta < 1 \), the operator \(f_\beta(A) \) is the infinitesimal generator of a \( C_0 \) semigroup on \( X \). It is assumed that \( f_\beta(A) \) satisfies condition (A\(^+\)) which by definition means that the following are satisfied: (i) \( \textup{Dom}(f_\beta(A)) \supseteq \textup{Dom}(A^2) \), and for some constant \( R \) independent of \( \beta \), there holds \( \Vert (-A + f_\beta(A)) x \Vert \le \beta R \Vert A^2 x \Vert \) for each \( x \in \textup{Dom}(A^2) \), and (ii) there exist \( \frac{\pi}{2} - \theta < \nu < \frac{\pi}{2} \) and some constant \( C \ge 0 \) independent of \( \beta \) such that the inequality \( \textup{Re} (-w + f_\beta(w) ) \le C \vert w \vert \) holds for each \( w \in S_\nu := \{0 \neq w \in \mathbb{C} : \vert \textup{arg} \, w \vert < \nu \} \). It is shown that under those conditions, the estimate \[ \Vert u(t) - v_\beta(t) \Vert \le c\beta^{1-\omega(t)} M^{\omega(t)}, \quad 0 < t < T, \] holds, where \( c \) and \( M \) denote some constants which depend on the smoothness both of the initial data \( \varphi \in X \) and the inhomogeneity \( h:[0,T] \to X \). In addition, \( \omega \) is some function satisfying \( 0 = \omega(0) \le \omega(t) < \omega(T) = 1 \) for each \( 0 \le t < T \), and \( u \) and \(…
reviewed by: Robert Plato
Mathematics Subject Classification ID: 47A52
Mathematics Subject Classification ID: 47D06
zbMATH DE Number: 7393032
zbMATH Keywords: ill-posed Cauchy problem
zbMATH Keywords: infinitesimal generator of a semigroup
zbMATH Keywords: holomorphic semigroup
zbMATH Keywords: \( C_0 \) semigroup
zbMATH Keywords: regularization
MaRDI profile type: Publication
full work available at URL: 978-3-030-51945-2 12
OpenAlex ID: W3143429128
cites work: Properties of solutions of ordinary differential equations in banach space
cites work: Structural stability for ill-posed problems in Banach space
cites work: Fractional powers of closed operators and the semigroups generated by them
cites work: Q5454875
cites work: Continuous dependence on modeling for nonlinear ill-posed problems
cites work: Regularization and convergence for ill-posed backward evolution equations in Banach spaces
cites work: Entire solutions of the abstract Cauchy problem
cites work: \(C\)-semigroups and the Cauchy problem
cites work: Q4039882
cites work: Q5409354
cites work: Logarithmic well-posed approximation of the backward heat equation in Banach space
cites work: Continuous Dependence on Modeling in Banach Space Using a Logarithmic Approximation
cites work: The functional calculus for sectorial operators
cites work: Modified quasi-reversibility method for final value problems in Banach spaces
cites work: Regularization for ill-posed Cauchy problems associated with generators of analytic semigroups
cites work: Regularization for a class of ill-posed Cauchy problems
cites work: Inverse problems. Tikhonov theory and algorithms
cites work: Q3020308
cites work: Approximation of a parabolic non-linear evolution equation backwards in time
cites work: An identification problem for a linear evolution equation in a Banach space and applications
cites work: General theory of the ill-posed Cauchy problem
cites work: Abstract Cauchy problems: three approaches
cites work: Q4765456
cites work: Logarithmic convexity results for holomorphic semigroups
cites work: Q5600361
cites work: Q4056957
cites work: Semigroups of linear operators and applications to partial differential equations
cites work: Q4256419
cites work: Q4066072
cites work: Q4340161
cites work: Variational methods in imaging
cites work: Regularization methods in Banach spaces.
cites work: The final value problem for evolution equations
cites work: Q3630646
cites work: On a backward parabolic problem with local Lipschitz source
cites work: Identification of the pollution source of a parabolic equation with the time-dependent heat conduction
description: scientific article; zbMATH DE number 7393032