Convergence of the Hilbert uniqueness method via Tikhonov regularization
type: Item
instance of: scholarly article
title: Convergence of the Hilbert uniqueness method via Tikhonov regularization
zbMATH Open document ID: 0943.65109
DOI: 10.1023/A:1021792225922
author: Stefan Kindermann
published in: Journal of Optimization Theory and Applications
publication date: 2000-09-05T00:00:00Z
review text: A parabolic differential equation on \([0,1]\times [0,T]\) with Dirichlet data and homogeneous initial data of the following form is considered \[ {\partial y\over\partial t}-{\partial\over\partial x} \Biggl(a(x){\partial y\over\partial x}\Biggr)= 0, \] \[ y(0,x)= 0,\quad y(t,0)= \nu_0(t),\quad y(t,1)= \nu_1(t), \] with diffusion coefficient \(a(x)\in C^1(0,1)\), \(a(x)\geq \mu>0\). For this problem, the author identifies the Hilbert uniqueness method with the calculation of the pseudoinverse. Because of its ill-posedness, it is approximated by a regularized Hilbert uniqueness method which is identical with Tikhonov regularization. The author finds sufficient conditions for convergence and convergence rates. Numerical experiments are given.
Mathematics Subject Classification ID: 65M30
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 35K05
Mathematics Subject Classification ID: 35R25
zbMATH DE Number: 1406369
zbMATH Keywords: ill-posed problem
zbMATH Keywords: numerical experiments
zbMATH Keywords: parabolic differential equation
zbMATH Keywords: Hilbert uniqueness method
zbMATH Keywords: pseudoinverse
zbMATH Keywords: Tikhonov regularization
zbMATH Keywords: convergence
reviewed by: Hans Benker
MaRDI profile type: Publication
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full work available at URL: a:1021792225922
OpenAlex ID: W185280565
description: scientific article; zbMATH DE number 1406369