Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top

2015 | Book

Iteratively Regularized Gauss-Newton Methods under Random Noise Read chapter Read first chapter

Inverse Problems and Applications

Editor: Larisa Beilina

Publisher: Springer International Publishing

Book Series : Springer Proceedings in Mathematics & Statistics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Login to get access
insite
SEARCH

About this book

​​This volume arose from the Third Annual Workshop on Inverse Problems, held in Stockholm on May 2-6, 2012. The proceedings present new analytical developments and numerical methods for solutions of inverse and ill-posed problems, which consistently pose complex challenges to the development of effective numerical methods. The book highlights recent research focusing on reliable numerical techniques for the solution of inverse problems, with relevance to a range of fields including acoustics, electromagnetics, optics, medical imaging, and geophysics. ​

Table of Contents

Frontmatter
Iteratively Regularized Gauss-Newton Methods under Random Noise
Abstract
We investigate a class of regularized Gauss–Newton type methods for solving irregular nonlinear equations with general smooth operators in a Hilbert space. Perturbations of right-hand sides of equations are modeled by Hilbert space valued random elements. We analyze two qualitatively different schemes of forming stochastic measurement data, establish approximation properties of the methods in the mean square sense and prove corresponding accuracy estimates The work is supported by the Russian Foundation for Basic Research (project no. 12–01–00239a).
Mikhail Yu. Kokurin, Anatoly B. Bakushinsky
Methods of Quantitative Reconstruction of Shapes and Refractive Indices from Experimental data
Abstract
In this chapter we summarize results of [5, 6, 14] and present new results of reconstruction of refractive indices and shapes of objects placed in the air from blind backscattered experimental data using two-stage numerical procedure of [4]. Data are collected using a microwave scattering facility which was built at the University of North Carolina at Charlotte.
On the first stage the approximately globally convergent method of [4] is applied to get a good first approximation for the exact solution. Results of this stage are presented in [5, 14]. On the second stage the local adaptive finite element method of [1] is applied to refine the solution obtained on the first stage. In this chapter we briefly describe methods and present new results for both stages.
Larisa Beilina, Nguyen Trung Thành, Michael V. Klibanov, John Bondestam Malmberg
A Posteriori Error Estimate in the Lagrangian Setting for an Inverse Problem Based on a New Formulation of Maxwell’s System
Abstract
In this paper we consider an inverse problem of determination of a dielectric permittivity function from a backscattered electromagnetic wave. The inverse problem is formulated as an optimal control problem for a certain partial differential equation derived from Maxwell’s system. We study a solution method based on finite element approximation and provide a posteriori error estimate for the use in an adaptive algorithm.
John Bondestam Malmberg
Determination of Permittivity from Propagation Constant Measurements in Optical Fibers
Abstract
We present a new method for determination of dielectric permittivity constant using measurements of fundamental mode of propagation constant in optical fiber’s. We first solve the forward spectral problem to compute the dispersion curve for the fundamental mode. Then using this curve we present an effective and accurate spline-collocation method for calculation of permittivity.
Evgenii Karchevskii, Alexandr Spiridonov, Larisa Beilina
Eigenmodes of Linearised Problems of Scattering and Generation of Oscillations on Cubically Polarisable Layers
Abstract
In the frequency domain, the resonant properties of nonlinear structures are determined by the proximity of the scattering/generation frequencies of the nonlinear structures to the complex eigenfrequencies of the corresponding homogeneous linear spectral problems with the induced nonlinear permeability of the medium. Here the case of cubically polarisable, canalising, and decanalising layers is considered.
Lutz Angermann, Yuri V. Shestopalov, Vasyl V. Yatsyk
Time Resolution in Transient Kinetics
Abstract
This study presents the mathematical background of deconvolution of concentration data in transient kinetic studies. In a case study with a flow reactor setup, it has been shown that the deconvolution algorithm results in a significant reduction in the time lag of an FTIR detector from approx. 23 s to approx. 3 s. This is an important achievement as otherwise the dynamic information of a reactive system (like the rate of adsorption or accumulation of surface species) would have been lost during that time interval. Using the regularizing theory of ill-posed, inverse problems, an algorithm for deconvolution of concentration measurements has been developed based on the discrepancy principle. Our software package, TranKin, can be easily adapted to various other laboratory reactor systems to enhance the time resolution of transient experiments.
Soheil Soltani, Ronnie Andersson, Bengt Andersson
Reconstruction of Dielectric Constants in a Cylindrical Waveguide
Abstract
We present the model of reconstruction of two constant complex dielectrics in a cylindrical waveguide using frequency domain measurements at the one point of a waveguide. We formulate two gradient-like methods to reconstruct these coefficients and present numerical examples showing performance of reconstruction.
Larisa Beilina, Anders Eriksson
Time-adaptive FEM for distributed parameter identification in mathematical model of HIV infection with drug therapy
Abstract
We propose a time-adaptive finite element method for the solution of a parameter identification problem for ODE system which describes dynamics of primary HIV infection with drug therapy. We present framework of a posteriori error estimate in the Tikhonov functional and in the Lagrangian. We also formulate the time-mesh refinement recommendation and an adaptive algorithm to find optimal values of the distributed parameter in our identification problem.
Larisa Beilina, Irina Gainova
The Layer-Stripping Algorithm for Reconstruction of Dielectrics in an Optical Fiber
Abstract
We present a new model of an approximate globally convergent method in a frequency domain for reconstruction of dielectric permittivity of a weakly guiding optical fiber. We consider data which are given only at the backscattered side of the medium which should be reconstructed. We formulate new approximately globally convergent algorithm for the reconstruction of dielectric permittivity function under the assumption that the magnetic permeability is a known constant.
Larisa Beilina, Evgenii Karchevskii
Simultaneous Reconstruction of Maxwell’s Coefficients from Backscattering Data
Abstract
We use the conjugade gradient method for the solution of an inverse problem for simultaneous reconstruction of the dielectric permittivity and magnetic permeability functions of the Maxwell's system in 3D using backscattering data. We show stability of our inverse problem using the Carleman estimates. Our numerical experiments show reliable reconstruction of both parameters using the optimization approach.
L. Beilina, M. Cristofol, K. Niinimäki
On the Solution of Forward and Inverse Problems of Voltammetry
Abstract
Inverse problem of voltammetry of simple electrolytes (aqueous solutions of electro-active components) is the coefficient inverse problem for the partial differential equation (PDE) of parabolic type. Since most algorithms for solution of such problems are based on the discrepancy minimization, their efficiency strongly depends on the way of calculation of the forward problem. In this paper we formulate forward and inverse problems of voltammetry, then consider three different algorithms of solution of the voltammetry equation, including developed by author semi-analytical method.
Nikolay Koshev
Metadata
Title
Inverse Problems and Applications
Editor
Larisa Beilina
Copyright Year
2015
Electronic ISBN
978-3-319-12499-5
Print ISBN
978-3-319-12498-8
DOI
https://doi.org/10.1007/978-3-319-12499-5

Premium Partner

    Image Credits