Inverse medium scattering for a nonlinear Helmholtz equation
type: Item
instance of: scholarly article
title: Inverse medium scattering for a nonlinear Helmholtz equation
zbMATH Open document ID: 1495.35210
author: Roland Griesmaier
author: Marvin Knöller
author: Rainer Mandel
published in: Journal of Mathematical Analysis and Applications
publication date: 2022-07-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/2202.05533
review text: In this paper, the authors discussed a direct and inverse scattering problem for a class of nonlinear Helmholtz equations in unbounded free space. Assuming that the intensities of the incident waves are sufficiently small relative to the size of the nonlinearity, they established existence and uniqueness of solutions to the direct and inverse scattering problem, and showed that the nonlinear index of refraction is uniquely determined based on linearization techniques and estimates for the linearization error. The authors also generalized two shape reconstruction methods, a factorization method and a monotonicity method, to recover the support of such nonlinear scattering objects. Numerical results were also provided to illustrate their theoretical findings.
reviewed by: Mengni Li
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 35P25
Mathematics Subject Classification ID: 35P30
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35J61
zbMATH DE Number: 7559877
zbMATH Keywords: inverse scattering
zbMATH Keywords: nonlinear Helmholtz equation
zbMATH Keywords: uniqueness
zbMATH Keywords: factorization method
zbMATH Keywords: monotonicity method
MaRDI profile type: Publication
OpenAlex ID: W4281555483
arXiv ID: 2202.05533
cites work: Monotonicity in inverse obstacle scattering on unbounded domains
cites work: Recovering a potential from Cauchy data in the two-dimensional case
cites work: A qualitative approach to inverse scattering theory
cites work: Complex Solutions and Stationary Scattering for the Nonlinear Helmholtz Equation
cites work: A simple method for solving inverse scattering problems in the resonance region
cites work: Inverse Acoustic and Electromagnetic Scattering Theory
cites work: An inverse problem for a semi-linear elliptic equation in Riemannian geometries
cites work: The direct and inverse scattering problem for the semilinear Schrödinger equation
cites work: Elliptic partial differential equations of second order
cites work: Monotonicity in Inverse Medium Scattering on Unbounded Domains
cites work: Non trivial \(L^q\) solutions to the Ginzburg-Landau equation
cites work: Three-dimensional Direct and Inverse Scattering for the Schrödinger Equation with a General Nonlinearity
cites work: Dimension Bounds in Monotonicity Methods for the Helmholtz Equation
cites work: Monotonicity and local uniqueness for the Helmholtz equation
cites work: Monotonicity-Based Shape Reconstruction in Electrical Impedance Tomography
cites work: Unique determination of potentials and semilinear terms of semilinear elliptic equations from partial Cauchy data
cites work: Global Uniqueness for a Two-Dimensional Semilinear Elliptic Inverse Problem
cites work: Global uniqueness for a semilinear elliptic inverse problem
cites work: Inverse problem for a nonlinear Helmholtz equation
cites work: Characterization of the shape of a scattering obstacle using the spectral data of the far field operator
cites work: Factorization of the far-field operator for the inhomogeneous medium case and an application in inverse scattering theory
cites work: New characterizations of solutions in inverse scattering theory
cites work: An Introduction to the Mathematical Theory of Inverse Problems
cites work: Q5441004
cites work: The mathematical theory of time-harmonic Maxwell's equations. Expansion-, integral-, and variational methods
cites work: Linear integral equations
cites work: Inverse problems for elliptic equations with power type nonlinearities
cites work: Explicit characterization of the support of non-linear inclusions
cites work: Uncountably many solutions for nonlinear Helmholtz and curl-curl equations
cites work: Q4459242
cites work: Reconstructions from boundary measurements
cites work: Multidimensional inverse spectral problem for the equation \(-\Delta \psi -(v(x)-Eu(x))\psi =0\)
cites work: Recovery of the potential from fixed-energy scattering data
cites work: Inverse Born approximation for the nonlinear two-dimensional Schrödinger operator
cites work: Inverse fixed energy scattering problem for the generalized nonlinear Schrödinger operator
cites work: A uniqueness theorem and reconstruction of singularities for a two-dimensional nonlinear Schrödinger equation
cites work: Direct and inverse scattering for nonlinear Schrödinger equation in 2D
cites work: Inverse backscattering problem for the generalized nonlinear Schrödinger operator in two dimensions
cites work: Q3563120
cites work: A new non-iterative inversion method for electrical resistance tomography
cites work: Q4496151
cites work: Q5332803
DOI: 10.1016/J.JMAA.2022.126356
description: scientific article