Abstract The vector problem of reconstruction of an unknown permittivity of an inhomogeneous body is considered. The original problem for Maxwell’s equations with an unknown permittivity and a given permeability is reduced to the system of integro-differential equations. The solution to the inverse problem is obtained in two steps. First, a solution to the vector integro-differential equation of the first kind is obtained from the given near-field data. The uniqueness of the solution to the integro-differential equation of the first kind is proved in the classes of piecewise constant functions. Second, the sought-for permittivity is straightforwardly calculated from the found solution and the total electric field. A series of test problems was solved using the two-step method. Procedures of approximate solutions’ refining were implemented. Comparison between the given permittivities and the found approximate solutions shows efficiency of the proposed method.
{"title":"Inverse vector problem of diffraction by inhomogeneous body with a piecewise smooth permittivity","authors":"M. Medvedik, Y. Smirnov, A. Tsupak","doi":"10.1515/jiip-2022-0060","DOIUrl":"https://doi.org/10.1515/jiip-2022-0060","url":null,"abstract":"Abstract The vector problem of reconstruction of an unknown permittivity of an inhomogeneous body is considered. The original problem for Maxwell’s equations with an unknown permittivity and a given permeability is reduced to the system of integro-differential equations. The solution to the inverse problem is obtained in two steps. First, a solution to the vector integro-differential equation of the first kind is obtained from the given near-field data. The uniqueness of the solution to the integro-differential equation of the first kind is proved in the classes of piecewise constant functions. Second, the sought-for permittivity is straightforwardly calculated from the found solution and the total electric field. A series of test problems was solved using the two-step method. Procedures of approximate solutions’ refining were implemented. Comparison between the given permittivities and the found approximate solutions shows efficiency of the proposed method.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46856106","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, we address stability results in determining the time-dependent scalar and vector potentials appearing in the convection-diffusion equation from the knowledge of the Cauchy data set. We prove Hölder-type stability estimates. The key tool used in this work is the geometric optics solution.
{"title":"Hölder stability estimates in determining the time-dependent coefficients of the heat equation from the Cauchy data set","authors":"Imen Rassas","doi":"10.1515/jiip-2021-0013","DOIUrl":"https://doi.org/10.1515/jiip-2021-0013","url":null,"abstract":"Abstract In this paper, we address stability results in determining the time-dependent scalar and vector potentials appearing in the convection-diffusion equation from the knowledge of the Cauchy data set. We prove Hölder-type stability estimates. The key tool used in this work is the geometric optics solution.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49069494","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Let X be a Banach space with norm ∥ ⋅ ∥ {|cdot|} . Let A : D ( A ) ⊂ X → X {A:D(A)subset Xrightarrow X} be an (possibly unbounded) operator that generates a uniformly bounded holomorphic semigroup. Suppose that ε > 0 {varepsilon>0} and T > 0 {T>0} are two given constants. The backward parabolic equation of finding a function u : [ 0 , T ] → X {u:[0,T]rightarrow X} satisfying u t + A u = 0 , 0 < t < T , ∥ u ( T ) - φ ∥ ⩽ ε , u_{t}+Au=0,quad 0
摘要设X为范数∥⋅∥{| cdot |}的Banach空间{。设A:D≠(A)∧X→X A:D(A) subset X rightarrow X}是一个产生一致有界全纯半群的(可能无界的)算子。假设ε >{varepsilon >}和T>{ T>}是两个给定的常数。求函数u的反抛物方程:[0,T]→X{ u:[0,T] rightarrow X}满足u T +A²u= 0,0
{"title":"Regularization of backward parabolic equations in Banach spaces by generalized Sobolev equations","authors":"N. Duc, D. Hào, M. Shishlenin","doi":"10.1515/JIP-2012-0046","DOIUrl":"https://doi.org/10.1515/JIP-2012-0046","url":null,"abstract":"Abstract Let X be a Banach space with norm ∥ ⋅ ∥ {|cdot|} . Let A : D ( A ) ⊂ X → X {A:D(A)subset Xrightarrow X} be an (possibly unbounded) operator that generates a uniformly bounded holomorphic semigroup. Suppose that ε > 0 {varepsilon>0} and T > 0 {T>0} are two given constants. The backward parabolic equation of finding a function u : [ 0 , T ] → X {u:[0,T]rightarrow X} satisfying u t + A u = 0 , 0 < t < T , ∥ u ( T ) - φ ∥ ⩽ ε , u_{t}+Au=0,quad 0<t<T,;|u(T)-varphi|leqslantvarepsilon, for φ in X, is regularized by the generalized Sobolev equation u α t + A α u α = 0 , 0 < t < T , u α ( T ) = φ , u_{alpha t}+A_{alpha}u_{alpha}=0,quad 0<t<T,;u_{alpha}(T)=varphi, where 0 < α < 1 {0<alpha<1} and A α = A ( I + α A b ) - 1 {A_{alpha}=A(I+alpha A^{b})^{-1}} with b ⩾ 1 {bgeqslant 1} . Error estimates of the method with respect to the noise level are proved.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/JIP-2012-0046","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46215460","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Let X be a Banach space with norm ∥⋅∥ {|cdot|} . Let A:D(A)⊂X→X {A:D(A)subset Xrightarrow X} be an (possibly unbounded) operator that generates a uniformly bounded holomorphic semigroup. Suppose that ε>0 {varepsilon>0} and T>0 {T>0} are two given constants. The backward parabolic equation of finding a function u:[0,T]→X {u:[0,T]rightarrow X} satisfying ut+Au=0,0<t<T,∥u(T)-φ∥⩽ε, u_{t}+Au=0,quad 0uαt+Aαuα=0,0<t<T,uα(T)
Abstract The paper considers the scattering problem for the first-order system of hyperbolic equations on the half-axis with a nonhomogeneous boundary condition. This problem models the phnomennon of wave propagation in a nonstationary medium where an incoming wave unaffected by a potential field. The scattering operator on the half-axis with a nonzero boundary condition is defined and the uniqueness of the inverse scattering problem (the problem of finding the potential with respect to scattering operator) is studied.
{"title":"Inverse scattering problem for nonstrict hyperbolic system on the half-axis with a nonzero boundary condition","authors":"M. Ismailov, T. Kal’menov","doi":"10.1515/jiip-2022-0027","DOIUrl":"https://doi.org/10.1515/jiip-2022-0027","url":null,"abstract":"Abstract The paper considers the scattering problem for the first-order system of hyperbolic equations on the half-axis with a nonhomogeneous boundary condition. This problem models the phnomennon of wave propagation in a nonstationary medium where an incoming wave unaffected by a potential field. The scattering operator on the half-axis with a nonzero boundary condition is defined and the uniqueness of the inverse scattering problem (the problem of finding the potential with respect to scattering operator) is studied.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43252501","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The non-convex α ∥ ⋅ ∥ ℓ 1 − β ∥ ⋅ ∥ ℓ 2 alphalVert,{cdot},rVert_{ell_{1}}-betalVert,{cdot},rVert_{ell_{2}} ( α ≥ β ≥ 0 alphageqbetageq 0 ) regularization is a new approach for sparse recovery. A minimizer of the α ∥ ⋅ ∥ ℓ 1 − β ∥ ⋅ ∥ ℓ 2 alphalVert,{cdot},rVert_{ell_{1}}-betalVert,{cdot},rVert_{ell_{2}} regularized function can be computed by applying the ST-( α ℓ 1 − β ℓ 2 alphaell_{1}-betaell_{2} ) algorithm which is similar to the classical iterative soft thresholding algorithm (ISTA). Unfortunately, It is known that ISTA converges quite slowly, and a faster alternative to ISTA is the projected gradient (PG) method. Nevertheless, the current applicability of the PG method is limited to linear inverse problems. In this paper, we extend the PG method based on a surrogate function approach to nonlinear inverse problems with the α ∥ ⋅ ∥ ℓ 1 − β ∥ ⋅ ∥ ℓ 2 alphalVert,{cdot},rVert_{ell_{1}}-betalVert,{cdot},rVert_{ell_{2}} ( α ≥ β ≥ 0 alphageqbetageq 0 ) regularization in the finite-dimensional space R n mathbb{R}^{n} . It is shown that the presented algorithm converges subsequentially to a stationary point of a constrained Tikhonov-type functional for sparsity regularization. Numerical experiments are given in the context of a nonlinear compressive sensing problem to illustrate the efficiency of the proposed approach.
{"title":"A projected gradient method for nonlinear inverse problems with 𝛼ℓ1 − 𝛽ℓ2 sparsity regularization","authors":"Zhuguang Zhao, Liang Ding","doi":"10.1515/jiip-2023-0010","DOIUrl":"https://doi.org/10.1515/jiip-2023-0010","url":null,"abstract":"Abstract The non-convex α ∥ ⋅ ∥ ℓ 1 − β ∥ ⋅ ∥ ℓ 2 alphalVert,{cdot},rVert_{ell_{1}}-betalVert,{cdot},rVert_{ell_{2}} ( α ≥ β ≥ 0 alphageqbetageq 0 ) regularization is a new approach for sparse recovery. A minimizer of the α ∥ ⋅ ∥ ℓ 1 − β ∥ ⋅ ∥ ℓ 2 alphalVert,{cdot},rVert_{ell_{1}}-betalVert,{cdot},rVert_{ell_{2}} regularized function can be computed by applying the ST-( α ℓ 1 − β ℓ 2 alphaell_{1}-betaell_{2} ) algorithm which is similar to the classical iterative soft thresholding algorithm (ISTA). Unfortunately, It is known that ISTA converges quite slowly, and a faster alternative to ISTA is the projected gradient (PG) method. Nevertheless, the current applicability of the PG method is limited to linear inverse problems. In this paper, we extend the PG method based on a surrogate function approach to nonlinear inverse problems with the α ∥ ⋅ ∥ ℓ 1 − β ∥ ⋅ ∥ ℓ 2 alphalVert,{cdot},rVert_{ell_{1}}-betalVert,{cdot},rVert_{ell_{2}} ( α ≥ β ≥ 0 alphageqbetageq 0 ) regularization in the finite-dimensional space R n mathbb{R}^{n} . It is shown that the presented algorithm converges subsequentially to a stationary point of a constrained Tikhonov-type functional for sparsity regularization. Numerical experiments are given in the context of a nonlinear compressive sensing problem to illustrate the efficiency of the proposed approach.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42176349","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose novel reverse time migration (RTM) methods for the imaging of periodic obstacles using only measurements from the lower or upper side of the obstacle arrays at a fixed frequency. We analyze the resolution of the lower side and upper side RTM methods in terms of propagating modes of the Rayleigh expansion, Helmholtz–Kirchhoff equation and the distance of the measurement surface to the obstacle arrays, where the periodic structure leads to novel analysis. We give some numerical experiments to justify the competitive efficiency of our imaging functionals and the robustness against noises. Further, numerical experiments show sharp images especially for the vertical part of the periodic obstacle in the lower-RTM case, which is not shared by results for imaging bounded compactly supported obstacles.
{"title":"Reverse time migration for imaging periodic obstacles with electromagnetic plane wave","authors":"Lide Cai, Junqing Chen","doi":"10.1515/jiip-2023-0039","DOIUrl":"https://doi.org/10.1515/jiip-2023-0039","url":null,"abstract":"We propose novel reverse time migration (RTM) methods for the imaging of periodic obstacles using only measurements from the lower or upper side of the obstacle arrays at a fixed frequency. We analyze the resolution of the lower side and upper side RTM methods in terms of propagating modes of the Rayleigh expansion, Helmholtz–Kirchhoff equation and the distance of the measurement surface to the obstacle arrays, where the periodic structure leads to novel analysis. We give some numerical experiments to justify the competitive efficiency of our imaging functionals and the robustness against noises. Further, numerical experiments show sharp images especially for the vertical part of the periodic obstacle in the lower-RTM case, which is not shared by results for imaging bounded compactly supported obstacles.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"24 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138537081","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We consider the inverse problems of restoring initial data and a source term depending on spatial variables and time in boundary value problems for the two-dimensional Burgers equation under a transverse diffusion in a rectangular and on a half-strip, like the Hopf–Cole transformation is applied to reduce Burgers’ equation to the heat equation with respect to the function that can be measured to obtain tomographic data. We prove the uniqueness of solutions in inverse problems with such additional data based on the Fourier representations and the Laplace transformation.
{"title":"On the uniqueness of solutions in inverse problems for Burgers’ equation under a transverse diffusion","authors":"A. Baev","doi":"10.1515/jiip-2022-0012","DOIUrl":"https://doi.org/10.1515/jiip-2022-0012","url":null,"abstract":"Abstract We consider the inverse problems of restoring initial data and a source term depending on spatial variables and time in boundary value problems for the two-dimensional Burgers equation under a transverse diffusion in a rectangular and on a half-strip, like the Hopf–Cole transformation is applied to reduce Burgers’ equation to the heat equation with respect to the function that can be measured to obtain tomographic data. We prove the uniqueness of solutions in inverse problems with such additional data based on the Fourier representations and the Laplace transformation.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"31 1","pages":"595 - 609"},"PeriodicalIF":1.1,"publicationDate":"2023-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45703685","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract This paper is concerned with the inverse scattering of the rough surfaces with multi-frequency phase and phaseless measurements. We present a high-order recursive iteration method based on the homotopy iteration technique to reconstruct the rough surfaces. The convergence for the multi-frequency homotopy iteration method is obtained under some conditions. Some numerical experiments show the effectiveness of the proposed algorithm.
{"title":"Rough surfaces reconstruction via phase and phaseless data by a multi-frequency homotopy iteration method","authors":"Shuqin Liu, Lei Zhang","doi":"10.1515/jiip-2021-0056","DOIUrl":"https://doi.org/10.1515/jiip-2021-0056","url":null,"abstract":"Abstract This paper is concerned with the inverse scattering of the rough surfaces with multi-frequency phase and phaseless measurements. We present a high-order recursive iteration method based on the homotopy iteration technique to reconstruct the rough surfaces. The convergence for the multi-frequency homotopy iteration method is obtained under some conditions. Some numerical experiments show the effectiveness of the proposed algorithm.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"0 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41541756","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract It is well known that the backward heat conduction problem of recovering the temperature u ( ⋅ , t ) {u(,cdot,,t)} at a time t ≥ 0 {tgeq 0} from the knowledge of the temperature at a later time, namely g := u ( ⋅ , τ ) {g:=u(,cdot,,tau)} for τ > t {tau>t} , is ill-posed, in the sense that small error in g can lead to large deviation in u ( ⋅ , t ) {u(,cdot,,t)} . However, in the case of a time fractional backward heat conduction problem (TFBHCP), the above problem is well-posed for t > 0 {t>0} and ill-posed for t = 0 {t=0} . We use this observation to obtain stable approximate solutions as solutions for t ∈ ( 0 , τ ] {tin(0,tau]} with t as regularization parameter for approximating the solution at t = 0 {t=0} , and derive error estimates under suitable source conditions. We shall also provide some numerical examples to illustrate the approximation properties of the regularized solutions.
摘要众所周知，{利用}{τ >t }{tau}{ >t的g:=}u≠({⋅，t) g:=u(， }{cdot}{ ，， }{}{tau}{)还原t≥0 t }{geq}{ 0时刻温度u(， }{cdot}{ ，，}t)的逆向热传导问题是不适定的，因为{g的小误差会导致u≠(⋅，t) u(， cdot ，，t)的大偏差}。然而，对于时间分数阶反向热传导问题(TFBHCP)，上述问题在t=0 t=0 t=0时是适定性{的，在t=0 t=0 }t=0时是不适定性{的}。我们利用这一观察结果，以t为{正则化参数 t in (0, tau)的稳定近似解，用于近似t=0 t=0处的解，}并在合适的源条件下得到误差估计。我们还将提供一些数值例子来说明正则解的近似性质。{}
{"title":"A new regularization for time-fractional backward heat conduction problem","authors":"M. Nair, P. Danumjaya","doi":"10.1515/jiip-2023-0043","DOIUrl":"https://doi.org/10.1515/jiip-2023-0043","url":null,"abstract":"Abstract It is well known that the backward heat conduction problem of recovering the temperature u ( ⋅ , t ) {u(,cdot,,t)} at a time t ≥ 0 {tgeq 0} from the knowledge of the temperature at a later time, namely g := u ( ⋅ , τ ) {g:=u(,cdot,,tau)} for τ > t {tau>t} , is ill-posed, in the sense that small error in g can lead to large deviation in u ( ⋅ , t ) {u(,cdot,,t)} . However, in the case of a time fractional backward heat conduction problem (TFBHCP), the above problem is well-posed for t > 0 {t>0} and ill-posed for t = 0 {t=0} . We use this observation to obtain stable approximate solutions as solutions for t ∈ ( 0 , τ ] {tin(0,tau]} with t as regularization parameter for approximating the solution at t = 0 {t=0} , and derive error estimates under suitable source conditions. We shall also provide some numerical examples to illustrate the approximation properties of the regularized solutions.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44667582","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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