M. Botchev, S. Kabanikhin, M. Shishlenin, E. Tyrtyshnikov
Abstract The horizontally diagonalize and fit (HDF) method is proposed to solve the ill-posed Cauchy problem for the three-dimensional Poisson equation with data given on the part of the boundary (a continuation problem). The HDF method consists in discretization over horizontal variables and transformation of the system of differential equations to a diagonal form. This allows to uncouple the original three-dimensional continuation problem into a moderate number of one-dimensional problems in the vertical dimension. The problem size reduction can be carried taking into account the noise level, so that the number k of one-dimensional problems appears to be a regularization parameter. Our experiments show that HDF is applicable to large-scale problems and for n ≤ 2500 {nleq 2500} is significantly more efficient than Landweber iteration.
{"title":"The Cauchy problem for the 3D Poisson equation: Landweber iteration vs. horizontally diagonalize and fit method","authors":"M. Botchev, S. Kabanikhin, M. Shishlenin, E. Tyrtyshnikov","doi":"10.1515/jiip-2022-0092","DOIUrl":"https://doi.org/10.1515/jiip-2022-0092","url":null,"abstract":"Abstract The horizontally diagonalize and fit (HDF) method is proposed to solve the ill-posed Cauchy problem for the three-dimensional Poisson equation with data given on the part of the boundary (a continuation problem). The HDF method consists in discretization over horizontal variables and transformation of the system of differential equations to a diagonal form. This allows to uncouple the original three-dimensional continuation problem into a moderate number of one-dimensional problems in the vertical dimension. The problem size reduction can be carried taking into account the noise level, so that the number k of one-dimensional problems appears to be a regularization parameter. Our experiments show that HDF is applicable to large-scale problems and for n ≤ 2500 {nleq 2500} is significantly more efficient than Landweber iteration.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"31 1","pages":"203 - 221"},"PeriodicalIF":1.1,"publicationDate":"2023-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47299316","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 study the well-posedness of the solution of an optimal control problem related to a multi-parameters identification problem in degenerate parabolic equations. Problems of this type have important applications in several fields of applied science. Unlike other inverse coefficient problems for classical parabolic equations, the mathematical model discussed in the paper is degenerate on both lateral boundaries of the domain. Moreover, the status of the two unknown coefficients are different, namely that the reconstruction of the source term is mildly ill-posed, while the inverse initial value problem is severely ill-posed. On the basis of optimal control framework, the problem is transformed into an optimization problem. The existence of the minimizer is proved and the necessary conditions which must be satisfied by the minimizer are also established. Due to the difference between ill-posedness degrees of the two unknown coefficients, the extensively used conjugate theory for parabolic equations cannot be directly applied for our problem. By carefully analyzing the necessary conditions and the direct problem, the uniqueness, stability and convergence of the minimizer are obtained. The results obtained in the paper are interesting and useful, and can be extended to more general parabolic equations with degenerate coefficients.
{"title":"Optimization method for a multi-parameters identification problem in degenerate parabolic equations","authors":"Liu Yang, Z. Deng","doi":"10.1515/jiip-2022-0038","DOIUrl":"https://doi.org/10.1515/jiip-2022-0038","url":null,"abstract":"Abstract In this paper, we study the well-posedness of the solution of an optimal control problem related to a multi-parameters identification problem in degenerate parabolic equations. Problems of this type have important applications in several fields of applied science. Unlike other inverse coefficient problems for classical parabolic equations, the mathematical model discussed in the paper is degenerate on both lateral boundaries of the domain. Moreover, the status of the two unknown coefficients are different, namely that the reconstruction of the source term is mildly ill-posed, while the inverse initial value problem is severely ill-posed. On the basis of optimal control framework, the problem is transformed into an optimization problem. The existence of the minimizer is proved and the necessary conditions which must be satisfied by the minimizer are also established. Due to the difference between ill-posedness degrees of the two unknown coefficients, the extensively used conjugate theory for parabolic equations cannot be directly applied for our problem. By carefully analyzing the necessary conditions and the direct problem, the uniqueness, stability and convergence of the minimizer are obtained. The results obtained in the paper are interesting and useful, and can be extended to more general parabolic equations with degenerate coefficients.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46513424","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 present novel reconstruction and stability analysis methodologies for two-dimensional, multi-coil MRI, based on analytic continuation ideas. We show that the 2-D, limited-data MRI inverse problem, whereby the missing parts of 𝐤 {mathbf{k}} -space (Fourier space) are lines parallel to either k1 {k_{1}} or k2 {k_{2}} (i.e., the 𝐤 {mathbf{k}} -space axis), can be reduced to a set of 1-D Fredholm type inverse problems. The Fredholm equations are then solved to recover the 2-D image on 1-D line profiles (“slice-by-slice” imaging). The technique is tested on a range of medical in vivo images (e.g., brain, spine, cardiac), and phantom data. Our method is shown to offer optimal performance, in terms of structural similarity, when compared against similar methods from the literature, and when the 𝐤 {mathbf{k}} -space data is sub-sampled at random so as to simulate motion corruption. In addition, we present a Singular Value Decomposition (SVD) and stability analysis of the Fredholm operators, and compare the stability properties of different 𝐤 {mathbf{k}} -space sub-sampling schemes (e.g., random vs uniform accelerated sampling).
{"title":"Multi-coil MRI by analytic continuation","authors":"James W. Webber","doi":"10.1515/jiip-2022-0046","DOIUrl":"https://doi.org/10.1515/jiip-2022-0046","url":null,"abstract":"Abstract We present novel reconstruction and stability analysis methodologies for two-dimensional, multi-coil MRI, based on analytic continuation ideas. We show that the 2-D, limited-data MRI inverse problem, whereby the missing parts of <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>𝐤</m:mi> </m:math> {mathbf{k}} -space (Fourier space) are lines parallel to either <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>k</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> {k_{1}} or <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>k</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> {k_{2}} (i.e., the <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>𝐤</m:mi> </m:math> {mathbf{k}} -space axis), can be reduced to a set of 1-D Fredholm type inverse problems. The Fredholm equations are then solved to recover the 2-D image on 1-D line profiles (“slice-by-slice” imaging). The technique is tested on a range of medical in vivo images (e.g., brain, spine, cardiac), and phantom data. Our method is shown to offer optimal performance, in terms of structural similarity, when compared against similar methods from the literature, and when the <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>𝐤</m:mi> </m:math> {mathbf{k}} -space data is sub-sampled at random so as to simulate motion corruption. In addition, we present a Singular Value Decomposition (SVD) and stability analysis of the Fredholm operators, and compare the stability properties of different <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>𝐤</m:mi> </m:math> {mathbf{k}} -space sub-sampling schemes (e.g., random vs uniform accelerated sampling).","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135794600","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 problem of time-harmonic acoustic waves from a penetrable cavity bounded by a layered structure and seeks to determine the shape and location of the cavity from interior near-field measurements. Of particular interest is that the near-field operator does not satisfy the main theorem of the factorization method, so we introduce a modified near-field operator and prove that it can be used to reconstruct the cavity. Numerical examples demonstrate the feasibility and effectiveness of our algorithm.
{"title":"The factorization method for a penetrable cavity scattering with interior near-field measurements","authors":"Qinghua Wu, Jun Guo, G. Yan","doi":"10.1515/jiip-2018-0111","DOIUrl":"https://doi.org/10.1515/jiip-2018-0111","url":null,"abstract":"Abstract This paper is concerned with the inverse scattering problem of time-harmonic acoustic waves from a penetrable cavity bounded by a layered structure and seeks to determine the shape and location of the cavity from interior near-field measurements. Of particular interest is that the near-field operator does not satisfy the main theorem of the factorization method, so we introduce a modified near-field operator and prove that it can be used to reconstruct the cavity. Numerical examples demonstrate the feasibility and effectiveness of our algorithm.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"31 1","pages":"19 - 30"},"PeriodicalIF":1.1,"publicationDate":"2023-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46554663","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 study the inverse problems of determining the unknown transverse shear force g ( t ) {g(t)} in a system governed by the damped Euler–Bernoulli equation ρ ( x ) u t t + μ ( x ) u t + ( r ( x ) u x x ) x x + ( κ ( x ) u x x t ) x x = 0 , ( x , t ) ∈ ( 0 , ℓ ) × ( 0 , T ] , rho(x)u_{tt}+mu(x)u_{t}+(r(x)u_{xx})_{xx}+(kappa(x)u_{xxt})_{xx}=0,quad(x,% t)in(0,ell)times(0,T], subject to the boundary conditions u ( 0 , t ) = 0 , u x ( 0 , t ) = 0 , [ r ( x ) u x x + κ ( x ) u x x t ] x = ℓ = 0 , - [ ( r ( x ) u x x + κ ( x ) u x x t ) x ] x = ℓ = g ( t ) , u(0,t)=0,quad u_{x}(0,t)=0,quad[r(x)u_{xx}+kappa(x)u_{xxt}]_{x=ell}=0,% quad-[(r(x)u_{xx}+kappa(x)u_{xxt})_{x}]_{x=ell}=g(t), for t ∈ [ 0 , T ] {tin[0,T]} , from the measured deflection ν ( t ) := u ( ℓ , t ) {nu(t):=u(ell,t)} , t ∈ [ 0 , T ] {tin[0,T]} , and from the bending moment ω ( t ) := - ( r ( 0 ) u x x ( 0 , t ) + κ ( 0 ) u x x t ( 0 , t ) ) , t ∈ [ 0 , T ] , omega(t):=-(r(0)u_{xx}(0,t)+kappa(0)u_{xxt}(0,t)),quad tin[0,T], where the terms ( κ ( x ) u x x t ) x x {(kappa(x)u_{xxt})_{xx}} and μ ( x ) u t {mu(x)u_{t}} account for the Kelvin–Voigt damping and external damping, respectively. The main purpose of this study is to analyze the Kelvin–Voigt damping effect on determining the unknown transverse shear force (boundary input) through the given boundary measurements. The inverse problems are transformed into minimization problems for Tikhonov functionals, and it is shown that the regularized functionals admit unique solutions for the inverse problems. By suitable regularity on the admissible class of shear force g ( t ) {g(t)} , we prove that these functionals are Fréchet differentiable, and the derivatives are expressed through the solutions of corresponding adjoint problems posed with measured data as boundary data associated with the direct problem. The solvability of these adjoint problems is obtained under the minimal regularity of the boundary data g ( t ) {g(t)} , which turns out to be the regularizing effect of the Kelvin–Voigt damping in the direct problem. Furthermore, using the Fréchet derivative of the more regularized Tikhonov functionals, we obtain remarkable Lipschitz stability estimates for the transverse shear force in terms of the given measurement by a feasible condition only on the Kelvin–Voigt damping coefficient.
摘要本文研究了由阻尼Euler–Bernoulli方程ρ，ℓ ) ×（0，T]，rho（x）u_｛tt｝+mu（x）u_｛T｝+（r（x）u-｛xx｝）_{xx｝+ℓ = 0，-[（r（x）u x x+κ（x）ux x t）x]x=ℓ = g（t），u（0，t）=0，quad u_｛x｝ℓ , t）｛nu（t）：=u（ell，t）｝，t∈[0，t]｛tin[0]，t]｝，并且从弯矩ω，其中项（κ（x）u x t）x｛（kappa（x）u_{xxt}）_{xx}｝和μ（x）ut｛mu（x）u_｛t｝｝分别说明了Kelvin-Voigt阻尼和外部阻尼。本研究的主要目的是通过给定的边界测量，分析Kelvin–Voigt阻尼对确定未知横向剪切力（边界输入）的影响。将Tikhonov泛函的逆问题转化为极小化问题，并证明正则化泛函允许逆问题的唯一解。通过剪切力容许类g（t）{g（t）}上的适当正则性，我们证明了这些泛函是Fréchet可微的，并且导数是通过将测量数据作为与直接问题相关的边界数据提出的相应伴随问题的解来表示的。这些伴随问题的可解性是在边界数据g（t）{g（t）}的最小正则性下获得的，这是直接问题中Kelvin–Voigt阻尼的正则化效应。此外，使用更正则化的Tikhonov泛函的Fréchet导数，我们仅在Kelvin–Voigt阻尼系数的可行条件下，就给定测量值而言，获得了横向剪切力的显著Lipschitz稳定性估计。
{"title":"Inverse problems of identifying the unknown transverse shear force in the Euler–Bernoulli beam with Kelvin–Voigt damping","authors":"Sakthivel Kumarasamy, A. Hasanov, Anjuna Dileep","doi":"10.1515/jiip-2022-0053","DOIUrl":"https://doi.org/10.1515/jiip-2022-0053","url":null,"abstract":"Abstract In this paper, we study the inverse problems of determining the unknown transverse shear force g ( t ) {g(t)} in a system governed by the damped Euler–Bernoulli equation ρ ( x ) u t t + μ ( x ) u t + ( r ( x ) u x x ) x x + ( κ ( x ) u x x t ) x x = 0 , ( x , t ) ∈ ( 0 , ℓ ) × ( 0 , T ] , rho(x)u_{tt}+mu(x)u_{t}+(r(x)u_{xx})_{xx}+(kappa(x)u_{xxt})_{xx}=0,quad(x,% t)in(0,ell)times(0,T], subject to the boundary conditions u ( 0 , t ) = 0 , u x ( 0 , t ) = 0 , [ r ( x ) u x x + κ ( x ) u x x t ] x = ℓ = 0 , - [ ( r ( x ) u x x + κ ( x ) u x x t ) x ] x = ℓ = g ( t ) , u(0,t)=0,quad u_{x}(0,t)=0,quad[r(x)u_{xx}+kappa(x)u_{xxt}]_{x=ell}=0,% quad-[(r(x)u_{xx}+kappa(x)u_{xxt})_{x}]_{x=ell}=g(t), for t ∈ [ 0 , T ] {tin[0,T]} , from the measured deflection ν ( t ) := u ( ℓ , t ) {nu(t):=u(ell,t)} , t ∈ [ 0 , T ] {tin[0,T]} , and from the bending moment ω ( t ) := - ( r ( 0 ) u x x ( 0 , t ) + κ ( 0 ) u x x t ( 0 , t ) ) , t ∈ [ 0 , T ] , omega(t):=-(r(0)u_{xx}(0,t)+kappa(0)u_{xxt}(0,t)),quad tin[0,T], where the terms ( κ ( x ) u x x t ) x x {(kappa(x)u_{xxt})_{xx}} and μ ( x ) u t {mu(x)u_{t}} account for the Kelvin–Voigt damping and external damping, respectively. The main purpose of this study is to analyze the Kelvin–Voigt damping effect on determining the unknown transverse shear force (boundary input) through the given boundary measurements. The inverse problems are transformed into minimization problems for Tikhonov functionals, and it is shown that the regularized functionals admit unique solutions for the inverse problems. By suitable regularity on the admissible class of shear force g ( t ) {g(t)} , we prove that these functionals are Fréchet differentiable, and the derivatives are expressed through the solutions of corresponding adjoint problems posed with measured data as boundary data associated with the direct problem. The solvability of these adjoint problems is obtained under the minimal regularity of the boundary data g ( t ) {g(t)} , which turns out to be the regularizing effect of the Kelvin–Voigt damping in the direct problem. Furthermore, using the Fréchet derivative of the more regularized Tikhonov functionals, we obtain remarkable Lipschitz stability estimates for the transverse shear force in terms of the given measurement by a feasible condition only on the Kelvin–Voigt damping coefficient.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43147303","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}
Pub Date : 2022-12-20DOI: 10.48550/arXiv.2212.10036
James W. Webber
Abstract We present novel reconstruction and stability analysis methodologies for two-dimensional, multi-coil MRI, based on analytic continuation ideas. We show that the 2-D, limited-data MRI inverse problem, whereby the missing parts of 𝐤 {mathbf{k}} -space (Fourier space) are lines parallel to either k 1 {k_{1}} or k 2 {k_{2}} (i.e., the 𝐤 {mathbf{k}} -space axis), can be reduced to a set of 1-D Fredholm type inverse problems. The Fredholm equations are then solved to recover the 2-D image on 1-D line profiles (“slice-by-slice” imaging). The technique is tested on a range of medical in vivo images (e.g., brain, spine, cardiac), and phantom data. Our method is shown to offer optimal performance, in terms of structural similarity, when compared against similar methods from the literature, and when the 𝐤 {mathbf{k}} -space data is sub-sampled at random so as to simulate motion corruption. In addition, we present a Singular Value Decomposition (SVD) and stability analysis of the Fredholm operators, and compare the stability properties of different 𝐤 {mathbf{k}} -space sub-sampling schemes (e.g., random vs uniform accelerated sampling).
{"title":"Multi-coil MRI by analytic continuation","authors":"James W. Webber","doi":"10.48550/arXiv.2212.10036","DOIUrl":"https://doi.org/10.48550/arXiv.2212.10036","url":null,"abstract":"Abstract We present novel reconstruction and stability analysis methodologies for two-dimensional, multi-coil MRI, based on analytic continuation ideas. We show that the 2-D, limited-data MRI inverse problem, whereby the missing parts of 𝐤 {mathbf{k}} -space (Fourier space) are lines parallel to either k 1 {k_{1}} or k 2 {k_{2}} (i.e., the 𝐤 {mathbf{k}} -space axis), can be reduced to a set of 1-D Fredholm type inverse problems. The Fredholm equations are then solved to recover the 2-D image on 1-D line profiles (“slice-by-slice” imaging). The technique is tested on a range of medical in vivo images (e.g., brain, spine, cardiac), and phantom data. Our method is shown to offer optimal performance, in terms of structural similarity, when compared against similar methods from the literature, and when the 𝐤 {mathbf{k}} -space data is sub-sampled at random so as to simulate motion corruption. In addition, we present a Singular Value Decomposition (SVD) and stability analysis of the Fredholm operators, and compare the stability properties of different 𝐤 {mathbf{k}} -space sub-sampling schemes (e.g., random vs uniform accelerated sampling).","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"31 1","pages":"1 - 17"},"PeriodicalIF":1.1,"publicationDate":"2022-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42054537","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, the existence and uniqueness of a strong solution of the inverse problem of determining a coefficient of right-hand side of the integro-differential Kelvin–Voigt equation are investigated. The unknown coefficient that we search defends on space variables. Additional information on a solution of the inverse problem is given here as an integral overdetermination condition. The original inverse problem is reduced to study an equivalent inverse problem with homogeneous initial condition. Then the equivalences of the last inverse problem to an operator equation of second kind is proved. We establish the sufficient conditions for the unique solvability of the operator equation of second kind.
{"title":"Inverse problem for integro-differential Kelvin–Voigt equations","authors":"Kh. Khompysh, Nursaule K. Nugymanova","doi":"10.1515/jiip-2020-0157","DOIUrl":"https://doi.org/10.1515/jiip-2020-0157","url":null,"abstract":"Abstract In this paper, the existence and uniqueness of a strong solution of the inverse problem of determining a coefficient of right-hand side of the integro-differential Kelvin–Voigt equation are investigated. The unknown coefficient that we search defends on space variables. Additional information on a solution of the inverse problem is given here as an integral overdetermination condition. The original inverse problem is reduced to study an equivalent inverse problem with homogeneous initial condition. Then the equivalences of the last inverse problem to an operator equation of second kind is proved. We establish the sufficient conditions for the unique solvability of the operator equation of second kind.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43328635","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 prove order-optimal convergence rates for the conjugate gradient method applied to linear ill-posed problems when not only the data are noisy but also when the operator is perturbed via discretization and modelling errors.
{"title":"Ill-posed problems and the conjugate gradient method: Optimal convergence rates in the presence of discretization and modelling errors","authors":"A. Neubauer","doi":"10.1515/jiip-2022-0039","DOIUrl":"https://doi.org/10.1515/jiip-2022-0039","url":null,"abstract":"Abstract In this paper, we prove order-optimal convergence rates for the conjugate gradient method applied to linear ill-posed problems when not only the data are noisy but also when the operator is perturbed via discretization and modelling errors.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"30 1","pages":"905 - 915"},"PeriodicalIF":1.1,"publicationDate":"2022-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42035689","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 study the inverse coefficient problem of identifying both the mass density ρ ( x ) > 0 rho(x)>0 and flexural rigidity r ( x ) > 0 r(x)>0 of a damped Euler–Bernoulli (cantilever) beam governed by the equation ρ ( x ) u t t + μ ( x ) u t + ( r ( x ) u x x ) x x = 0 rho(x)u_{tt}+mu(x)u_{t}+(r(x)u_{xx})_{xx}=0 , ( x , t ) ∈ ( 0 , ℓ ) × ( 0 , T ) (x,t)in(0,ell)times(0,T) , subject to boundary conditions u ( 0 , t ) = u x ( 0 , t ) = 0 u(0,t)=u_{x}(0,t)=0 , u x x ( ℓ , t ) = 0 u_{xx}(ell,t)=0 , - ( r ( x ) u x x ( x , t ) ) x | x = ℓ = g ( t ) -(r(x)u_{xx}(x,t))_{x}|_{x=ell}=g(t) , from the available measured boundary deflection ν ( t ) := u ( ℓ , t ) nu(t):=u(ell,t) and rotation θ ( t ) := u x ( ℓ , t ) theta(t):=u_{x}(ell,t) at the free end of the beam. The distinctive feature of the considered inverse coefficient problem is that not one, but two Neumann-to-Dirichlet operators have to be formally defined. The inverse problem is hence formulated as a system of nonlinear Neumann-to-Dirichlet operator equations with the right-hand sides consisting of the measured outputs. As a natural consequence of this approach, a vector-form Tikhonov functional is introduced whose components are squares of the L 2 L^{2} -norm differences between predicted and measured outputs. We then prove existence of a quasi-solution of the inverse problem and derive explicit gradient formulae for the Fréchet derivatives of both components of the Tikhonov functional. These results are instrumental to any gradient based algorithms for reconstructing the two unknown coefficients of the considered damped Euler–Bernoulli beam.
摘要本文研究了质量密度ρ∑(x) >的反系数问题 rho一个阻尼欧拉-伯努利(悬臂)梁的弯曲刚度r(x)>0 r(x)>0 r(x)>0由方程ρ (x)减去u t减去t + μ (x)减去u t + (r(x)减去u x减去x) x减去x = 0 rho(x)u{tt}+mu(x)u{t}+(r(x)u{xx}){xx}=0， (x,t)∈(0,r) x (0,t) (x,t)in(0;ell）times(0,T)，满足边界条件u∑(0,T) =u x∑(0,T) = 0 u(0, T) =u＿{x}(0,t)=0, u x乘以x乘以(r,t)等于0{xx}（ell，t)=0， -(r减去(x)减去u x减去x减去x减去(x,t)) x | x = r = g减去(t{xx}(x,t))＿{x}|＿{x=ell}=g(t)，从可测边界位移ν∑(t):= u∑(r, t) nu(t):=u(ell，t)和旋转θ∑(t):= u x∑(r,t) theta(t):=u＿{x}（ell，t)在梁的自由端。所考虑的逆系数问题的显著特征是，不是一个，而是两个诺伊曼-狄利克雷算子必须被正式定义。因此，反问题被表述为一个非线性诺伊曼-狄利克雷算子方程系统，其右侧由测量输出组成。作为这种方法的自然结果，引入了一个矢量形式的吉洪诺夫泛函，其分量是l2l ^的平方{2} -预测输出和测量输出之间的规范差异。然后证明了逆问题的拟解的存在性，并推导出Tikhonov泛函两个分量的fr导数的显式梯度公式。这些结果有助于任何基于梯度的算法重建考虑阻尼欧拉-伯努利梁的两个未知系数。
{"title":"Simultaneous determination of mass density and flexural rigidity of the damped Euler–Bernoulli beam from two boundary measured outputs","authors":"C. Sebu","doi":"10.1515/jiip-2022-0044","DOIUrl":"https://doi.org/10.1515/jiip-2022-0044","url":null,"abstract":"Abstract In this paper, we study the inverse coefficient problem of identifying both the mass density ρ ( x ) > 0 rho(x)>0 and flexural rigidity r ( x ) > 0 r(x)>0 of a damped Euler–Bernoulli (cantilever) beam governed by the equation ρ ( x ) u t t + μ ( x ) u t + ( r ( x ) u x x ) x x = 0 rho(x)u_{tt}+mu(x)u_{t}+(r(x)u_{xx})_{xx}=0 , ( x , t ) ∈ ( 0 , ℓ ) × ( 0 , T ) (x,t)in(0,ell)times(0,T) , subject to boundary conditions u ( 0 , t ) = u x ( 0 , t ) = 0 u(0,t)=u_{x}(0,t)=0 , u x x ( ℓ , t ) = 0 u_{xx}(ell,t)=0 , - ( r ( x ) u x x ( x , t ) ) x | x = ℓ = g ( t ) -(r(x)u_{xx}(x,t))_{x}|_{x=ell}=g(t) , from the available measured boundary deflection ν ( t ) := u ( ℓ , t ) nu(t):=u(ell,t) and rotation θ ( t ) := u x ( ℓ , t ) theta(t):=u_{x}(ell,t) at the free end of the beam. The distinctive feature of the considered inverse coefficient problem is that not one, but two Neumann-to-Dirichlet operators have to be formally defined. The inverse problem is hence formulated as a system of nonlinear Neumann-to-Dirichlet operator equations with the right-hand sides consisting of the measured outputs. As a natural consequence of this approach, a vector-form Tikhonov functional is introduced whose components are squares of the L 2 L^{2} -norm differences between predicted and measured outputs. We then prove existence of a quasi-solution of the inverse problem and derive explicit gradient formulae for the Fréchet derivatives of both components of the Tikhonov functional. These results are instrumental to any gradient based algorithms for reconstructing the two unknown coefficients of the considered damped Euler–Bernoulli beam.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"30 1","pages":"917 - 930"},"PeriodicalIF":1.1,"publicationDate":"2022-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46019258","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 a problem of dynamic 2D vector tomography, i.e. the object under investigation changes during the data acquisition. More precisely, we consider the case when the object motion is a combination of rotation and shifting. The task is then to reconstruct the searched-for vector field by known values of the dynamic ray transforms. In order to solve this dynamic inverse problem, we first study properties of the dynamic ray transforms operators. In particular, the singular value decompositions of the operators are constructed using classic orthogonal polynomials. Following from this study, a numerical algorithm for solving the dynamic problem is proposed based on the truncated singular value decomposition method.
{"title":"A numerical solution of the dynamic vector tomography problem using the truncated singular value decomposition method","authors":"A. Polyakova, I. Svetov","doi":"10.1515/jiip-2022-0019","DOIUrl":"https://doi.org/10.1515/jiip-2022-0019","url":null,"abstract":"Abstract We consider a problem of dynamic 2D vector tomography, i.e. the object under investigation changes during the data acquisition. More precisely, we consider the case when the object motion is a combination of rotation and shifting. The task is then to reconstruct the searched-for vector field by known values of the dynamic ray transforms. In order to solve this dynamic inverse problem, we first study properties of the dynamic ray transforms operators. In particular, the singular value decompositions of the operators are constructed using classic orthogonal polynomials. Following from this study, a numerical algorithm for solving the dynamic problem is proposed based on the truncated singular value decomposition method.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"0 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41651521","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：