Journal of Inverse and Ill-Posed Problems最新文献

{"title":"A partial inverse problem for non-self-adjoint Sturm–Liouville operators with a constant delay","authors":"Yu Ping Wang, B. Keskin, Chung‐Tsun Shieh","doi":"10.1515/jiip-2020-0058","DOIUrl":"https://doi.org/10.1515/jiip-2020-0058","url":null,"abstract":"Abstract In this paper we study a partial inverse spectral problem for non-self-adjoint Sturm–Liouville operators with a constant delay and show that subspectra of two boundary value problems with one common boundary condition are sufficient to determine the complex potential. We developed the Horváth’s method in [M. Horváth, On the inverse spectral theory of Schrödinger and Dirac operators, Trans. Amer. Math. Soc. 353 2001, 10, 4155–4171] for the self-adjoint Sturm–Liouville operator without delay into the non-self-adjoint Sturm–Liouville differential operator with a constant delay.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"31 1","pages":"479 - 486"},"PeriodicalIF":1.1,"publicationDate":"2023-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46310282","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}

{"title":"A note on the degree of ill-posedness for mixed differentiation on the d-dimensional unit cube","authors":"B. Hofmann, Hans-Jürgen Fischer","doi":"10.48550/arXiv.2303.14473","DOIUrl":"https://doi.org/10.48550/arXiv.2303.14473","url":null,"abstract":"Abstract Numerical differentiation of a function over the unit interval of the real axis, which is contaminated with noise, by inverting the simple integration operator J mapping in L 2 {L^{2}} is discussed extensively in the literature. The complete singular system of the compact operator J is explicitly given with singular values σ n ⁢ ( J ) {sigma_{n}(J)} asymptotically proportional to 1 n {frac{1}{n}} . This indicates a degree one of ill-posedness for the associated inverse problem of differentiation. We recall the concept of the degree of ill-posedness for linear operator equations with compact forward operators in Hilbert spaces. In contrast to the one-dimensional case, there is little specific material available about the inverse problem of mixed differentiation, where the d-dimensional analog J d {J_{d}} to J, defined over unit d-cube, is to be inverted. In this note, we show for that problem that the degree of ill-posedness stays at one for all dimensions d ∈ ℕ {din{mathbb{N}}} . Some more discussion refers to the two-dimensional case in order to characterize the range of the operator J 2 {J_{2}} .","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41936631","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}

{"title":"Adaptive Runge–Kutta regularization for a Cauchy problem of a modified Helmholtz equation","authors":"Fadhel Jday, H. Omri","doi":"10.1515/jiip-2020-0014","DOIUrl":"https://doi.org/10.1515/jiip-2020-0014","url":null,"abstract":"Abstract In this paper, we investigate the Cauchy problem for the modified Helmholtz equation. We consider the data completion problem in a bounded cylindrical domain on which the Neumann and the Dirichlet conditions are given in a part of the boundary. Since this problem is ill-posed, we reformulate it as an optimal control problem with an appropriate cost function. The method of factorization of boundary value problems is used to immediately obtain an approximation of the missing boundary data. In order to regularize this problem, we firstly scrutinize two classical regularizations for the cost function. Then we propose a new numerical regularization named “adaptive Runge–Kutta regularization”, which does not require any penalization term. Finally, we compare them numerically.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"31 1","pages":"351 - 374"},"PeriodicalIF":1.1,"publicationDate":"2023-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46293684","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}

{"title":"The mean field games system: Carleman estimates, Lipschitz stability and uniqueness","authors":"M. Klibanov","doi":"10.1515/jiip-2023-0023","DOIUrl":"https://doi.org/10.1515/jiip-2023-0023","url":null,"abstract":"Abstract An overdetermination is introduced in an initial condition for the second order mean field games system (MFGS). This makes the resulting problem close to the classical ill-posed Cauchy problems for PDEs. Indeed, in such a problem an overdetermination in boundary conditions usually takes place. A Lipschitz stability estimate is obtained. This estimate implies uniqueness. A new Carleman estimate is derived. This latter estimate is called “quasi-Carleman estimate”, since it contains two test functions rather than a single one in conventional Carleman estimates. These two estimates play the key role. Carleman estimates were not applied to the MFGS prior to the recent work of Klibanov and Averboukh in [M. V. Klibanov and Y. Averboukh, Lipschitz stability estimate and uniqueness in the retrospective analysis for the mean field games system via two Carleman estimates, preprint 2023, https://arxiv.org/abs/2302.10709].","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"31 1","pages":"455 - 466"},"PeriodicalIF":1.1,"publicationDate":"2023-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42465597","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}

{"title":"Solution of the backward problem for the space-time fractional diffusion equation related to the release history of a groundwater contaminant","authors":"A. H. Salehi Shayegan, A. Zakeri, Adib Salehi Shayegan","doi":"10.1515/jiip-2022-0054","DOIUrl":"https://doi.org/10.1515/jiip-2022-0054","url":null,"abstract":"Abstract Finding the history of a groundwater contaminant plume from final measurements is an ill-posed problem and, consequently, its solution is extremely sensitive to errors in the input data. In this paper, we study this problem mathematically. So, firstly, existence and uniqueness theorems of a quasi-solution in an appropriate class of admissible initial data are given. Secondly, in order to overcome the ill-posedness of the problem and also approximate the quasi-solution, two approaches (computational and iterative algorithms) are provided. In the computational algorithm, the finite element method and TSVD regularization are applied. This method is tested by two numerical examples. The results reveal the efficiency and applicability of the proposed method. Also, in order to construct the iterative methods, an explicit formula for the gradient of the cost functional J is given. This result helps us to construct two iterative methods, i.e., the conjugate gradient algorithm and Landweber iteration algorithm. We prove the Lipschitz continuity of the gradient of the cost functional, monotonicity and convergence of the iterative methods. At the end of the paper, a numerical example is given to show the validation of the iterative algorithms.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42391961","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}

{"title":"Uniqueness of the potential in a time-fractional diffusion equation","authors":"X. Jing, Jigen Peng","doi":"10.1515/jiip-2020-0046","DOIUrl":"https://doi.org/10.1515/jiip-2020-0046","url":null,"abstract":"Abstract This article concerns the uniqueness of an inverse coefficient problem of identifying a spatially varying potential in a one-dimensional time-fractional diffusion equation. The input sources are given by a complete system in L 2 ⁢ ( 0 , 1 ) {L^{2}(0,1)} , and measurements are observed at the end point of the spatial interval. Firstly, we provide the positive lower bound of the Green function for the differential operator with different boundary conditions. Then, based on the positive lower bound estimation of the Green function, the relationship between the Green function, the solution of the forward problem, and the potential, such measurements uniquely determine the potential on the entire interval under different boundary conditions.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"31 1","pages":"467 - 477"},"PeriodicalIF":1.1,"publicationDate":"2023-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48904028","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}

{"title":"Inverse problems for equations of a mixed parabolic-hyperbolic type with power degeneration in finding the right-hand parts that depend on time","authors":"K. Sabitov, S. Sidorov","doi":"10.1515/jiip-2020-0154","DOIUrl":"https://doi.org/10.1515/jiip-2020-0154","url":null,"abstract":"Abstract For the equation of a mixed parabolic-hyperbolic type with a power degeneration on the type change line, the inverse problems to determine the time-dependent factors of right-hand sides are studied. Based on the formula for solving a direct problem, the solution of inverse problems is equivalently reduced to the solvability of loaded integral equations. Using the theory of integral equations, the corresponding theorems for the existence and uniqueness of the solutions of the stated inverse problems are proved and the explicit formulas for the solution have been given.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47552324","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}

{"title":"Robin–Dirichlet alternating iterative procedure for solving the Cauchy problem for Helmholtz equation in an unbounded domain","authors":"Pauline Achieng, Fredrik Berntsson, Vladimir Kozlov","doi":"10.1515/jiip-2020-0133","DOIUrl":"https://doi.org/10.1515/jiip-2020-0133","url":null,"abstract":"Abstract We consider the Cauchy problem for the Helmholtz equation with a domain in <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>ℝ</m:mi> <m:mi>d</m:mi> </m:msup> </m:math> {mathbb{R}^{d}} , <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>d</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> {dgeq 2} with N cylindrical outlets to infinity with bounded inclusions in <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mi>ℝ</m:mi> <m:mrow> <m:mi>d</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mo>.</m:mo> </m:mrow> </m:math> {mathbb{R}^{d-1}.} Cauchy data are prescribed on the boundary of the bounded domains and the aim is to find solution on the unbounded part of the boundary. In 1989, Kozlov and Maz’ya [14] proposed an alternating iterative method for solving Cauchy problems associated with elliptic, self-adjoint and positive-definite operators in bounded domains. Different variants of this method for solving Cauchy problems associated with Helmholtz-type operators exists. We consider the variant proposed by Berntsson, Kozlov, Mpinganzima and Turesson (2018) [4] for bounded domains and derive the necessary conditions for the convergence of the procedure in unbounded domains. For the numerical implementation, a finite difference method is used to solve the problem in a simple rectangular domain in <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>ℝ</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> {mathbb{R}^{2}} that represent a truncated infinite strip. The numerical results shows that by appropriate truncation of the domain and with appropriate choice of the Robin parameters <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>μ</m:mi> <m:mn>0</m:mn> </m:msub> </m:math> {mu_{0}} and <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>μ</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> {mu_{1}} , the Robin–Dirichlet alternating iterative procedure is convergent.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"70 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135533216","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}

{"title":"Stability of determination of Riemann surface from its DN-map in terms of holomorphic immersions","authors":"M. Belishev, D. Korikov","doi":"10.1515/jiip-2022-0024","DOIUrl":"https://doi.org/10.1515/jiip-2022-0024","url":null,"abstract":"Abstract Suppose that ( M , g ) {(M,g)} is a compact Riemann surface with metric g and boundary ∂ ⁡ M {partial M} , and Λ is its DN-map. Let M ′ {M^{prime}} be diffeomorphic to M, let ∂ ⁡ M = ∂ ⁡ M ′ {partial M=partial M^{prime}} and let Λ ′ {Lambda^{prime}} be the DN-map of ( M ′ , g ′ ) {(M^{prime},g^{prime})} . Put ( M ′ , g ′ ) ∈ 𝕄 t {(M^{prime},g^{prime})inmathbb{M}_{t}} if ∥ Λ ′ - Λ ∥ H 1 ⁢ ( ∂ ⁡ M ) → L 2 ⁢ ( ∂ ⁡ M ) ⩽ t {lVertLambda^{prime}-LambdarVert_{H^{1}(partial M)to L^{2}(partial M)}% leqslant t} holds. We show that, for any holomorphic immersion ℰ : M → ℂ n {mathscr{E}:Mtomathbb{C}^{n}} ( n ⩾ 1 {ngeqslant 1} ), the relation sup M ′ ∈ 𝕄 t ⁡ inf ℰ ′ ⁡ d H ⁢ ( ℰ ′ ⁢ ( M ′ ) , ℰ ⁢ ( M ) ) → t → 0 0 sup_{M^{prime}inmathbb{M}_{t}}inf_{mathscr{E}^{prime}}d_{H}(mathscr{E}% ^{prime}(M^{prime}),mathscr{E}(M))xrightarrow{tto 0}0 holds, where d H {d_{H}} is the Hausdorff distance in ℂ n {mathbb{C}^{n}} and the infimum is taken over all holomorphic immersions ℰ ′ : M ′ ↦ ℂ n {mathscr{E}^{prime}:M^{prime}mapstomathbb{C}^{n}} . As it is known, Λ determines not the surface ( M , g ) {(M,g)} but its conformal class { ( M , ρ ⁢ g ) ⁢ ∣ ρ > ⁢ 0 , ρ | ∂ ⁡ M = 1 } , bigl{{}(M,rho g)midrho>0,,rho|_{partial M}=1bigr{}}, while holomorphic immersions are determined by this class. In the mean time, ( M , g ) {(M,g)} is conformally equivalent to ℰ ⁢ ( M ) {mathscr{E}(M)} , and ( M ′ , g ′ ) {(M^{prime},g^{prime})} is conformally equivalent to ℰ ′ ⁢ ( M ′ ) {mathscr{E}^{prime}(M^{prime})} . Thus, the closeness of the surfaces ℰ ′ ⁢ ( M ′ ) {mathscr{E}^{prime}(M^{prime})} and ℰ ⁢ ( M ) {mathscr{E}(M)} in ℂ n {mathbb{C}^{n}} reflects the closeness of the corresponding conformal classes for close DN-maps.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"31 1","pages":"159 - 176"},"PeriodicalIF":1.1,"publicationDate":"2023-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48263039","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}

{"title":"Block Toeplitz Inner-Bordering method for the Gelfand–Levitan–Marchenko equations associated with the Zakharov–Shabat system","authors":"S. Medvedev, I. Vaseva, M. Fedoruk","doi":"10.1515/jiip-2022-0072","DOIUrl":"https://doi.org/10.1515/jiip-2022-0072","url":null,"abstract":"Abstract We propose a generalized method for solving the Gelfand–Levitan–Marchenko equation (GLME) based on the block version of the Toeplitz Inner-Bordering (TIB). The method works for the signals containing both the continuous and the discrete spectra. The method allows us to calculate the potential at an arbitrary point and does not require small spectral data. Using this property, we can perform calculations to the right and to the left of the selected starting point. For the discrete spectrum, the procedure of cutting off exponentially growing matrix elements is suggested to avoid the numerical instability and perform calculations for soliton solutions spaced apart in the time domain.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"31 1","pages":"191 - 202"},"PeriodicalIF":1.1,"publicationDate":"2023-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46694032","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}