| 1. |
- Achieng, Pauline, 1990-, et al.
(författare)
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Robin-Dirichlet alternating iterative procedure for solving the Cauchy problem for Helmholtz equation in an unbounded domain
- 2023
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Ingår i: Journal of Inverse and Ill-Posed Problems. - : WALTER DE GRUYTER GMBH. - 0928-0219 .- 1569-3945. ; 31:5
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Tidskriftsartikel (refereegranskat)abstract
- We consider the Cauchy problem for the Helmholtz equation with a domain in with N cylindrical outlets to infinity with bounded inclusions in . 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 Mazya proposed an alternating iterative method for solving Cauchy problems associated with elliptic, selfadjoint 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) 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 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 mu(0) and mu(1), the Robin-Dirichlet alternating iterative procedure is convergent.
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| 2. |
- Agranovsky, Mark, et al.
(författare)
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Research biography of a distinguished expert in the field of inverse problems : Professor Eric Todd Quinto
- 2022
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Ingår i: Journal of Inverse and Ill-Posed Problems. - : Walter de Gruyter GmbH. - 0928-0219 .- 1569-3945. ; 30:4, s. 613-617
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Tidskriftsartikel (refereegranskat)abstract
- This article gives a brief overview of the research in microlocal analysis, tomography, and integral geometry of Professor Eric Todd Quinto, Robinson Professor of Mathematics at Tufts University, along with the collaborators and colleagues who influenced his work.
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| 3. |
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| 4. |
- Beilina, Larisa, 1970, et al.
(författare)
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A new approximate mathematical model for global convergence for a coefficient inverse problem with backscattering data
- 2012
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Ingår i: Journal of Inverse and Ill-Posed Problems. - : Walter de Gruyter GmbH. - 0928-0219 .- 1569-3945. ; 20:4, s. 513-565
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Tidskriftsartikel (refereegranskat)abstract
- An approximately globally convergent numerical method for a 3d coefficient inverse problem for a hyperbolic equation with backscattering data is presented. A new approximate mathematical model is presented as well. An approximation is used only on the first iteration and amounts to the truncation of a certain asymptotic series. A significantly new element of the convergence analysis is that the so-called "tail functions" are estimated. Numerical results in 2d and 3d cases are discussed, including the one for a quite heterogeneous medium.
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| 5. |
- Beilina, Larisa, 1970, et al.
(författare)
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Synthesis of global convergence and adaptivity for a hyperbolic coefficient inverse problem in 3D
- 2010
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Ingår i: Journal of Inverse and Ill-Posed Problems. - 0928-0219 .- 1569-3945. ; 18:1, s. 85-132
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Tidskriftsartikel (refereegranskat)abstract
- A globally convergent numerical method for a 3-Dimensional Coefficient Inverse Problem for a hyperbolic equation is presented. A new globally convergent theorem is proven. It is shown that this technique provides a good first guess for the Finite Element Adaptive method (adaptivity) method. This leads to a synthesis of both approaches. Numerical results are presented.
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| 6. |
- Boman, Jan
(författare)
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A hypersurface containing the support of a Radon transform must be an ellipsoid. II : The general case
- 2021
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Ingår i: Journal of Inverse and Ill-Posed Problems. - : Walter de Gruyter GmbH. - 0928-0219 .- 1569-3945. ; 29:3, s. 351-367
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Tidskriftsartikel (refereegranskat)abstract
- If the Radon transform of a compactly supported distribution f not equal 0 in R-n is supported on the set of tangent planes to the boundary partial derivative D of a bounded convex domain D, then partial derivative D must be an ellipsoid. The special case of this result when the domain D is symmetric was treated in [J. Boman, A hypersurface containing the support of a Radon transform must be an ellipsoid. I: The symmetric case, J. Geom. Anal. (2020), DOI 10.1007/s12220-020-00372-8]. Here we treat the general case.
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| 7. |
- Cheng, Xiaoliang, et al.
(författare)
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A modified coupled complex boundary method for an inverse chromatography problem
- 2018
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Ingår i: Journal of Inverse and Ill-Posed Problems. - : Walter de Gruyter. - 0928-0219 .- 1569-3945. ; 26:1, s. 33-49
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Tidskriftsartikel (refereegranskat)abstract
- Adsorption isotherms are the most important parameters in rigorous models of chromatographic processes. In this paper, in order to recover adsorption isotherms, we consider a coupled complex boundary method (CCBM), which was previously proposed for solving an inverse source problem [2]. With CCBM, the original boundary fitting problem is transferred to a domain fitting problem. Thus, this method has advantages regarding robustness and computation in reconstruction. In contrast to the traditional CCBM, for the sake of the reduction of computational complexity and computational cost, the recovered adsorption isotherm only corresponds to the real part of the solution of a forward complex initial boundary value problem. Furthermore, we take into account the position of the profiles and apply the momentum criterion to improve the optimization progress. Using Tikhonov regularization, the well-posedness, convergence properties and regularization parameter selection methods are studied. Based on an adjoint technique, we derive the exact Jacobian of the objective function and give an algorithm to reconstruct the adsorption isotherm. Finally, numerical simulations are given to show the feasibility and efficiency of the proposed regularization method.
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| 8. |
- Djehiche, Boualem, 1962-, et al.
(författare)
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A functional Hodrick-Prescott filter
- 2017
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Ingår i: Journal of Inverse and Ill-Posed Problems. - New York : WALTER DE GRUYTER GMBH. - 0928-0219 .- 1569-3945. ; 25:2, s. 135-148
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Tidskriftsartikel (refereegranskat)abstract
- We propose a functional version of the Hodrick-Prescott filter for functional data which take values in an infinite-dimensional separable Hilbert space. We further characterize the associated optimal smoothing operator when the associated linear operator is compact and the underlying distribution of the data is Gaussian.
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| 9. |
- Feng, Xiao-Li, et al.
(författare)
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A quasi-boundary-value method for the Cauchy problem for elliptic equations with nonhomogeneous Neumann data
- 2010
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Ingår i: Journal of Inverse and Ill-Posed Problems. - : Walter de Gruyter. - 0928-0219 .- 1569-3945. ; 18:6, s. 617-645
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Tidskriftsartikel (refereegranskat)abstract
- A Cauchy problem for elliptic equations with nonhomogeneous Neumann datain a cylindrical domain is investigated in this paper. For the theoretical aspect the a-prioriand a-posteriori parameter choice rules are suggested and the corresponding error estimatesare obtained. About the numerical aspect, for a simple case results given by twomethods based on the discrete Sine transform and the finite difference method are presented;an idea of left-preconditioned GMRES (Generalized Minimum Residual) methodis proposed to deal with the high dimensional case to save the time; a view of dealingwith a general domain is suggested. Some ill-posed problems regularized by the quasiboundary-value method are listed and some rules of this method are suggested.
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| 10. |
- Feng, Xiao-Li, et al.
(författare)
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Stability and regularization of a backward parabolic PDE with variable coefficients
- 2010
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Ingår i: Journal of Inverse and Ill-Posed Problems. - 0928-0219 .- 1569-3945. ; 18, s. 217-243
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Tidskriftsartikel (refereegranskat)abstract
- We consider a backward parabolic partial differential equation (BPPDE) withvariable coefficient a.x; t / in time. A new modification is used on the logarithmic convexitymethod to obtain a conditional stability estimate. Based on a formal solution, wereveal the essence of the ill-posedness and propose a simple regularization method. Moreover,we apply the regularization method to two representative cases. The results of boththeoretical and numerical performance show the validity of our method.
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