32 results
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2. New iterative reconstruction methods for fan-beam tomography.
- Author
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Kazantsev, Daniil and Pickalov, Valery
- Subjects
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ITERATIVE methods (Mathematics) , *GEOMETRIC tomography , *COORDINATE transformations , *INTERPOLATION , *ALGORITHMS - Abstract
In this paper, we present a novel class of iterative reconstruction methods for severely angular undersampled or/and limited-view tomographic problems with fan-beam scanning geometry. The proposed algorithms are based on a new analytical transform which generalizes Fourier-slice theorem to divergent-beam scanning geometries. Using a non-rigid coordinate transform, divergent rays can be reorganized into parallel ones. Therefore, one can employ a simpler parallel-beam projection model instead of more complicated divergent-beam geometries. Various existing iterative reconstruction techniques for divergent-beam geometries can be easily adapted to the proposed framework. The significant advantage of this formulation is the possibility of exploiting efficient Fourier-based recovery methods without rebinning of the projections. In case of highly sparse measurements (few-view data), rebinning methods are not suitable due to error-prone angular interpolation involved. In this work, three new methods based on the novel analytical framework for fan-beam geometry are presented: the Gerchberg-Papoulis algorithm, the Neumann decomposition method and its total variation regularized version. Presented numerical experiments demonstrate that the methods can be competitive in reconstructing from few-view noisy tomographic measurements. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
3. A STUDY ON REGULARIZATION FOR DISCRETE INVERSE PROBLEMS WITH MODEL-DEPENDENT NOISE.
- Author
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BENVENUTO, FEDERICO
- Subjects
- *
MATHEMATICAL regularization , *DISCRETE systems , *INVERSE problems , *OPERATOR theory , *ALGORITHMS , *ITERATIVE methods (Mathematics) - Abstract
In this paper we consider discrete inverse problems for which noise becomes negligible compared to data with increasing model norm. We introduce two novel definitions of regularization for characterizing inversion methods which provide approximations of ill-conditioned inverse operators consistent with such noisy data. In particular, these d efinitions, respectively, require that the reconstruction error computed from normalized data (p-asymptotic regularization) and the relative reconstruction error (p-relative regularization) go to zero as the model norm tends to infinity, 0 ≤ p < 1 being a parameter controlling the increase rate of the noise level. We investigate the relationship between these two definitions and we prove that t hey are all equivalent for positively homogeneous iterative algorithms with suitable stopping rules. This result has as a crucial consequence that such iterative algorithms realize regularization independently of the noise model. Then we give sufficient conditions for such methods to be p-asymptotic and p-relative regularizations in a discrete setting and we prove that the classical expectation maximization algorithm for Poisson data and the Landweber algorithm, if suitably stopped, are regu larization methods in this sense. We perform numerical simulations in the case of image deconvolution and computerized tomography to show that, in the presence of model-dependent noise, the reconst ructions provided by the above mentioned methods improve with increasing model norm as required by the p-asymptotic and p-relative regularization properties. More extensive studies on the p-asym ptotic and p-relative regularizations for Tikhonov-type methods will be the object of future work. [ABSTRACT FROM AUTHOR]
- Published
- 2017
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4. The analysis study on nonlinear iterative methods for inverse problems.
- Author
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Du, Lingling, Li, Jing, and Wang, Jinping
- Subjects
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INVERSE problems , *ITERATIVE methods (Mathematics) , *NONLINEAR theories , *ALGORITHMS , *STOCHASTIC convergence - Abstract
In this paper, the nonlinear iterative methods, which are different from the classical algorithms, to solve inverse problems are presented. Our methods by denoting some parameters and some properties of the algorithm in both noise and noiseless cases are studied. Finally, the convergence of the sequence generated by the algorithm without noise is discussed. [ABSTRACT FROM AUTHOR]
- Published
- 2017
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5. Approximation of zeros of inverse strongly monotone operators in Banach spaces
- Author
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Saejung, Satit and Yotkaew, Pongsakorn
- Subjects
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APPROXIMATION theory , *INVERSE problems , *MONOTONE operators , *BANACH spaces , *ALGORITHMS , *ITERATIVE methods (Mathematics) , *MATHEMATICAL sequences , *STOCHASTIC convergence , *VARIATIONAL inequalities (Mathematics) - Abstract
Abstract: In this paper, we consider the projection algorithm studied by Iiduka and Takahashi (2008) for finding a solution of the variational inequality problem for an inverse strongly monotone operator in a Banach space. We first remark that, under the assumptions imposed on the operator in their paper, the iterative sequence converges weakly to a zero of the operator, not just a solution of the variational inequality problem. In our proof, slightly modified from the original, we do not assume the uniform smoothness of a space as was the case there. Finally, using Halpern’s type method, we modify this algorithm to obtain the strong convergence to a zero of an inverse strongly monotone operator which is nearest to the initial element of the algorithm in the sense of the Bergman distance associated with the function . [Copyright &y& Elsevier]
- Published
- 2012
- Full Text
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6. An iterative thresholding algorithm for linear inverse problems with multi-constraints and its applications
- Author
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Khoramian, Saman
- Subjects
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ALGORITHMS , *INVERSE problems , *NUMERICAL analysis , *MATHEMATICAL analysis , *ITERATIVE methods (Mathematics) , *PROBLEM solving - Abstract
Abstract: In this paper, we will present a generalization for a minimization problem from I. Daubechies, M. Defrise, and C. DeMol (2004) . This generalization is useful for solving many practical problems in which more than one constraint are involved. In this regard, we will conclude the findings of many papers (most of which are on image processing) from this generalization. It is hoped that the approach proposed in this paper will be a suitable reference for some applied works where multi-frames, multi-wavelets, or multi-constraints are present in linear inverse problems. [Copyright &y& Elsevier]
- Published
- 2012
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7. KRASNOSELSKI--MANN ITERATION FOR HIERARCHICAL FIXED POINTS AND EQUILIBRIUM PROBLEM.
- Author
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Marino, Giuseppe, Colao, Vittorio, Muglia, Luigi, and Yonghong Yao
- Subjects
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FIXED point theory , *ITERATIVE methods (Mathematics) , *MATHEMATICAL functions , *HILBERT space , *NONEXPANSIVE mappings , *ALGORITHMS , *NUMERICAL analysis , *INVERSE problems , *STOCHASTIC convergence , *EQUILIBRIUM - Abstract
We give an explicit Krasnoselski-Mann type method for finding common solutions of the following system of equilibrium and hierarchical fixed points: {G(x*,y) ≥0, ∀x ϵFix(T), find x* ϵ Fix(T) such that ⟨X* - ƒ (x*), x - x*⟨ ≥ 0, ∀x ϵFix(T), where C is a closed convex subset of a Hilbert space H, G : C × C →ℝ is an equilibrium function, T : C → C is a nonexpansive mapping with Fix(T) its set of fixed points and ƒ : C → C is a ρ-contraction. Our algorithm is constructed and proved using the idea of the paper of [Y. Yao and Y.-C. Liou, 'Weak and strong convergence of Krasnosel'skiĭ-Mann iteration for hierarchical fixed point problems', Inverse Problems 24 (2008), 501-508], in which only the variational inequality problem of finding hierarchically a fixed point of a nonexpansive mapping T with respect to a ρ-contraction ƒ was considered. The paper follows the lines of research of corresponding results of Moudafi and Théra. [ABSTRACT FROM AUTHOR]
- Published
- 2009
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8. Bregman iterative algorithms for 2D geosounding inversion.
- Author
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Hidalgo-Silva, Hugo and Gómez-Treviño, E.
- Subjects
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INVERSION (Geophysics) , *ITERATIVE methods (Mathematics) , *INVERSE problems , *STOCHASTIC convergence , *ALGORITHMS - Abstract
Bregman iterative algorithms have been extensively used forand total variation regularization problems, allowing to obtain simple, fast and effective algorithms. In this paper, three already-available algorithms for geosounding inversion are modified by including them in a Bregman iterative procedure. The resulting algorithms are easy to implement and do not require any optimization package. Modelling results are presented for synthetic and field data, observing better convergence properties than the original versions, avoiding the need of any continuation descent procedure. [ABSTRACT FROM AUTHOR]
- Published
- 2015
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9. Iterative solutions of the inverse problems of frequency sounding and electrical prospecting of layered media.
- Author
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Timonov, A.
- Subjects
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NUMERICAL analysis , *INVERSE problems , *ITERATIVE methods (Mathematics) , *ALGORITHMS , *COEFFICIENTS (Statistics) , *ELECTROMAGNETISM , *MATHEMATICAL models - Abstract
The paper presents the iterative solutions of two coefficient inverse problems (CIPs) arising in frequency sounding and electrical prospecting. An iterative algorithm is constructed to obtain such solutions. Exploiting the Beilina–Klibanov approach to CIPs, this algorithm possesses the new iterative and refinement procedures. These features enhance significantly both the spatial and contrast resolutions of reconstructed coefficients. The computational effectiveness of the proposed numerical technique is demonstrated in computational experiments with two applied CIPs: electromagnetic or acoustic frequency sounding and electrical prospecting of layered media. The Slichter–Langer–Tikhonov formulation is exploited as a mathematical model of the latter. [ABSTRACT FROM PUBLISHER]
- Published
- 2014
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10. To solve the inverse Cauchy problem in linear elasticity by a novel Lie-group integrator.
- Author
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Liu, Chein-Shan
- Subjects
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INVERSE problems , *NUMERICAL solutions to the Cauchy problem , *LINEAR systems , *LIE groups , *ITERATIVE methods (Mathematics) , *NUMERICAL analysis , *ALGORITHMS - Abstract
In this paper, we propose a simple, iteration free and easy-to-implement numerical algorithm for the solution of inverse Cauchy problem in linear or nonlinear elasticity. The bottom of a finite rectangular plate is imposed by overspecified boundary data, and we seek unknown data on the top side. A spring-damping transform method (SDTM) is introduced to the Navier equations, such that after a discretization by the differential quadrature method, we can apply a novel Lie-group integrator, namely the mixed group-preserving scheme (MGPS), to solve them as an initial value problem. Several numerical examples including nonlinear ones are examined to show that the MGPS can overcome the ill-posed behaviour of the inverse Cauchy problem in elasticity, which has good efficiency and stability against the noisy disturbance, even with an intensity large up toand. [ABSTRACT FROM AUTHOR]
- Published
- 2014
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11. Generalized Tikhonov regularization method for large-scale linear inverse problems.
- Author
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Di Zhang and Ting-Zhu Huang
- Subjects
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TIKHONOV regularization , *INVERSE problems , *PARAMETER estimation , *NUMERICAL analysis , *ITERATIVE methods (Mathematics) , *LANCZOS method , *ALGORITHMS - Abstract
In this paper we propose a regularization of general Tikhonov type for large-scale ill-posed problems. We introduce the projection method of iterative bidiagonalization and show that the regularization parameter can be chosen without prior knowledge of the noise variance by using the method of balancing principle. An algorithm implicate the efficient numerical realization of the new choice rule. Numerical experiments for severely ill-show benchmark inverse problems show that new method is effective compared with other criterions. [ABSTRACT FROM AUTHOR]
- Published
- 2013
12. A new iterative firm-thresholding algorithm for inverse problems with sparsity constraints.
- Author
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Voronin, Sergey and Woerdeman, Hugo J.
- Subjects
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ITERATIVE methods (Mathematics) , *ALGORITHMS , *INVERSE problems , *CONSTRAINT satisfaction , *APPROXIMATION theory , *MATHEMATICAL regularization - Abstract
Abstract: In this paper we propose a variation of the soft-thresholding algorithm for finding sparse approximate solutions of the equation , where as the sparsity of the iterate increases the penalty function changes. In this approach, sufficiently large entries in a sparse iterate are left untouched. The advantage of this approach is that a higher regularization constant can be used, leading to a significant reduction of the total number of iterations. Numerical experiments for sparse recovery problems, also with noisy data, are included. [Copyright &y& Elsevier]
- Published
- 2013
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13. Algorithms for Non-Negatively Constrained Maximum Penalized Likelihood Reconstruction in Tomographic Imaging.
- Author
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Jun Ma
- Subjects
- *
IMAGE reconstruction algorithms , *COMPUTER algorithms , *DIAGNOSTIC imaging , *PIXELS , *ITERATIVE methods (Mathematics) , *INVERSE problems - Abstract
Image reconstruction is a key component in many medical imaging modalities. The problem of image reconstruction can be viewed as a special inverse problem where the unknown image pixel intensities are estimated from the observed measurements. Since the measurements are usually noise contaminated, statistical reconstruction methods are preferred. In this paper we review some non-negatively constrained simultaneous iterative algorithms for maximum penalized likelihood reconstructions, where all measurements are used to estimate all pixel intensities in each iteration. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF
14. Variable metric quasi-Fejér monotonicity
- Author
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Combettes, Patrick L. and Vũ, Bằng C.
- Subjects
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MONOTONIC functions , *PROOF theory , *STOCHASTIC convergence , *ALGORITHMS , *NONLINEAR analysis , *ITERATIVE methods (Mathematics) , *MATRIX norms - Abstract
Abstract: The notion of quasi-Fejér monotonicity has proven to be an efficient tool to simplify and unify the convergence analysis of various algorithms arising in applied nonlinear analysis. In this paper, we extend this notion in the context of variable metric algorithms, whereby the underlying norm is allowed to vary at each iteration. Applications to convex optimization and inverse problems are demonstrated. [Copyright &y& Elsevier]
- Published
- 2013
- Full Text
- View/download PDF
15. Levenberg–Marquardt iterative regularization for the pulse-type impact-force reconstruction
- Author
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Gunawan, Fergyanto E.
- Subjects
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ITERATIVE methods (Mathematics) , *IMPACT (Mechanics) , *ALGORITHMS , *PARAMETER estimation , *MEASUREMENT , *INVERSE problems - Abstract
Abstract: In this paper, we study the Levenberg–Marquardt algorithm with the trust region strategy to iteratively solve the ill-posed impact-force reconstruction problem. This particular problem is interesting because the impact-force particularly those of the pulse-type are difficult to be measured directly. In the present approach, the necessity of regularization is enforced by means of the trust region approach. This study mainly contributes a systematic approach to locate the optimal solution by tracking the Levenberg–Marquardt parameter. The proposed method is evaluated by solving two typical problem existed in the inverse problem of the impact-force reconstruction. Reasonable accurate impact-forces were produced from the both evaluations. [Copyright &y& Elsevier]
- Published
- 2012
- Full Text
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16. An alternating direction algorithm for two-phase flow visualization using gamma computed tomography.
- Author
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Xue, Qian, Wang, Huaxiang, Cui, Ziqiang, and Yang, Chengyi
- Subjects
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TOMOGRAPHY , *GAMMA rays , *IMAGING systems , *ALGORITHMS , *INVERSE problems , *ITERATIVE methods (Mathematics) - Abstract
In order to build high-speed imaging systems with low cost and low radiation leakage, the number of radioactive sources and detectors in the multiphase flow computed tomography (CT) system has to be limited. Moreover, systematic and random errors are inevitable in practical applications. The limited and corrupted measurement data have made the tomographic inversion process the most critical part in multiphase flow CT. Although various iterative reconstruction algorithms have been developed based on least squares minimization, the imaging quality is still inadequate for the reconstruction of relatively complicated bubble flow. This paper extends an alternating direction method (ADM), which is originally proposed in compressed sensing, to image two-phase flow using a low-energy γ-CT system. An l1 norm-based regularization technique is utilized to treat the ill-posedness of the inverse problem, and the image reconstruction model is reformulated into one having partially separable objective functions, thereafter a dual-based ADM is adopted to solve the resulting problem. The feasibility is demonstrated in prototype experiments. Comparisons between the ADM and the conventional iterative algorithms show that the former has obviously improved the space resolution in reasonable time. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
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17. On convergence rates for iteratively regularized procedures with linear penalty terms.
- Author
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Smirnova, Alexandra
- Subjects
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ITERATIVE methods (Mathematics) , *ALGORITHMS , *INVERSE problems , *PROBLEM solving , *SMOOTHNESS of functions , *MATHEMATICAL analysis , *STOCHASTIC convergence - Abstract
The impact of this paper is twofold. First, we study convergence rates of the iteratively regularized Gauss-Newton (IRGN) algorithm with a linear penalty term under a generalized source assumption and show how the regularizing properties of new iterations depend on the solution smoothness. Secondly, we introduce an adaptive IRGN procedure, which is investigated under a relaxed smoothness condition. The introduction and analysis of a more general penalty term are of great importance since, apart from bringing stability to the numerical scheme designed for solving a large class of applied inverse problems, it allows us to incorporate various types of a priori information available on the model. Both a priori and a posteriori stopping rules are investigated. For the a priori stopping rule, optimal convergence rates are derived. A numerical example illustrating convergence rates is considered [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
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18. A Parametric Level-Set Approach to Simultaneous Object Identification and Background Reconstruction for Dual-Energy Computed Tomography.
- Author
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Semerci, Oguz and Miller, Eric L.
- Subjects
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TOMOGRAPHY , *IMAGE reconstruction , *LEVEL set methods , *ATTENUATION (Physics) , *IMAGE processing , *ITERATIVE methods (Mathematics) , *ALGORITHMS - Abstract
Dual-energy computerized tomography has gained great interest because of its ability to characterize the chemical composition of a material rather than simply providing relative attenuation images as in conventional tomography. The purpose of this paper is to introduce a novel polychromatic dual-energy processing algorithm, with an emphasis on detection and characterization of piecewise constant objects embedded in an unknown cluttered background. Physical properties of the objects, particularly the Compton scattering and photoelectric absorption coefficients, are assumed to be known with some level of uncertainty. Our approach is based on a level-set representation of the characteristic function of the object and encompasses a number of regularization techniques for addressing both the prior information we have concerning the physical properties of the object and the fundamental physics-based limitations associated with our ability to jointly recover the Compton scattering and photoelectric absorption properties of the scene. In the absence of an object with appropriate physical properties, our approach returns a null characteristic function and, thus, can be viewed as simultaneously solving the detection and characterization problems. Unlike the vast majority of methods that define the level-set function nonparametrically, i.e., as a dense set of pixel values, we define our level set parametrically via radial basis functions and employ a Gauss–Newton-type algorithm for cost minimization. Numerical results show that the algorithm successfully detects objects of interest, finds their shape and location, and gives an adequate reconstruction of the background. [ABSTRACT FROM PUBLISHER]
- Published
- 2012
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19. Sparsity reconstruction in electrical impedance tomography: An experimental evaluation
- Author
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Gehre, Matthias, Kluth, Tobias, Lipponen, Antti, Jin, Bangti, Seppänen, Aku, Kaipio, Jari P., and Maass, Peter
- Subjects
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ELECTRICAL impedance tomography , *IMAGE reconstruction , *INVERSE problems , *ALGORITHMS , *SMOOTHING (Numerical analysis) , *ITERATIVE methods (Mathematics) - Abstract
Abstract: We investigate the potential of sparsity constraints in the electrical impedance tomography (EIT) inverse problem of inferring the distributed conductivity based on boundary potential measurements. In sparsity reconstruction, inhomogeneities of the conductivity are a priori assumed to be sparse with respect to a certain basis. This prior information is incorporated into a Tikhonov-type functional by including a sparsity-promoting -penalty term. The functional is minimized with an iterative soft shrinkage-type algorithm. In this paper, the feasibility of the sparsity reconstruction approach is evaluated by experimental data from water tank measurements. The reconstructions are computed both with sparsity constraints and with a more conventional smoothness regularization approach. The results verify that the adoption of -type constraints can enhance the quality of EIT reconstructions: in most of the test cases the reconstructions with sparsity constraints are both qualitatively and quantitatively more feasible than that with the smoothness constraint. [Copyright &y& Elsevier]
- Published
- 2012
- Full Text
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20. Algebraic reconstruction of the general-order poles of a meromorphic function.
- Author
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Nara, Takaaki
- Subjects
- *
MEROMORPHIC functions , *INVERSE problems , *ITERATIVE methods (Mathematics) , *ALGORITHMS , *NUMERICAL analysis , *NUMERICAL solutions to equations , *MATHEMATICAL analysis - Abstract
This paper presents a method for identifying the general-order poles of a meromorphic function algebraically from its values on the unit circle, which has various applications in inverse source problems in potential analysis. First, we derive a system of Dth-degree equations for N distinct poles zn of order Dn, where n = 1, 2, . . . ,N and D = max1≤n≤N{Dn}. Then, we transform these equations into linear equations for the coefficients of the Nth-degree equation whose roots are zn so that the poles are obtained algebraically from data. The obtained poles can be used as an initial solution for iterative algorithms. A method for estimating the order Dn of each pole is also proposed and is numerically verified. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
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21. An iterative algorithm for large size least-squares constrained regularization problems
- Author
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Piccolomini, E. Loli and Zama, F.
- Subjects
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ITERATIVE methods (Mathematics) , *INVERSE problems , *ALGORITHMS , *LEAST squares , *CONSTRAINED optimization , *APPROXIMATION theory , *NUMERICAL analysis - Abstract
Abstract: In this paper we propose an iterative algorithm to solve large size linear inverse ill posed problems. The regularization problem is formulated as a constrained optimization problem. The dual Lagrangian problem is iteratively solved to compute an approximate solution. Before starting the iterations, the algorithm computes the necessary smoothing parameters and the error tolerances from the data. The numerical experiments performed on test problems show that the algorithm gives good results both in terms of precision and computational efficiency. [Copyright &y& Elsevier]
- Published
- 2011
- Full Text
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22. On epicardial potential reconstruction using regularization schemes with the L1-norm data term.
- Author
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Guofa Shou, Ling Xia, Feng Liu, Mingfeng Jiang, and Stuart Crozier
- Subjects
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ELECTROCARDIOGRAPHY , *INVERSE problems , *NP-complete problems , *BODY surface mapping , *ALGORITHMS , *ITERATIVE methods (Mathematics) , *MEASUREMENT errors - Abstract
The electrocardiographic (ECG) inverse problem is ill-posed and usually solved by regularization schemes. These regularization methods, such as the Tikhonov method, are often based on the L2-norm data and constraint terms. However, L2-norm-based methods inherently provide smoothed inverse solutions that are sensitive to measurement errors, and also lack the capability of localizing and distinguishing multiple proximal cardiac electrical sources. This paper presents alternative regularization schemes employing the L1-norm data term for the reconstruction of epicardial potentials (EPs) from measured body surface potentials (BSPs). During numerical implementation, the iteratively reweighted norm algorithm was applied to solve the L1-norm-related schemes, and measurement noises were considered in the BSP data. The proposed L1-norm data term-based regularization schemes (with L1 and L2 penalty terms of the normal derivative constraint (labelled as L1TV and L1L2)) were compared with the L2-norm data terms (Tikhonov with zero-order and normal derivative constraints, labelled as ZOT and FOT, and the total variation method labelled as L2TV). The studies demonstrated that, with averaged measurement noise, the inverse solutions provided by the L1L2 and FOT algorithms have less relative error values. However, when larger noise occurred in some electrodes (for example, signal lost during measurement), the L1TV and L1L2 methods can obtain more accurate EPs in a robust manner. Therefore the L1-norm data term-based solutions are generally less perturbed by measurement noises, suggesting that the new regularization scheme is promising for providing practical ECG inverse solutions. [ABSTRACT FROM AUTHOR]
- Published
- 2011
- Full Text
- View/download PDF
23. Generalized inverse problems for part symmetric matrices on a subspace in structural dynamic model updating
- Author
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Liu, Xian-xia, Li, Jiao-fen, and Hu, Xi-Yan
- Subjects
- *
INVERSE problems , *SYMMETRIC matrices , *CONJUGATE gradient methods , *ITERATIVE methods (Mathematics) , *EIGENVALUES , *ALGORITHMS , *PERTURBATION theory - Abstract
Abstract: An matrix is said to be -symmetric if for all , where is given. In this paper, by extending the idea of the conjugate gradient least squares (CGLS) method, we construct an iterative method for solving a generalized inverse eigenvalue problem: minimizing where is the Frobenius norm, and are given, and is a -symmetric matrix to be solved. Our algorithm produces a suitable such that within finite iteration steps in the absence of roundoff errors, if such an exists. We show that the algorithm is stable any case, and we give results of numerical experiments that support this claim. [ABSTRACT FROM AUTHOR]
- Published
- 2011
- Full Text
- View/download PDF
24. Iterative solvers for Tikhonov regularization of dense inverse problems.
- Author
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Popa, Constantin
- Subjects
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ITERATIVE methods (Mathematics) , *INVERSE problems , *ORTHOGONALIZATION , *ALGORITHMS , *FREDHOLM equations , *INTEGRAL equations , *STOCHASTIC convergence - Abstract
According to the special demands arising from the development of science and technology, in the last decades appeared a special class of problems that are inverse to the classical direct ones. Such an inverse problem is concerned with the opposite way, usually followed by a direct one: finding the cause of a given effect or finding the law of evolution given the cause and effect. Very frequently, such inverse problems are modelled by Fredholm first-kind integral equations that give rise after discretization to (very) ill-conditioned linear systems, in classical or least squares formulation. Then, an efficient numerical solution can be obtained by using the Tikhonov regularization technique. In this respect, in the present paper, we propose three Kovarik-like algorithms for numerical solution of the regularized problem. We prove convergence for all three methods and present numerical experiments on a mathematical model of an inverse problem concerned with the determination of charge distribution generating a given electric field. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
25. Adaptive wavelet methods and sparsity reconstruction for inverse heat conduction problems.
- Author
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Bonesky, Thomas, Dahlke, Stephan, Maass, Peter, and Raasch, Thorsten
- Subjects
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HEAT conduction , *NUMERICAL analysis , *INVERSE problems , *WAVELETS (Mathematics) , *RECONSTRUCTION (Graph theory) , *ITERATIVE methods (Mathematics) , *IRON ores , *ALGORITHMS , *NUMERICAL solutions to partial differential equations - Abstract
This paper is concerned with the numerical treatment of inverse heat conduction problems. In particular, we combine recent results on the regularization of ill-posed problems by iterated soft shrinkage with adaptive wavelet algorithms for the forward problem. The analysis is applied to an inverse parabolic problem that stems from the industrial process of melting iron ore in a steel furnace. Some numerical experiments that confirm the applicability of our approach are presented. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
26. A parallel evolutionary strategy based simulation–optimization approach for solving groundwater source identification problems
- Author
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Mirghani, Baha Y., Mahinthakumar, Kumar G., Tryby, Michael E., Ranjithan, Ranji S., and Zechman, Emily M.
- Subjects
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PARALLEL computers , *EVOLUTIONARY computation , *COMPUTER simulation , *MATHEMATICAL optimization , *GROUNDWATER , *INVERSE problems , *TRANSPORT theory , *ITERATIVE methods (Mathematics) , *ALGORITHMS - Abstract
Abstract: Groundwater characterization involves the resolution of unknown system characteristics from observation data, and is often classified as an inverse problem. Inverse problems are difficult to solve due to natural ill-posedness and computational intractability. Here we adopt the use of a simulation–optimization approach that couples a numerical pollutant-transport simulation model with evolutionary search algorithms for solution of the inverse problem. In this approach, the numerical transport model is solved iteratively during the evolutionary search. This process can be computationally intensive since several hundreds to thousands of forward model evaluations are typically required for solution. Given the potential computational intractability of such a simulation–optimization approach, parallel computation is employed to ease and enable the solution of such problems. In this paper, several variations of a groundwater source identification problem is examined in terms of solution quality and computational performance. The computational experiments were performed on the TeraGrid cluster available at the National Center for Supercomputing Applications. The results demonstrate the performance of the parallel simulation–optimization approach in terms of solution quality and computational performance. [Copyright &y& Elsevier]
- Published
- 2009
- Full Text
- View/download PDF
27. A sparse-sparse iteration for computing a sparse incomplete factorization of the inverse of an SPD matrix
- Author
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Salkuyeh, Davod Khojasteh and Toutounian, Faezeh
- Subjects
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ITERATIVE methods (Mathematics) , *FACTORIZATION , *SYMMETRIC matrices , *INVERSE problems , *LINEAR systems , *ALGORITHMS , *CONJUGATE gradient methods - Abstract
Abstract: In this paper, a method via sparse-sparse iteration for computing a sparse incomplete factorization of the inverse of a symmetric positive definite matrix is proposed. The resulting factorized sparse approximate inverse is used as a preconditioner for solving symmetric positive definite linear systems of equations by using the preconditioned conjugate gradient algorithm. Some numerical experiments on test matrices from the Harwell–Boeing collection for comparing the numerical performance of the presented method with one available well-known algorithm are also given. [Copyright &y& Elsevier]
- Published
- 2009
- Full Text
- View/download PDF
28. An Iterative Algorithm for the Backward Heat Conduction Problem Based on Variable Relaxation Factors.
- Author
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Jourhmane, M. and Mera, N. S.
- Subjects
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ALGORITHMS , *HEAT conduction , *ITERATIVE methods (Mathematics) , *INVERSE problems , *DIFFERENTIAL equations - Abstract
In this paper, an iterative algorithm is proposed for solving the backward heat conduction problem (BHCP). The algorithm is based on allowing variable relaxation factors for an iterative algorithm proposed by Kozlov and Maz'ya [1]. The convergence of the relaxation algorithm is analysed both theoretically and numerically. The boundary element method (BEM) is used to implement numerically the algorithm and to show that the ill-posed BHCP is regularized by using an appropriate stopping criterion. [ABSTRACT FROM AUTHOR]
- Published
- 2002
- Full Text
- View/download PDF
29. AN EFFICIENT LINEAR SOLVER FOR NONLINEAR PARAMETER IDENTIFICATION PROBLEMS.
- Author
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Yee Lo Keung and Jun Zou
- Subjects
- *
ELLIPTIC functions , *LINEAR algebra , *ITERATIVE methods (Mathematics) , *NUMERICAL analysis , *ALGORITHMS , *INVERSE problems , *DIFFERENTIAL equations , *ELLIPTIC differential equations , *MATHEMATICS - Abstract
In this paper, we study some efficient numerical methods for parameter identifications in elliptic systems. The proposed numerical methods are conducted iteratively and each iteration involves only solving positive definite linear algebraic systems, although the original inverse problems are ill-posed and highly nonlinear. The positive definite systems can be naturally preconditioned with their corresponding block diagonal matrices. Numerical experiments are presented to illustrate the efficiency of the proposed algorithms. [ABSTRACT FROM AUTHOR]
- Published
- 2000
30. Inverse problem of time-dependent heat sources numerical reconstruction
- Author
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Yang, Liu, Dehghan, Mehdi, Yu, Jian-Ning, and Luo, Guan-Wei
- Subjects
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INVERSE problems , *OPERATOR equations , *ALGORITHMS , *GREEN'S functions , *NUMERICAL analysis , *ITERATIVE methods (Mathematics) - Abstract
Abstract: This work studies the inverse problem of reconstructing a time-dependent heat source in the heat conduction equation using the temperature measurement specified at an internal point. Problems of this type have important applications in several fields of applied science. By the Green’s function method, the inverse problem is reduced to an operator equation of the first kind which is known to be ill-posed. The uniqueness of the solution for the inverse problem is obtained by the contraction mapping principle. A numerical algorithm on the basis of the Landweber iteration is designed to deal with the operator equation and some typical numerical experiments are also performed in the paper. The numerical results show that the proposed method is stable and the unknown heat source is recovered very well. [Copyright &y& Elsevier]
- Published
- 2011
- Full Text
- View/download PDF
31. An iterative approach to the solution of an inverse problem in linear elasticity
- Author
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Ellabib, A. and Nachaoui, A.
- Subjects
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ITERATIVE methods (Mathematics) , *ALGORITHMS , *INVERSE problems , *DIFFERENTIAL equations , *BOUNDARY element methods - Abstract
Abstract: This paper presents an iterative alternating algorithm for solving an inverse problem in linear elasticity. A relaxation procedure is developed in order to increase the rate of convergence of the algorithm and two selection criteria for the variable relaxation factors are provided. The boundary element method is used in order to implement numerically the constructing algorithm. We discuss this implementation, mention the use of Krylov methods to solve the obtained linear algebraic systems of equations and investigate the convergence and the stability when the data is perturbed by noise. [Copyright &y& Elsevier]
- Published
- 2008
- Full Text
- View/download PDF
32. An algorithm for total variation regularization in high-dimensional linear problems.
- Author
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Michel Defrise, Christian Vanhove, and Xuan Liu
- Subjects
- *
ALGORITHMS , *INVERSE problems , *ITERATIVE methods (Mathematics) , *STOCHASTIC convergence , *MATHEMATICAL optimization , *LINEAR differential equations - Abstract
This paper describes an iterative algorithm for high-dimensional linear inverse problems, which is regularized by a differentiable discrete approximation of the total variation (TV) penalty. The algorithm is an interlaced iterative method based on optimization transfer with a separable quadratic surrogate for the TV penalty. The surrogate cost function is optimized using the block iterative regularized algebraic reconstruction technique (RSART). A proof of convergence is given and convergence is illustrated by numerical experiments with simulated parallel-beam computerized tomography (CT) data. The proposed method provides a block-iterative and convergent, hence efficient and reliable, algorithm to investigate the effects of TV regularization in applications such as CT. [ABSTRACT FROM AUTHOR]
- Published
- 2011
- Full Text
- View/download PDF
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