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Journal articles on the topic 'Tikhonov regularization method'

Author: Grafiati

Published: 4 June 2021

Last updated: 27 July 2025

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1

Sjöberg, L. "Solutions to Linear Inverse Problems on the Sphere by Tikhonov Regularization, Wiener filtering and Spectral Smoothing and Combination — A Comparison." Journal of Geodetic Science 2, no. 1 (2012): 31–37. http://dx.doi.org/10.2478/v10156-011-0021-z.

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Solutions to Linear Inverse Problems on the Sphere by Tikhonov Regularization, Wiener filtering and Spectral Smoothing and Combination — A ComparisonSolutions to linear inverse problems on the sphere, common in geodesy and geophysics, are compared for Tikhonov's method of regularization, Wiener filtering and spectral smoothing and combination as well as harmonic analysis. It is concluded that Wiener and spectral smoothing, although based on different assumptions and target functions, yield the same estimator. Also, provided that the extra information on the signal and error degree variances is

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2

Zeng, Xiaoniu, Tianyou Liu, Xiaofeng Tan, Hongru Li, and Shengjie Luo. "An improved Tikhonov regularization method combined with the exponential filter function." Journal of Computational Methods in Sciences and Engineering 25, no. 2 (2024): 1114–22. https://doi.org/10.1177/14727978241295283.

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Tikhonov regularization is one of the most popular methods for solving linear discrete ill-posed problems. This approach involves transforming the original problem into a penalized least-squares problem, yielding a solution that exhibits greater robustness against data inaccuracies and computational errors that may occur during the solving process. The choice of the regularization matrix significantly influences the accuracy of the resulting solution. In this paper, we propose a novel method for selecting the regularization matrix based on exponential filter functions, which have a unique conn

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3

Al-Mahdawi, H. K., Farah Hatem Khorsheed, Ali Subhi Alhumaima, Ali J. Ramadhan, Kilan M. Hussien, and Hussein Alkattan. "Intelligent Particle Swarm Optimization Method for Parameter Selecting in Regularization Method for Integral Equation." BIO Web of Conferences 97 (2024): 00039. http://dx.doi.org/10.1051/bioconf/20249700039.

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We use the Tikhonov method as a regularization technique for solving the integral equation of the first kind with noisy and noise-free data. Following that, we go over how to choose the Tikhonov regularization parameter by implementing the Intelligent Piratical Swarm Optimization (IPOS) technique. The effectiveness of combining these two approaches IPOS and Tikhonov is demonstrated to be highly practicable.

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Hämarik, Uno, Reimo Palm, and Toomas Raus. "EXTRAPOLATION OF TIKHONOV REGULARIZATION METHOD." Mathematical Modelling and Analysis 15, no. 1 (2010): 55–68. http://dx.doi.org/10.3846/1392-6292.2010.15.55-68.

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We consider regularization of linear ill‐posed problem Au = f with noisy data fδ, ¦fδ - f¦≤ δ . The approximate solution is computed as the extrapolated Tikhonov approximation, which is a linear combination of n ≥ 2 Tikhonov approximations with different parameters. If the solution u* belongs to R((A*A) n ), then the maximal guaranteed accuracy of Tikhonov approximation is O(δ 2/3) versus accuracy O(δ 2n/(2n+1)) of corresponding extrapolated approximation. We propose several rules for choice of the regularization parameter, some of these are also good in case of moderate over‐ and underestimat

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Yang, Xiao-Juan, and Li Wang. "A modified Tikhonov regularization method." Journal of Computational and Applied Mathematics 288 (November 2015): 180–92. http://dx.doi.org/10.1016/j.cam.2015.04.011.

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6

Fuhry, Martin, and Lothar Reichel. "A new Tikhonov regularization method." Numerical Algorithms 59, no. 3 (2011): 433–45. http://dx.doi.org/10.1007/s11075-011-9498-x.

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7

Fang, Sheng, Kui Ying, Jianping Cheng, et al. "Parallel magnetic resonance imaging using wavelet-based multivariate regularization." Journal of X-Ray Science and Technology: Clinical Applications of Diagnosis and Therapeutics 18, no. 2 (2010): 145–55. http://dx.doi.org/10.3233/xst-2010-025000250.

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The parallel imaging technique reduces the scan time at the expense of increased noise, due to its ill-conditioned system matrix. Tikhonov regularization has been proposed for SENSE to reduce the noise. However, Tikhonov regularized images suffer from residual aliasing aritfacts or image blurring when a low resolution prior image is used. This study used wavelet-based multivariate regularization to overcome this problem, while maintaining the computational efficiency of Tikhonov regularization. In this method, SENSE is formularized as a multilevel-structured problem in the wavelet domain. Regu

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Xue, Xuemin, and Xiangtuan Xiong. "A Posteriori Fractional Tikhonov Regularization Method for the Problem of Analytic Continuation." Mathematics 9, no. 18 (2021): 2255. http://dx.doi.org/10.3390/math9182255.

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In this paper, the numerical analytic continuation problem is addressed and a fractional Tikhonov regularization method is proposed. The fractional Tikhonov regularization not only overcomes the difficulty of analyzing the ill-posedness of the continuation problem but also obtains a more accurate numerical result for the discontinuity of solution. This article mainly discusses the a posteriori parameter selection rules of the fractional Tikhonov regularization method, and an error estimate is given. Furthermore, numerical results show that the proposed method works effectively.

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Yang, Suhua, Xingjun Luo, Chunmei Zeng, Zhihai Xu, and Wenyu Hu. "On the Parameter Choice in the Multilevel Augmentation Method." Computational Methods in Applied Mathematics 20, no. 3 (2020): 555–71. http://dx.doi.org/10.1515/cmam-2018-0189.

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AbstractIn this paper, we apply the multilevel augmentation method for solving ill-posed Fredholm integral equations of the first kind via iterated Tikhonov regularization method. The method leads to fast solutions of the discrete regularization methods for the equations. The convergence rates of iterated Tikhonov regularization are achieved by using a modified parameter choice strategy. Finally, numerical experiments are given to illustrate the efficiency of the method.

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10

Throne, Robert, and Lorraine Olson. "The Steady Inverse Heat Conduction Problem: A Comparison of Methods With Parameter Selection." Journal of Heat Transfer 123, no. 4 (2001): 633–44. http://dx.doi.org/10.1115/1.1372193.

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In the past we have developed the Generalized Eigensystem GESL techniques for solving inverse boundary value problems in steady heat conduction, and found that these vector expansion methods often give superior results to those obtained with standard Tikhonov regularization methods. However, these earlier comparisons were based on the optimal results for each method, which required that we know the true solution to set the value of the regularization parameter (t) for Tikhonov regularization and the number of mode clusters Nclusters for GESL. In this paper we introduce a sensor sensitivity met

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Zeng, Xiaoniu, Xihai Li, Juan Su, Daizhi Liu, and Hongxing Zou. "An adaptive iterative method for downward continuation of potential-field data from a horizontal plane." GEOPHYSICS 78, no. 4 (2013): J43—J52. http://dx.doi.org/10.1190/geo2012-0404.1.

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We have developed an improved adaptive iterative method based on the nonstationary iterative Tikhonov regularization method for performing a downward continuation of the potential-field data from a horizontal plane. Our method uses the Tikhonov regularization result as initial value and has an incremental geometric choice of the regularization parameter. We compared our method with previous methods (Tikhonov regularization, Landweber iteration, and integral-iteration method). The downward-continuation performance of these methods in spatial and wavenumber domains were compared with the aspects

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12

Eskandari, J. Hasani, and H. Zareamoghaddam. "dam** Different Tikhonov Regularization Operators for Elementary Residual Method." SIJ Transactions on Computer Science Engineering & its Applications (CSEA) 02, no. 01 (2014): 19–24. http://dx.doi.org/10.9756/sijcsea/v2i1/-0201060101.

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13

Jiang, Jinhui, Hongzhi Tang, M. Shadi Mohamed, Shuyi Luo, and Jianding Chen. "Augmented Tikhonov Regularization Method for Dynamic Load Identification." Applied Sciences 10, no. 18 (2020): 6348. http://dx.doi.org/10.3390/app10186348.

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We introduce the augmented Tikhonov regularization method motivated by Bayesian principle to improve the load identification accuracy in seriously ill-posed problems. Firstly, the Green kernel function of a structural dynamic response is established; then, the unknown external loads are identified. In order to reduce the identification error, the augmented Tikhonov regularization method is combined with the Green kernel function. It should be also noted that we propose a novel algorithm to determine the initial values of the regularization parameters. The initial value is selected by finding a

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14

Dong, Wen, and Tao Sun. "Comparison of Tikhonov Regularization and Adaptive Regularization for III-Posed Problems." Applied Mechanics and Materials 380-384 (August 2013): 1193–96. http://dx.doi.org/10.4028/www.scientific.net/amm.380-384.1193.

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nverse problems are important interdisciplinary subject, which receive more and more attention in recent years in the areas of mathematics, computer science, information science and other applied natural sciences. There is close relationship between inverse problems and ill-posedness. Regularization is an important strategy when computing the ill-posed problems to maintain the stability of the computation.This paper compares a new regularization method,which is called Adaptive regularization, with the traditional Tikhonov regularization method. The conclusion that Adaptive regularization metho

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Guo, Hua, Guolin Liu, and Luyao Wang. "An Improved Tikhonov-Regularized Variable Projection Algorithm for Separable Nonlinear Least Squares." Axioms 10, no. 3 (2021): 196. http://dx.doi.org/10.3390/axioms10030196.

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In this work, we investigate the ill-conditioned problem of a separable, nonlinear least squares model by using the variable projection method. Based on the truncated singular value decomposition method and the Tikhonov regularization method, we propose an improved Tikhonov regularization method, which neither discards small singular values, nor treats all singular value corrections. By fitting the Mackey–Glass time series in an exponential model, we compare the three regularization methods, and the numerically simulated results indicate that the improved regularization method is more effectiv

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16

He, Yiran. "The Tikhonov Regularization Method for Set-Valued Variational Inequalities." Abstract and Applied Analysis 2012 (2012): 1–10. http://dx.doi.org/10.1155/2012/172061.

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This paper aims to establish the Tikhonov regularization theory for set-valued variational inequalities. For this purpose, we firstly prove a very general existence result for set-valued variational inequalities, provided that the mapping involved has the so-called variational inequality property and satisfies a rather weak coercivity condition. The result on the Tikhonov regularization improves some known results proved for single-valued mapping.

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Jahanjooy, Saber, Mohammad Ali Riahi, and Hamed Ghanbarnejad Moghanloo. "Blind inversion of multidimensional seismic data using sequential Tikhonov and total variation regularizations." GEOPHYSICS 87, no. 1 (2021): R53—R61. http://dx.doi.org/10.1190/geo2020-0692.1.

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The acoustic impedance (AI) model is key data for seismic interpretation, usually obtained from its nonlinear relation with seismic reflectivity. Common approaches use initial geologic and seismic information to constrain the AI model estimation. When no accurate prior information is available, these approaches may dictate false results at some parts of the model. The regularization of ill-posed underdetermined problems requires some constraints to restrict the possible results. Available seismic inversion methods mostly use Tikhonov or total variation (TV) regularizations with some adjustment

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Du, Wei, and Yangyang Zhang. "The Calculation of High-Order Vertical Derivative in Gravity Field by Tikhonov Regularization Iterative Method." Mathematical Problems in Engineering 2021 (May 8, 2021): 1–13. http://dx.doi.org/10.1155/2021/8818552.

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In mathematics, statistics, and computer science, particularly in the fields of machine learning and inverse problems, regularization is a process of introducing additional information in order to solve an ill-posed problem or to prevent overfitting. The Tikhonov regularization method is widely used to solve complex problems in engineering. The vertical derivative of gravity can highlight the local anomalies and separate the horizontal superimposed abnormal bodies. The higher the order of the vertical derivative is, the stronger the resolution is. However, it is generally considered that the c

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Azikri de Deus, Hilbeth P., Claudio R. Ávila S. Jr., Ivan Moura Belo, and André T. Beck. "The Tikhonov regularization method in elastoplasticity." Applied Mathematical Modelling 36, no. 10 (2012): 4687–707. http://dx.doi.org/10.1016/j.apm.2011.11.086.

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QIAN, AI-LIN, and JIAN-FENG MAO. "OPTIMAL ERROR BOUND AND A GENERALIZED TIKHONOV REGULARIZATION METHOD FOR IDENTIFYING AN UNKNOWN SOURCE IN THE POISSON EQUATION." International Journal of Wavelets, Multiresolution and Information Processing 12, no. 01 (2013): 1450004. http://dx.doi.org/10.1142/s0219691314500040.

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In this note we prove a stability estimate for an inverse heat source problem. Based on the obtained stability estimate, we present a generalized Tikhonov regularization method and obtain the error estimate. Numerical experiment shows that the generalized Tikhonov regularization works well.

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Faghidian, S. Ali. "A regularized approach to linear regression of fatigue life measurements." International Journal of Structural Integrity 7, no. 1 (2016): 95–105. http://dx.doi.org/10.1108/ijsi-12-2014-0071.

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Purpose – The linear regression technique is widely used to determine empirical parameters of fatigue life profile while the results may not continuously depend on experimental data. Thus Tikhonov-Morozov method is utilized here to regularize the linear regression results and consequently reduces the influence of measurement noise without notably distorting the fatigue life distribution. The paper aims to discuss these issues. Design/methodology/approach – Tikhonov-Morozov regularization method would be shown to effectively reduce the influences of measurement noise without distorting the fati

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Liu, Kui, Jieqing Tan, and Benyue Su. "An Adaptive Image Denoising Model Based on Tikhonov and TV Regularizations." Advances in Multimedia 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/934834.

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To avoid the staircase artifacts, an adaptive image denoising model is proposed by the weighted combination of Tikhonov regularization and total variation regularization. In our model, Tikhonov regularization and total variation regularization can be adaptively selected based on the gradient information of the image. When the pixels belong to the smooth regions, Tikhonov regularization is adopted, which can eliminate the staircase artifacts. When the pixels locate at the edges, total variation regularization is selected, which can preserve the edges. We employ the split Bregman method to solve

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Xu, Xiaowei, and Ting Bu. "An Adaptive Parameter Choosing Approach for Regularization Model." International Journal of Pattern Recognition and Artificial Intelligence 32, no. 08 (2018): 1859013. http://dx.doi.org/10.1142/s0218001418590139.

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The choice of regularization parameters is a troublesome issue for most regularization methods, e.g. Tikhonov regularization method, total variation (TV) method, etc. An appropriate parameter for a certain regularization approach can obtain fascinating results. However, general methods of choosing parameters, e.g. Generalized Cross Validation (GCV), cannot get more precise results in practical applications. In this paper, we consider exploiting the more appropriate regularization parameter within a possible range, and apply the estimated parameter to Tikhonov model. In the meanwhile, we obtain

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Chen, Tianyi, Jürgen Kusche, Yunzhong Shen, and Qiujie Chen. "A Combined Use of TSVD and Tikhonov Regularization for Mass Flux Solution in Tibetan Plateau." Remote Sensing 12, no. 12 (2020): 2045. http://dx.doi.org/10.3390/rs12122045.

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Limited by the Gravity Recovery and Climate Experiment (GRACE) and GRACE Follow-On (GRACE-FO) measurement principle and sensors, the spatial resolution of mass flux solutions is about 2–3° in mid-latitudes at monthly intervals. To retrieve a mass flux solution in the Tibetan Plateau (TP) with better visual spatial resolution, we combined truncated singular value decomposition (TSVD) and Tikhonov regularization to solve for a mascon modeling. The monthly mass flux parameters resolved at 1° are smoothed to about 2° by truncating the eigen-spectrum of the normal equation (i.e., using the TSVD app

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Wang, Lin Jun, You Xiang Xie, and Hai Hua Wu. "A New Computational Inverse Method and Application to the Identification of Dynamic Loads." Advanced Materials Research 631-632 (January 2013): 1298–302. http://dx.doi.org/10.4028/www.scientific.net/amr.631-632.1298.

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In this paper, we propose a new computational inverse method for solving the identification of multi-source dynamic loads acting on a simply supported plate. Using a priori choosing appropriate regularization parameter, the present method can obtain higher optimum asymptotic order of the regularized solution than ordinary Tikhonov regularization method. In the numerical simulations, the identification problem of multi-source dynamic loads on a surface of simply supported plate is successfully solved by the present method. Meanwhile, most of its performances are better than ordinary Tikhonov re

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Sumin, Mikhail I. "On the role of Lagrange multipliers and duality in ill-posed problems for constrained extremum. To the 60th anniversary of the Tikhonov regularization method." Russian Universities Reports. Mathematics, no. 144 (2023): 414–35. http://dx.doi.org/10.20310/2686-9667-2023-28-144-414-435.

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The important role of Lagrange multipliers and duality in the theory of ill-posed problems for a constrained extremum is discussed. The central attention is paid to the problem of stable approximate finding of a normal (minimum in norm) solution of the operator equation of the first kind Az=u, z∈D⊆Z, where A:Z→U is a linear bounded operator, u∈U is a given element, D⊆Z is a convex closed set, Z,U are Hilbert spaces. As is known, this problem is classical for the theory of ill-posed problems. We consider two problems equivalent to it (from the point of view of the simultaneous existence of thei

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Gong, Rongfang, Xiaoliang Cheng, and Weimin Han. "A New Coupled Complex Boundary Method for Bioluminescence Tomography." Communications in Computational Physics 19, no. 1 (2016): 226–50. http://dx.doi.org/10.4208/cicp.230115.150615a.

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AbstractIn this paper, we introduce and study a new method for solving inverse source problems, through a working model that arises in bioluminescence tomography (BLT). In the BLT problem, one constructs quantitatively the bioluminescence source distribution inside a small animal from optical signals detected on the animal's body surface. The BLT problem possesses strong ill-posedness and often the Tikhonov regularization is used to obtain stable approximate solutions. In conventional Tikhonov regularization, it is crucial to choose a proper regularization parameter for trade off between the a

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Luo, Xingjun, Zhaofu Ouyang, Chunmei Zeng, and Fanchun Li. "Multiscale Galerkin methods for the nonstationary iterated Tikhonov method with a modified posteriori parameter selection." Journal of Inverse and Ill-posed Problems 26, no. 1 (2018): 109–20. http://dx.doi.org/10.1515/jiip-2017-0009.

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AbstractIn this paper, we consider a fast multiscale Galerkin method with compression technique for solving Fredholm integral equations of the first kind via the nonstationary iterated Tikhonov regularization. A modified a posteriori regularization parameter choice strategy is established, which leads to optimal convergence rates.

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Sofieva, V. F., J. Tamminen, H. Haario, E. Kyrölä, and M. Lehtinen. "Ozone profile smoothness as a priori information in the inversion of limb measurements." Annales Geophysicae 22, no. 10 (2004): 3411–20. http://dx.doi.org/10.5194/angeo-22-3411-2004.

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Abstract. In this work we discuss inclusion of a priori information about the smoothness of atmospheric profiles in inversion algorithms. The smoothness requirement can be formulated in the form of Tikhonov-type regularization, where the smoothness of atmospheric profiles is considered as a constraint or in the form of Bayesian optimal estimation (maximum a posteriori method, MAP), where the smoothness of profiles can be included as a priori information. We develop further two recently proposed retrieval methods. One of them - Tikhonov-type regularization according to the target resolution - d

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Novati, Paolo, and Maria Rosaria Russo. "A GCV based Arnoldi-Tikhonov regularization method." BIT Numerical Mathematics 54, no. 2 (2013): 501–21. http://dx.doi.org/10.1007/s10543-013-0447-z.

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Mekoth, Chitra, Santhosh George, and P. Jidesh. "Fractional Tikhonov regularization method in Hilbert scales." Applied Mathematics and Computation 392 (March 2021): 125701. http://dx.doi.org/10.1016/j.amc.2020.125701.

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Li, Yan Qiu, and Shi Liu. "Image Reconstruction Algorithm for Ultrasound Tomography Based on Transmission Mode." Key Engineering Materials 474-476 (April 2011): 754–59. http://dx.doi.org/10.4028/www.scientific.net/kem.474-476.754.

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One of the key issues in ultrasonic tomography is the calculation of sensitivity and image reconstruction based on the sensitivity. A method is developed to calculate the sensitivity of ultrasonic tomography in rectangular area in this paper. On this basis, numerical modelling calculation of different substances distribution in a rectangular area is made of by using LBP algorithm, the standard Tikhonov regularization method and the Landweber iteration method respectively. Reconstruction simulation experiments of changing the quantity of pixel is conducted. The results show that imaging results

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Groetsch, C. W. "Uniform convergence of regularization methods for Fredholm equations of the first kind." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 39, no. 2 (1985): 282–86. http://dx.doi.org/10.1017/s1446788700022539.

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AbstractFor Fredholm equations of the first kind with continuous kernels we investigate the uniform convergence of a general class of regularization methods. Applications are made to Tikhonov regularization and Landweber's iteration method.

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LU, S., S. V. PEREVERZEV, and U. TAUTENHAHN. "Dual Regularized Total Least Squares And Multi-Parameter Regularization." Computational Methods in Applied Mathematics 8, no. 3 (2008): 253–62. http://dx.doi.org/10.2478/cmam-2008-0018.

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AbstractIn this paper we continue our study of solving ill-posed problems with a noisy right-hand side and a noisy operator. Regularized approximations are obtained by Tikhonov regularization with differential operators and by dual regularized total least squares (dual RTLS) which can be characterized as a special multi-parameter regularization method where one of the two regularization parameters is negative. We report on order optimality results for both regularized approximations, discuss compu-tational aspects, provide special algorithms and show by experiments that dual RTLS is competitiv

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Liu, Tang Wei, He Hua Xu, Xue Lin Qiu, and Xiao Bin Shi. "Multiscale Parameter Identification Method for Three Dimension Steady Heat Transfer Model of Composite Materials." Advanced Materials Research 706-708 (June 2013): 152–57. http://dx.doi.org/10.4028/www.scientific.net/amr.706-708.152.

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In this paper, for heat conductivity identification of three dimension steady heat transfer model of composite materials, a new hybrid Tikhonov regularization mixed multiscale finite-element method is present. First the mathematical models of the forward and the coefficient inverse problems are discussed. Then the forward model is solved by mixed multiscale FEM which utilizes the effects of fine-scale heterogeneities through basis functions formulation computed from local heat transfer problems. At last the numerical approximation of inverse coefficient problem is obtained by Tikhonov regulari

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Likassa, Habte Tadesse, Wen Xian, and Xuan Tang. "New Robust Regularized Shrinkage Regression for High-Dimensional Image Recovery and Alignment via Affine Transformation and Tikhonov Regularization." International Journal of Mathematics and Mathematical Sciences 2020 (November 6, 2020): 1–10. http://dx.doi.org/10.1155/2020/1286909.

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In this work, a new robust regularized shrinkage regression method is proposed to recover and align high-dimensional images via affine transformation and Tikhonov regularization. To be more resilient with occlusions and illuminations, outliers, and heavy sparse noises, the new proposed approach incorporates novel ideas affine transformations and Tikhonov regularization into high-dimensional images. The highly corrupted, distorted, or misaligned images can be adjusted through the use of affine transformations and Tikhonov regularization term to ensure a trustful image decomposition. These novel

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Baba, K., and M. Ochi. "Monitoring of Transient Temperature Distribution in Piping." Journal of Pressure Vessel Technology 116, no. 4 (1994): 419–22. http://dx.doi.org/10.1115/1.2929610.

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A numerical method for monitoring temperature distribution in which boundary flux and initial state are unknown is presented. Regularizations based on Tikhonov and Beck’s method are employed. Then, regularization parameters are evaluated by the L-curve. The method is applied to an actual piping problem in a steam power plant and compared with measured data, and it is also applied to a two-dimensional thermal shock problem.

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Reginska, Teresa. "DISCREPANCY SETS FOR COMBINED LEAST SQUARES PROJECTION AND TIKHONOV REGULARIZATION." Mathematical Modelling and Analysis 22, no. 2 (2017): 202–12. http://dx.doi.org/10.3846/13926292.2017.1289987.

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To solve a linear ill-posed problem, a combination of the finite dimensional least squares projection method and the Tikhonov regularization is considered. The dimension of the projection is treated as the second parameter of regularization. A two-parameter discrepancy principle defines a discrepancy set for any data error bound. The aim of the paper is to describe this set and to indicate its subset such that for regularization parameters from this subset the related regularized solution has the same order of accuracy as the Tikhonov regularization with the standard discrepancy principle but

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QIAN, AILIN, and YUJIANG WU. "OPTIMAL ERROR BOUND AND APPROXIMATION METHODS FOR A CAUCHY PROBLEM OF THE MODIFIED HELMHOLTZ EQUATION." International Journal of Wavelets, Multiresolution and Information Processing 09, no. 02 (2011): 305–15. http://dx.doi.org/10.1142/s0219691311004080.

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We consider a Cauchy problem for a modified Helmholtz equation, especially when we give the optimal error bound for this problem. Some spectral regularization methods and a revised Tikhonov regularization method are used to stabilize the problem from the viewpoint of general regularization theory. Hölder-type stability error estimates are provided for these regularization methods. According to the optimality theory of regularization, the error estimates are order optimal.

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Ma, Chao Zhong, Yong Wei Gu, Ji Fu, Yuan Lu Du, and Qing Ming Gui. "The Regularization Method Based on Complex Collinearity Diagnostics and Metrics." Applied Mechanics and Materials 416-417 (September 2013): 1393–98. http://dx.doi.org/10.4028/www.scientific.net/amm.416-417.1393.

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In a large number of measurement data processing, the ill-posed problem is widespread. For such problems, this paper introduces the solution of ill-posed problem of the unity of expression and Tikhonov regularization method, and then to re-collinearity diagnostics and metrics based on proposed based on complex collinearity diagnostics and the metric regularization method is given regularization matrix selection methods and regularization parameter determination formulas. Finally, it uses a simulation example to verify the effectiveness of the method.

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Li, Shouxiao, Huaxiang Wang, Tonghai Liu, et al. "A fast Tikhonov regularization method based on homotopic mapping for electrical resistance tomography." Review of Scientific Instruments 93, no. 4 (2022): 043709. http://dx.doi.org/10.1063/5.0077483.

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Electrical resistance tomography (ERT) is considered a novel sensing technique for monitoring conductivity distribution. Image reconstruction of ERT is an ill-posed inverse problem. In this paper, an improved regularization reconstruction method is presented to solve this issue. We adopted homotopic mapping to choose the regularization parameter of the iterative Tikhonov algorithm. The standard normal distribution function was used to continuously adjust the regularization parameter. Subsequently, the resultant image vector was deployed as the initial value of the iterative Tikhonov algorithm

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Landi, Germana, Fabiana Zama, and Villiam Bortolotti. "A New Hybrid Inversion Method for 2D Nuclear Magnetic Resonance Combining TSVD and Tikhonov Regularization." Journal of Imaging 7, no. 2 (2021): 18. http://dx.doi.org/10.3390/jimaging7020018.

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This paper is concerned with the reconstruction of relaxation time distributions in Nuclear Magnetic Resonance (NMR) relaxometry. This is a large-scale and ill-posed inverse problem with many potential applications in biology, medicine, chemistry, and other disciplines. However, the large amount of data and the consequently long inversion times, together with the high sensitivity of the solution to the value of the regularization parameter, still represent a major issue in the applicability of the NMR relaxometry. We present a method for two-dimensional data inversion (2DNMR) which combines Tr

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Corral, Luis, and Pablo E. Román. "Computation analysis of regularization methods and parameter selection for acoustics radiation modes source reconstruction of vibrating plates." INTER-NOISE and NOISE-CON Congress and Conference Proceedings 263, no. 2 (2021): 4295–302. http://dx.doi.org/10.3397/in-2021-2655.

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Source localization and power estimation is a topic of great interest in acoustics and vibration. Acoustic source radiation modes reconstruction is a method of particular interest. Solutions leads to determinate sound/vibration power and surface velocity distribution from sparse acoustics samples. It has been shown that the quality of the results depends on Tikhonov regularization parameter. This inverse method is based on the radiation resistance matrix and we show that a single instruction multiple threads computing approach for graphics processing unit device exhibit better speed performanc

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He, Wei, Bing Li, Zheng Xu, Haijun Luo, and Peng Ran. "A COMBINED REGULARIZATION ALGORITHM FOR ELECTRICAL IMPEDANCE TOMOGRAPHY SYSTEM USING RECTANGULAR ELECTRODES ARRAY." Biomedical Engineering: Applications, Basis and Communications 24, no. 04 (2012): 313–22. http://dx.doi.org/10.4015/s1016237212500263.

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A novel Electrical Impedance Tomography system with rectangular electrodes array and back electrode is proposed. This system could reconstruct a deeper target and is easy to operate. By studying different reconstructed algorithms: Tikhonov regularization and the Newton's One-step Error Reconstructor (NOSER), a combined regularization algorithm is proposed. The L-curve and posteriori method are used to choose Tikhonov and NOSER regularization parameter. Two evaluation parameters of reconstructed algorithm: normalization mean square distance criterion (NMSD), normalized mean absolute distance cr

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Yu, Kai, Benxue Gong, and Zhenyu Zhao. "A modified Tikhonov regularization for unknown source in space fractional diffusion equation." Open Mathematics 20, no. 1 (2022): 1309–19. http://dx.doi.org/10.1515/math-2022-0513.

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Abstract In this article, we consider the identification of an unknown steady source in a class of fractional diffusion equations. A modified Tikhonov regularization method based on Hermite expansion is presented to deal with the ill-posedness of the problem. By using the properties of Hermitian functions, we construct a modified penalty term for the Tikhonov functional. It can be proved that the method can adaptively achieve the order optimal results when we choose the regularization parameter by the discrepancy principle. Some examples are also provided to verify the effectiveness of the met

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Tao Yong, 陶勇, 王肖霞 Wang Xiaoxia, 闫国庆 Yan Guoqing, and 杨风暴 Yang Fengbao. "Computational Ghost Imaging Method Based on Tikhonov Regularization." Laser & Optoelectronics Progress 57, no. 2 (2020): 021016. http://dx.doi.org/10.3788/lop57.021016.

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Cornelis, J., N. Schenkels, and W. Vanroose. "Projected Newton method for noise constrained Tikhonov regularization." Inverse Problems 36, no. 5 (2020): 055002. http://dx.doi.org/10.1088/1361-6420/ab7d2b.

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Joachimiak, Magda, Michał Ciałkowski, and Andrzej Frąckowiak. "Stable method for solving the Cauchy problem with the use of Chebyshev polynomials." International Journal of Numerical Methods for Heat & Fluid Flow 30, no. 3 (2019): 1441–56. http://dx.doi.org/10.1108/hff-05-2019-0416.

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Purpose The purpose of this paper is to present the method for solving the inverse Cauchy-type problem for the Laplace’s equation. Calculations were made for the rectangular domain with the target temperature on three boundaries and, additionally, on one of the boundaries, the heat flux distribution was selected. The purpose of consideration was to determine the distribution of temperature on a section of the boundary of the investigated domain (boundary Γ1) and find proper method for the problem regularization. Design/methodology/approach The solution of the direct and the inverse (Cauchy-typ

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Bourgeois, Laurent, and Arnaud Recoquillay. "A mixed formulation of the Tikhonov regularization and its application to inverse PDE problems." ESAIM: Mathematical Modelling and Numerical Analysis 52, no. 1 (2018): 123–45. http://dx.doi.org/10.1051/m2an/2018008.

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This paper is dedicated to a new way of presenting the Tikhonov regularization in the form of a mixed formulation. Such formulation is well adapted to the regularization of linear ill-posed partial differential equations because when it comes to discretization, the mixed formulation enables us to use some standard finite elements. As an application of our theory, we consider an inverse obstacle problem in an acoustic waveguide. In order to solve it we use the so-called “exterior approach”, which couples the mixed formulation of Tikhonov regularization and a level set method. Some 2d numerical

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Cheng, Jiaju, and Jianwen Luo. "Tikhonov-regularization-based projecting sparsity pursuit method for fluorescence molecular tomography reconstruction." Chinese Optics Letters 18, no. 1 (2020): 011701. http://dx.doi.org/10.3788/col202018.011701.

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