Journal articles: 'Inverse Tikhonov Regularization (ITR)'

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Francisco, Casesnoves. "MATHEMATICAL 3D OPTIMIZATION GNU-OCTAVE WITH REVIEW OF 2D/3D CRITICAL TEMPERATURE OPTIMIZATION MOLECULAR EFFECT MODEL IN HIGH TEMPERATURE SUPERCONDUCTORS THALLIUM CLASS [TC > 0°] AND SUPERCONDUCTING MULTIFUNCTIONAL TRANSMISSION LINE INVENT." INTERNATIONAL JOURNAL OF MATHEMATICS AND COMPUTER RESEARCH 10, no. 06 (2022): 2750–59. https://doi.org/10.5281/zenodo.6719543.

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GNU-Octave   System   software   was   designed   to   obtain   3D/2DGraphical   Optimization   for Molecular   EffectModel   (MEM)in   [   Tl-Sn-Pb-Ba-Si-Mn-Mg-Cu-O   ]High   Temperature Superconductors  (HTSCs)  class [exclusively  forTC >  0°  ]. The  study  results  show  two  parts, namely, GNU-Octave 3D Graphical Optimization and 2D/3D MEM optimization review. Series of 2D   MEM   GNU-Octave   graphs   with   increasing   ILS   polynomial   degree   are   developed   to demonstrate the equaling of the MEM for this HTSCs class.Comparisons to Matlab solutionsfromprevious  contributions  were  also  presented.  Results comprise Inverse Tikhonov  Regularization algorithms and mathematical methods for this HTSCs class. Solutions prove Numerical/Graphical coincidence  between  MEM  and  experimental  data  is  proven.  ILS  norm-residuals for  MEM show be acceptable and  low. Electronics  Physics  applications  for  Superconductors  and  HTSCs and optimal TCselection/constraints  are explained.Invention  of  Superconducting  Multifunctional Transmission Line is proposed [Casesnoves, 2021]

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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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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 method for estimating appropriate values of Nclusters for GESL. We compare those results with Tikhonov regularization using the Combined Residual and Smoothing Operator (CRESO) to estimate the appropriate values of t. We find that both methods are quite effective at estimating the appropriate parameters, and that GESL often gives superior results to Tikhonov regularization even when Nclusters is estimated from measured data.

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DE VITO, ERNESTO, LORENZO ROSASCO, and ANDREA CAPONNETTO. "DISCRETIZATION ERROR ANALYSIS FOR TIKHONOV REGULARIZATION." Analysis and Applications 04, no. 01 (2006): 81–99. http://dx.doi.org/10.1142/s0219530506000711.

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We study the discretization of inverse problems defined by a Carleman operator. In particular, we develop a discretization strategy for this class of inverse problems and we give a convergence analysis. Learning from examples, as well as the discretization of integral equations, can be analyzed in our setting.

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Nguyen, Hai, Jonathan Wittmer, and Tan Bui-Thanh. "DIAS: A Data-Informed Active Subspace Regularization Framework for Inverse Problems." Computation 10, no. 3 (2022): 38. http://dx.doi.org/10.3390/computation10030038.

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This paper presents a regularization framework that aims to improve the fidelity of Tikhonov inverse solutions. At the heart of the framework is the data-informed regularization idea that only data-uninformed parameters need to be regularized, while the data-informed parameters, on which data and forward model are integrated, should remain untouched. We propose to employ the active subspace method to determine the data-informativeness of a parameter. The resulting framework is thus called a data-informed (DI) active subspace (DIAS) regularization. Four proposed DIAS variants are rigorously analyzed, shown to be robust with the regularization parameter and capable of avoiding polluting solution features informed by the data. They are thus well suited for problems with small or reasonably small noise corruptions in the data. Furthermore, the DIAS approaches can effectively reuse any Tikhonov regularization codes/libraries. Though they are readily applicable for nonlinear inverse problems, we focus on linear problems in this paper in order to gain insights into the framework. Various numerical results for linear inverse problems are presented to verify theoretical findings and to demonstrate advantages of the DIAS framework over the Tikhonov, truncated SVD, and the TSVD-based DI approaches.

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He, Ya Li, Shi Qiu Zheng, and Yan Mei Yang. "Tikhonov Regularization Solve Partial Differential Equations Inverse Problem." Applied Mechanics and Materials 50-51 (February 2011): 459–62. http://dx.doi.org/10.4028/www.scientific.net/amm.50-51.459.

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This article discussed the partial differential equation inverse problem. Because the partial differential equation inverse is the misalignment improperly posed problem, therefore analyzed has had the improperly posed reason. To process the partial differential equation inverse correctly not well-posed ness this difficulty, obtains relies on continuously the data stable approximate solution, has drawn support from the regularization related concept and the regularization general theory.

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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 method is a stronger regularization method than the traditional Tikhonov regularization method can be made by computing some numerical examples.

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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 criterion (NMAD) are used to evaluate the result's precision of inverse problem quantificationally. The comparison among Tikhonov regularization, NOSER and the combined regularization shows that the ill-condition and the error of inverse problem are reduced. This new algorithm can decrease condition number by 70%, NMSD by 51%, and NMAD by 41% at least. Simulate results show that the combined regularization algorithm could reconstructed the target image in the depth from 10–40 mm. The experimental results show that a 15 mm × 9 mm × 9 mm cuboids whose depth is 35 mm could be distinguished. The performance of this system and the combined regularization algorithm demonstrate significantly better spatial resolution and minor reconstructed error.

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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 regularization method.

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Faiyaz, Chowdhury Abrar, Pabel Shahrear, Rakibul Alam Shamim, Thilo Strauss, and Taufiquar Khan. "Comparison of Different Radial Basis Function Networks for the Electrical Impedance Tomography (EIT) Inverse Problem." Algorithms 16, no. 10 (2023): 461. http://dx.doi.org/10.3390/a16100461.

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This paper aims to determine whether regularization improves image reconstruction in electrical impedance tomography (EIT) using a radial basis network. The primary purpose is to investigate the effect of regularization to estimate the network parameters of the radial basis function network to solve the inverse problem in EIT. Our approach to studying the efficacy of the radial basis network with regularization is to compare the performance among several different regularizations, mainly Tikhonov, Lasso, and Elastic Net regularization. We vary the network parameters, including the fixed and variable widths for the Gaussian used for the network. We also perform a robustness study for comparison of the different regularizations used. Our results include (1) determining the optimal number of radial basis functions in the network to avoid overfitting; (2) comparison of fixed versus variable Gaussian width with or without regularization; (3) comparison of image reconstruction with or without regularization, in particular, no regularization, Tikhonov, Lasso, and Elastic Net; (4) comparison of both mean square and mean absolute error and the corresponding variance; and (5) comparison of robustness, in particular, the performance of the different methods concerning noise level. We conclude that by looking at the R2 score, one can determine the optimal number of radial basis functions. The fixed-width radial basis function network with regularization results in improved performance. The fixed-width Gaussian with Tikhonov regularization performs very well. The regularization helps reconstruct the images outside of the training data set. The regularization may cause the quality of the reconstruction to deteriorate; however, the stability is much improved. In terms of robustness, the RBF with Lasso and Elastic Net seem very robust compared to Tikhonov.

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Luo, Xue-ping. "Tikhonov regularization methods for inverse variational inequalities." Optimization Letters 8, no. 3 (2013): 877–87. http://dx.doi.org/10.1007/s11590-013-0643-4.

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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 regularization method.

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Peng, Ya Mian, Kai Li Wang, and Huan Cheng Zhang. "Improved Tikhonov Regularization Research and Applied in Inverse Problem." Applied Mechanics and Materials 50-51 (February 2011): 447–50. http://dx.doi.org/10.4028/www.scientific.net/amm.50-51.447.

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The regularization which constructs with the first filter function is precisely the Tikhonov regularization. This article has proven the Tikhonov functional minimization problem is decides suitably, namely satisfies the solution the existence, the solution unique reconciliation to rely on continuously the data stability; and this minimization problem in solves the first class equation equally the normal equation. The numerical simulation experiment's result indicated that distinguishes the inverse with the regular reduction solution parameter to have the numerical precision to be high and the stability is good and convergence rate quick characteristic.

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Hinestroza, Doris, and Luisa Fernanda Vargas. "A Generalized Tikhonov Regularization Using Two Parameters Applied to Linear Inverse Ill-Posed Problems." Revista de Ciencias 11 (November 14, 2011): 16–25. http://dx.doi.org/10.25100/rc.v11i0.527.

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De Cezaro, A. "On a Level-Set Method for Ill-Posed Problems with Piecewise Nonconstant Coefficients." Journal of Applied Mathematics 2013 (2013): 1–15. http://dx.doi.org/10.1155/2013/123643.

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We investigate a level-set-type method for solving ill-posed problems, with the assumption that the solutions are piecewise, but not necessarily constant functions with unknown level sets and unknown level values. In order to get stable approximate solutions of the inverse problem, we propose a Tikhonov-type regularization approach coupled with a level-set framework. We prove the existence of generalized minimizers for the Tikhonov functional. Moreover, we prove convergence and stability for regularized solutions with respect to the noise level, characterizing the level-set approach as a regularization method for inverse problems. We also show the applicability of the proposed level-set method in some interesting inverse problems arising in elliptic PDE models.

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Fang, Ximing. "A hybrid regularization model for linear inverse problems." Filomat 36, no. 8 (2022): 2739–48. http://dx.doi.org/10.2298/fil2208739f.

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For the ill-posed linear inverse problem, we propose a hybrid regularization model, which possesses the characters of Tikhonov regularization and TV regularization to some extent. Through transformation, the hybrid regularization is reformulated as an equivalent minimization problem. To solve the minimization problem, we present two modified iterative shrinkage-thresholding algorithms (MISTA) based on the fast iterative shrinkage-thresholding algorithm (FISTA) and the iterative shrinkagethresholding algorithm (ISTA). The numerical experiments are performed to show the effectiveness and superiority of the regularization model and the presented algorithms.

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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 experiments show the feasibility of our approach.

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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 available, the standard Tikhonov method is inferior to the other methods, which, in contrast to Tikhonov's approach, match the spectral errors and signals in an optimum way. We show that the corresponding Tikhonov matrix for optimum regularization can only be determined approximately. Moreover, as Tikhonov's method solves an integral equation, it is less computationally efficient than the other methods, which use forward integration. Also harmonic analysis uses direct integration and is not hampered, as previous methods, with spectral leakage. Spectral combination, in addition to filtering, has the advantage of combining different data sets by least squares spectral weighting.

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Weissmann, Simon, Neil K. Chada, Claudia Schillings, and Xin T. Tong. "Adaptive Tikhonov strategies for stochastic ensemble Kalman inversion." Inverse Problems 38, no. 4 (2022): 045009. http://dx.doi.org/10.1088/1361-6420/ac5729.

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Abstract Ensemble Kalman inversion (EKI) is a derivative-free optimizer aimed at solving inverse problems, taking motivation from the celebrated ensemble Kalman filter. The purpose of this article is to consider the introduction of adaptive Tikhonov strategies for EKI. This work builds upon Tikhonov EKI (TEKI) which was proposed for a fixed regularization constant. By adaptively learning the regularization parameter, this procedure is known to improve the recovery of the underlying unknown. For the analysis, we consider a continuous-time setting where we extend known results such as well-posedness and convergence of various loss functions, but with the addition of noisy observations for the limiting stochastic differential equations (i.e. stochastic TEKI). Furthermore, we allow a time-varying noise and regularization covariance in our presented convergence result which mimic adaptive regularization schemes. In turn we present three adaptive regularization schemes, which are highlighted from both the deterministic and Bayesian approaches for inverse problems, which include bilevel optimization, the maximum a posteriori formulation and covariance learning. We numerically test these schemes and the theory on linear and nonlinear partial differential equations, where they outperform the non-adaptive TEKI and EKI.

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Slagel, J. Tanner, Julianne Chung, Matthias Chung, David Kozak, and Luis Tenorio. "Sampled Tikhonov regularization for large linear inverse problems." Inverse Problems 35, no. 11 (2019): 114008. http://dx.doi.org/10.1088/1361-6420/ab2787.

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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 conversion of the high-order vertical derivative is unstable. In this paper, based on Tikhonov regularization for solving the high-order vertical derivatives of gravity field and combining with the iterative method for successive approximation, the Tikhonov regularization method for calculating the vertical high-order derivative in gravity field is proposed. The recurrence formula of Tikhonov regularization iterative method is obtained. Through the analysis of the filtering characteristics of this method, the high-order vertical derivative of gravity field calculated by this method is stable. Model tests and practical data processing also show that the method is of important theoretical significance and practical value.

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Liu, Songshu. "Recovering a Space-Dependent Source Term in the Fractional Diffusion Equation with the Riemann–Liouville Derivative." Mathematics 10, no. 17 (2022): 3213. http://dx.doi.org/10.3390/math10173213.

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This research determines an unknown source term in the fractional diffusion equation with the Riemann–Liouville derivative. This problem is ill-posed. Conditional stability for the inverse source problem can be given. Further, a fractional Tikhonov regularization method was applied to regularize the inverse source problem. In the theoretical results, we propose a priori and a posteriori regularization parameter choice rules and obtain the convergence estimates.

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Florens, Jean-Pierre, and Anna Simoni. "REGULARIZING PRIORS FOR LINEAR INVERSE PROBLEMS." Econometric Theory 32, no. 1 (2014): 71–121. http://dx.doi.org/10.1017/s0266466614000796.

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This paper proposes a new Bayesian approach for estimating, nonparametrically, functional parameters in econometric models that are characterized as the solution of a linear inverse problem. By using a Gaussian process prior we propose the posterior mean as an estimator and prove frequentist consistency of the posterior distribution. The latter provides the frequentist validation of our Bayesian procedure. We show that the minimax rate of contraction of the posterior distribution can be obtained provided that either the regularity of the prior matches the regularity of the true parameter or the prior is scaled at an appropriate rate. The scaling parameter of the prior distribution plays the role of a regularization parameter. We propose a new data-driven method for optimally selecting in practice this regularization parameter. We also provide sufficient conditions such that the posterior mean, in a conjugate-Gaussian setting, is equal to a Tikhonov-type estimator in a frequentist setting. Under these conditions our data-driven method is valid for selecting the regularization parameter of the Tikhonov estimator as well. Finally, we apply our general methodology to two leading examples in econometrics: instrumental regression and functional regression estimation.

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Huang, Rong. "The inverse problem of bimorph mirror tuning on a beamline." Journal of Synchrotron Radiation 18, no. 6 (2011): 930–37. http://dx.doi.org/10.1107/s0909049511036648.

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One of the challenges of tuning bimorph mirrors with many electrodes is that the calculated focusing voltages can be different by more than the safety limit (such as 500 V for the mirrors used at 17-ID at the Advanced Photon Source) between adjacent electrodes. A study of this problem at 17-ID revealed that the inverse problem of the tuningin situ, using X-rays, became ill-conditioned when the number of electrodes was large and the calculated focusing voltages were contaminated with measurement errors. Increasing the number of beamlets during the tuning could reduce the matrix condition number in the problem, but obtaining voltages with variation below the safety limit was still not always guaranteed and multiple iterations of tuning were often required. Applying Tikhonov regularization and using the L-curve criterion for the determination of the regularization parameter made it straightforward to obtain focusing voltages with well behaved variations. Some characteristics of the tuning results obtained using Tikhonov regularization are given in this paper.

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Yan, Xiong Bin, and Ting Wei. "Inverse space-dependent source problem for a time-fractional diffusion equation by an adjoint problem approach." Journal of Inverse and Ill-posed Problems 27, no. 1 (2019): 1–16. http://dx.doi.org/10.1515/jiip-2017-0091.

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AbstractIn this paper, we consider an inverse space-dependent source problem for a time-fractional diffusion equation by an adjoint problem approach; that is, to determine the space-dependent source term from a noisy final data. Based on the series expression of the solution for the direct problem, we improve the regularity of the weak solution for the direct problem under strong conditions, and we provide the existence and uniqueness for the adjoint problem. Further, we use the Tikhonov regularization method to solve the inverse source problem and provide a conjugate gradient algorithm to find an approximation to the minimizer of the Tikhonov regularization functional. Numerical examples in one-dimensional and two-dimensional cases are provided to show the effectiveness of the proposed method.

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Antholzer, Stephan, and Markus Haltmeier. "Discretization of Learned NETT Regularization for Solving Inverse Problems." Journal of Imaging 7, no. 11 (2021): 239. http://dx.doi.org/10.3390/jimaging7110239.

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Deep learning based reconstruction methods deliver outstanding results for solving inverse problems and are therefore becoming increasingly important. A recently invented class of learning-based reconstruction methods is the so-called NETT (for Network Tikhonov Regularization), which contains a trained neural network as regularizer in generalized Tikhonov regularization. The existing analysis of NETT considers fixed operators and fixed regularizers and analyzes the convergence as the noise level in the data approaches zero. In this paper, we extend the frameworks and analysis considerably to reflect various practical aspects and take into account discretization of the data space, the solution space, the forward operator and the neural network defining the regularizer. We show the asymptotic convergence of the discretized NETT approach for decreasing noise levels and discretization errors. Additionally, we derive convergence rates and present numerical results for a limited data problem in photoacoustic tomography.

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Chen, Yu Yan, Xu Wang, Dan Yang, and Yi Lv. "A New Hybrid Image Reconstruction Algorithm for Magnetic Induction Tomography." Advanced Materials Research 532-533 (June 2012): 1706–10. http://dx.doi.org/10.4028/www.scientific.net/amr.532-533.1706.

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A new hybrid algorithm is presented in this paper, which solves the ill-posed inverse problem of magnetic induction tomography (MIT) and improves the quality of reconstructed image. The hybrid algorithm firstly produces the preliminary image region using Tikhonov regularization algorithm, and then it obtains the final reconstructed image using variation regularization algorithm. The hybrid algorithm, compared with the Tikhonov regularization algorithm and the variation regularization algorithm, overcomes the numerical instability of MIT image reconstruction and accelerates the convergence speed of image reconstruction, and it also improves the resolving power of targets conductor and the quality of the reconstructed image. Simulation results show that the quality of the reconstructed image obtained using the hybrid algorithm is enhanced, so an effective algorithm for magnetic induction tomography (MIT) is introduced.

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Joachimiak, Magda, and Michał Ciałkowski. "Influence of the heat source location on the stability of the solution to the Cauchy problem." E3S Web of Conferences 323 (2021): 00016. http://dx.doi.org/10.1051/e3sconf/202132300016.

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In this paper the solution to the Cauchy-type inverse problem for the Laplace’s equation is presented. A modified Tikhonov regularization was applied here. The regularization parameter was chosen using the Morozov principle. The relation between the location of the heat source (function singularity) and the stability of the solution to the inverse problem was analyzed. Variable thermal loads in the area were simulated by changing the location of heat sources along two boundaries of the rectangle calculation domain.

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Gao, Jinghuai, Dehua Wang, and Jigen Peng. "A Tikhonov-Type Regularization Method for Identifying the Unknown Source in the Modified Helmholtz Equation." Mathematical Problems in Engineering 2012 (2012): 1–13. http://dx.doi.org/10.1155/2012/878109.

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An inverse source problem in the modified Helmholtz equation is considered. We give a Tikhonov-type regularization method and set up a theoretical frame to analyze the convergence of such method. A priori and a posteriori choice rules to find the regularization parameter are given. Numerical tests are presented to illustrate the effectiveness and stability of our proposed method.

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Chen, Jiawei, Xingxing Ju, Elisabeth Köbis, and Yeong-Cheng Liou. "Tikhonov type regularization methods for inverse mixed variational inequalities." Optimization 69, no. 2 (2019): 401–13. http://dx.doi.org/10.1080/02331934.2019.1607339.

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Popa, Constantin. "Iterative solvers for Tikhonov regularization of dense inverse problems." International Journal of Computer Mathematics 87, no. 14 (2010): 3199–208. http://dx.doi.org/10.1080/00207160902971558.

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Rastogi, Abhishake. "Nonlinear Tikhonov regularization in Hilbert scales for inverse learning." Journal of Complexity 82 (June 2024): 101824. http://dx.doi.org/10.1016/j.jco.2024.101824.

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Moura, Henrique Gomes, Edson Costa Junior, Arcanjo Lenzi, and Vinicius Carvalho Rispoli. "On a Stochastic Regularization Technique for Ill-Conditioned Linear Systems." Open Engineering 9, no. 1 (2019): 52–60. http://dx.doi.org/10.1515/eng-2019-0008.

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AbstractKnowledge about the input–output relations of a system can be very important in many practical situations in engineering. Linear systems theory comes from applied mathematics as an efficient and simple modeling technique for input–output systems relations. Many identification problems arise from a set of linear equations, using known outputs only. It is a type of inverse problems, whenever systems inputs are sought by its output only. This work presents a regularization method, called random matrix method, which is able to reduce errors on the solution of ill-conditioned inverse problems by introducing modifications into the matrix operator that rules the problem. The main advantage of this approach is the possibility of reducing the condition number of the matrix using the probability density function that models the noise in the measurements, leading to better regularization performance. The method described was applied in the context of a force identification problem and the results were compared quantitatively and qualitatively with the classical Tikhonov regularization method. Results show the presented technique provides better results than Tikhonov method when dealing with high-level ill-conditioned inverse problems.

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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 performance than common approaches to achieve the solution. We compare four regularization and three estimating methods for regularization parameters. We use a similarity measure to the simulated cases in three frequencies. Tikhonov regularization exhibits best reconstruction results. However, truncated singular vector decomposition also shows good performance with the advantage of not using a regularization parameter. Graphics processing unit implementation reduce reconstruction's computing time at least in a half.

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Fan, Yuchuan, Chunyu Zhao та Hongye Yu. "Research on Dynamic Load Identification Based on Explicit Wilson-θ and Improved Regularization Algorithm". Shock and Vibration 2019 (25 березня 2019): 1–15. http://dx.doi.org/10.1155/2019/8756546.

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In the research of dynamic load identification, the method of obtaining kernel function matrix is usually rather cumbersome. To solve this problem, an explicit dynamic load identification algorithm based on the Wilson-θ (DLIAEW) method is proposed to easily obtain the kernel function matrix as long as the parameters of the system are known. To aim at the ill-posed problem in the inverse problem, this paper improves the Tikhonov regularization, proposes an improved regularization algorithm (IRA), and introduces the U-curve method to determine the regularization parameters. In the numeric simulation investigation of a four dofs vibrating system, effects of the sampling frequency and the noise level on the regularization parameters and the identification errors of impact and harmonic loads for the IRA are discussed in comparison with the Tikhonov regularization. Finally, the experiments of a cantilever beam excited by impact and harmonic loads are carried out to verify the advantages of the IRA.

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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 to improve the image quality. Finally, the improved method was combined with a projection algorithm based on the Krylov subspace, which was also effective in reducing the computational time. Both simulation and experimental results indicated that the new algorithm could improve the real-time performance and imaging quality.

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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 Truncated Singular Value Decomposition and Tikhonov regularization in order to accelerate the inversion time and to reduce the sensitivity to the value of the regularization parameter. The Discrete Picard condition is used to jointly select the SVD truncation and Tikhonov regularization parameters. We evaluate the performance of the proposed method on both simulated and real NMR measurements.

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Cheng, Xiaoliang, Lele Yuan, and Kewei Liang. "Inverse source problem for a distributed-order time fractional diffusion equation." Journal of Inverse and Ill-posed Problems 28, no. 1 (2020): 17–32. http://dx.doi.org/10.1515/jiip-2019-0006.

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AbstractThis paper studies an inverse source problem for a time fractional diffusion equation with the distributed order Caputo derivative. The space-dependent source term is recovered from a noisy final data. The uniqueness, ill-posedness and a conditional stability for this inverse source problem are obtained. The inverse problem is formulated into a minimization functional with Tikhonov regularization method. Further, based on the series representation of the regularized solution, we give convergence rates of the regularized solution under an a-priori and an a-posteriori regularization parameter choice rule. With an adjoint technique for computing the gradient of the regularization functional, the conjugate gradient method is applied to reconstruct the space-dependent source term. Two numerical examples illustrate the effectiveness of the proposed method.

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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 accuracy and stability of approximate solutions. The new method is based on a combination of the boundary condition and the boundary measurement in a parameter-dependent single complex Robin boundary condition, followed by the Tikhonov regularization. By properly adjusting the parameter in the Robin boundary condition, we achieve two important properties for our new method: first, the regularized solutions are uniformly stable with respect to the regularization parameter so that the regularization parameter can be chosen based solely on the consideration of the solution accuracy; second, the convergence order of the regularized solutions reaches one with respect to the noise level. Then, the finite element method is used to compute numerical solutions and a new finite element error estimate is derived for discrete solutions. These results improve related results found in the existing literature. Several numerical examples are provided to illustrate the theoretical results.

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Power, J. F., and M. C. Prystay. "Nondestructive Optical Depth Profiling in Thin Films through Robust Inversion of the Laser Photopyroelectric Effect Impulse Response." Applied Spectroscopy 49, no. 6 (1995): 725–46. http://dx.doi.org/10.1366/0003702953964570.

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The laser photopyroelectric effect measures an optical absorption depth profile in a thin film through the spatial dependence of a heat flux source established below the film surface by light absorption from a short optical pulse. In this work, inverse depth profile reconstruction was achieved by means of an inverse method based on the expectation-minimum principle (as reported in a companion paper), applied in conjunction with a constrained least-squares minimization, to invert the photopyroelectric theory. This method and zero-order Tikhonov regularization were applied to the inversion of experimental photopyroelectric data obtained from samples with a variety of discrete and continuous depth dependences of optical absorption. While both methods were found to deliver stable and accurate performance under experimental conditions, the method based on the constrained expectation-minimum principle was found to exhibit improved resolution and robustness over zero-order Tikhonov regularization.

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Zhang, Xinming. "An Inverse Problem Solution Scheme for Solving the Optimization Problem of Drug-Controlled Release from Multilaminated Devices." Computational and Mathematical Methods in Medicine 2020 (August 1, 2020): 1–15. http://dx.doi.org/10.1155/2020/8380691.

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The optimization problem of drug release based on the multilaminated drug-controlled release devices has been solved in this paper under the inverse problem solution scheme. From the viewpoint of inverse problem, the solution of optimization problem can be regarded as the solution problem of a Fredholm integral equation of first kind. The solution of the Fredholm integral equation of first kind is a well-known ill-posed problem. In order to solve the severe ill-posedness, a modified regularization method is presented based on the Tikhonov regularization method and the truncated singular value decomposition method. The convergence analysis of the modified regularization method is also given. The optimization results of the initial drug concentration distribution obtained by the modified regularization method demonstrate that the inverse problem solution scheme proposed in this paper has the advantages of the numerical accuracy and antinoise property.

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Chen, Yong-Gang, Fan Yang, Xiao-Xiao Li, and Dun-Gang Li. "The Fractional Tikhonov Regularization Method to Identify the Initial Value of the Nonhomogeneous Time-Fractional Diffusion Equation on a Columnar Symmetrical Domain." Symmetry 14, no. 8 (2022): 1633. http://dx.doi.org/10.3390/sym14081633.

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In this paper, the inverse problem for identifying the initial value of a time fractional nonhomogeneous diffusion equation in a columnar symmetric region is studied. This is an ill-posed problem, i.e., the solution does not depend continuously on the data. The fractional Tikhonov regularization method is applied to solve this problem and obtain the regularization solution. The error estimations between the regularization solution and the exact solution are also obtained under the priori and the posteriori regularization parameter choice rules, respectively. Some examples are given to show this method’s effectiveness.

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Kumar, Nishant, Lukas Krause, Thomas Wondrak, Sven Eckert, Kerstin Eckert, and Stefan Gumhold. "Robust Reconstruction of the Void Fraction from Noisy Magnetic Flux Density Using Invertible Neural Networks." Sensors 24, no. 4 (2024): 1213. http://dx.doi.org/10.3390/s24041213.

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Electrolysis stands as a pivotal method for environmentally sustainable hydrogen production. However, the formation of gas bubbles during the electrolysis process poses significant challenges by impeding the electrochemical reactions, diminishing cell efficiency, and dramatically increasing energy consumption. Furthermore, the inherent difficulty in detecting these bubbles arises from the non-transparency of the wall of electrolysis cells. Additionally, these gas bubbles induce alterations in the conductivity of the electrolyte, leading to corresponding fluctuations in the magnetic flux density outside of the electrolysis cell, which can be measured by externally placed magnetic sensors. By solving the inverse problem of the Biot–Savart Law, we can estimate the conductivity distribution as well as the void fraction within the cell. In this work, we study different approaches to solve the inverse problem including Invertible Neural Networks (INNs) and Tikhonov regularization. Our experiments demonstrate that INNs are much more robust to solving the inverse problem than Tikhonov regularization when the level of noise in the magnetic flux density measurements is not known or changes over space and time.

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Nair, M. Thamban, and Devika Shylaja. "Conforming and nonconforming finite element methods for biharmonic inverse source problem." Inverse Problems 38, no. 2 (2021): 025001. http://dx.doi.org/10.1088/1361-6420/ac3ec5.

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Abstract This paper deals with the numerical approximation of the biharmonic inverse source problem in an abstract setting in which the measurement data is finite-dimensional. This unified framework in particular covers the conforming and nonconforming finite element methods (FEMs). The inverse problem is analysed through the forward problem. Error estimate for the forward solution is derived in an abstract set-up that applies to conforming and Morley nonconforming FEMs. Since the inverse problem is ill-posed, Tikhonov regularization is considered to obtain a stable approximate solution. Error estimate is established for the regularized solution for different regularization schemes. Numerical results that confirm the theoretical results are also presented.

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Atifi, Khalid, Idriss Boutaayamou, Hamed Ould Sidi, and Jawad Salhi. "An Inverse Source Problem for Singular Parabolic Equations with Interior Degeneracy." Abstract and Applied Analysis 2018 (December 9, 2018): 1–16. http://dx.doi.org/10.1155/2018/2067304.

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The main purpose of this work is to study an inverse source problem for degenerate/singular parabolic equations with degeneracy and singularity occurring in the interior of the spatial domain. Using Carleman estimates, we prove a Lipschitz stability estimate for the source term provided that additional measurement data are given on a suitable interior subdomain. For the numerical solution, the reconstruction is formulated as a minimization problem using the output least squares approach with the Tikhonov regularization. The Fréchet differentiability of the Tikhonov functional and the Lipschitz continuity of the Fréchet gradient are proved. These properties allow us to apply gradient methods for numerical solution of the considered inverse source problem.

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Shen, Yu, and Xiangtuan Xiong. "Identifying the Heat Source in Radially Symmetry and Axis-Symmetry Problems." Symmetry 16, no. 2 (2024): 134. http://dx.doi.org/10.3390/sym16020134.

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This paper solves the inverse source problem of heat conduction in which the source term only varies with time. The application of the discrete regularization method, a kind of effective radial symmetry and axisymmetric heat conduction problem source identification that does not involve the grid integral numerical method, is put forward. Taking the fundamental solution as the fundamental function, the classical Tikhonov regularization method combined with the L-curve criterion is used to select the appropriate regularization parameters, so the problem is transformed into a class of ill-conditioned linear algebraic equations to solve with an optimal solution. Several numerical examples of inverse source problems are given. Simultaneously, a few numerical examples of inverse source problems are given, and the effectiveness and superiority of the method is shown by the results.

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Fu, Chu-Li, Hong-Fang Li, Xiang-Tuan Xiong, and Peng Fu. "Optimal Tikhonov approximation for a sideways parabolic equation." International Journal of Mathematics and Mathematical Sciences 2005, no. 8 (2005): 1221–37. http://dx.doi.org/10.1155/ijmms.2005.1221.

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We consider an inverse heat conduction problem with convection term which appears in some applied subjects. This problem is ill posed in the sense that the solution (if it exists) does not depend continuously on the data. A generalized Tikhonov regularization method for this problem is given, which realizes the best possible accuracy.

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Kabanikhin, Sergey Igorevich, M. A. Shishlenin, D. B. Nurseitov, A. T. Nurseitova, and S. E. Kasenov. "Comparative Analysis of Methods for Regularizing an Initial Boundary Value Problem for the Helmholtz Equation." Journal of Applied Mathematics 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/786326.

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We consider an ill-posed initial boundary value problem for the Helmholtz equation. This problem is reduced to the inverse continuation problem for the Helmholtz equation. We prove the well-posedness of the direct problem and obtain a stability estimate of its solution. We solve numerically the inverse problem using the Tikhonov regularization, Godunov approach, and the Landweber iteration. Comparative analysis of these methods is presented.

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Polyakova, R. V., and I. P. Yudin. "MATHEMATICAL MODELLING OF THE MAGNETIC SYSTEM BY A.N. TIKHONOV REGULARIZATION METHOD." Chronos 7, no. 9(71) (2022): 42–51. http://dx.doi.org/10.52013/2658-7556-71-9-9.

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A nonlinear magnetostatic inverse problem is investigated for a case when it is needed to create a required magnetic field using conductors which coordinates vary on condition that current value is equal in all the conductors. It is known that the same problems fall under the category of noncorrect problems. Mathematical statement of this type of nonlinear magnetostatic inverse problems is given. The proposed numerical algorithm using the regularization method by A.N.Tikhonov permits to overcome comparatively easily the difficulties connected with badly conditioned equation systems to which usually the magnetostatic inverse problems reduce. This algorithm allows one to calculate the existing winding geometry of superconducting dipole ironless megnet which ensures a magnetic field homogeneity up to 10-5 within a rectangular aperture.

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Rastogi, Abhishake. "Tikhonov regularization with oversmoothing penalty for nonlinear statistical inverse problems." Communications on Pure & Applied Analysis 19, no. 8 (2020): 4111–26. http://dx.doi.org/10.3934/cpaa.2020183.

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Journal articles on the topic 'Inverse Tikhonov Regularization (ITR)'

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