Relevant bibliographies by topics / Inverse Tikhonov Regularization (ITR)

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Academic literature on the topic 'Inverse Tikhonov Regularization (ITR)'

Author: Grafiati
Published: 5 June 2025
Last updated: 24 June 2025

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Journal articles on the topic "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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Dissertations / Theses on the topic "Inverse Tikhonov Regularization (ITR)"

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Anzengruber, Stephan W., Bernd Hofmann, and Peter Mathé. "Regularization properties of the discrepancy principle for Tikhonov regularization in Banach spaces." Universitätsbibliothek Chemnitz, 2012. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-99353.

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The stable solution of ill-posed non-linear operator equations in Banach space requires regularization. One important approach is based on Tikhonov regularization, in which case a one-parameter family of regularized solutions is obtained. It is crucial to choose the parameter appropriately. Here, a variant of the discrepancy principle is analyzed. In many cases such parameter choice exhibits the feature, called regularization property below, that the chosen parameter tends to zero as the noise tends to zero, but slower than the noise level. Here we shall show such regularization property under two natural assumptions. First, exact penalization must be excluded, and secondly, the discrepancy principle must stop after a finite number of iterations. We conclude this study with a discussion of some consequences for convergence rates obtained by the discrepancy principle under the validity of some kind of variational inequality, a recent tool for the analysis of inverse problems.
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Bondarenko, Oleksandr. "Optimal Control for an Impedance Boundary Value Problem." Thesis, Virginia Tech, 2010. http://hdl.handle.net/10919/36136.

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We consider the analysis of the scattering problem. Assume that an incoming time harmonic wave is scattered by a surface of an impenetrable obstacle. The reflected wave is determined by the surface impedance of the obstacle. In this paper we will investigate the problem of choosing the surface impedance so that a desired scattering amplitude is achieved. We formulate this control problem within the framework of the minimization of a Tikhonov functional. In particular, questions of the existence of an optimal solution and the derivation of the optimality conditions will be addressed.<br>Master of Science
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Slagel, Joseph Tanner. "Row-Action Methods for Massive Inverse Problems." Diss., Virginia Tech, 2019. http://hdl.handle.net/10919/90377.

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Numerous scientific applications have seen the rise of massive inverse problems, where there are too much data to implement an all-at-once strategy to compute a solution. Additionally, tools for regularizing ill-posed inverse problems are infeasible when the problem is too large. This thesis focuses on the development of row-action methods, which can be used to iteratively solve inverse problems when it is not possible to access the entire data-set or forward model simultaneously. We investigate these techniques for linear inverse problems and for separable, nonlinear inverse problems where the objective function is nonlinear in one set of parameters and linear in another set of parameters. For the linear problem, we perform a convergence analysis of these methods, which shows favorable asymptotic and initial convergence properties, as well as a trade-off between convergence rate and precision of iterates that is based on the step-size. These row-action methods can be interpreted as stochastic Newton and stochastic quasi-Newton approaches on a reformulation of the least squares problem, and they can be analyzed as limited memory variants of the recursive least squares algorithm. For ill-posed problems, we introduce sampled regularization parameter selection techniques, which include sampled variants of the discrepancy principle, the unbiased predictive risk estimator, and the generalized cross-validation. We demonstrate the effectiveness of these methods using examples from super-resolution imaging, tomography reconstruction, and image classification.<br>Doctor of Philosophy<br>Numerous scientific problems have seen the rise of massive data sets. An example of this is super-resolution, where many low-resolution images are used to construct a high-resolution image, or 3-D medical imaging where a 3-D image of an object of interest with hundreds of millions voxels is reconstructed from x-rays moving through that object. This work focuses on row-action methods that numerically solve these problems by repeatedly using smaller samples of the data to avoid the computational burden of using the entire data set at once. When data sets contain measurement errors, this can cause the solution to get contaminated with noise. While there are methods to handle this issue, when the data set becomes massive, these methods are no longer feasible. This dissertation develops techniques to avoid getting the solution contaminated with noise, even when the data set is immense. The methods developed in this work are applied to numerous scientific applications including super-resolution imaging, tomography, and image classification.
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Onal, Murat. "Evaulation Of Spatial And Spatio-temporal Regularization Approaches In Inverse Problem Of Electrocardiography." Master's thesis, METU, 2008. http://etd.lib.metu.edu.tr/upload/2/12610045/index.pdf.

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Conventional electrocardiography (ECG) is an essential tool for investigating cardiac disorders such as arrhythmias or myocardial infarction. It consists of interpretation of potentials recorded at the body surface that occur due to the electrical activity of the heart. However, electrical signals originated at the heart suffer from attenuation and smoothing within the thorax, therefore ECG signal measured on the body surface lacks some important details. The goal of forward and inverse ECG problems is to recover these lost details by estimating the heart&amp<br>#8217<br>s electrical activity non-invasively from body surface potential measurements. In the forward problem, one calculates the body surface potential distribution (i.e. torso potentials) using an appropriate source model for the equivalent cardiac sources. In the inverse problem of ECG, one estimates cardiac electrical activity based on measured torso potentials and a geometric model of the torso. Due to attenuation and spatial smoothing that occur within the thorax, inverse ECG problem is ill-posed and the forward model matrix is badly conditioned. Thus, small disturbances in the measurements lead to amplified errors in inverse solutions. It is difficult to solve this problem for effective cardiac imaging due to the ill-posed nature and high dimensionality of the problem. Tikhonov regularization, Truncated Singular Value Decomposition (TSVD) and Bayesian MAP estimation are some of the methods proposed in literature to cope with the ill-posedness of the problem. The most common approach in these methods is to ignore temporal relations of epicardial potentials and to solve the inverse problem at every time instant independently (column sequential approach). This is the fastest and the easiest approach<br>however, it does not include temporal correlations. The goal of this thesis is to include temporal constraints as well as spatial constraints in solving the inverse ECG problem. For this purpose, two methods are used. In the first method, we solved the augmented problem directly. Alternatively, we solve the problem with column sequential approach after applying temporal whitening. The performance of each method is evaluated.
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Hofmann, Bernd, and Romy Krämer. "Maximum entropy regularization for calibrating a time-dependent volatility function." Universitätsbibliothek Chemnitz, 2004. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200401213.

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We investigate the applicability of the method of maximum entropy regularization (MER) including convergence and convergence rates of regularized solutions to the specific inverse problem (SIP) of calibrating a purely time-dependent volatility function. In this context, we extend the results of [16] and [17] in some details. Due to the explicit structure of the forward operator based on a generalized Black-Scholes formula the ill-posedness character of the nonlinear inverse problem (SIP) can be verified. Numerical case studies illustrate the chances and limitations of (MER) versus Tikhonov regularization (TR) for smooth solutions and solutions with a sharp peak.
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Cho, Taewon. "Computational Advancements for Solving Large-scale Inverse Problems." Diss., Virginia Tech, 2021. http://hdl.handle.net/10919/103772.

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For many scientific applications, inverse problems have played a key role in solving important problems by enabling researchers to estimate desired parameters of a system from observed measurements. For example, large-scale inverse problems arise in many global problems and medical imaging problems such as greenhouse gas tracking and computational tomography reconstruction. This dissertation describes advancements in computational tools for solving large-scale inverse problems and for uncertainty quantification. Oftentimes, inverse problems are ill-posed and large-scale. Iterative projection methods have dramatically reduced the computational costs of solving large-scale inverse problems, and regularization methods have been critical in obtaining stable estimations by applying prior information of unknowns via Bayesian inference. However, by combining iterative projection methods and variational regularization methods, hybrid projection approaches, in particular generalized hybrid methods, create a powerful framework that can maximize the benefits of each method. In this dissertation, we describe various advancements and extensions of hybrid projection methods that we developed to address three recent open problems. First, we develop hybrid projection methods that incorporate mixed Gaussian priors, where we seek more sophisticated estimations where the unknowns can be treated as random variables from a mixture of distributions. Second, we describe hybrid projection methods for mean estimation in a hierarchical Bayesian approach. By including more than one prior covariance matrix (e.g., mixed Gaussian priors) or estimating unknowns and hyper-parameters simultaneously (e.g., hierarchical Gaussian priors), we show that better estimations can be obtained. Third, we develop computational tools for a respirometry system that incorporate various regularization methods for both linear and nonlinear respirometry inversions. For the nonlinear systems, blind deconvolution methods are developed and prior knowledge of nonlinear parameters are used to reduce the dimension of the nonlinear systems. Simulated and real-data experiments of the respirometry problems are provided. This dissertation provides advanced tools for computational inversion and uncertainty quantification.<br>Doctor of Philosophy<br>For many scientific applications, inverse problems have played a key role in solving important problems by enabling researchers to estimate desired parameters of a system from observed measurements. For example, large-scale inverse problems arise in many global problems such as greenhouse gas tracking where the problem of estimating the amount of added or removed greenhouse gas at the atmosphere gets more difficult. The number of observations has been increased with improvements in measurement technologies (e.g., satellite). Therefore, the inverse problems become large-scale and they are computationally hard to solve. Another example of an inverse problem arises in tomography, where the goal is to examine materials deep underground (e.g., to look for gas or oil) or reconstruct an image of the interior of the human body from exterior measurements (e.g., to look for tumors). For tomography applications, there are typically fewer measurements than unknowns, which results in non-unique solutions. In this dissertation, we treat unknowns as random variables with prior probability distributions in order to compensate for a deficiency in measurements. We consider various additional assumptions on the prior distribution and develop efficient and robust numerical methods for solving inverse problems and for performing uncertainty quantification. We apply our developed methods to many numerical applications such as greenhouse gas tracking, seismic tomography, spherical tomography problems, and the estimation of CO2 of living organisms.
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Pasha, Mirjeta. "Krylov subspace type methods for the computation of non-negative or sparse solutions of ill-posed problems." Kent State University / OhioLINK, 2020. http://rave.ohiolink.edu/etdc/view?acc_num=kent1586459362313778.

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Bürger, Steven, and Bernd Hofmann. "About a deficit in low order convergence rates on the example of autoconvolution." Universitätsbibliothek Chemnitz, 2013. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-130630.

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We revisit in L2-spaces the autoconvolution equation x ∗ x = y with solutions which are real-valued or complex-valued functions x(t) defined on a finite real interval, say t ∈ [0,1]. Such operator equations of quadratic type occur in physics of spectra, in optics and in stochastics, often as part of a more complex task. Because of their weak nonlinearity deautoconvolution problems are not seen as difficult and hence little attention is paid to them wrongly. In this paper, we will indicate on the example of autoconvolution a deficit in low order convergence rates for regularized solutions of nonlinear ill-posed operator equations F(x)=y with solutions x† in a Hilbert space setting. So for the real-valued version of the deautoconvolution problem, which is locally ill-posed everywhere, the classical convergence rate theory developed for the Tikhonov regularization of nonlinear ill-posed problems reaches its limits if standard source conditions using the range of F (x† )∗ fail. On the other hand, convergence rate results based on Hölder source conditions with small Hölder exponent and logarithmic source conditions or on the method of approximate source conditions are not applicable since qualified nonlinearity conditions are required which cannot be shown for the autoconvolution case according to current knowledge. We also discuss the complex-valued version of autoconvolution with full data on [0,2] and see that ill-posedness must be expected if unbounded amplitude functions are admissible. As a new detail, we present situations of local well-posedness if the domain of the autoconvolution operator is restricted to complex L2-functions with a fixed and uniformly bounded modulus function.
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Rückert, Nadja. "Studies on two specific inverse problems from imaging and finance." Doctoral thesis, Universitätsbibliothek Chemnitz, 2012. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-91587.

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This thesis deals with regularization parameter selection methods in the context of Tikhonov-type regularization with Poisson distributed data, in particular the reconstruction of images, as well as with the identification of the volatility surface from observed option prices. In Part I we examine the choice of the regularization parameter when reconstructing an image, which is disturbed by Poisson noise, with Tikhonov-type regularization. This type of regularization is a generalization of the classical Tikhonov regularization in the Banach space setting and often called variational regularization. After a general consideration of Tikhonov-type regularization for data corrupted by Poisson noise, we examine the methods for choosing the regularization parameter numerically on the basis of two test images and real PET data. In Part II we consider the estimation of the volatility function from observed call option prices with the explicit formula which has been derived by Dupire using the Black-Scholes partial differential equation. The option prices are only available as discrete noisy observations so that the main difficulty is the ill-posedness of the numerical differentiation. Finite difference schemes, as regularization by discretization of the inverse and ill-posed problem, do not overcome these difficulties when they are used to evaluate the partial derivatives. Therefore we construct an alternative algorithm based on the weak formulation of the dual Black-Scholes partial differential equation and evaluate the performance of the finite difference schemes and the new algorithm for synthetic and real option prices.
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CENTORRINO, SAMUELE. "Essays in Nonparamentric Estimation with Instrumental Variables." Doctoral thesis, Università degli Studi di Milano-Bicocca, 2014. http://hdl.handle.net/10281/109031.

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This thesis deals with the broad problem of causality and endogeneity in econometrics when the function of interest is estimated nonparametrically. It explores this problem in two separate frameworks. In the cross sectional, iid setting, it considers the estimation of a nonlinear additively separable model, in which the regression function depends on an endogenous explanatory variable. Endogeneity is, in this case, broadly defined. It can relate to reverse causality (the dependent variable can also affects the independent regressor) or to simultaneity (the error term contains information that can be related to the explanatory variable). Identification and estimation of the regression function is performed using the method of instrumental variables. In the time series context, it studies the implications of the assumption of exogeneity in a regression type model in continuous time. In this model, the state variable depends on its past values, but also on some external covariates and the researcher is interested in the nonparametric estimation of both the conditional mean and the conditional variance functions. This first chapter deals with the latter topic. In particular, we give sufficient conditions under which the researcher can make meaningful inference in such a model. It shows that noncausality is a sufficient condition for exogeneity if the researcher is not willing to make any assumption on the dynamics of the covariate process. However, if the researcher is willing to assume that the covariate process follows a simple stochastic differential equation, then the assumption of noncausality becomes irrelevant. Chapters two to four are instead completely devoted to the simple iid model. The function of interest is known to be the solution of an inverse problem. In the second chapter, this estimation problem is considered when the regularization is achieved using a penalization on the L2-norm of the function of interest (so-called Tikhonov regularization). We derive the properties of a leave-one-out cross validation criterion in order to choose the regularization parameter. In the third chapter, coauthored with Jean-Pierre Florens, we extend this model to the case in which the dependent variable is not directly observed, but only a binary transformation of it. We show that identification can be obtained via the decomposition of the dependent variable on the space spanned by the instruments, when the residuals in this reduced form model are taken to have a known distribution. We finally show that, under these assumptions, the consistency properties of the estimator are preserved. Finally, chapter four, coauthored with Frédérique Fève and Jean-Pierre Florens, performs a numerical study, in which the properties of several regularization techniques are investigated. In particular, we gather data-driven techniques for the sequential choice of the smoothing and the regularization parameters and we assess the validity of wild bootstrap in nonparametric instrumental regressions.
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Books on the topic "Inverse Tikhonov Regularization (ITR)"

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Gockenbach, Mark S. Linear Inverse Problems and Tikhonov Regularization. American Mathematical Society, 2016.

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Huber, Richard. Variational Regularization for Systems of Inverse Problems: Tikhonov Regularization with Multiple Forward Operators. Springer Spektrum, 2019.

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Book chapters on the topic "Inverse Tikhonov Regularization (ITR)"

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Engl, Heinz W., Martin Hanke, and Andreas Neubauer. "Tikhonov Regularization." In Regularization of Inverse Problems. Springer Netherlands, 2000. http://dx.doi.org/10.1007/978-94-009-1740-8_5.

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Flemming, Jens. "Tikhonov Regularization." In Variational Source Conditions, Quadratic Inverse Problems, Sparsity Promoting Regularization. Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-95264-2_5.

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Engl, Heinz W., Martin Hanke, and Andreas Neubauer. "Tikhonov Regularization of Nonlinear Problems." In Regularization of Inverse Problems. Springer Netherlands, 2000. http://dx.doi.org/10.1007/978-94-009-1740-8_10.

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Doicu, Adrian, Thomas Trautmann, and Franz Schreier. "Tikhonov regularization for linear problems." In Numerical Regularization for Atmospheric Inverse Problems. Springer Berlin Heidelberg, 2010. http://dx.doi.org/10.1007/978-3-642-05439-6_3.

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Doicu, Adrian, Thomas Trautmann, and Franz Schreier. "Tikhonov regularization for nonlinear problems." In Numerical Regularization for Atmospheric Inverse Problems. Springer Berlin Heidelberg, 2010. http://dx.doi.org/10.1007/978-3-642-05439-6_6.

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Bertero, Mario, Patrizia Boccacci, and Christine De MoI. "Quadratic Tikhonov regularization and filtering." In Introduction to Inverse Problems in Imaging, 2nd ed. CRC Press, 2021. http://dx.doi.org/10.1201/9781003032755-4.

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Huber, Richard. "General Tikhonov Regularisation." In Variational Regularization for Systems of Inverse Problems. Springer Fachmedien Wiesbaden, 2019. http://dx.doi.org/10.1007/978-3-658-25390-5_2.

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Xiao, Tingyan, Yuan Zhao, and Guozhong Su. "Extrapolation Techniques of Tikhonov Regularization." In Optimization and Regularization for Computational Inverse Problems and Applications. Springer Berlin Heidelberg, 2010. http://dx.doi.org/10.1007/978-3-642-13742-6_5.

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Engl, H. W., M. Hanke, and A. Neubauer. "Tikhonov Regularization of Nonlinear Differential-Algebraic Equations." In Inverse Problems and Theoretical Imaging. Springer Berlin Heidelberg, 1990. http://dx.doi.org/10.1007/978-3-642-75298-8_12.

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Artamonov, Boris, Ekaterina Koptelova, Elena Shimanovskaya, and Anatoly G. Yagola. "Tikhonov Regularization for Gravitational Lensing Research." In Optimization and Regularization for Computational Inverse Problems and Applications. Springer Berlin Heidelberg, 2010. http://dx.doi.org/10.1007/978-3-642-13742-6_14.

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Conference papers on the topic "Inverse Tikhonov Regularization (ITR)"

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Jeyamohan, Kunaratnam, Tommy H. T. Chan, Khac-Duy Nguyen, and David P. Thambiratnam. "Computation-Effective Method for Prestress Force and Moving Force Identification." In IABSE Symposium, Tokyo 2025: Environmentally Friendly Technologies and Structures: Focusing on Sustainable Approaches. International Association for Bridge and Structural Engineering (IABSE), 2025. https://doi.org/10.2749/tokyo.2025.2336.

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&lt;p&gt;Structural health monitoring (SHM) of existing prestressed concrete bridges (PCBs) is of paramount importance due to the direct impact of failures on the safety of the bridge users. Numerous studies have been conducted to identify the existing prestress force and moving force, however, most of the methods face practical challenges and suffer from inherent ill‐conditioning. Considering these issues, this study proposes a load shape function (LSF)‐based approach for the synergic identification of prestress force and moving force in continuously supported PCBs. Through numerical studies, it is demonstrated that LSF approach exhibits remarkable accuracy in estimating prestress force and moving force using measured displacements. The integration of the Tikhonov regularization technique with a tri‐diagonal regularization matrix (&amp;#119819;&amp;#120785; ) into the inverse synergic identification method reduces the ill‐posed characteristics of the prestressed concrete bridge‐vehicle system. This approach facilitates the identification of well‐posed solutions with reduced computational effort. Ultimately, it promotes environmental sustainability by enabling more efficient SHM through timely maintenance of PCBs.&lt;/p&gt;
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Olson, Lorraine G., and Robert D. Throne. "Regularization Matrices for Inverse Electrocardiography." In ASME 1998 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 1998. http://dx.doi.org/10.1115/imece1998-0223.

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Abstract In a recent series of papers we proposed a new class of methods, the generalized eigensystem (GES) methods, for solving the inverse problem of electrocardiography. In this paper, we compare zero, first, and second order regularized GES methods to zero, first, and second order Tikhonov methods. Results from higher order regularization depend heavily on the exact form of the regularization operator, and operators generated by finite element techniques give the most accurate and consistent results. The GES techniques always produce smaller average relative errors than the Tikhonov techniques, but as the regularization order increases the difference in average relative errors between the two techniques becomes less pronounced.
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Wu, Menglu, and Xiaolin Chen. "Tikhonov Regularization Methods for the Inverse Scalp Electroencephalography." In ASME 2009 International Mechanical Engineering Congress and Exposition. ASMEDC, 2009. http://dx.doi.org/10.1115/imece2009-10538.

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Electroencephalography (EEG) source localization of brain activity is of high diagnostic value. Noninvasive numerical procedures can be developed to help reconstruct the cortical brain activities from the low-spatial-resolution scalp EEG measurement. In this paper, Tikhonov regularization methods are employed to tackle the solution difficulty associated with the ill-posed reconstruction problem. Three different techniques, namely the L-curve method, the generalized cross validation (GCV) and the discrepancy principle (DP), are implemented to help identify an optimum parameter for the numerical regularization. The numerical procedures are verified by comparing reconstruction results with available theoretical potential solutions for a simplified concentric sphere head model. All three parameter selection methods achieve good results and the L-curve method produces the best regularization effect among the three when the noise level is high in the contaminated scalp data input. More studies are performed on a computational model of an anatomically realistic human head. Our results show that the combination of Tikhonov regularization with the L-curve parameter selection method can effectively regularize the ill-posed inverse EEG problem for brain potential reconstruction.
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Lamm, Patricia K. "On the Local Regularization of Inverse Problems of Volterra Type." In ASME 1995 Design Engineering Technical Conferences collocated with the ASME 1995 15th International Computers in Engineering Conference and the ASME 1995 9th Annual Engineering Database Symposium. American Society of Mechanical Engineers, 1995. http://dx.doi.org/10.1115/detc1995-0664.

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Abstract We consider a local regularization method for the solution of first-kind Volterra integral equations with convolution kernel. The local regularization is based on a splitting of the original Volterra operator into “local” and “global” parts, and a use of Tikhonov regularization to stabilize the inversion of the local operator only. The regularization parameters for the local procedure include the standard Tikhonov parameter, as well as a parameter that represents the length of the local regularization interval. We present a convergence theory for the infinite-dimensional regularization problem and show that the regularized solutions converge to the true solution as the regularization parameters go to zero (in a prescribed way). In addition, we show how numerical implementation of the ideas of local regularization can lead to the notion of “sequential Tikhonov regularization” for Volterra problems; this approach has been shown in (Lamm and Eldén, 1995) to be just as effective as Tikhonov regularization, but to be much more efficient computationally.
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Okamoto, Kei, and Ben Q. Li. "Optimal Regularization Methods for Inverse Heat Transfer Problems." In ASME 2004 Heat Transfer/Fluids Engineering Summer Conference. ASMEDC, 2004. http://dx.doi.org/10.1115/ht-fed2004-56395.

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The Tikhonov regularization method has been used to find the unknown heat flux distribution along the boundary when the temperature measurements are known in the interior of a sample. Mathematically, the inverse problem is ill-posed, though physically correct, and prone to instability. This paper discusses the fundamental issues concerning the selection of optimal regularization parameters for inverse heat transfer calculations. Towards this end, a finite-element-based inverse algorithm is developed. Five different methods, that is, the maximum likelihood (ML), the ordinary cross-validation (OCV), the generalized cross-validation (GCV), the L-curve method, and the discrepancy principle, are evaluated for the purpose of determining optimal regularization parameters. An assessment of these methods is made using 1-D and 2-D inverse steady heat conduction problems where analytical solutions are available. The optimal regularization method is also compared with the Levenberg-Marquardt method for inverse heat transfer calculations. Results show that in general the Tikhonov regularization method is superior over the Levenberg-Marquardt method when the input data errors are noisy. With the appropriately determined regularization parameter, the inverse algorithm is applied to estimate the heat flux of spray cooling of a 3-D microelectronic component with an embedded heating source.
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Rastogi, Abhishake. "Tikhonov regularization with oversmoothing penalty for linear statistical inverse learning problems." In THIRD INTERNATIONAL CONFERENCE OF MATHEMATICAL SCIENCES (ICMS 2019). AIP Publishing, 2019. http://dx.doi.org/10.1063/1.5136221.

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Daun, K. J., K. A. Thomson, F. Liu, and G. J. Smallwood. "Solution of Abel’s Integral Equation Using Tikhonov Regularization." In ASME 2005 International Mechanical Engineering Congress and Exposition. ASMEDC, 2005. http://dx.doi.org/10.1115/imece2005-81430.

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This paper presents a method based on Tikhonov regularization for solving one-dimensional inverse tomography problems that arise in combustion applications. In this technique, Tikhonov regularization transforms the ill-conditioned set of equations generated by onion-peeling deconvolution into a well-conditioned set that is more stable to measurement errors that arise in experimental settings. The performance of this method is compared to that of onion-peeling and Abel three-point deconvolution by solving for a known field variable distribution from projected data contaminated with artificially-generated error. The results show that Tikhonov deconvolution provides a more accurate field distribution than onion-peeling and Abel three-point deconvolution, and is more stable than the other two methods as the distance between projected data points decreases.
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Pasadas, Dario, A. Lopes Ribeiro, Helena Ramos, and Tiago Rocha. "ECT image analysis applying an inverse problem algorithm with Tikhonov/TV Regularization." In 2015 IEEE International Instrumentation and Measurement Technology Conference (I2MTC). IEEE, 2015. http://dx.doi.org/10.1109/i2mtc.2015.7151396.

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Shen, Jin, Yanting Cheng, Qiuyan Han, Wei Liu, and Jingling Song. "Influence of noise to PCS particle sizing with Tikhonov regularization inverse algorithm." In International Conference of Optical Instrument and Technology, edited by Yunlong Sheng, Yongtian Wang, and Lijiang Zeng. SPIE, 2008. http://dx.doi.org/10.1117/12.810810.

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Dong, Xinwen, Shuhan Zhuang, Yuhan Xu, and Sheng Fang. "Regularization Parameter Identification for Nonlinear Inverse Modeling of Atmospheric Emissions." In ASME 2023 International Conference on Environmental Remediation and Radioactive Waste Management. American Society of Mechanical Engineers, 2023. http://dx.doi.org/10.1115/icem2023-107995.

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Abstract Regularized inverse modeling techniques formulate the rate estimation problem of airborne pollutant emission with the residual and regularization terms. In iterative minimization, the first term purposes to reduce the discrepancies between the atmospheric dispersion simulations and the environmental observations, and the second term ensures a stable solution and shapes the temporal features of emissions. The regularization parameter specifies the proportion of two terms, indirectly affecting the behavior of the solution. However, an identification approach may not suit all forms of regularized inverse models, for instance, the nonlinear Tikhonov method and the total variation (TV) method. In this paper, we have investigated three identification approaches that couple with nonlinear inverse models, a total of six combinations, to estimate the Cesium-137 emission rate following the Fukushima accident. Among them, the L-curve approach always provides the largest parameter value. The regularization works even with a small parameter, especially the TV method. The GCV approach brings the best Tikhonov estimation of the total amount, but the cost-α simultaneous minimization results in nearly the best metrics of concentration simulations in all combinations. Using the same identification approach, the TV estimate provides both the peak and constant emissions that is closer to the well-regarded subjective estimate.
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