A study on regularization parameter choice for interior tomography based on truncated Hilbert transform

2011 ◽  
Author(s):  
Junfeng Wu ◽  
Xuanqin Mou ◽  
Shaojie Tang
Keyword(s):  
Hilbert Transform ◽  
Regularization Parameter ◽  
Parameter Choice

scholarly journals Regularization of backward time-fractional parabolic equations by Sobolev-type equations

2020 ◽  
Vol 28 (5) ◽  
pp. 659-676
Author(s):  
Dinh Nho Hào ◽  
Nguyen Van Duc ◽  
Nguyen Van Thang ◽  
Nguyen Trung Thành
Keyword(s):  
Error Estimates ◽  
Parabolic Equations ◽  
Regularization Parameter ◽  
A Priori ◽  
Sobolev Type ◽  
A Posteriori ◽  
Parameter Choice ◽  
Numerical Tests ◽  
Ill Posed ◽  
Parameter Choice Rules

AbstractThe problem of determining the initial condition from noisy final observations in time-fractional parabolic equations is considered. This problem is well known to be ill-posed, and it is regularized by backward Sobolev-type equations. Error estimates of Hölder type are obtained with a priori and a posteriori regularization parameter choice rules. The proposed regularization method results in a stable noniterative numerical scheme. The theoretical error estimates are confirmed by numerical tests for one- and two-dimensional equations.

scholarly journals The Impact of the Discrepancy Principle on the Tikhonov-Regularized Solutions with Oversmoothing Penalties

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 331
Author(s):  
Bernd Hofmann ◽  
Christopher Hofmann
Keyword(s):  
Case Studies ◽  
Convergence Rates ◽  
Regularization Parameter ◽  
A Priori ◽  
Discrepancy Principle ◽  
Parameter Choice ◽  
Operator Equations ◽  
Analytical Results ◽  
Ill Posed ◽  
The Impact

This paper deals with the Tikhonov regularization for nonlinear ill-posed operator equations in Hilbert scales with oversmoothing penalties. One focus is on the application of the discrepancy principle for choosing the regularization parameter and its consequences. Numerical case studies are performed in order to complement analytical results concerning the oversmoothing situation. For example, case studies are presented for exact solutions of Hölder type smoothness with a low Hölder exponent. Moreover, the regularization parameter choice using the discrepancy principle, for which rate results are proven in the oversmoothing case in in reference (Hofmann, B.; Mathé, P. Inverse Probl. 2018, 34, 015007) is compared to Hölder type a priori choices. On the other hand, well-known analytical results on the existence and convergence of regularized solutions are summarized and partially augmented. In particular, a sketch for a novel proof to derive Hölder convergence rates in the case of oversmoothing penalties is given, extending ideas from in reference (Hofmann, B.; Plato, R. ETNA. 2020, 93).

On robust estimation of discrete Hilbert transform of noisy data

Geophysics ◽  
2013 ◽  
Vol 78 (6) ◽  
pp. V239-V249 ◽  
Author(s):  
Indrajit G. Roy
Keyword(s):  
Conformal Mapping ◽  
Robust Estimation ◽  
Hilbert Transform ◽  
Regularization Parameter ◽  
Noisy Data ◽  
Geophysical Data ◽  
Fourier Domain ◽  
Waveform Data ◽  
Discrete Hilbert Transform ◽  
Noise Contamination

We developed a novel technique of robust estimation of the discrete Hilbert transform (DHT) of noisy geophysical data. The technique used the sinc method, in which the data were transformed via conformal mapping and the sinc bases were determined by solving a linear matrix equation. A transformation rule was presented for selecting a suitable conformal mapping function that would transform the class of geophysical data set in an appropriate interval range. A novel regularization technique was designed to obtain a robust solution of sinc bases when the data contained noise, in which an optimal regularization parameter was obtained in an automated way using a 1D optimization scheme. The technique of selecting the optimal value of the regularization parameter required no a priori knowledge about the level of noise contamination in the data. Numerical experiments were conducted on synthetically generated and published field data sets with a varying level of noise contamination to test the performance of the scheme. The results obtained using the proposed technique of DHT and those obtained by a standard Fourier domain technique were compared, and it was established that the proposed scheme of discrete Hilbert transformation performed better than that of the standard Fourier domain technique, for noise free and noisy data. The scheme was applied successfully on potential field and infrasound waveform data and also in estimating instantaneous frequency of nonstationary ultrasonic waveform data, which suggested applicability of the scheme to a wide class of geophysical data.

scholarly journals ON NUMERICAL REALIZATION OF QUASIOPTIMAL PARAMETER CHOICES IN (ITERATED) TIKHONOV AND LAVRENTIEV REGULARIZATION

2009 ◽  
Vol 14 (1) ◽  
pp. 99-108 ◽  
Author(s):  
Toomas Raus ◽  
Uno Hämarik
Keyword(s):  
Hilbert Spaces ◽  
Noise Level ◽  
Regularization Parameter ◽  
Numerical Realization ◽  
A Posteriori ◽  
Parameter Choice ◽  
Linear Combinations ◽  
Choice Alternative ◽  
Lavrentiev Regularization ◽  
Ill Posed

We consider linear ill‐posed problems in Hilbert spaces with noisy right hand side and given noise level. For approximation of the solution the Tikhonov method or the iterated variant of this method may be used. In self‐adjoint problems the Lavrentiev method or its iterated variant are used. For a posteriori choice of the regularization parameter often quasioptimal rules are used which require computing of additionally iterated approximations. In this paper we propose for parameter choice alternative numerical schemes, using instead of additional iterations linear combinations of approximations with different parameters.

A wavelet method for numerical fractional derivative with noisy data

2016 ◽  
Vol 14 (05) ◽  
pp. 1650038 ◽  
Author(s):  
Xiangtuan Xiong ◽  
Qiang Cheng ◽  
Yanfeng Kong ◽  
Jin Wen
Keyword(s):  
Fractional Derivative ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Large Change ◽  
Stability Estimates ◽  
Parameter Choice ◽  
Wavelet Regularization ◽  
Ill Posed ◽  
Meyer Wavelet ◽  
Parameter Choice Rules

Numerical fractional differentiation is a classical ill-posed problem in the sense that a small perturbation in the data can cause a large change in the fractional derivative. In this paper, we consider a wavelet regularization method for solving a reconstruction problem for numerical fractional derivative with noise. A Meyer wavelet projection regularization method is given, and the Hölder-type stability estimates under both apriori and aposteriori regularization parameter choice rules are obtained. Some numerical examples show that the method works well.

scholarly journals The Truncation Regularization Method for Identifying the Initial Value on Non-Homogeneous Time-Fractional Diffusion-Wave Equations

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1007 ◽  
Author(s):  
Fan Yang ◽  
Qu Pu ◽  
Xiao-Xiao Li ◽  
Dun-Gang Li
Keyword(s):  
Wave Equations ◽  
Regularization Parameter ◽  
A Priori ◽  
Initial Boundary ◽  
Fractional Diffusion ◽  
Value Functions ◽  
Parameter Choice ◽  
Initial Value ◽  
Diffusion Wave ◽  
Initial Boundary Value

In the essay, we consider an initial value question for a mixed initial-boundary value of time-fractional diffusion-wave equations. This matter is an ill-posed problem; the solution relies discontinuously on the measured information. The truncation regularization technique is used for restoring the initial value functions. The convergence estimations are given in a priori regularization parameter choice regulations and a posteriori regularization parameter choice regulations. Numerical examples are given to demonstrate this is effective and practicable.

scholarly journals An optimal order yielding discrepancy principle for simplified regularization of ill-posed problems in Hilbert scales

2003 ◽  
Vol 2003 (39) ◽  
pp. 2487-2499 ◽  
Author(s):  
Santhosh George ◽  
M. Thamban Nair
Keyword(s):  
Approximate Solution ◽  
Hilbert Spaces ◽  
Regularization Parameter ◽  
Optimal Order ◽  
Discrepancy Principle ◽  
Parameter Choice ◽  
Bounded Linear ◽  
Choice Strategy ◽  
Ill Posed ◽  
Parameter Choice Strategy

Recently, Tautenhahn and Hämarik (1999) have considered a monotone rule as a parameter choice strategy for choosing the regularization parameter while considering approximate solution of an ill-posed operator equationTx=y, whereTis a bounded linear operator between Hilbert spaces. Motivated by this, we propose a new discrepancy principle for the simplified regularization, in the setting of Hilbert scales, whenTis a positive and selfadjoint operator. When the datayis known only approximately, our method provides optimal order under certain natural assumptions on the ill-posedness of the equation and smoothness of the solution. The result, in fact, improves an earlier work of the authors (1997).

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