Estimating Solution Smoothness and Data Noise with Tikhonov Regularization

Abstract In this paper, a Cauchy problem for the Laplace equation is considered. We develop a modified Tikhonov regularization method based on Hermite expansion to deal with the ill posed-ness of the problem. The regularization parameter is determined by a discrepancy principle. For various smoothness conditions, the solution process of the method is uniform and the convergence rate can be obtained self-adaptively. Numerical tests are also carried out to verify the effectiveness of the method.

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Bathymetry inversion using the modified gravity-geologic method: application of the rectangular prism model and Tikhonov regularization

Applied Geophysics ◽

10.1007/s11770-020-0821-y ◽

2020 ◽

Vol 17 (3) ◽

pp. 377-389

Author(s):

Jian Xing ◽

Xin-Xi Chen ◽

Long Ma

Keyword(s):

Tikhonov Regularization ◽

Modified Gravity ◽

Rectangular Prism ◽

Prism Model

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A New Hybrid Inversion Method for 2D Nuclear Magnetic Resonance Combining TSVD and Tikhonov Regularization

Journal of Imaging ◽

10.3390/jimaging7020018 ◽

2021 ◽

Vol 7 (2) ◽

pp. 18

Author(s):

Germana Landi ◽

Fabiana Zama ◽

Villiam Bortolotti

Keyword(s):

Nuclear Magnetic Resonance ◽

Magnetic Resonance ◽

Tikhonov Regularization ◽

Large Scale ◽

Regularization Parameter ◽

High Sensitivity ◽

Inversion Method ◽

Nmr Relaxometry ◽

Value Decomposition ◽

Nuclear Magnetic

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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A fractional Tikhonov regularization method for an inverse backward and source problems in the time-space fractional diffusion equations

Chaos Solitons & Fractals ◽

10.1016/j.chaos.2021.111127 ◽

2021 ◽

Vol 150 ◽

pp. 111127

Author(s):

Smina Djennadi ◽

Nabil Shawagfeh ◽

Omar Abu Arqub

Keyword(s):

Tikhonov Regularization ◽

Regularization Method ◽

Fractional Diffusion ◽

Diffusion Equations ◽

Tikhonov Regularization Method ◽

Fractional Diffusion Equations ◽

Time Space

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The Tikhonov regularization for vector equilibrium problems

Computational Optimization and Applications ◽

10.1007/s10589-020-00258-z ◽

2021 ◽

Author(s):

Lam Quoc Anh ◽

Tran Quoc Duy ◽

Le Dung Muu ◽

Truong Van Tri

Keyword(s):

Tikhonov Regularization ◽

Equilibrium Problems ◽

Vector Equilibrium Problems ◽

Vector Equilibrium

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Computation of distribution of relaxation times by Tikhonov regularization for Li ion batteries: usage of L-curve method

Scientific Reports ◽

10.1038/s41598-021-91871-3 ◽

2021 ◽

Vol 11 (1) ◽

Author(s):

T. Paul ◽

P. W. Chi ◽

Phillip M. Wu ◽

M. K. Wu

Keyword(s):

Tikhonov Regularization ◽

Relaxation Times ◽

Detection Methods ◽

Similar Process ◽

Fredholm Integral Equations ◽

Li Ion Batteries ◽

Impedance Data ◽

Curve Method ◽

Li Ion ◽

Process Detection

AbstractIn this paper, the distribution of relaxation times (DRTs) functions are calculated numerically in Matlab for synthetic impedance data from single parallel $$RC$$ RC circuit and two parallel $$RC$$ RC circuits connected in series, experimental impedance data from supercapacitors and α-LiFeO2 anode based Li ion batteries. The quality of the impedance data is checked with the Kramers–Krönig (KK) relations. The DRTs are calculated within the KK compatible regime for all the systems using Tikhonov regularization (TR) method. Here we use a fast and simple L-curve method to estimate the TR parameter (λ) for regularization of the Fredholm integral equations of first kind in impedance. Estimation of the regularization parameters are performed effectively from the offset of the global corner of the L-curve rather than simply using the global corner. The physical significances of DRT peaks are also discussed by calculating the effective resistances and capacitances coupled with peak fitting program. For instance, two peaks in the DRTs justify the electrical double layer capacitance and ion diffusion phenomena for supercapacitors in low to intermediate frequencies respectively. Moreover, the surface film effect, Li/electrolyte and electrode/electrolyte charge transfer related processes are identified for α-LiFeO2 anode based Li-ion batteries. This estimation of the offset of the global corner extends the L-curve approach coupled with the Tikhonov regularization in the field of electrochemistry and can also be applied in similar process detection methods.

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