• Geophysics and Geophysical Exploration
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• v.10 no.2
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• pp.124-130
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• 2007

Least squares ($l^2$ norm) solutions of seismic inversion tend to be very sensitive to data points with large errors. The $l^1$ norm minimization gives more robust solutions, but usually with higher computational cost. Iteratively reweighted least squares (IRLS) method gives efficient approximate solutions of these $l^1$ norm problems. I propose an efficient implementation of the IRLS method for a hybrid $l^1/l^2$ minimization problem that behaves as $l^2$ norm fit for small residual and $l^1$ norm fit for large residuals. The proposed algorithm shows more robust characteristics to the decision of the threshold value than the l1 norm IRLS inversion does with respect to the threshold value to avoid singularity.

• 한국지구물리탐사학회:학술대회논문집
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• 2007.06a
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• pp.124-129
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• 2007

For seismic imaging and inversion, the inverted image depends on how we define the objective function. ${\ell}^1$-norm is more robust than ${\ell}^2$-norm. However, it is difficult to apply the Newton-type algorithm directly because the partial derivative for ${\ell^1$-norm has a singularity. In our paper, to overcome the difficulties of singularities, Huber function given by hybrid ${\ell}^1/{\ell}^2$-norm is used. We tested the robustness of our new object function with several noisy data set. Numerical results show that the new objective function is more robust to band limited spiky noise than the conventional object function.

• Structural Engineering and Mechanics
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• v.85 no.1
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• pp.119-133
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• 2023

Impact event is the key factor influencing the operational state of the mechanical equipment. Additionally, nonlinear factors existing in the complex mechanical equipment which are currently attracting more and more attention. Therefore, this paper proposes a novel hybrid-separate identification strategy to solve the force identification problem of the nonlinear structure under impact excitation. The 'hybrid' means that the identification strategy contains both l1-norm (sparse) and l2-norm regularization methods. The 'separate' means that the nonlinear response part only generated by nonlinear force needs to be separated from measured response. First, the state-of-the-art two-step iterative shrinkage/thresholding (TwIST) algorithm and sparse representation with the cubic B-spline function are developed to solve established normalized sparse regularization model to identify the accurate impact force and accurate peak value of the nonlinear force. Then, the identified impact force is substituted into the nonlinear response separation equation to obtain the nonlinear response part. Finally, a reduced transfer equation is established and solved by the classical Tikhonove regularization method to obtain the wave profile (variation trend) of the nonlinear force. Numerical and experimental identification results demonstrate that the novel hybrid-separate strategy can accurately and efficiently obtain the nonlinear force and impact force for the nonlinear structure.

• Journal of applied mathematics & informatics
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• v.24 no.1_2
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• pp.179-190
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• 2007

In this paper, we consider the general variational inequality GVI(F, g, C), where F and g are mappings from a Hilbert space into itself and C is the fixed point set of a nonexpansive mapping. We suggest and analyze a new modified hybrid steepest-descent method of type method $u_{n+l}=(1-{\alpha}+{\theta}_{n+1})Tu_n+{\alpha}u_n-{\theta}_{n+1g}(Tu_n)-{\lambda}_{n+1}{\mu}F(Tu_n),\;n{\geq}0$. for solving the general variational inequalities. The sequence $\{x_n}\$ is shown to converge in norm to the solutions of the general variational inequality GVI(F, g, C) under some mild conditions. Application to constrained generalized pseudo-inverse is included. Results proved in the paper can be viewed as an refinement and improvement of previously known results.

• The Transactions of The Korean Institute of Electrical Engineers
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• v.56 no.10
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• pp.1859-1864
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• 2007

Existing barcode signal restoration algorithms are not robust to unmodeled outliers that may exist in scanned barcode images due to scratches, dirts, etc. In this paper, we describe a robust barcode signal restoration algorithm that uses the hybrid $L_1-L_2$ norm as a similarity measure. To optimze the similarity measure, we propose a modified iterative reweighted least squares algorithm based on the one step minimization of a quadratic surrogate function. In the simulations and experiments with barcode images, the proposed method showed better robustness than the conventional MSE based method. In addition, the proposed method converged quickly during optimization process.