• Geophysics and Geophysical Exploration
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• v.7 no.3
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• pp.191-196
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• 2004

Seismic waveform inversion can be solved by using the classical Gauss-Newton method, which needs to construct the huge Hessian by the directly computed Jacobian. The property of Hessian mainly depends upon a source and receiver aperture, a velocity model, an illumination Bone and a frequency content of source wavelet. In this paper, we try to invert the Marmousi seismic data by controlling the huge Hessian appearing in the Gauss-Newton method. Wemake the two kinds of he approximate Hessian. One is the banded Hessian and the other is the approximate Hessian with automatic gain function. One is that the 1st updated velocity model from the banded Hessian is nearly the same of the result from the full approximate Hessian. The other is that the stability using the automatic gain function is more improved than that without automatic gain control.

• 한국지구물리탐사학회:학술대회논문집
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• 2007.06a
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• pp.329-336
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• 2007

For scaling of the gradient of misfit function, we develop a new pseudo-Hessian matrix constructed by combining amplitude field and pseudo-Hessian matrix. Since pseudo- Hessian matrix neglects the calculation of the zero-lag auto-correlation of impulse responses in the approximate Hessian matrix, the pseudo-Hessian matrix has a limitation to scale the gradient of misfit function compared to the approximate Hessian matrix. To validate the new pseudo- Hessian matrix, we perform frequency-domain elastic full waveform inversion using this Hessian matrix. By synthetic experiments, we show that the new pseudo-Hessian matrix can give better convergence to the true model than the old one does. Furthermore, since the amplitude fields are intrinsically obtained in forward modeling procedure, we do not have to pay any extra cost to compute the new pseudo-Hessian. We think that the new pseudo-Hessian matrix can be used as an alternative of the approximate Hessian matrix of the Gauss-Newton method.

• Communications of the Korean Mathematical Society
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• v.22 no.3
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• pp.419-428
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• 2007

In this paper, we will investigate the Hessian geometry of the homogeneous domain over the hypersurface given by a function F : $\mathbb{R}^n{\rightarrow}\mathbb{R}$ with ${\mid}{\det}\;DdF\mid=1$.

• East Asian mathematical journal
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• v.39 no.5
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• pp.529-536
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• 2023

This paper introduces the Frobenius endomophisms on the binary Hessian curves. It provides an efficient and computable homomorphism for computing point multiplication on binary Hessian curves. As an application, it is possible to construct the GLV method combined with the Frobenius endomorphism to accelerate scalar multiplication over the curve.

• Geophysics and Geophysical Exploration
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• v.22 no.4
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• pp.195-201
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• 2019

The logarithmic objective function used in waveform inversion minimizes the logarithmic differences between the observed and modeled data. Laplace-domain waveform inversions usually adopt the logarithmic objective function and the diagonal elements of the pseudo-Hessian for optimization. In this case, we apply the Levenberg-Marquardt algorithm to prevent the diagonal elements of the pseudo-Hessian from being zero or near-zero values. In this study, we analyzed the diagonal elements of the pseudo-Hessian of the logarithmic objective function and showed that there is no zero or near-zero value in the diagonal elements of the pseudo-Hessian for acoustic waveform inversion in the Laplace domain. Accordingly, we do not need to apply the Levenberg-Marquardt algorithm when we regularize the gradient direction using the pseudo-Hessian of the logarithmic objective function. Numerical examples using synthetic and field datasets demonstrate that we can obtain inversion results without applying the Levenberg-Marquardt method.

• Journal of the Korean Institute of Telematics and Electronics
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• v.14 no.4
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• pp.22-31
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• 1977

A general equation by which the Hessian matrix of an error function can be determined directly, has been derived. It was verified to be useful in optimization processes that include the Hessian matrix. A few design examples had shown that this method had accelerated the processes of finding the minimums. The advantage of this technique is the possibility of optimizing functions that composed of both the phases and magnitudes.

• Proceedings of the KSEEG Conference
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• 2002.04a
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• pp.159-162
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• 2002

We tried to obtain an initial velocity model for prestack depth migration via waveform inversion. For application of any field data we chose a smooth background layered velocity model (v=v0 + k x z) as an initial velocity model. Newton type waveform inversion needs to invert huge Hessian matrix. In order to compute full Hessian matrix arising from full aperture data and full illumination zone, we meet insurmountable difficulties of paying astronomical computing cost. For the layered media, approximate Hessian emerging from single shot aperture data can be used repeatedly for split spread source configuration. In our work of using this Hessian characteristic of layered media we attempted to obtain the approximate velocity model as close as possible to the true velocity model in first iteration.

• The Journal of the Korea institute of electronic communication sciences
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• v.5 no.2
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• pp.138-144
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• 2010

This paper presents the experimental analysis of a F-Hessian SIFT-Based Railroad Level-Crossing Safety Vision System. Region of surveillance, region of interests, data matching based on extracting feature points has been examined under the laboratory condition by the model rig on a small scale. Real-time system were observed by using SIFT based on F-Hessian feature tracking method and other common algorithm.

• Bulletin of the Korean Mathematical Society
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• v.61 no.4
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• pp.999-1017
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• 2024

In this paper, we study the Neumann problem for the complex Hessian quotient equation ${\frac{{\sigma}_k({\tau}{\Delta}uI+{\partial}{\bar{\partial}u)}}{{\sigma}_l({\tau}{\Delta}uI+{\partial}{\bar{\partial}u)}}}={\psi}$ with 0 ≤ 𝑙 < k ≤ n. We prove a priori estimate and global C¹ estimates, in particular, we use the double normal second derivatives on the boundary to establish the global C² estimates and prove the existence and the uniqueness for the Neumann problem of the above complex Hessian quotient equation.

• Journal of the Korean Mathematical Society
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• v.59 no.1
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• pp.53-69
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• 2022

In this paper, we are concerned with a class of mixed Hessian curvature equations with non-degeneration. By using the maximum principle and constructing an auxiliary function, we obtain the interior gradient estimate of (k - 1)-admissible solutions.