Conjugate Gradient Method for the Solution of Inverse Problems: Application in Linear Seismic Tomography

Brufati, Tuanny Elyz Brandeleiro; Oliveira, Saulo P.; Bassrei, Amin

doi:10.5540/tema.2015.016.03.0185

Cited by 3 publications

(2 citation statements)

References 17 publications

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“…The condition number depends on the properties of a particular integral operator and can indeed be large in many applications such as image processing [14][15][16][17], geophysics [18][19][20][21] and high energy physics [22][23][24][25][26]. However, as it is shown in section 2, the condition vector norms are used in eq.…”

Section: Proposed Methodsmentioning

confidence: 99%

A new method for obtaining a Born cross section using visible cross section data from e+e− colliders

Gribanov

Попов

2021

J. High Energ. Phys.

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In this paper, we propose a new method for obtaining a Born cross section using visible cross section data. It is assumed that the initial state radiation is taken into account in a visible cross section, while in a Born cross section this effect is ommited. Since the equation that connects Born and visible cross sections is an integral equation of the first kind, the problem of finding its numerical solution is ill-posed. Various regularization-based approaches are often used to solve ill-posed problems, since direct methods usually do not lead to an acceptable result. However, in this paper it is shown that a direct method can be successfully used to numerically solve the considered equation under the condition of a small beam energy spread and uncertainty. This naive method is based on finding a numerical solution to the integral equation by reducing it to a system of linear equations. The naive method works well because the kernel of the integral operator is a rapidly decreasing function of the variable x. This property of the kernel leads to the fact that the condition number of the matrix of the system of linear equations is of the order of unity, which makes it possible to neglect the ill-posedness of the problem when the above condition is satisfied. The advantages of the naive method are its model independence and the possibility of obtaining the covariance matrix of a Born cross section in a simple way.It should be noted that there are already a number of methods for obtaining a Born cross section using visible cross section data, which are commonly used in e+e− experiments. However, at least some of these methods have various disadvantages, such as model dependence and relative complexity of obtaining a Born cross section covariance matrix. It should be noted that this paper focuses on the naive method, while conventional methods are hardly covered. The paper also discusses solving the problem using the Tikhonov regularization, so that the reader can better understand the difference between regularized and non-regularized solutions. However, it should be noted that, in contrast to the naive method, regularization methods can hardly be used for precise obtaining of a Born cross section. The reason is that the regularized solution is biased and the covariance matrix of this solution do not represent the correct covariance matrix of a Born cross section.

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Section: Proposed Methodsmentioning

confidence: 99%

A new method for obtaining a Born cross section using visible cross section data from e+e− colliders

Gribanov

Попов

2021

J. High Energ. Phys.

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show abstract

Section: Proposed Methodsmentioning

confidence: 99%

A new method for obtaining a Born cross section using visible cross section data from $e^+e^-$ colliders

Gribanov,

Popov

2021

Preprint

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In this article, we propose a new method for obtaining a Born cross section using visible cross section data. It is assumed that the initial state radiation is taken into account in a visible cross section, while in a Born cross section this effect is ommited. Since the equation that connects Born and visible cross sections is an integral equation of the first kind, the problem of finding its numerical solution is ill-posed. Various regularizationbased approaches are often used to solve ill-posed problems, since direct methods usually do not lead to an acceptable result. However, in this article it is shown that a direct method can be successfully used to numerically solve the considered equation under the condition of a small beam energy spread and uncertainty. This naive method is based on finding a numerical solution to the integral equation by reducing it to a system of linear equations. The naive method works well because the kernel of the integral operator is a rapidly decreasing function of the variable x. This property of the kernel leads to the fact that the condition number of the matrix of the system of linear equations is of the order of unity, which makes it possible to neglect the ill-posedness of the problem when the above condition is satisfied. The advantages of the naive method are its model independence and the possibility of obtaining the covariance matrix of a Born cross section in a simple way.It should be noted that there are already a number of methods for obtaining a Born cross section using visible cross section data, which are commonly used in e + e − experiments. However, at least some of these methods have various disadvantages, such as model dependence and relative complexity of obtaining a Born cross section covariance matrix. It should be noted that this article focuses on the naive method, while conventional methods are hardly covered. The article also discusses solving the problem using the Tikhonov regularization, so that the reader can better understand the difference between regularized and non-regularized solutions. However, it should be noted that, in contrast to the naive method, regularization methods can hardly be used for precise obtaining of a Born cross section. The reason is that the regularized solution is biased and the covariance matrix of this solution do not represent the correct covariance matrix of a Born cross section.

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Generative models based on eigendecomposition for dense ray tracing

Oliveira

Castelán

Baltazar

2022

The Journal of the Acoustical Society of America

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In this work, we present an algorithm capable of emulating ray trajectories that obey the least action principle. The method is based on spectral decomposition of geometric shapes taken from a set of raypaths. As the proposed work relies on shape analysis, it is agnostic on the underlying physics of raypath generation. As such, it is independent of the ray tracing method used to generate the training paths. In cases of mildly heterogeneous media or scenarios with a limited number of geometrical scatters, we show that the algorithm is capable of efficiently populating a given scenario with a dense array of emulated rays whose trajectories are in close agreement with actual rays. We argue that the algorithm also serves as an effective method capable of detecting regions where ray variation is high, such as when possible shadow zones are present.

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Conjugate Gradient Method for the Solution of Inverse Problems: Application in Linear Seismic Tomography

Cited by 3 publications

References 17 publications

A new method for obtaining a Born cross section using visible cross section data from e+e− colliders

A new method for obtaining a Born cross section using visible cross section data from e+e− colliders

A new method for obtaining a Born cross section using visible cross section data from $e^+e^-$ colliders

Generative models based on eigendecomposition for dense ray tracing

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