An approximate solution of a Fredholm integral equation of the first kind by the residual method

Танана, В. П.; Vishnyakov, E. Yu.; Sidikova, A. I.

doi:10.1134/s1995423916010080

Cited by 13 publications

(7 citation statements)

References 4 publications

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“…The practicability of the optimal regularization parameter is verified based on the two-dimensional wave equation. Tanana et al [33] propose a variational regularization method with a regularization parameter from the residual principle and reducing the problem to a system of linear algebraic equations. The accuracy of the approximate solution is estimated with allowance for the error of the finite dimensional approximation of the problem.…”

Section: Regularization Methodsmentioning

confidence: 99%

An overview of numerical methods for the first kind Fredholm integral equation

Zhang

2019

SN Appl. Sci.

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In the field of engineering technology, many problems can be transformed into the first kind Fredholm integral equation, which has a prominent feature called "ill-posedness". This property makes it difficult to find the analytical solution of first kind Fredholm integral equation. Therefore, how to find the numerical solution of first kind Fredholm integral equation has been a common concern of domestic and overseas scholars in recent years. In this article, various numerical solution methods of first kind Fredholm integral equation are introduced in detail. First, the existence and convergence of the solution of the integral equation are given. Second, the current mainstream numerical methods, such as regularization method, wavelet analysis method and multilevel iteration method are introduced in detail. Finally, we presented a concise overview of the numerical method of first kind Fredholm integral equation.

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Section: Regularization Methodsmentioning

confidence: 99%

An overview of numerical methods for the first kind Fredholm integral equation

Zhang

2019

SN Appl. Sci.

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“…Violations of stability are much harder to remedy because they imply that a small disturbance in the data leads to a large disturbance in the estimated solution [2] , [3] , [4] , [5] . Various methods and algorithms for solving IP “inverse problems” have been explained and used in [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] . The success of these methods and algorithms is largely based on understanding and analyzing the mathematical problems related to the declarations of the properties of these IP “inverse problems” and identifying specific difficulties in solving them [17] , [18] , [19] , [20] , [21] , [22] .…”

Section: Introductionmentioning

confidence: 99%

Parallel multigrid method for solving inverse problems

Al-Mahdawi

Sidikova²,

Alkattan³

et al. 2022

MethodsX

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“…For certain subclasses of this problem, such as density deconvolution (Delaigle 2008) good methods exist and can achieve optimal convergence rates as the number of observations increases (Carroll and Hall 1988). However, generally applicable approaches which do not assume a particular form of g typically require discretization of the domain, X, which restricts their applications to low-dimensional scenarios, and often assume a piecewiseconstant solution (Ma 2011;Koenker and Mizera 2014;Tanana, Vishnyakov, and Sidikova 2016;Burger, Resmerita, and Benning 2019;Yang et al 2020). This is the case for the popular expectation-maximization smoothing (EMS) scheme (Silverman et al 1990), a smoothed version of the infinite-dimensional expectation-maximization algorithm of Kondor (1983).…”

Section: Introductionmentioning

confidence: 99%

A Particle Method for Solving Fredholm Equations of the First Kind

Crucinio

Doucet

Johansen

2021

Journal of the American Statistical Association

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Fredholm integral equations of the first kind are the prototypical example of ill-posed linear inverse problems. They model, among other things, reconstruction of distorted noisy observations and indirect density estimation and also appear in instrumental variable regression. However, their numerical solution remains a challenging problem. Many techniques currently available require a preliminary discretization of the domain of the solution and make strong assumptions about its regularity. For example, the popular expectation-maximization smoothing (EMS) scheme requires the assumption of piecewise constant solutions which is inappropriate for most applications. We propose here a novel particle method that circumvents these two issues. This algorithm can be thought of as a Monte Carlo approximation of the EMS scheme which not only performs an adaptive stochastic discretization of the domain but also results in smooth approximate solutions. We analyze the theoretical properties of the EMS iteration and of the corresponding particle algorithm. Compared to standard EMS, we show experimentally that our novel particle method provides state-of-the-art performance for realistic systems, including motion deblurring and reconstruction of cross-sectional images of the brain from positron emission tomography.

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An approximate solution of a Fredholm integral equation of the first kind by the residual method

Cited by 13 publications

References 4 publications

An overview of numerical methods for the first kind Fredholm integral equation

An overview of numerical methods for the first kind Fredholm integral equation

Parallel multigrid method for solving inverse problems

A Particle Method for Solving Fredholm Equations of the First Kind

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