Solving second order non-linear hyperbolic PDEs using generalized finite difference method (GFDM)

Ureña, Francisco; Gavete, L.; García, A.; Benito, Juan José; Vargas, A.M.

doi:10.1016/j.cam.2019.05.028

Cited by 20 publications

(7 citation statements)

References 28 publications

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“…Therefore, the finite difference method [ 19 , 20 ]] is used as the discrete differential operator to solve Eq (10) , and the discrete expression of image enhancement is obtained as shown in Eq (11) : Where, ∇ N ( I t ) = I x,y−1 − I x,y , ∇ S ( I t ) = I x,y+1 − I x,y , ∇ E ( I t ) = I x−1,y+1 − I x,y , ∇ W ( I t ) = I x+1,y − I x,y is four divergence formulas, whose values are the partial derivatives of the current pixel in the directions of southeast, northwest, and northwest, respectively. cN x,y = exp(−‖∇ N ( I t )‖ 2 / k 2 ), cS x,y = exp(−‖∇ S ( I t )‖ 2 / k 2 ), cE x,y = exp(−‖∇ E ( I t )‖ 2 / k 2 ), cW x,y = exp(−‖∇ W ( I t )‖ 2 / k 2 ) is the thermal conductivity in the four directions of southeast, northwest, I is the current image.…”

Section: Methodsmentioning

confidence: 99%

“…Where div is the divergence operator, r is the gradient operator, t represents the iteration time, and ρ is the diffusion coefficient function of the image at time t, which is used to control the anisotropic characteristics of the diffusion process. Therefore, the finite difference method [19,20]] is used as the discrete differential operator to solve Eq (10), and the discrete expression of image enhancement is obtained as shown in Eq (11):…”

Section: Plos Onementioning

confidence: 99%

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Low illumination fog noise image denoising method based on ACE-GPM

Li,

Zhou,

Wang

2024

PLoS ONE

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The Perona-Malik (P-M) model exhibits deficiencies such as noise amplification, new noise introduction, and significant gradient effects when processing noisy images. To address these issues, this paper proposes an image-denoising algorithm, ACE-GPM, which integrates an Automatic Color Equalization (ACE) algorithm with a gradient-adjusted P-M model. Initially, the ACE algorithm is employed to enhance the contrast of low-light images obscured by fog and noise. Subsequently, the Otsu method, a technique to find the optimal threshold based on between-class variance, is applied for precise segmentation, enabling more accurate identification of different regions within the image. After that, distinct gradients enhance the image’s foreground and background via an enhancement function that accentuates edge and detailed information. The denoising process is finalized by applying the gradient P-M model, employing a gradient descent approach to further emphasize image edges and details. Experimental evidence indicates that the proposed ACE-GPM algorithm not only elevates image contrast and eliminates noise more effectively than other denoising methods but also preserves image details and texture information, evidenced by an average increase of 0.42 in the information entropy value. Moreover, the proposed solution achieves these outcomes with reduced computational resource expenditures while maintaining high image quality.

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Section: Methodsmentioning

confidence: 99%

Section: Plos Onementioning

confidence: 99%

Low illumination fog noise image denoising method based on ACE-GPM

Li,

Zhou,

Wang

2024

PLoS ONE

View full text Add to dashboard Cite

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“…Many works have been devoted to the numerical solutions of semilinear elliptic problems such as finite element method (FEM) [2,3], finite difference method [4], finite volume element method [5] and discontinuous Galerkin method [6]. Recently, some collocation meshless (or meshfree) methods [7,8], Galerkintype meshless method [8] and generalized finite difference method [9,10] have been developed to solve the semilinear PDEs. Unlike mesh-based numerical methods, the shape functions used in the meshless methods [11][12][13][14] are linkage with nodes (or particles) scattered in the underlying computational domain, which reduces the dependence on the mesh.…”

Section: Introductionmentioning

confidence: 99%

A Nitsche-Based Element-Free Galerkin Method for Semilinear Elliptic Problems

Li¹

2024

AAMM

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A Nitsche-based element-free Galerkin (EFG) method for solving semilinear elliptic problems is developed and analyzed in this paper. The existence and uniqueness of the weak solution for semilinear elliptic problems are proved based on a condition that the nonlinear term is an increasing Lipschitz continuous function of the unknown function. A simple iterative scheme is used to deal with the nonlinear integral term. We proved the existence, uniqueness and convergence of the weak solution sequence for continuous level of the simple iterative scheme. A commonly used assumption for approximate space, sometimes called inverse assumption, is proved. Optimal order error estimates in L 2 and H 1 norms are proved for the linear and semilinear elliptic problems. In the actual numerical calculation, the characteristic distance h does not appear explicitly in the parameter β introduced by the Nitsche method. The theoretical results are confirmed numerically.

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“…The use of second-order approximations with the GFDM has received attention in the last two decades and has been successfully consolidated. Its applications vary among many types of problems, such us: seismic wave propagation, 7 hyperbolic nonlinear equations, 8 transient heat flow in anisotropic composites, 9 mathematical models of tumor growth, 10 inverse heat conduction problem, 11 discontinuous crack-faces, 12 plate bending problems, 13 etc.…”

Section: Introductionmentioning

confidence: 99%

A strategy to avoid ill‐conditioned stars in the generalized finite difference method for solving one‐dimensional problems

Ferreira

Ureña

Ramos

2021

Comp and Math Methods

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In this paper, we solve linear boundary value problems of second-order in ordinary differential equations with the generalized finite difference method and compare the numerical accuracy for different orders of approximations. We develop a strategy for dealing with ill-conditioned stars based on the condition number of the matrix of derivatives. In addition, we consider a scheme implemented with parallel processing for the formation of the stars and the calculation of the derivatives. We present some examples with high gradients in irregular discretizations exaggerated on purpose, to highlight the efficiency of the proposed strategy. K E Y W O R D Sfourth-order approximations, generalized finite difference method, ill-conditioned stars, parallel processing 1 Miguel Ureña, PhD. contributed to this article in his personal capacity. The views expressed are his own and do not necessarily represent the views of Statistics Spain.

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Solving second order non-linear hyperbolic PDEs using generalized finite difference method (GFDM)

Cited by 20 publications

References 28 publications

Low illumination fog noise image denoising method based on ACE-GPM

Low illumination fog noise image denoising method based on ACE-GPM

A Nitsche-Based Element-Free Galerkin Method for Semilinear Elliptic Problems

A strategy to avoid ill‐conditioned stars in the generalized finite difference method for solving one‐dimensional problems

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