A greedy sparse meshless method for solving heat conduction problems

Fadaei, Yasin; Moghadam, Mahmoud Mohseni

doi:10.1007/s00366-016-0494-6

Cited by 3 publications

(5 citation statements)

References 31 publications

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“…In related literature, a number of techniques are proposed to determine an optimal set of collocation nodes, often summarized under the umbrella term greedy sparse methods. 58,59 However, these methods are, in many cases, iterative and are likely to require excessively high computational costs in practice. Therefore, we propose a different approach that: (i) proves to be straightforward to implement, (ii) improves the numerical stability of the simulation, and (iii) reduces computational effort.…”

Section: Numerical Implementationmentioning

confidence: 99%

“…Moreover, a minimum number of nodes with adequate distances to one another is preferable, as it improves the conditioning of the moment matrices in (48) and thus ensures the numerical stability of the simulation. In related literature, a number of techniques are proposed to determine an optimal set of collocation nodes, often summarized under the umbrella term greedy sparse methods 58,59 . However, these methods are, in many cases, iterative and are likely to require excessively high computational costs in practice.…”

Section: Solution Strategymentioning

confidence: 99%

“…Remark To enhance the numerical stability of the collocation procedure, we suggest a practical and straightforward method to decrease the density of cloud nodes by iteratively reducing the number of nodes along the cloud coordinate axes by a factor of 2, as conceptually illustrated in Figure 7. While this may not necessarily yield an optimal choice of collocation nodes as described in References 58 and 59, it has been found to effectively improve the conditioning of the moment matrices in (47). To ensure numerical stability prior to simulation, we recommend monitoring the condition numbers of the moment matrices.…”

Section: Solution Strategymentioning

confidence: 99%

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Dirichlet‐type absorbing boundary conditions for peridynamic scalar waves in two‐dimensional viscous media

Hermann

Shojaei

Seleson

et al. 2023

Numerical Meth Engineering

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Construction of absorbing boundary conditions (ABCs) for nonlocal models is generally challenging, primarily due to the fact that nonlocal operators are commonly associated with volume constrained boundary conditions. Moreover, application of Fourier and Laplace transforms, which are essential for the majority of available methods for ABCs, to nonlocal models is complicated. In this paper, we propose a simple method to construct accurate ABCs for peridynamic scalar wave‐type problems in viscous media. The proposed ABCs are constructed in the time and space domains and are of Dirichlet type. Consequently, their implementation is relatively simple, since no derivatives of the wave field are required. The proposed ABCs are derived at the continuum level, from a semi‐analytical solution of the exterior domain using harmonic exponential basis functions in space and time (plane‐wave modes). The numerical implementation is done using a meshfree collocation approach employed within a boundary layer adjacent to the interior domain boundary. The modes satisfy the peridynamic numerical dispersion relation, resulting in a compatible solution of the interior region (near‐field) with that of the exterior region (far‐field). The accuracy and stability of the proposed ABCs are demonstrated with several numerical examples in two‐dimensional unbounded domains.

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Section: Numerical Implementationmentioning

confidence: 99%

Section: Solution Strategymentioning

confidence: 99%

Section: Solution Strategymentioning

confidence: 99%

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Dirichlet‐type absorbing boundary conditions for peridynamic scalar waves in two‐dimensional viscous media

Hermann

Shojaei

Seleson

et al. 2023

Numerical Meth Engineering

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“…Recently, some meshfree greedy algorithm is used to find a solution of different kinds of PDEs. 26,27 Because of some difficulty, there is not a general convergence analysis for the greedy algorithms. Some authors have tried to find convergence analysis of greedy algorithm.…”

Section: Literature On Greedy Algorithmmentioning

confidence: 99%

“…In Mirzaei, a local Petrov‐Galerkin method based on greedy algorithm is used. Recently, some meshfree greedy algorithm is used to find a solution of different kinds of PDEs …”

Section: Introductionmentioning

confidence: 99%

A greedy algorithm for partition of unity collocation method in pricing American options

Fadaei

Khan

Akgül

2019

Math Methods in App Sciences

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A greedy algorithm in combination with radial basis functions partition of unity collocation (GRBF‐PUC) scheme is used as a locally meshless method for American option pricing. The radial basis function partition of unity method (RBF‐PUM) is a localization technique. Because of having interpolation matrices with large condition numbers, global approximants and some local ones suffer from instability. To overcome this, a greedy algorithm is added to RBF‐PUM. The greedy algorithm furnishes a subset of best nodes among the points X. Such nodes are then used as points of trial in a locally supported RBF approximant for each partition. Using of greedy selected points leads to decreasing the condition number of interpolation matrices and reducing the burdensome in pricing American options.

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Numerical investigation based on radial basis function–finite-difference (RBF–FD) method for solving the Stokes–Darcy equations

Safarpoor

Shirzadi

2019

Engineering with Computers

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A greedy sparse meshless method for solving heat conduction problems

Cited by 3 publications

References 31 publications

Dirichlet‐type absorbing boundary conditions for peridynamic scalar waves in two‐dimensional viscous media

Dirichlet‐type absorbing boundary conditions for peridynamic scalar waves in two‐dimensional viscous media

A greedy algorithm for partition of unity collocation method in pricing American options

Numerical investigation based on radial basis function–finite-difference (RBF–FD) method for solving the Stokes–Darcy equations

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