Publication Date: 2012
Engineering Analysis with Boundary Elements (9557997)36(4)pp. 511-519
The meshless local boundary integral equation (MLBIE) method with an efficient technique to deal with the time variable are presented in this article to analyze the transient heat conduction in continuously nonhomogeneous functionally graded materials (FGMs). In space, the method is based on the local boundary integral equations and the moving least squares (MLS) approximation of the temperature and heat flux. In time, again the MLS approximates the equivalent Volterra integral equation derived from the heat conduction problem. It means that, the MLS is used for approximation in both time and space domains, and we avoid using the finite difference discretization or Laplace transform methods to overcome the time variable. Finally the method leads to a single generalized Sylvester equation rather than some (many) linear systems of equations. The method is computationally attractive, which is shown in couple of numerical examples for a finite strip and a hollow cylinder with an exponential spatial variation of material parameters. © 2011 Elsevier Ltd. ALl Rights Reserved.
Publication Date: 2012
IMA Journal of Numerical Analysis (2724979)32(3)pp. 983-1000
The moving least squares (MLS) method provides an approximation û of a function u based solely on values u(xj) of u on scattered 'meshless' nodes xj. Derivatives of u are usually approximated by derivatives of û. In contrast to this, we directly estimate derivatives of u from the data, without any detour via derivatives of û. This is a generalized MLS technique, and we prove that it produces diffuse derivatives as introduced by Nyroles et al. (1992, Generalizing the finite element method: diffuse approximation and diffuse elements. Comput. Mech., 10, 307-318). Consequently, these turn out to be efficient direct estimates of the true derivatives, without anything 'diffuse' about them, and we prove optimal rates of convergence towards the true derivatives. Numerical examples confirm this, and we finally show how the use of shifted and scaled polynomials as basis functions in the generalized and standard MLS approximation stabilizes the algorithm. © 2010 The author. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
Publication Date: 2018
Acta Mechanica (15970)229(6)pp. 2657-2673
This paper concerns a new and fast meshfree method for the linear coupled thermoelasticity problem. The resulting algorithm provides an attractive alternative to existing mesh-based and meshfree methods. Compared with mesh-based methods, the proposed technique inherits the advantages of meshfree methods allowing the use of scattered points instead of a predefined mesh. Compared with the existing meshfree methods, the proposed technique is truly meshless, requiring no background mesh for both trial and test spaces and, more importantly, numerical integrations are done over low-degree polynomials rather than complicated shape functions. In fact, this method mimics the known advantages of both meshless and finite element methods, where in the former triangulation is not required for approximation and in the latter the stiffness and mass matrices are set up by integration against simple polynomials. The numerical results of the present work concern the thermal and mechanical shocks in a finite domain considering classical coupled theory of thermoelasticity. © 2018, Springer-Verlag GmbH Austria, part of Springer Nature.
Publication Date: 2018
SIAM Journal on Numerical Analysis (361429)56(1)pp. 274-295
In this paper, a numerical solution of partial differential equations on the unit sphere is given by using a kernel trial approximation in combination with a special Petrov–Galerkin test discretization. The solvability of the scheme is proved, and the error bounds are obtained for functions in appropriate Sobolev spaces. The condition number of the final system is estimated in terms of discretization parameters. The method is meshless because in the trial side the numerical solution parameterizes entirely in terms of scattered points and in the test side everything breaks down to simple numerical integrations over independent spherical caps. This means that no connected background mesh is required for either approximation or integration. © 2018 Society for Industrial and Applied Mathematics.
Publication Date: 2017
BIT Numerical Mathematics (63835)57(4)pp. 1041-1063
In this paper a direct approximation method on the sphere, constructed by generalized moving least squares, is presented and analyzed. It is motivated by numerical solution of partial differential equations on spheres and other manifolds. The new method generalizes the finite difference methods, someway, for scattered data points on each local subdomain. As an application, the Laplace–Beltrami equation is solved and the theoretical and experimental results are given. The new approach eliminates some drawbacks of the previous methods. © 2017, Springer Science+Business Media Dordrecht.
Publication Date: 2016
Numerical Methods for Partial Differential Equations (0749159X)32(3)pp. 847-861
The meshless local Petrov-Galerkin (MLPG) method with global radial basis functions (RBF) as trial approximation leads to a full final linear system and a large condition number. This makes MLPG less efficient when the number of data points is increased. We can overcome this drawback if we avoid using more points from the data site than absolutely necessary. In this article, we equip the MLPG method with the greedy sparse approximation technique of (Schaback, Numercail Algorithms 67 (2014), 531-547) and use it for numerical solution of partial differential equations. This scheme uses as few neighbor nodal values as possible and allows to control the consistency error by explicit calculation. Whatever the given RBF is, the final system is sparse and the algorithm is well-conditioned. © 2015 Wiley Periodicals, Inc.
Publication Date: 2016
Journal of Computational and Applied Mathematics (3770427)294pp. 93-101
This paper provides the error estimates for generalized moving least squares (GMLS) derivatives approximations of a Sobolev function in Lp norms and extends them for local weak forms of DMLPG methods. Sometimes they are called diffuse or uncertain derivatives, but precisely they are direct approximants of exact derivatives which possess the optimal rates of convergence. GMLS derivatives approximations are different from the standard derivatives of MLS approximation. While they are much easier to evaluate at considerably lower cost, in this paper the same orders of convergence with comparison to the standard derivatives are obtained for them. © 2015 Elsevier B.V. All rights reserved.
Publication Date: 2016
Acta Mechanica (15970)227(3)pp. 619-632
This article describes a new and fast meshfree method based on a generalized moving least squares (GMLS) approximation and the local weak forms for vibration analysis in solids. In contrast to the meshless local Petrov–Galerkin method, GMLS directly approximates the local weak forms from meshless nodal values, which shifts the local integrations over the low-degree polynomial basis functions rather than over the complicated MLS shape functions. Besides, if the method is set up properly, all local integrals have the same value if all local subdomains have the same shape. These features reduce the computational costs, remarkably. The new technique is called direct meshless local Petrov–Galerkin (DMLPG) method. In DMLPG, the stiff and mass matrices are constructed by integration against polynomials. This overcomes the main drawback of meshfree methods in comparison with the finite element methods (FEM). The Newmark scheme is adapted as a time integration method, and numerical results are presented for various dynamic problems. The results are compared with the exact solutions, if available, and the FEM solutions. © 2015, Springer-Verlag Wien.
Publication Date: 2015
Engineering Analysis with Boundary Elements (9557997)59pp. 36-42
The aim of this work is application of the direct meshless local Petrov-Galerkin (DMLPG) method for solving a two-dimensional time fractional advection-diffusion equation. This method is based on the generalized moving least squares (GMLS) approximation, and makes a considerable reduction in the cost of numerical integrations in weak forms. In fact, DMLPG shifts the integrals over the close form polynomials rather than the complicated MLS shape functions. Moreover, the values of integrals on subdomains with the same shapes are equal. Thus DMLPG is a weak-based meshless technique in the cost-level of collocation or integration-free methods. In time domain, a simple and suitable finite difference approximation is employed. Some examples show the advantages of the new method in comparison with the traditional MLPG method. © 2015 Elsevier Ltd. All rights reserved.
Publication Date: 2015
Journal of Computational and Applied Mathematics (3770427)282pp. 237-250
In this article the error estimation of the moving least squares approximation is provided for functions in fractional order Sobolev spaces. The analysis presented in this paper extends the previous estimations and explains some unnoticed mathematical details. An application to Galerkin method for partial differential equations is also supplied. © 2015 Elsevier B.V. All rights reserved.
Publication Date: 2014
IET Communications (17518628)(10)pp. 1837-1849
A finite set system (FSS) is a pair (V, B) where V is a finite set whose members are called points, equipped with a finite collection of its subsets B whose members are called blocks. In this paper, FSSs are used to define a class of quasi-cyclic low-density parity-check (LDPC) codes, called FSS codes, such that the constructed codes possess large girth and arbitrary column-weight distributions. Especially, the constructed column weight-2 FSS codes have higher rates than the column weight-2 geometric and cylinder-type codes with the same girths. To find the maximum girth of FSS codes based on (V, B), inevitable walks are defined in B such that the maximum girth is determined by the smallest length of the inevitable walks in B. Simulation results show that the constructed FSS codes have very good performance over the additive white Gaussian noise channel with iterative decoding and achieve significantly large coding gains compared with the random-like LDPC codes of the same lengths and rates. © The Institution of Engineering and Technology 2014.
Publication Date: 2014
Numerical Algorithms (10171398)65(2)pp. 275-291
As an improvement of the Meshless Local Petrov–Galerkin (MLPG), the Direct Meshless Local Petrov–Galerkin (DMLPG) method is applied here to the numerical solution of transient heat conduction problem. The new technique is based on direct recoveries of test functionals (local weak forms) from values at nodes without any detour via classical moving least squares (MLS) shape functions. This leads to an absolutely cheaper scheme where the numerical integrations will be done over low–degree polynomials rather than complicated MLS shape functions. This eliminates the main disadvantage of MLS based methods in comparison with finite element methods (FEM), namely the costs of numerical integration. © Springer Science+Business Media New York 2013.
Publication Date: 2022
2025 29th International Computer Conference, Computer Society of Iran, CSICC 2025pp. 266-273
Following the Bitcoin model, many modern blockchains reward their miners in two ways: (i) a base reward for each block that is mined, and (ii) the transaction fees of those transactions that are included in the mined block. The base reward is fixed by the respective blockchain's protocol and is not under the miner's control. Hence, for a miner who wishes to maximize earnings, the fundamental problem is to form a valid block with maximal total transaction fees and then try to mine it. Moreover, in many protocols, including Bitcoin itself, the base reward halves at predetermined intervals, hence increasing the importance of maximizing transaction fees and mining an optimal block. This problem is further complicated by the fact that transactions can be prerequisites of each other or have conflicts (in case of double-spending). In this work, we consider the problem of forming an optimal block, i.e. a valid block with maximal total transaction fees, given a set of unmined transactions. The problem is known to be NP-hard. As such, there is no hope in solving it efficiently for general instances. However, we observe that its real-world instances are quite sparse, i.e. the transactions have very few dependencies and conflicts. Using this fact, and exploiting a well-known graph sparsity parameter, namely pathwidth, we present an exact linear-time parameterized algorithm that is applicable to the real-world instances and obtains optimal results. We also provide an experimental evaluation demonstrating that our approach outperforms current Bitcoin miners in practice, obtaining a significant increase in transaction fee revenues. © 2022 IEEE.
Publication Date: 2021
Computers and Mathematics with Applications (8981221)103pp. 1-11
We consider a least-squares variational kernel-based method for numerical solution of second order elliptic partial differential equations on a multi-dimensional domain. In this setting it is not assumed that the differential operator is self-adjoint or positive definite as it should be in the Rayleigh-Ritz setting. However, the new scheme leads to a symmetric and positive definite algebraic system of equations. Moreover, the resulting method does not rely on certain subspaces satisfying the boundary conditions. The trial space for discretization is provided via standard kernels that reproduce the Sobolev spaces as their native spaces. The error analysis of the method is given, but it is partly subjected to an inverse inequality on the boundary which is still an open problem. The condition number of the final linear system is approximated in terms of the smoothness of the kernel and the discretization quality. Finally, the results of some computational experiments support the theoretical error bounds. © 2021 Elsevier Ltd
Publication Date: 2021
Advances in Computational Mathematics (10197168)47(5)
In this paper, we present a rational RBF interpolation method to approximate multivariate functions with poles or other singularities on or near the domain of approximation. The method is based on scattered point layouts and is flexible with respect to the geometry of the problem’s domain. Despite the existing rational RBF-based techniques, the new method allows the use of conditionally positive definite kernels as basis functions. In particular, we use polyharmonic kernels and prove that the rational polyharmonic interpolation is scalable. The scaling property results in a stable algorithm provided that the method be implemented in a localized form. To this aim, we combine the rational polyharmonic interpolation with the partition of unity method. Sufficient number of numerical examples in one, two and three dimensions are given to show the efficiency and the accuracy of the method. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
Publication Date: 2020
Numerical Methods for Partial Differential Equations (0749159X)36(6)pp. 1682-1698
This paper concerns a numerical solution for the diffusion equation on the unit sphere. The given method is based on the spherical basis function approximation and the Petrov–Galerkin test discretization. The method is meshless because spherical triangulation is not required neither for approximation nor for numerical integration. This feature is achieved through the spherical basis function approximation and the use of local weak forms instead of a global variational formulation. The local Petrov–Galerkin formulation allows to compute the integrals on small independent spherical caps without any dependence on a connected background mesh. Experimental results show the accuracy and the efficiency of the new method. © 2020 Wiley Periodicals LLC
Publication Date: 2020
Computers and Mathematics with Applications (8981221)79(9)pp. 2624-2643
In this paper, the idea of direct discretization via radial basis functions (RBFs) is applied on a local Petrov–Galerkin test space of a partial differential equation (PDE). This results to a weak-based RBF-generated finite difference (RBF–FD) scheme that possesses some useful properties. The error and stability issues are considered. When the PDE solution or the basis function has low smoothness, the new method gives more accurate results than the already well-established strong-based collocation methods. Although the method uses a Galerkin formulation, it still remains meshless because not only the approximation process relies on scattered point layouts but also integrations are done over non-connected, independent and well-shaped subdomains. Some applications to potential and elasticity problems on scattered data points support the theoretical analysis and show the efficiency of the proposed method. © 2019 Elsevier Ltd
Publication Date: 2020
Applied Numerical Mathematics (1689274)150pp. 222-232
In this paper the error analysis of the kernel collocation method for partial differential equations on the unit sphere is presented. A simple analysis is given when the true solutions lie in arbitrary Sobolev spaces. This also extends the previous studies for true solutions outside the associated native spaces. Finally, some experimental results support the theoretical error bounds. © 2019 IMACS
Publication Date: 2019
Journal of Scientific Computing (8857474)79(1)pp. 493-516
In this paper a numerical simulation based on radial basis functions is presented for the time-dependent Allen–Cahn equation on surfaces with no boundary. In order to approximate the temporal variable, a first-order time splitting technique is applied. The error analysis is given when the true solution lies on appropriate Sobolev spaces defined on surfaces. The method only requires a set of scattered points on a given surface and an approximation to the surface normal vectors at these points. Besides, the approach is based on Cartesian coordinates and thus any coordinate singularity has been omitted. Some numerical results are given to illustrate the ability of the technique on sphere, torus and red blood cell as three well-known surfaces. © 2018, Springer Science+Business Media, LLC, part of Springer Nature.
Publication Date: 2023
JOURNAL OF COMPUTATIONAL PHYSICS (00219991)479
In this paper a new direct RBF partition of unity (D-RBF-PU) method is developed for numerical solution of partial differential equations defined on smooth orientable surfaces which are discretized with sets of scattered nodes and with approximations to normal vectors at each of the nodes. The accuracy, stability and efficiency of the new method are studied through some theoretical and experimental results. This method is a localized RBF based technique, results in a perfectly sparse final linear system, uses only scattered nodes on the surface rather than a connected mesh, and is applicable for a large class of PDEs on manifolds. Applications to some biological and chemical reaction-diffusion models are also given. Results show that the new method outperforms other comparable techniques for surface PDEs.(c) 2023 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).
Publication Date: 2023
COMPUTERS & OPERATIONS RESEARCH (03050548)152
The objective functions in optimization models of the sum-of-squares clustering problem reflect intra-cluster similarity and inter-cluster dissimilarities and in general, optimal values of these functions can be considered as appropriate measures for compactness of clusters. However, the use of the objective function alone may not lead to the finding of separable clusters. To address this shortcoming in existing models for clustering, we develop a new optimization model where the objective function is represented as a sum of two terms reflecting the compactness and separability of clusters. Based on this model we develop a two-phase incremental clustering algorithm. In the first phase, the clustering function is minimized to find compact clusters and in the second phase, a new model is applied to improve the separability of clusters. The Davies-Bouldin cluster validity index is applied as an additional measure to compare the compactness of clusters and silhouette coefficients are used to estimate the separability of clusters. The performance of the proposed algorithm is demonstrated and compared with that of four other algorithms using synthetic and real-world data sets. Numerical results clearly show that in comparison with other algorithms the new algorithm is able to find clusters with better separability and similar compactness.
Publication Date: 2020
AKCE INTERNATIONAL JOURNAL OF GRAPHS AND COMBINATORICS (09728600)(3)pp. 870-876
A dominating set in a graph G is a subset of vertices D such that every vertex in V\D is a neighbor of some vertex of D. The domination number of G is the minimum size of a dominating set of G and it is denoted by gamma (G). A dominating set with cardinality gamma (G) is called optimal dominating set. Also, a subset D of a graph G is a [1, 2]-set if, each vertex v is an element of V\D is adjacent to either one or two vertices in D and the minimum cardinality of [1, 2]-dominating set of G, is denoted by gamma [1,2](G). Chang's conjecture says that for every 16 <= m <= n,gamma (Gm,n)=(n+2)(m+2)5-4 and this conjecture has been proven by Goncalves et al. This paper presents an explicit constructing method to find an optimal dominating set for grid graph G(m)(,)(n) where m,n >= 16 in O (size of answer). In addition, we will show that gamma (Gm,n)=gamma [1,2](Gm,n) where m,n >= 16 holds in response to an open question posed by Chellali et al.
Publication Date: 2025
International Journal of Geometric Methods in Modern Physics (02198878)22(8)
In this paper, we aim to interpret the background gravitational effects appearing in quantum field theory on curved space-time by studying the Brownian motion of quantum states along with the Hamilton–Perelman Ricci flow. It has been shown that the Wiener measure automatically contains the Einstein–Hilbert action and the path-integral formulation of the scalar quantum field theory on curved space-time at the first order of local approximations. This provides a well-defined formulation of the path-integral measure for quantum field theory in the presence of gravity. However, we establish that the emergence of Einstein–Hilbert action is independent of the matter field interactions and is a merely entropic/geometric effect stemming from the nature of the Ricci flow of the universe geometry. We also extract an explicit formula for the cosmological constant in terms of the Ricci flow and Hamilton’s theorem for 3-manifolds. Then, we discuss the cosmological features of the FLRW solution in ΛCDM Model via the derived equations of the Ricci flow. We also argue the correlation between our formulations and the entropic aspects of gravity. Finally, we provide some theoretical evidence that proves the second law of thermodynamics is the basic source of gravity and probably a more fundamental concept. © 2025 World Scientific Publishing Company.
Publication Date: 2024
International Journal of Geometric Methods in Modern Physics (02198878)21(12)
This paper aims to provide a consistent, finite-valued, and mathematically well-defined reformulation of Feynman's path-integral measure for quantum fields obtained by studying the Wiener stochastic process in the infinite-dimensional Hilbert space of quantum states. This reformulation will undoubtedly have a crucial role in formulating quantum gravity within a mathematically well-defined framework. In fact, this study is fundamentally different from any previous research on the relationship between Feynman's path-integral and the Wiener stochastic process. In this research, we focus on the fact that the classic Wiener measure is no longer applicable in infinite-dimensional Hilbert spaces due to fundamental differences between displacements in low and extremely high dimensions. Thus, an analytic norm motivated by the role of the fractal functions in the Wilsonian renormalization approach is worked out to properly characterize Brownian motion in the Hilbert space of quantum states on a compact flat manifold. This norm, the so-called fractal norm, pushes the rougher functions (physically the quantum states with higher energies) to the farther points of the Hilbert space until the fractal functions as the roughest ones are moved to infinity. Implementing the Wiener stochastic process with the fractal norm, results in a modified form of the Wiener measure called the Wiener fractal measure, which is a well-defined measure for Feynman's path-integral formulation of quantum fields. Wiener fractal measure has a complicated formula of nonlocal terms but produces the Klein-Gordon action at the first order of approximation. Using complex integrals to compensate for the removal of non-local terms appearing in higher orders of approximation, the Wiener fractal measure turns into a complex measure and generates Feynman's path-integral formulation of scalar quantum fields. This brings us to the main objective of this study. Finally, some various significant aspects of quantum field theory (such as renormalizability, RG flow, Wick rotation, regularization, etc.) are revisited by means of the analytical aspects of the Wiener fractal measure. © 2024 World Scientific Publishing Company.
Publication Date: 2023
International Journal of Geometric Methods in Modern Physics (02198878)20(10)
A differential geometric statement of the noncommutative topological index theorem is worked out for covariant star products on noncommutative vector bundles. To start, a noncommutative manifold is considered as a product space X = Y × Z, wherein Y is a closed manifold, and Z is a flat Calabi-Yau m-fold. Also, a semi-conformally flat metric is considered for X which leads to a dynamical noncommutative spacetime from the viewpoint of noncommutative gravity. Based on the Kahler form of Z, the noncommutative star product is defined covariantly on vector bundles over X. This covariant star product leads to the celebrated Groenewold-Moyal product for trivial vector bundles and their flat connections, such as C∞(X). Hereby, the noncommutative characteristic classes are defined properly and the noncommutative Chern-Weil theory is established by considering the covariant star product and the superconnection formalism. Finally, the index of the ?-noncommutative version of elliptic operators is studied and the noncommutative topological index theorem is stated accordingly. © 2023 World Scientific Publishing Company.
Publication Date: 2022
International Journal of Geometric Methods in Modern Physics (02198878)19(1)
The entire geometric formulations of the BRST and the anti-BRST structures are worked out in presence of the Nakanishi-Lautrup field. It is shown that in the general form of gauge fixing mechanisms within the Faddeev-Popov quantization approach, the anti-BRST invariance reflects thoroughly the classical symmetry of the Yang-Mills theories with respect to gauge fixing methods. The Nakanishi-Lautrup field is also defined and worked out as a geometric object. This formulation helps us to introduce two absolutely new topological invariants of quantized Yang-Mills theories, so-called the Nakanishi-Lautrup invariants. The cohomological structure of the anti-BRST symmetry is also studied and the anti-BRST topological index is derived accordingly. © 2022 World Scientific Publishing Company.
