Discrete Operator Research Articles

A simple and efficient numerical method for the Allen–Cahn equation on effective symmetric triangular meshes

<abstract><p>In this paper, we propose a novel, simple, efficient, and explicit numerical method for the Allen–Cahn (AC) equation on effective symmetric triangular meshes. First, we compute the net vector of all vectors starting from each node point to its one-ring neighbor vertices and virtually adjust the neighbor vertices so that the net vector is zero. Then, we define the values at the virtually adjusted nodes using linear and quadratic interpolations. Finally, we define a discrete Laplace operator on triangular meshes. We perform several computational experiments to demonstrate the performance of the proposed numerical method for the Laplace operator, the diffusion equation, and the AC equation on triangular meshes.</p></abstract>

Non-perturbative positivity and weak Hölder continuity of Lyapunov exponent for some discrete multivariable Jacobi operators

Non-perturbative positivity and weak Hölder continuity of Lyapunov exponent for some discrete multivariable Jacobi operators

Two Integral Models and Applications of Hesitant Fuzzy Information Fusion

The existing fuzzy multiattribute decision making methods rely too much on discrete information fusion operators. In the context of big data, it cannot meet the needs of rapid fusion of massive data, nor can it reflect the continuous change of decision-makers thinking over time. In this article, we study how to integrate large-scare continuous hesitant fuzzy information more efficiently. We define the concepts of continuous hesitant fuzzy sets and continuous hesitant fuzzy functions. Based on these, we further explore definite line integrals of continuous hesitant fuzzy functions and their related properties. Two continuous hesitant fuzzy information integration models based on the hesitant fuzzy calculus are proposed. We also commit to revealing the internal connections between the first line integral model, the second line integral model, and the hesitant fuzzy weighted operator. We state why it is necessary to introduce novel aggregation models based on definite line integrals, and then provide applications of the proposed models to prove their validity and rationality. Finally, we compare the proposed models with other aggregation operators to further demonstrate their effectiveness and rationality.

The discrete convolution for fractional cosine-sine series and its application in convolution equations

<abstract><p>The fractional sine series (FRSS) and the fractional cosine series (FRCS) were defined. Three types of discrete convolution operations for FRCS and FRSS were introduced, along with a detailed investigation into their corresponding convolution theorems. The interrelationship between these convolution operations was also discussed. Additionally, as an application of the presented results, two forms of discrete convolution equations based on the proposed convolution theorems were examined, resulting in explicit solutions for these equations. Furthermore, numerical simulations were provided to demonstrate that our proposed solution can be easily implemented with low computational complexity.</p> </abstract>

Essential Spectrum of Discrete Laplacian – Revisited

Consider the discrete Laplacian operator A acting on l2(Z). It is well known from the classical literature that the essential spectrum of A is a compact interval. In this article, we give an elementary proof for this result, using the finite-dimensional truncations An of A. We do not rely on symbol analysis or any infinite-dimensional arguments. Instead, we consider the eigenvalue-sequences of the truncations An and make use of the filtration techniques due to Arveson. Usage of such techniques to the discrete Schrödinger operator and to the multi-dimensional settings will be interesting future problems.

On linear models for discrete operator inference in time dependent problems

Numerical methods such as finite elements or finite differences to solve ordinary (ODEs) or partial differential equations (PDEs) are essential to computing in science and engineering. These discrete methods approximate the differential operators as matrices with non-zero coefficients for close-by degrees of freedom — the so-called stencil. For the computation of the stencil coefficients, the symbolic differential equation is generally required beforehand.In a recent work, we have demonstrated the usage of linear regression models for the accurate inference of stencil coefficients from spatial field data of linear, one- and two-dimensional PDEs with constant coefficients.In this paper, we introduce the propagation error, which may be used to evaluate regressed stencil coefficients with respect to the numerical stability of the induced system, which is a critical aspect for applications. Based on the propagation error, we continue by comparing several optimization techniques for variable selection that allow us to extend the applicability of linear models in stencil regression to non-linear PDEs. Finally, we highlight how a sliding-window approach may be utilized on spatio-temporal data to consider PDEs with non-constant coefficients.In summary, the analyses provided in this paper extend the use of linear regression models for the inference of stencil coefficients to a much broader class of applications.All source code associated with this study is publicly available athttps://github.com/HSU-HPC/ESCO_OperatorInference_TimeDependentProblems.

Massively parallelized interpolated factored Green function method

This paper presents the first parallel implementation of the novel “Interpolated Factored Green Function” (IFGF) method introduced recently for the accelerated evaluation of discrete integral operators arising in wave scattering and other areas (Bauinger and Bruno, Jour. Computat. Phys., 2021). On the basis of the hierarchical IFGF interpolation strategy, the proposed (hybrid MPI-OpenMP) parallel implementation results in efficient data communication, and it scales up to large numbers of cores—without any hard limitations on the number of cores efficiently employed. Moreover, on any given number of cores, the proposed parallel approach preserves the O(Nlog⁡N) computing cost inherent in the sequential version of the IFGF algorithm. Unlike other approaches, the IFGF method does not utilize the Fast Fourier Transform (FFT), and it is thus better suited for efficient parallelization in distributed-memory computer systems. In particular, the IFGF method relies on a “peer-to-peer” strategy wherein, at every level, field propagation is directly enacted via “exchanges” between “peer” polynomials of constant degree, without data accumulation in large-scale “telephone-central” mathematical constructs such as those used in the Fast Multipole Method (FMM) and pure FFT-based approaches. A variety of numerical results presented in this paper illustrate the character of the proposed parallel algorithm, in particular demonstrating scaling from 1 to all 1,680 cores available in the High Performance Computing cluster used, and for problems of up to 4,096 wavelengths in acoustic size.

Robust high-order unfitted finite elements by interpolation-based discrete extension

In this work, we propose a novel formulation for the solution of partial differential equations using finite element methods on unfitted meshes. The proposed formulation relies on the discrete extension operator proposed in the aggregated finite element method. This formulation is robust with respect to the location of the boundary/interface within the cell. One can prove enhanced stability results, not only on the physical domain, but on the whole active mesh. However, the stability constants grow exponentially with the polynomial order being used, since the underlying extension operators are defined via extrapolation. To address this issue, we introduce a new variant of aggregated finite elements, in which the extension in the physical domain is an interpolation for polynomials of order higher than two. As a result, the stability constants only grow at a polynomial rate with the order of approximation. We demonstrate that this approach enables robust high-order approximations with the aggregated finite element method. The proposed method is consistent, optimally convergent, and with a condition number that scales optimally for high order approximation.

Approximation properties of a discrete operator

Abstract In the present research, we estimate some approximation properties for the generalization of a sequence of positive linear operators considered by Holhoş [6] with the following operators available in literature as special examples: Szàsz operator, Ismail-May operator associated with x(1 + x)2. We estimate its recurrence relation, moments and verify the same for its particular cases. Finally, we graphically compare the convergence of this new general operator for a continuous function, examine the convergence for each instance, and come to a conclusion regarding which scenario provides the best approximation.

High-Order Directional Fields

We introduce a framework for representing face-based directional fields of an arbitrary piecewise-polynomial order. Our framework is based on a primal-dual decomposition of fields, where the exact component of a field is the gradient of piecewise-polynomial conforming function, and the coexact component is defined as the adjoint of a dimensionally-consistent discrete curl operator. Our novel formulation sidesteps the difficult problem of constructing high-order non-conforming function spaces, and makes it simple to harness the flexibility of higher-order finite elements for directional-field processing. Our representation is structure-preserving, and draws on principles from finite-element exterior calculus. We demonstrate its benefits for applications such as Helmholtz-Hodge decomposition, smooth PolyVector fields, the vector heat method, and seamless parameterization.

A weak Galerkin finite element method for Allen–Cahn equation with a nonuniform two‐step backward differentiation formula scheme

AbstractIn this article, we consider a weak Galerkin finite element method and a nonuniform two‐step backward differentiation formula scheme for solving the Allen–Cahn equation. A discrete weak gradient operator on discontinuous piecewise polynomials is used in the numerical scheme. It is well‐known that the Allen–Cahn equations have a nonlinear stability property, that is, the free‐energy functional decreases with respect to time. A corresponding energy stability analysis of the discretized system is developed. Optimal order of L2‐norm error estimates are derived. Numerical experiments are preformed to support the theoretical results.

Maximum of the Membrane Model on Regular Trees

The discrete membrane model is a Gaussian random interface whose inverse covariance is given by the discrete biharmonic operator on a graph. In literature almost all works have considered the field as indexed over {{,mathrm{{mathbb {Z}}},}}^d, and this enabled one to study the model using methods from partial differential equations. In this article we would like to investigate the dependence of the membrane model on a different geometry, namely trees. The covariance is expressed via a random walk representation which was first determined by Vanderbei in (Ann Probab 12:311–314, 1984). We exploit this representation on m-regular trees and show that the infinite volume limit on the infinite tree exists when mge 3. Further we determine the behavior of the maximum under the infinite and finite volume measures.

Adaptive evolutionary jellyfish search algorithm based optimal photovoltaic array reconfiguration under partial shading condition for maximum power extraction

This paper proposes an adaptive evolutionary jellyfish search algorithm (AEJSA) to optimally reconfigure photovoltaic (PV) array under partial shading condition (PSC) for real-time maximum power extraction. Jellyfish search algorithm (JSA) is selected owing to its effectiveness for real-time optimization. Besides, a series of discrete operations are performed on JSA to solve the discrete optimization problem of PV array reconfiguration. Due to the inherent drawback of JSA that it is easy to trap at the local optimal solution, an adaptive threshold for changing search mechanism is adopted to balance the local exploration and global exploitation. If the number of times that the value of objective function keeps unchanged exceeds this threshold, three operations (exchange, moving, and inver-over) will be implemented on the whole population for a wide global exploitation. In addition, to verify the feasibility of the hardware implementation of AEJSA, a hardware-in-the-loop test on a RTLAB platform is employed. Eleven meta-heuristic algorithms are applied and compared to AEJSA under objective PSC and subjective PSC to evaluate the optimized performance of AEJSA under various shadow conditions. The simulation results show that the mismatched power loss obtained by AEJSA is smallest, which reduced by 7.26% against gravitational search algorithm.

High Order Geometric Methods With Splines: An Analysis of Discrete Hodge-Star Operators

A new kind of spline geometric method approach is presented. Its main ingredient is the use of well established spline spaces forming a discrete de Rham complex to construct a primal sequence $\{X^k_h\}^n_{k=0}$, starting from splines of degree $p$, and a dual sequence $\{\tilde{X}^k_h\}_{k=0}^n$, starting from splines of degree $p-1$. By imposing homogeneous boundary conditions to the spaces of the primal sequence, the two sequences can be isomorphically mapped into one another. Within this setup, many familiar second order partial differential equations can be finally accommodated by explicitly constructing appropriate discrete versions of constitutive relations, called Hodge--star operators. Several alternatives based on both global and local projection operators between spline spaces will be proposed. The appeal of the approach with respect to similar published methods is twofold: firstly, it exhibits high order convergence. Secondly, it does not rely on the geometric realization of any (topologically) dual mesh. Several numerical examples in various space dimensions will be employed to validate the central ideas of the proposed approach and compare its features with the standard Galerkin approach in Isogeometric Analysis.

On an approach for evaluating certain trigonometric character sums using the discrete time heat kernel

In this article we develop a general method by which one can explicitly evaluate certain sums of nth powers of products of d≥1 elementary trigonometric functions evaluated at m=(m1,…,md)-th roots of unity. Our approach is to first identify the individual terms in the expression under consideration as eigenvalues of a discrete Laplace operator associated to a graph whose vertices form a d-dimensional discrete torus Gm which depends on m. The sums in question are then related to the nth step of a Markov chain on Gm. The Markov chain admits the interpretation as a particular random walk, also viewed as a discrete time and discrete space heat diffusion, so then the sum in question is related to special values of the associated heat kernel. Our evaluation follows by deriving a combinatorial expression for the heat kernel, which is obtained by periodizing the heat kernel on the infinite lattice Zd which covers Gm.

Fractons on graphs and complexity

We introduce two exotic lattice models on a general spatial graph. The first one is a matter theory of a compact Lifshitz scalar field, while the second one is a certain rank-2 $U(1)$ gauge theory of fractons. Both lattice models are defined via the discrete Laplacian operator on a general graph. We unveil an intriguing correspondence between the physical observables of these lattice models and graph theory quantities. For instance, the ground state degeneracy of the matter theory equals the number of spanning trees of the spatial graph, which is a common measure of complexity in graph theory ("GSD = complexity"). The discrete global symmetry is identified as the Jacobian group of the graph. In the gauge theory, superselection sectors of fractons are in one-to-one correspondence with the divisor classes in graph theory. In particular, under mild assumptions on the spatial graph, the fracton immobility is proven using a graph-theoretic Abel-Jacobi map.

Solving the Flexible Job Shop Scheduling Problem Using a Discrete Improved Grey Wolf Optimization Algorithm

The flexible job shop scheduling problem (FJSP) is of great importance for realistic manufacturing, and the problem has been proven to be NP-hard (non-deterministic polynomial time) because of its high computational complexity. To optimize makespan and critical machine load of FJSP, a discrete improved grey wolf optimization (DIGWO) algorithm is proposed. Firstly, combined with the random Tent chaotic mapping strategy and heuristic rules, a hybrid initialization strategy is presented to improve the quality of the original population. Secondly, a discrete grey wolf update operator (DGUO) is designed by discretizing the hunting process of grey wolf optimization so that the algorithm can solve FJSP effectively. Finally, an adaptive convergence factor is introduced to improve the global search ability of the algorithm. Thirty-five international benchmark problems as well as twelve large-scale FJSPs are used to test the performance of the proposed DIGWO. Compared with the optimization algorithms proposed in recent literature, DIGWO shows better solution accuracy and convergence performance in FJSPs at different scales.

Combining Max pooling-Laplacian theory and k-means clustering for novel camouflage pattern design.

Camouflage is the main means of anti-optical reconnaissance, and camouflage pattern design is an extremely important step in camouflage. Many scholars have proposed many methods for generating camouflage patterns. k-means algorithm can solve the problem of generating camouflage patterns quickly and accurately, but k-means algorithm is prone to inaccurate convergence results when dealing with large data images leading to poor camouflage effects of the generated camouflage patterns. In this paper, we improve the k-means clustering algorithm based on the maximum pooling theory and Laplace's algorithm, and design a new camouflage pattern generation method independently. First, applying the maximum pooling theory combined with discrete Laplace differential operator, the maximum pooling-Laplace algorithm is proposed to compress and enhance the target background to improve the accuracy and speed of camouflage pattern generation; combined with the k-means clustering principle, the background pixel primitives are processed to iteratively calculate the sample data to obtain the camouflage pattern mixed with the background. Using color similarity and shape similarity for evaluation, the results show that the combination of maximum pooling theory with Laplace algorithm and k-means algorithm can effectively solve the problem of inaccurate results of k-means algorithm in processing large data images. The new camouflage pattern generation method realizes the design of camouflage patterns for different backgrounds and achieves good results. In order to verify the practical application value of the design method, this paper produced test pieces based on the designed camouflage pattern generation method and tested the camouflage effect of camouflage pattern in sunny and cloudy days respectively, and the final test results were good.

Hybrid PSO (SGPSO) with the Incorporation of Discretization Operator for Training RBF Neural Network and Optimal Feature Selection

Hybrid PSO (SGPSO) with the Incorporation of Discretization Operator for Training RBF Neural Network and Optimal Feature Selection

Bivariate discrete Mittag-Leffler functions with associated discrete fractional operators

Based on the recently published article about bivariate Mittag-Leffler function Eα,β,γδ(x) by Fernandez et al. (2020), we introduce the bivariate discrete Mittag-Leffler function, denoted by Eα,β,γ¯δ(λ1,λ2;x), as a discrete version of the results of Fernandez et al. under some constraints in this study. We establish this new definition to find a fractional difference equation. Then, we employ the fractional sum and differences formulas to get the results with respect to the bivariate discrete Mittag-Leffler function. Moreover, we give the discrete Laplace transform of the corresponding discrete function and build up the discrete sum operator to show up the semigroup property on some constraints. Also, left inverse of the discrete sum operator is given. Finally, we end the paper by two examples and conclusion.