Efficient Numerical Algorithm Research Articles

An efficient numerical algorithm for solving nonlinear fractional Volterra integro-differential equation

An efficient numerical algorithm for solving nonlinear fractional Volterra integro-differential equation

A fast and efficient numerical algorithm for image segmentation and denoising

<abstract><p>Image segmentation is the process of partitioning an image into homogenous regions, and represents one of the most fundamental and important procedures in image processing. Image denoising is a process to remove unwanted noise from a digital image, enhancing its visual quality. Various algorithms, like non-local means and deep learning-based approaches, have been developed to remove noise while preserving important image details. Currently, the prevalent application of pattern recognition technology is achieved through the implementation of image segmentation algorithms. In this study, we present a new, highly efficient, and fast computational scheme specifically developed for a phase-field mathematical model of image segmentation. The numerical methodology is based on an operator splitting method (OSM). The split operators are solved by using closed-form analytic solutions and a finite difference method (FDM) with an alternating direction explicit (ADE) method. To show the notable efficiency and rapid computational performance of the proposed computational algorithm, we conduct a series of numerical experiments. Through these computational tests, we confirm a significant contribution to the advancement of methodologies employed in the critical domain of image processing.</p></abstract>

Numerical Computation of Neumann Controls for the Heat Equation on a Finite Interval

This paper presents a new numerical method which approximates Neumann type null controls for the heat equation and is based on the Fokas method. This is a direct method for solving problems originating from the control theory, which allows the realisation of an efficient numerical algorithm that requires small computational effort for determining the null control with exponentially small error. Furthermore, the unified character of the Fokas method makes the extension of the numerical algorithm to a wide range of other linear PDEs and different type of boundary conditions straightforward.

Revisiting Diffusive Representations for Enhanced Numerical Approximation of Fractional Integrals

This study reexamines diffusive representations for fractional integrals with the goal of pioneering new variants of such representations. These variants aim to offer highly efficient numerical algorithms for the approximate computation of fractional integrals. The approach seamlessly aligns with established techniques used in addressing problems involving integer-order operators, contributing to a unified framework for numerical solutions.

Novel Variants of Diffusive Representation of Fractional Integrals: Construction and Numerical Computation

In this paper, we revisit the diffusive representations of fractional integrals established in Diethelm (2023a) to explore novel variants of such representations which provide highly efficient numerical algorithms for the approximate numerical evaluation of fractional integrals.

Simulation of transient heat transfer in multilayer walls with photovoltaic cell and air by using efficient numerical methods

This paper focuses on simulating nonlinear transient heat transfer in multilayer walls under varying heat loads, using efficient numerical algorithms. The algorithms employed are completely explicit and unconditionally stable for the linear conduction-convection case. The study investigates heat transfer through building walls, considering different wall geometries and heat load scenarios, encompassing both cooling and heating. Our primary objective is to analyse how heat transfer depends on the wall materials and evaluate algorithm performance in cases involving heat transfer between solid surfaces and fluid (convection) on the outdoor surface, particularly across an air gap between insulation and Photovoltaic Cells (PVC). The results reveal that insulation prevents heat from entering the building, maintaining a comfortable indoor environment. Forced convection significantly enhances heat dissipation, especially during cooling operations to protect PVC with limited working temperature. Furthermore, the simulations highlight the air gap's efficiency in cooling PVC and reducing maximum temperatures on the insulation's outer surface, especially under forced convection conditions. The test results show that the leapfrog hopscotch algorithm offers the best solution for this highly stiff system, followed by the asymmetric and shifted hopscotch algorithms. This study offers insights into optimizing heat transfer within multilayer walls, with implications for building energy efficiency and thermal comfort.

Analysis of MHD Richardson Flow Past An Exponentially Stretched Infinite Plate with Suction and Cross-Diffusion Effects

The present paper investigate the effects of magnetic field (MHD), Richardson and suction on an exponentially expanded infinite plate by studying the convective heat and mass transfer of a non-Newtonian incompressible viscous and electrically conducting fluid. Cross-diffusion impacts are also taken into consideration. The governing partial differential equations (PDEs) are transformed into ordinary differential equations through the application of well-posed similarity transformation variables (STVs). Thus, the transformed dimensionless equations are solved analytically by integrating factor approach and the resulting solutions are simulated with an efficient stability numerical algorithm known as Mathematica. The results are displayed in tabular and graphical forms while the effects of various parameters on the velocity, temperature, concentration, skin–friction coefficient, Nusselt and Sherwood numbers are discussed in details. It was found that velocity falls when magnetic field and suction parameters increase. Also, the temperature and nanoparticle concentration decreases as suction number rises but are enhanced as diffusion-thermo and thermal-diffusivity parameters rise. An increase in Richardson and Prandtl numbers leads to a decrease in skin-friction and upsurge in the rate of heat transportation. The results of this study can be used to advance the design, operation, and performance of various systems encountered in industrial and scientific applications.

New algorithms for approximating oscillatory Bessel integrals with Cauchy-type singularities

In this paper, we present an efficient numerical algorithm for approximating integrals involving highly oscillatory Bessel functions with Cauchy-type singularities. By employing the technique of complex line integration, the highly oscillatory Bessel integrals are transformed into oscillatory integrals with a Fourier kernel. When the integration interval does not contain zeros, we use Cauchy’s theorem to transform the integration path to the complex plane and then use the Gaussian–Laguerre formula to compute the integral. For cases in which the integration interval contains zeros, we decompose the integral into two parts: the ordinary and the singular integral. We give a stable recursive formula based on Chebyshev polynomials and Bessel functions for ordinary integrals. For singular integrals, we utilize the MeijerG function for efficient computation. Numerical examples verify the effectiveness of the new algorithm and the fast convergence.

Neumann series of Bessel functions for inverse coefficient problems

Consider the Sturm–Liouville equation with a real‐valued potential . Let be its solution satisfying certain initial conditions for a number of , where , and are some complex numbers. Denote , where . We propose a method for solving the inverse problem of the approximate recovery of the potential and number from the following data . In general, the problem is ill‐posed; however, it finds numerous practical applications. Such inverse problems as the recovery of the potential from a Weyl function or the inverse two‐spectra Sturm–Liouville problem are its special cases. Moreover, the inverse problem of determining the shape of a human vocal tract also reduces to the considered inverse problem. The proposed method is based on special Neumann series of Bessel functions representations for solutions of Sturm–Liouville equations. With their aid the problem is reduced to the classical inverse Sturm–Liouville problem of recovering from two spectra, which is solved again with the help of the same representations. The overall approach leads to an efficient numerical algorithm for solving the inverse problem. Its numerical efficiency is illustrated by several examples.

An efficient fractional step numerical algorithm for time-delayed singularly perturbed 2D convection-diffusion–reaction problem with two small parameters

An efficient fractional step numerical algorithm for time-delayed singularly perturbed 2D convection-diffusion–reaction problem with two small parameters

Wavelet representation of hardcore bosons

We consider the one-dimensional Tonks–Girardeau gas with a space-dependent potential out of equilibrium. We derive the exact dynamics of the system when divided into n boxes and decomposed into energy eigenstates within each box. It is a representation of the wave function that is a mixture between real space and momentum space, with basis elements consisting of plane waves localized in a box, giving rise to the term ‘wavelet’. Using this representation, we derive the emergence of generalized hydrodynamics in appropriate limits without assuming local relaxation. We emphasize that a generalized hydrodynamic behaviour emerges in a high-momentum and short-time limit, in addition to the more common large-space and late-time limit, which is akin to a semi-classical expansion. In this limit, conserved charges do not require numerous particles to be described by generalized hydrodynamics. We also show that this wavelet representation provides an efficient numerical algorithm for a complete description of the out-of-equilibrium dynamics of hardcore bosons.

Transportation of cuttings by drilling mud in horizontal wells. Part 2. Numerical results of drilling at optimal rheological properties of mud

The relevance. Ensuring technological equipment operation efficiency when drilling wells with an arbitrary orientation of their wellbores. In particular, the search for productive hydrocarbon reservoirs requires drilling deeper wells with extended horizontal sections, for which the removal of the cuttings bed is the key problem of modern drilling. The aim. Study the features and establish the patterns of cuttings movement through the annulus eccentric space formed by a drill string and well walls under various modes (in-situ) and drilling conditions within the framework of modern mathematical models of hydrodynamics, numerical algorithms and approaches implemented in ANSYS CFD. Optimize the parameters that determine drilling, such as an inclination angle, mass-average velocity and morphology of the inlet rheologically complex viscous flow to form the maximum allowable drilling velocity for various combinations of mud effective viscosity and mixture flow rate. Give practical recommendations for engineers to reduce time, costs and material resources for maintenance and optimization of well cleaning. The object. A well with an eccentric core and an extended horizontal section operating in modes and conditions close to real drilling. The methods. Complex of physical and mathematical analysis and numerical modeling of hydrodynamics and mass transfer in homogeneous and heterogeneous continuous media. The media were widely tested on the class of internal rheological complex flows, the verification of which was carried out under appropriate conditions, performed by other authors when solving problems of drilling and cleaning wells. The results. The paper presents the results of a numerical study of hydrodynamics of a dispersed flow of a rheological complex viscous mixture of a drilling fluid with the properties of a Herschel–Bulkley type fluid and sand particles in wells with an arbitrary generatrix of its wellbore in an eccentric space. The results are required to clarify the features that accompany direct-flow and swirling flow (the moving wall method), identify the patterns in such modes, accounting for which allows for effective cleaning of the annulus. The calculations were performed within the framework of modern mathematical models of the RANS-method and the Euler–Lagrangian approach to describing the motion of heterogeneous media, implemented in the ANSYS CFD software and efficient numerical algorithms for determining intra- and interfacial processes of mass and momentum transfer in mixtures. The authors have determined that when drilling a vertical section, the most effective parameter in terms of cleaning control is the effective viscosity of the mud; the most problematic configurations for cleaning are wells with sloping sections at angles close to the vertical. When drilling a horizontal section, the eccentricity of the drill string creates “the dead” zone of flow and complicates cleaning. This zone should be removed by creating swirling flow conditions through relatively weak rotation of a drill string (30–60 rpm). When drilling inclined sections, it is necessary to reduce a solution viscosity (preferably to the level of fresh water) and increase the flow rate (in this case, to 2 m/s or higher). When drilling a horizontal section, a mud with a viscosity close to fresh water is recommended at flow rates of about 2 m/s (or 640 gpm) at the well inlet and a maximum drilling velocity of 9 m/h.

Numerical analysis of non-proportional biaxial reverse experiments with a two-surface anisotropic cyclic plasticity-damage approach

This paper deals with the numerical analysis of ductile damage and fracture behavior under non-proportional biaxial reverse loading conditions. A two-surface anisotropic cyclic elastic–plastic-damage continuum model is adequately presented, which takes into account the Bauschinger effect, the stress-differential effect, and the change of hardening rate after reverse loading. An efficient Euler explicit numerical integration algorithm, based on the inelastic (plastic or plastic-damage) predictor-elastic corrector approach, is utilized to analyze the stress and finite strain loading histories. Detailed discussions are provided on different numerical integration-related consistent tangent operators that achieve convergence within the global Newton–Raphson scheme. The proposed continuum model is implemented into the commercial software Ansys as a user-defined subroutine (UMAT). Furthermore, the novel non-proportional biaxial tensile reverse experiments are performed to validate the proposed continuum model. The associated numerical simulations investigate the stability and accuracy of the proposed algorithm and material model.

The extended block generalized Störmer–Cowell methods for second-order nonlinear delay-differential–algebraic equations with index-1

This paper is concerned with the numerical solutions of second-order nonlinear delay-differential–algebraic equations (SNDDAEs) with index-1. By extending block generalized Störmer–Cowell methods (BGSCMs) for second-order ordinary differential equations, a class of efficient numerical algorithms for solving index-1 SNDDAEs are derived. It is proved that the extended BGSCMs are uniquely solvable and convergent of order p, where p is the consistence order of the methods. Numerical example is presented to illustrate the computational effectiveness of the extended BGSCMs and the correctness of the corresponding theoretical results.

Flexible filaments in applied fields

We propose an efficient numerical algorithm for 2D simulations of flexible filaments. We apply it to calculate the metastable states that arise from the buckling instability of a ferromagnetic filament during magnetic field inversion as well as the eigenfrequencies of the buckling deformations at subcritical values of the magnetoelastic number. To demonstrate the efficiency of the algorithm we also analyze the dynamics of a flexible filament with an attached Quincke particle and illustrate its bifurcations between oscillating and rotating states.

Algorithms on low energy spectra of the Hubbard model pertinent to molecular nanomagnets

SummaryComputer supported design of materials with tailored properties is an important part of computational science so that developments of new effective algorithms is a key issue. Molecular magnetism deals with complex objects described by the quantum physics and their simulations come up against the computational constraints. In this work, we consider a numerical version of a recently developed hybrid approach that combines the ab initio DFT method with the exact diagonalization of the microscopic Hubbard model (HM). The latter avoids the prior perturbative framework but increases computing resources needed for solving the eigenvalue problem of the matrix representation of HM. We demonstrate that in the case of the ring‐shape molecular nanomagnets the computational demands can be curbed due to efficient numerical matrix construction algorithms provided. These algorithms pertain both to the calculation of the single nonzero matrix elements and to their localization in the final sparse matrix that is subsequently diagonalized. Their implementation executed on a simple six‐core‐computer system and the Mathematica environment leads to a significant gain in the computing time for the exemplary chromium‐based molecule denoted Cr, running the code sequentially. We claim, referring to the numerical diagonalization of the spin Hamiltonian matrices, that the parallelization flaws related to the high‐level programming approach can be all removed, porting the code to the appropriate HPC environment. A prospective implementation of the code, based on some dedicated and optimized mathematical libraries, will ultimately open a window for further applications of the Hubbard model in molecular magnetism.

An Efficient and Fast Sparse Grid Algorithm for High-Dimensional Numerical Integration

This paper is concerned with developing an efficient numerical algorithm for the fast implementation of the sparse grid method for computing the d-dimensional integral of a given function. The new algorithm, called the MDI-SG (multilevel dimension iteration sparse grid) method, implements the sparse grid method based on a dimension iteration/reduction procedure. It does not need to store the integration points, nor does it compute the function values independently at each integration point; instead, it reuses the computation for function evaluations as much as possible by performing the function evaluations at all integration points in a cluster and iteratively along coordinate directions. It is shown numerically that the computational complexity (in terms of CPU time) of the proposed MDI-SG method is of polynomial order O(d3Nb)(b≤2) or better, compared to the exponential order O(N(logN)d−1) for the standard sparse grid method, where N denotes the maximum number of integration points in each coordinate direction. As a result, the proposed MDI-SG method effectively circumvents the curse of dimensionality suffered by the standard sparse grid method for high-dimensional numerical integration.

A Weibull gradient prior for image restoration

In image restoration, image priors with a heavy-tailed gradient distribution found in natural images have been successfully applied to address ill-posed problems such as denoising, deblurring, and super-resolution. However, existing literature has limitations in fitting well with image gradient magnitudes using widely used distributions. This paper introduces the Weibull distribution as a potential model that can better describe image gradient magnitudes compared to existing methods. Specifically, the Weibull distribution provides the best fit for describing small gradient magnitudes. Within the framework of maximum a posteriori estimation, a recovery model based on the Weibull gradient prior is introduced. The shape parameter of the Weibull prior is automatically estimated during the iterative process using the method of moments, allowing the model to better conform to the real images’ gradient distribution. The proposed model is computed using the alternating direction method of multipliers. Moreover, the properties of the solution to the sub-problem related to the Weibull prior are analyzed, and an efficient numerical algorithm for solving the multivariate proximal operator is proposed. Experimental results validate that the proposed model exhibits superior performance and is particularly well-suited for piecewise constant images compared to competing methods.

MRI reconstruction via multi transforms learning and logarithm ratio for vectorized groups

Compressed sensing (CS) has been demonstrated to substantially reduce the scan time of magnetic resonance imaging (MRI) by undersampling k-space data. The desirable reconstruction of MR image depends on a reasonable sparse representation model that synthesizes the prior information of image as much as possible. Since learning transform is expected to provide a sparser representation, which is associated with lower reconstruction error, in this work, a multi group transforms learning scheme is proposed, which combines the adaptive sparsifying process with the utilization of nonlocal self-similarity, to generate the highly sparse coefficients of groups composed of similar patches. In addition, a non-convex logarithm ratio function is defined to better approximate l0 norm, and thus employed as the regularizer function to enforce sparsity over image coefficients. Furthermore, based on the alternating direction method of multipliers (ADMM), an efficient numerical algorithm is deduced to optimize the proposed CS-MRI model. Within this framework, the obtained coefficients and the undersampled k-space data are incorporated in a least squares problem, which is efficiently solved to reconstruct a clear MR image. The learned multi group transforms overcome the restriction of Kronecker product structure and global transform, thus resulting in obvious performance improvement. The proposed iterative formula to calculate the proximal mapping of logarithm ratio guarantees the convergence towards the unique optimal solution, also brings a certain performance gain. Extensive experimental results reveal that the proposed method not only results in better visual quality but achieves higher PSNR and SSIM than competing methods for all sampling rates and noise intensities.

Highly Efficient Numerical Algorithm for Nonlinear Space Variable-Order Fractional Reaction–Diffusion Models

In this paper, we present a new highly efficient numerical algorithm for nonlinear variable-order space fractional reaction–diffusion equations. The algorithm is based on a new method developed by using the Gaussian quadrature pole rational approximation. A splitting technique is used to address the issues related to computational efficiency and the stability of the method. Two linear systems need to be solved using the same real-valued discretization matrix. The stability and convergence of the method are discussed analytically and demonstrated through numerical experiments by solving test problems from the literature. The variable-order diffusion effects on the solution profiles are illustrated through graphs. Finally, numerical experiments demonstrate the superiority of the presented method in terms of computational efficiency, accuracy, and reliability.