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In this study, we develop new efficient parallel techniques for solving both distinct and multiple roots of nonlinear problems at the same time. The parallel techniques represent an innovative contribution to the discipline, with local convergence of the ninth order. Theoretical research shows the rapid convergence and effectiveness of the proposed parallel schemes. To assess the suggested scheme’s stability and consistency, we look at certain biomedical engineering applications, such as osteoporosis in Chinese women, blood rheology, and differential equations. Overall, detailed analyses of convergence behavior, memory utilization, computational time, and percentage computational efficiency show that the novel parallel techniques outperform the traditional methods. The proposed methods would be more suitable for large-scale computational problems in biomedical applications due to their advantages in memory efficiency, CPU time, and error reduction.

Nonlinear equations are essential in research and engineering because they simulate complicated processes such as fluid dynamics, chemical reactions, and population growth. The development of advanced methods to address them becomes essential for scientific and applied research enhancements, as their resolution influences innovations by aiding in the proper prediction or optimization of the system. In this research, we develop a novel biparametric family of inverse parallel techniques designed to improve stability and accelerate convergence in parallel iterative algorithm. Bifurcation and chaos theory were used to find the best parameter regions that increase the parallel method’s effectiveness and stability. Our newly developed biparametric family of parallel techniques is more computationally efficient than current approaches, as evidenced by significant reductions in the number of iterations and basic operations each iterations step for solving nonlinear equations. Engineering applications examined with rough beginning data demonstrate high accuracy and superior convergence compared to existing classical parallel schemes. Analysis of global convergence further shows that the proposed methods outperform current methods in terms of error control, computational time, percentage convergence, number of basic operations per iteration, and computational order. These findings indicate broad usage potential in engineering and scientific computation.

In this article, we constructed a derivative-free family of iterative techniques for extracting simultaneously all the distinct roots of a non-linear polynomial equation. Convergence analysis is discussed to show that the proposed family of iterative method has fifth order convergence. Nonlinear test models including fractional conversion, predator-prey, chemical reactor and beam designing models are included. Also many other interesting results concerning symmetric problems with application of group symmetry are also described. The simultaneous iterative scheme is applied starting with the initial estimates to get the exact roots within the given tolerance. The proposed iterative scheme requires less function evaluations and computation time as compared to existing classical methods. Dynamical planes are exhibited in CAS-MATLAB (R2011B) to show how the simultaneous iterative approach outperforms single roots finding methods that might confine the divergence zone in terms of global convergence. Furthermore, convergence domains, namely basins of attraction that are symmetrical through fractal-like edges, are analyzed using the graphical tool. Numerical results and residual graphs are presented in detail for the simultaneous iterative method. An extensive study has been made for the newly developed simultaneous iterative scheme, which is found to be efficient, robust and authentic in its domain.

Articular cartilage (AC) is essential for minimizing friction in the human knee, but its healthy function is highly influenced by biomechanical factors such as weight bearing. Non-invasive biomechanical and numerical simulations are widely used to study AC but often require complex and costly numerical approximations. Machine learning (ML) provides a more efficient alternative and uses data from these numerical methods for training. Hybrid ML models (HML) complemented by reduced-order numerical models can achieve similar outcomes with minimal data input but may have problems with generalizability across different scales. In this study, we present an extended HML framework (EHML) for developing a multiscale surrogate model specifically tailored for knee cartilage simulations. Our approach is based on integrating hybrid graph neural networks (GNNs) with tissue-scale data and aims to achieve remarkable few-shot learning and potential zero-shot generalizability for large-scale analysis. The main proposed idea is a physics-constrained data augmentation (DA) technique coupled with a set of pre-processing and customization algorithms to bridge the scales. Specifically, we integrate feature transformations, resampling, and cost-sensitive functions to manage the observed data imbalances, all within a customized, memory-efficient training framework. Our rigorous testing using an advanced multi-physics cartilage model demonstrates the viability of our approach. Comparative analyses underscore the significant role of pre-processing and DA methods in enhancing generalizability and efficiency. They helped reduce the normalized mean squared errors to 0.1 or less (compared to the ablated model with its error of 2 or higher). Therefore, this work represents an important step towards addressing the challenges of limited generalizability and efficiency of existing ML-based surrogate models and opens new possibilities for their application in more complex simulations.

Nonlinear problems, which often arise in various scientific and engineering disciplines, typically involve nonlinear equations or functions with multiple solutions. Analytical solutions to these problems are often impossible to obtain, necessitating the use of numerical techniques. This research proposes an efficient and stable Caputo-type inverse numerical fractional scheme for simultaneously approximating all roots of nonlinear equations, with a convergence order of 2ψ+2. The scheme is applied to various nonlinear problems, utilizing dynamical analysis to determine efficient initial values for a single root-finding Caputo-type fractional scheme, which is further employed in inverse fractional parallel schemes to accelerate convergence rates. Several sets of random initial vectors demonstrate the global convergence behavior of the proposed method. The newly developed scheme outperforms existing methods in terms of accuracy, consistency, validation, computational CPU time, residual error, and stability.

This article proposes a variable stepsize implementation of an improved one-step block method (OBM) that uses two intermediate points to solve Burgers' equation. The model's problems under consideration are semi-discretized by reducing them into an equivalent differential system using the method of lines. The OBM is improved with an embedding-like method executed in variable stepsize mode. A few of Burgers' real-life model problems are numerically solved to test the accuracy of our error estimation and mesh selection strategy. The numerical results confirmed that the variable stepsize implementation used in this manuscript performed better than the fixed stepsize implementation.

Cobalt–Chromium–Molybdenum (Co–Cr–Mo) alloys are commonly used for artificial hip and knee joint metallic implants. These components are subjected to repetitive loads during service. Therefore, materials used for such applications must exhibit a high fatigue crack resistance. In this research, Co–28Cr–6Mo (wt.-%) powder was utilized as a feedstock in a laser powder bed fusion process to produce test coupons. The coupons were then subjected to load-controlled cyclic material tests in the low cycle fatigue regime to study mechanical response and microstructural changes of the material. With the progressing number of cycles, a continuous increase in macroscopic plastic strain was observed. The electron backscattered diffraction analysis revealed that cyclic loading caused deformation-induced face-centered cubic (FCC) → hexagonal close-packed (HCP) phase transformation. In addition, the phase transition generated an accumulation of plastic strain at the FCC/HCP interface giving rise to crack nucleation. The crack propagation path along HCP orientation variants with high mechanical work and strain hardening mechanism is discussed.

The PageRank model is widely utilized for analyzing a variety of scientific issues beyond its original application in modeling web search engines. In recent years, considerable research effort has focused on developing high-performance iterative methods to solve this model, particularly when the dimension is exceedingly large. However, due to the ever-increasing extent and size of data networks in various applications, the computational requirements of the PageRank model continue to grow. This has led to the development of new techniques that aim to reduce the computational complexity required for the solution. In this paper, we present a recursive 5-type lumping algorithm combined with a two-stage elimination strategy that leverage characteristics about the nonzero structure of the underlying network and the nonzero values of the PageRank coefficient matrix. This method reduces the initial PageRank problem to the solution of a remarkably smaller and sparser linear system. As a result, it leads to significant cost reductions for computing PageRank solutions, particularly in scenarios involving large and/or multiple damping factors. Numerical experiments conducted on over 50 real-world networks demonstrate that the proposed methods can effectively exploit characteristics of PageRank problems for efficient computations.

In scientific and engineering disciplines, vectorial problems involving systems of equations or functions with multiple variables frequently arise, often defying analytical solutions and necessitating numerical techniques. This research introduces an efficient numerical scheme capable of simultaneously approximating all roots of nonlinear equations with a convergence order of ten, specifically designed for vectorial problems. Random initial vectors are employed to assess the global convergence behavior of the proposed scheme. The newly developed method surpasses methods in the existing literature in terms of accuracy, consistency, computational CPU time, residual error, and stability. This superiority is demonstrated through numerical experiments tackling engineering problems and solving heat equations under various diffusibility parameters and boundary conditions. The findings underscore the efficacy of the proposed approach in addressing complex nonlinear systems encountered in diverse applied scenarios.

This paper presents and successfully applies an optimized hybrid block technique using a variable stepsize implementation to integrate a type of singularly perturbed parabolic convection–diffusion problems. The problem under consideration is semi-discretized by utilizing the method of lines. A few numerical experiments have been presented to ascertain the proposed error estimation and adaptive stepsize strategy. Furthermore, the comparison of the proposed method with other techniques in the literature is conducted via numerical experiments, and the results show that our method outperforms other existing methods.

In this article, we construct an efficient family of simultaneous methods for finding all the distinct as well as multiple roots of polynomial equations. Convergence analysis proves that the order of convergence of newly constructed family of simultaneous methods is seventeen. Fractal-based simultaneous iterative algorithms are thoroughly examined. Using self-similar features, fractal-based simultaneous schemes can converge to solutions faster, saving computational time and resources necessary for solving nonlinear equations. Fractals analysis illustrates the newly developed method’s global convergence behavior when compared to single root-finding procedures for solving fractional order polynomials that arise in complex engineering applications. Some real problems from various branches of engineering along with some higher degree polynomials are considered as test examples to show the global convergence property of simultaneous methods, performance and efficiency of the proposed family of methods. Further computational efficiencies, CPU time and residual graphs are also drawn to validate the efficiency, robustness of the newly introduced family of methods as compared to the existing methods in the literature.

Quantum calculus can provide new insights into the nonlinear behaviour of functions and equations, addressing problems that may be difficult to tackle by classical calculus due to high nonlinearity. Iterative methods for solving nonlinear equations can benefit greatly from the mathematical theory and tools provided by quantum calculus, e.g., using the concept of q-derivatives, which extends beyond classical derivatives. In this paper, we develop parallel numerical root-finding algorithms that approximate all distinct roots of nonlinear equations by utilizing q-analogies of the function derivative. Furthermore, we utilize neural networks to accelerate the convergence rate by providing accurate initial guesses for our parallel schemes. The global convergence of the q-parallel numerical techniques is demonstrated using random initial approximations on selected biomedical applications, and the efficiency, stability, and consistency of the proposed hybrid numerical schemes are analyzed.

We proposed and analyzed the fractional simultaneous technique for approximating all the roots of nonlinear equations in this research study. The newly developed fractional Caputo-type simultaneous scheme’s order of convergence is 3ς+5, according to convergence analysis. Engineering-related numerical test problems are taken into consideration to demonstrate the efficiency and stability of fractional numerical schemes when compared to previously published numerical iterative methods. The newly developed fractional simultaneous approach converges on random starting guess values at random times, demonstrating its global convergence behavior. Although the newly developed method shows global convergent behavior when all starting guess values are distinct, the method diverges otherwise. The total computational time, number of iterations, error graphs and maximum residual error all clearly illustrate the stability and consistency of the developed scheme. The rate of convergence increases as the fractional parameter’s value rises from 0.1 to 1.0.

This paper presents an efficient two-step hybrid block method (ETHBM) to obtain an approximate solution to the FitzHugh–Nagumo problem. The considered partial differential equation model problems are semi-discretized, reducing them to equivalent ordinary differential equations using the method of lines. In order to evaluate the effectiveness of the proposed ETHBM, three numerical examples are presented and compared with the results obtained through existing methods. The results demonstrate that the proposed ETHBM produces more efficient results than some other numerical approaches in the literature.

Finding all the roots of a nonlinear equation is an important and difficult task that arises naturally in numerous scientific and engineering applications. Sequential iterative algorithms frequently use a deflating strategy to compute all the roots of the nonlinear equation, as rounding errors have the potential to produce inaccurate results. On the other hand, simultaneous iterative parallel techniques require an accurate initial estimation of the roots to converge effectively. In this paper, we propose a new class of global neural network-based root-finding algorithms for locating real and complex polynomial roots, which exploits the ability of machine learning techniques to learn from data and make accurate predictions. The approximations computed by the neural network are used to initialize two efficient fractional Caputo-inverse simultaneous algorithms of convergence orders ς+2 and 2ς+4, respectively. The results of our numerical experiments on selected engineering applications show that the new inverse parallel fractional schemes have the potential to outperform other state-of-the-art nonlinear root-finding methods in terms of both accuracy and elapsed solution time.

This paper presents a new hybrid block method formulated in variable stepsize mode to solve some first-order initial value problems of ODEs and time-dependent partial differential equations in applied sciences and engineering. The proposed method is implemented considering an adaptive stepsize strategy to maintain the estimated error in each step within a specified tolerance. In order to evaluate the performance and usefulness of the proposed technique in real-world applications, several differential problems from applied sciences and engineering, such as the SIR model, Jacobi elliptic function problem, and chemical reactions problems, are solved numerically. The results of numerical simulations in this work demonstrate that the proposed method is more efficient than other existing numerical methods used for comparisons.

The vast majority of compressed digital data that flows nowadays on modern high-speed networks is directly related to human activity. It describes what we do, what we see and photograph, where we go, whom we meet, and specifically every moment of our lives. This brings up issues and concerns regarding the necessity to safeguard user privacy as well as to protect the digital multimedia contents that are delivered to offer new experiences. In this paper, we explore a unified approach to compression and privacy by considering different types of digital data (text, images, sound, and hyperspectral images).

In this paper, we propose a semi-implicit difference scheme for solving one-dimensional nonlinear space-fractional diffusion equations. The method is first-order accurate in time and second-order accurate in space. It uses a fractional central difference formula and the backward Euler method to approximate its space and time derivatives, respectively. Stability and convergence properties of the proposed scheme are proved with the help of a discrete Grönwall inequality. Moreover, we extend the method to the solution of two-dimensional nonlinear models. A fast matrix-free implementation based on preconditioned Krylov subspace methods is presented for solving the discretized linear systems. The resulting fast preconditioned semi-implicit difference scheme reduces the memory requirement of conventional semi-implicit difference schemes from O(Ns2) to O(Ns) and the computational complexity from O(Ns3) to O(NslogNs) in each iterative step, where Ns is the number of space grid points. Experiments with two numerical examples are shown to support the theoretical findings and to illustrate the efficiency of our proposed method.

The increase in memory and in network traffic used and caused by new sequenced biological data has recently deeply grown. Genomic projects such as HapMap and 1000 Genomes have contributed to the very large rise of databases and network traffic related to genomic data and to the development of new efficient technologies. The large-scale sequencing of samples of DNA has brought new attention and produced new research, and thus the interest in the scientific community for genomic data has greatly increased. In a very short time, researchers have developed hardware tools, analysis software, algorithms, private databases, and infrastructures to support the research in genomics. In this paper, we analyze different approaches for compressing digital files generated by Next-Generation Sequencing tools containing nucleotide sequences, and we discuss and evaluate the compression performance of generic compression algorithms by confronting them with a specific system designed by Jones et al. specifically for genomic file compression: Quip. Moreover, we present a simple but effective technique for the compression of DNA sequences in which we only consider the relevant DNA data and experimentally evaluate its performances.

Air-borne and space-borne acquired hyperspectral images are used to recognize objects and to classify materials on the surface of the earth. The state of the art compressor for lossless compression of hyperspectral images is the Spectral oriented Least SQuares (SLSQ) compressor (see [1–7]). In this paper we discuss hyperspectral image compression: we show how to visualize each band of a hyperspectral image and how this visualization suggests that an appropriate band ordering can lead to improvements in the compression process. In particular, we consider two important distance measures for band ordering: Pearson’s Correlation and Bhattacharyya distance, and report on experimental results achieved by a Java-based implementation of SLSQ.

If we can use previous knowledge of the source (or the knowledge of a source that is correlated to the one we want to compress) to exploit the compression process then we can have significant gains in compression. By doing this in the fundamental source coding theorem we can substitute entropy with conditional entropy and we have a new theoretical limit that allows for better compression. To do this, when data compression is used for data transmission, we can assume some degree of interaction between the compressor and the decompressor that can allow a more efficient usage of the previous knowledge they both have of the source. In this paper we review previous work that applies interactive approaches to data compression and discuss this possibility.

Airborne and spaceborne hyperspectral sensors collect information which is derived from the electromagnetic spectrum of an observed area. Hyperspectral data are used in several studies and they are an important aid in different real-life applications (e.g., mining and geology applications, ecology, surveillance, etc.). A hyperspectral image has a three-dimensional structure (a sort of datacube): it can be considered as a sequence of narrow and contiguous spectral channels (bands). The objective of this paper is to present a framework permits the efficient storage/transmission of an input hyperspectral image, and its protection. The proposed framework relies on a reversible invisible watermarking scheme and an efficient lossless compression algorithm. The reversible watermarking scheme is used in conjunction with digital signature techniques in order to permit the verification of the integrity of a hyperspectral image by the receiver.

In this work, we present a scheme for the lossy compression of image sequences, based on the Adaptive Vector Quantization (AVQ) algorithm. The AVQ algorithm is a lossy compression algorithm for grayscale images, which processes the input data in a single-pass, by using the properties of the vector quantization to approximate data. First, we review the key aspects of the AVQ algorithm and, subsequently, we outline the basic concepts and the design choices behind the proposed scheme. Finally, we report the experimental results, which highlight an improvement in compression performances when our scheme is compared with the AVQ algorithm.

The Biconjugate A-Orthogonal Residual (BiCOR) method carried out in finite precision arithmetic by means of the biconjugate A-orthonormalization procedure may possibly tend to suffer from two sources of numerical instability, known as two kinds of breakdowns, similarly to those of the Biconjugate Gradient (BCG) method. This paper naturally exploits the composite step strategy employed in the development of the composite step BCG (CSBCG) method into the BiCOR method to cure one of the breakdowns called as pivot breakdown. Analogously to the CSBCG method, the resulting interesting variant, with only a minor modification to the usual implementation of the BiCOR method, is able to avoid near pivot breakdowns and compute all the well-defined BiCOR iterates stably on the assumption that the underlying biconjugate A-orthonormalization procedure does not break down. Another benefit acquired is that it seems to be a viable algorithm providing some further practically desired smoothing of the convergence history of the norm of the residuals, which is justified by numerical experiments. In addition, the exhibited method inherits the promising advantages of the empirically observed stability and fast convergence rate of the BiCOR method over the BCG method so that it outperforms the CSBCG method to some extent.

Hyperspectral images are widely used in several real-life applications. In this paper, we investigate on the compression of hyperspectral images by considering different aspects, including the optimization of the computational complexity in order to allow implementations on limited hardware (i.e., hyperspectral sensors, etc.). We present an approach that relies on a three-dimensional predictive structure. Our predictive structure, 3D-MBLP, uses one or more previous bands as references to exploit the redundancies among the third dimension. The achieved results are comparable, and often better, with respect to the other state-of-art lossless compression techniques for hyperspectral images.