Gaussian Elimination Algorithm Research Articles

Efficient learning-driven data transmission algorithm for cloud-to-thing continuum

In the cloud-to-thing continuum paradigm, the efficient transmission of learning-driven data is critical for facilitating real-time decision-making and continuous learning processes. Fountain codes have gained popularity as Forward Error Correction (FEC) schemes in application layer networking due to their ability to recover lost data packets. Gaussian Elimination (GE) algorithms, which form the core of Fountain codes, can be optimized to improve transmission efficiency. We introduces techniques that leverage GE algorithms to enhance the transmission of learning-driven data within the cloud-to-thing continuum. A fast matrix element architecture is developed to improve computational speed. Additionally, a pipelined normalization component based on binary trees is introduced to enable parallel processing. A pipelined elimination component utilizing binary trees is also presented to further boost transmission speeds and efficiency. These techniques are implemented using Field Programmable Gate Array (FPGA) technology. Their performance is evaluated against existing methods, demonstrating significant gains in transmission efficiency. By effectively managing increasing Internet traffic loads and optimizing the flow of learning-driven data, these approaches facilitate more efficient communication between cloud-based AI systems and edge devices. Ultimately, this research supports enhanced real-time decision-making and continuous learning processes across the full cloud-to-thing continuum.

An advanced computational method for nonlinear inelastic thermo-mechanical analysis of spatial steel frames in fires

A novel advanced computational method incorporating the stability functions and the distributed plasticity model into the fiber beam-column element is proposed for the first time to predict the nonlinear inelastic thermo-mechanical behavior of spatial steel frames in fires. The space spread of plasticity is captured by tracing the uniaxial temperature-stress–strain relation of each fiber on sections at integration points. The element stiffness matrix is integrated via the Gauss-Lobatto numerical integration scheme. The nonlinear material behavior is captured by adopting the distributed plasticity model with the force-based interpolation function whereas the geometric nonlinearity of P-δ and P-Δ effects are considered by using the stability functions and a geometric matrix, respectively. The shear deformation and residual stresses which normally exist in steel frames are also considered. For the implementation, a nonlinear thermal incremental-iterative solution scheme based on the Newton-Raphson and Gaussian Elimination algorithm is also developed to address the nonlinear problems due to thermal expansion and material degradation. By taking advantage of the computational benefits of stability functions and the force-based interpolation function, the proposed method offers a novel computational framework for nonlinear structural analysis in a fire with excellent computational efficiency. The reliability and accuracy of the proposed program are verified by comparing the obtained results with results from test data, existing studies, and Abaqus. In addition, the computational performance of the proposed numerical program is about 10 times more effective than the Abaqus commercial software, thus would offer an extremely valuable tool for practical fire resistance designs.

Haar wavelet method for solution of variable order linear fractional integro-differential equations

<abstract><p>In this paper, we developed a computational Haar collocation scheme for the solution of fractional linear integro-differential equations of variable order. Fractional derivatives of variable order is described in the Caputo sense. The given problem is transformed into a system of algebraic equations using the proposed Haar technique. The results are obtained by solving this system with the Gauss elimination algorithm. Some examples are given to demonstrate the convergence of Haar collocation technique. For different collocation points, maximum absolute and mean square root errors are computed. The results demonstrate that the Haar approach is efficient for solving these equations.</p></abstract>

Fast Soft Decision Decoding Algorithm for Linear Block Codes Using Permuted Generator Matrices

The Gaussian elimination algorithm is an essential part of the ordered statistics-based decoding (OSD); the algorithm is required to be executed at least once for a typical OSD algorithm, with its computational complexity serving as the lower bound of the total computational complexity. When the signal-to-noise ratio (SNR) is relatively low, the computational complexity of the Gaussian elimination algorithm can be ignored as the decoding process is relatively complex. However, with an increase in the SNR, this cannot be ignored. In this letter, we propose a fast soft decision decoding algorithm using templates that are precalculated and permuted generator matrices, enabling us to decode a received vector without having to perform the OSD algorithm. In particular, if the Hamming distance between the received vector and the candidate codeword generated by one of the templates is less than a certain threshold, we can terminate the decoding process without executing the typical OSD algorithm. This aids in reducing the computational complexity in high SNR regimes without compromising the decoding performance.

Overview of Adaptive and Low-Rank Approximation Algorithms for Modeling of The Influence of Electromagnetic Waves Generated by The Cell Phone Antenna on The Human Head

This paper presents an overview of formulations and algorithms dedicated to modeling the influence of electromagnetic waves on the human head. We start from the three-dimensional MRI scan of the human head. We approximate the MRI scan by the continuous approximation span over three-dimensional h adaptive mesh with quadratic polynomials. Next, we introduce time-harmonic Maxwell equations with a 1.8 GHz cell-phone antenna. We solve the problem of the propagation of electromagnetic waves on the human head. We compute the specific absorption rate used as the heat source for the Pennes bioheat equation. Finally, we introduce the Pennes bio-heat equation modeling the heat generated by the electromagnetic waves propagating through the skull, tissue, and air layers in the human head. We discuss the discretization and time-stepping algorithm for the Pennes equation’s solution over the human head. Namely, we focus on the Crank-Nicolson time integration scheme, to solve the bioheat transfer equations. We employ the hp finite elements with hierarchical shape functions and hp adaptive algorithm in three-dimensions. We propose an adaptive algorithm mixed with time-stepping iterations, where we simultaneously adapt the computational mesh, solve the Maxwell and Pennes equations, and we iterative with time steps. We employ the sparse Gaussian elimination algorithm with low-rank compression of the off-diagonal matrix blocks for the factorization of matrices. We conclude with the statement that 15 minutes of talk with a 1.8 GHz antenna of 1 Wat power results in increased brain tissue temperature up to 38.4 Celsius degree.

THEORETICAL AND COMPUTATIONAL RESULTS FOR MIXED TYPE VOLTERRA–FREDHOLM FRACTIONAL INTEGRAL EQUATIONS

In this paper, we develop a numerical method for the solutions of mixed type Volterra–Fredholm fractional integral equations (FIEs). The proposed algorithm is based on Haar wavelet collocation technique (HWCT). Under certain conditions, we prove the existence and uniqueness of the solution. Also, some stability results are given of Hyers–Ulam (H–U) type. With the help of the HWCT, the considered problem is transformed into a system of algebraic equations which is then solved for the required results by using Gauss elimination algorithm. Some numerical examples for convergence of the proposed technique are taken from the literature. Maximum absolute and root mean square errors are calculated for different collocation points (CPs). The results show that the HWCT is an effective method for solving FIEs. The convergence rate for different CPS is also calculated, which is nearly equal to [Formula: see text].

Gaussian Elimination versus Greedy Methods for the Synthesis of Linear Reversible Circuits

Linear reversible circuits represent a subclass of reversible circuits with many applications in quantum computing. These circuits can be efficiently simulated by classical computers and their size is polynomially bounded by the number of qubits, making them a good candidate to deploy efficient methods to reduce computational costs. We propose a new algorithm for synthesizing any linear reversible operator by using an optimized version of the Gaussian elimination algorithm coupled with a tuned LU factorization. We also improve the scalability of purely greedy methods. Overall, on random operators, our algorithms improve the state-of-the-art methods for specific ranges of problem sizes: The custom Gaussian elimination algorithm provides the best results for large problem sizes (n > 150), while the purely greedy methods provide quasi optimal results when n < 30. On a benchmark of reversible functions, we manage to significantly reduce the CNOT count and the depth of the circuit while keeping other metrics of importance (T-count, T-depth) as low as possible.

A Hyper-Heuristic Heterogeneous Multisensor Node Scheme for Energy Efficiency in Larger Wireless Sensor Networks Using DEEC-Gaussian Algorithm

A wireless sensor network (WSN) is an intellect-sustainable network that comprises multiple spatially distributed sensor nodes and several sink nodes that collect data from sensors. WSNs remain an active research area in the literature due to challenging factors such as the selection of sensor location according to a given premise, finding optimal routing algorithm, and ensuring energy efficiency and consumption. Minimizing energy and prolonging the network lifetime in the WSNs are the focus of this research work. In the literature, a clustering approach is used in grouping sensor nodes into clusters and is seen as an effective technique used in optimizing energy consumption in WSNs. Hence, in this paper, we put forward a novel clustering-based approach by amalgamating the Gaussian elimination method with the Distributed Energy-Efficient Clustering to produce DEEC_Gaussian (DEEC_Gaus) to stabilize energy efficiency optimization in WSNs. We took the advantages of DEEC and Gaussian elimination algorithms to resolve energy efficiency problems in WSNs. DEEC presents attributes such as increased heterogeneity performance level, clustering stability in operation, and energy efficiency which helps to prolong network lifetime while the Gaussian elimination algorithm added an additional advantage to improve and optimize energy efficiency, to aggregate packets of operations performed in the network lifestyle of energy efficiency in WSNs. The simulations were carried out using MATLAB software with 1000 to 1500 nodes. The performance of the proposed work was compared with state-of-the-art algorithms such as DEEC, DDEEC, and EDEEC_E. The simulated results presented show that the proposed DEEC-Gauss outperformed the three other conventional algorithms in terms of network lifetime, first node dead, tenth node dead, alive nodes, and the overall timing of the packets received at the base station. The results showed that the proposed hyper-heuristic heterogeneous multisensor DEEC-Gauss algorithm presented an average percentage of 3.0% improvement for the tenth node dead (TND) and further improvement of 4.8% for the first node dead (FND). When the performance was compared to the state-of-the-art algorithms in larger networks, the overall delivery was greatly improved and optimized.

Second‐Order Delay Differential Equations to Deal the Experimentation of Internet of Industrial Things via Haar Wavelet Approach

In this article, an efficient numerical approach for the solution of second‐order delay differential equations to deal with the experimentation of the Internet of Industrial Things (IIoT) is presented. With the help of the Haar wavelet technique, the considered problem is transformed into a system of algebraic equations which is then solved for the required results by using Gauss elimination algorithm. Some numerical examples for convergence of the proposed technique are taken from the literature. Maximum absolute and root mean square errors are calculated for various collocation points. The results show that the Haar wavelet method is an effective method for solving delay differential equations of second order. The convergence rate is also measured for various collocation points, which is almost equal to 2.

Fast scalable construction of ([compressed] static | minimal perfect hash) functions

Recent advances in the analysis of random linear systems on finite fields have paved the way for the construction of constant-time data structures representing static functions and minimal perfect hash functions using less space with respect to existing techniques. The main obstacle for any practical application of these results is the time required to solve such linear systems: despite they can be made very small, the computation is still too slow to be feasible.In this paper, we describe in detail a number of heuristics and programming techniques to speed up the solution of these systems by orders of magnitude, making the overall construction competitive with the standard and widely used MWHC technique, which is based on hypergraph peeling. In particular, we introduce broadword programming techniques for fast equation manipulation and a lazy Gaussian elimination algorithm. We also describe a number of technical improvements to the data structure which further reduce space usage and improve lookup speed.Our implementation of these techniques yields a minimal perfect hash function data structure occupying 2.24 bits per element, compared to 2.68 for MWHC-based ones, and a static function data structure which reduces the multiplicative overhead from 1.23 to 1.024. For functions whose output has low entropy, we are able to implement feasibly for the first time the Hreinsson–Krøyer–Pagh approach, which makes it possible, for example, to store a function with an output of 106 values distributed following a power law of exponent 2 in just 2.76 bits per key instead of 20.

Rank revealing Gaussian elimination by the maximum volume concept

A Gaussian elimination algorithm is presented that reveals the numerical rank of a matrix and identifies a square nonsingular submatrix of maximum dimension. The bounds on the singular values of the submatrix and its Schur complement are similar to the best known bounds for rank revealing LU factorization, but in contrast to existing methods the algorithm does not make use of the normal matrix. An implementation for dense matrices is described whose computational cost is roughly twice the cost of an LU factorization with complete pivoting. Because of its flexibility in choosing pivot elements, the algorithm is amenable to implementation with blocked memory access and for sparse matrices.

Cyber-attack Detection Strategy Based on Distribution System State Estimation

Cyber-attacks that tamper with measurement information threaten the security of state estimation for the current distribution system. This paper proposes a cyber-attack detection strategy based on distribution system state estimation (DSSE). The uncertainty of the distribution network is represented by the interval of each state variable. A three-phase interval DSSE model is proposed to construct the interval of each state variable. An improved iterative algorithm (IIA) is developed to solve the interval DSSE model and to obtain the lower and upper bounds of the interval. A cyber-attack is detected when the value of the state variable estimated by the traditional DSSE is out of the corresponding interval determined by the interval DSSE. To validate the proposed cyber-attack detection strategy, the basic principle of the cyber-attack is studied, and its general model is formulated. The proposed cyber-attack model and detection strategy are conducted on the IEEE 33-bus and 123-bus systems. Comparative experiments of the proposed IIA, Monte Carlo simulation algorithm, and interval Gauss elimination algorithm prove the validation of the proposed method.

Incremental Gaussian Elimination Approach to Implement OMP for Sparse Signal Measurement

An efficient architecture of the orthogonal matching pursuit (OMP) algorithm is proposed to recover signals compressively measured at the sub-Nyquist rate. The proposed architecture is implemented on the field-programmable gate array (FPGA) for performance validation. In the place of matrix factorization-based pseudoinverse computation, Gaussian elimination (GE) is used to compute the signal estimate. A novel incremental Gaussian elimination (IGE) algorithm is proposed and used in the OMP algorithm. The proposed design is targeted to the Virtex6 FPGA device to compare with other reported works for $K=256$ , $N=1024$ , and $m=36$ , where $N$ is the number of samples, $K$ is the measurement vector length, and $m$ is the signal sparsity level. The recovery signal-to-noise ratio (RSNR) of 23.98 dB is achieved. The proposed work is validated by implementing it on the Artix7 FPGA device by taking compressed measurements from an analog to information converter (AIC). The input signal is synthesized as a random combination of sine waves with different frequencies. The proposed architecture is hardware-efficient and faster, and consumes low dynamic power than other existing designs. The proposed design is hardware-efficient even for the higher value of $m/K$ .

Fast Analytic Electromigration Analysis for General Multisegment Interconnect Wires

Electromigration (EM) is considered to be one of the most important reliability issues for current and future ICs in 10-nm technology and below. In this article, we propose a fast analytic solution to compute the stress evolution in the confined multisegment interconnect wires. The new method, called the accelerated separation of variables (ASOV) method, aims to find the analytic solutions of the partial differential equations of stress in confined interconnect metals based on the SOV method. It offers several improvements over the existing plain SOV-based method. First, we show that the accuracy of the solution depends on the structure of the interconnects. As a result, the number of required eigenvalues is structure and problem dependent, instead of fixed numbers used by the existing SOV method. Second, for the straight line multisegment and star-structured multiterminal interconnects, analytical expressions are formulated to calculate the eigenvalues directly instead of using numerical methods as in the existing SOV method. Third, we propose a linear Gaussian elimination (GE) algorithm by exploiting the banded structure with the serrated-edge form of the transcendental matrix, which can significantly speed up GE process, and is the key computing step in the SOV-based solution framework. Fourth, instead of using the simple bisection search, we propose to use an enhanced determinant-based secant iterative method to find the eigenvalues of the transcendental matrix. Numerical results show that a good agreement is achieved between analytical and numerical results on two special cases, and the resulting algorithm can lead to 3–5X speedup over the existing plain SOV-based solution on a number of multisegment interconnects benchmarks.

MRB Decoding of LT Codes Over AWGN Channels

In this letter we propose a novel decoder for Luby transform codes over noisy channel based on Gaussian elimination algorithm. The proposed soft-on-the-fly Gaussian elimination algorithm permits to perform most reliable basis decoding while distributing the complexity all along the symbols reception, and reducing the latency due to new decoding attempts in case of decoding failures.

Performance Analysis and Design of LT Codes Under the Gaussian Elimination Algorithm in Wireless Sensor Networks

With the growing demand, wireless sensor networks (WSNs) play an increasingly important role in many fields. The focus has been on deploying Luby transform (LT) codes in WSNs because of their inherent advantages in erasure channels. Since most data transmitted by WSNs are small, the Gaussian elimination (GE) decoding algorithm is highly attractive. In this paper, we investigate the performance analysis and code design methods for LT codes under the GE algorithm. First, the analytical expressions of the data erasure probability are detailed for two simple kinds of LT codes. Then, a characteristic indicator of the data erasure probability is introduced to evaluate the performance of general LT codes. Finally, we propose novel design methods by optimizing degree distributions from three goals based on the characteristic indicator. The numerical results validate the proposed performance analysis and code design methods in comparison with the conventional method.

A Comparison of Gauss Elimination Method for Dense Linear Systems on Hypercube and Mesh Parallel Architectures

The problem of solving a system of N linear equations Ax = b (where A is a known N x N matrix, and b and x are, respectively, the known and unknown N x 1 vectors) is frequently encountered in many fields of science and engineering. Efficient numerical methods for solving this problem on uniprocessor systems have been developed. The Gaussian elimination method which is inherently serial is one of the most popular methods for solving the system of linear algebraic equations. In this paper we compare the well-known Gaussian elimination algorithm for solving dense linear systems Ax = b on two well known architectures: Hypercube and two dimensional mesh. Our objective is to analyse the complexity of Gauss elimination algorithm on Hypercube and to present an algorithm for Gauss elimination and analyse its complexity on mesh parallel architecture.

A Semi-Analytical Pressure-Transient Model to Detect Inter-well Interference of Multi-Well-Pad-Production Scheme in Shale Gas Reservoirs

Recently, Multi-Well-Pad-Production (MWPP) scheme has been in the center of attention as a promising technology to improve Shale Gas (SG) recovery. However, Inter-Well Pressure Interference (IWPI) induced by MWPP scheme severely distorts flow regimes, which strongly challenges the traditional pressure-transient analysis methods, which focus on Single Multi-Fractured Horizontal Wells (SMFHW) without IWPI. Therefore, a methodology to identify pressure-transient response of MWPP scheme without and with IWPI is urgent. To fill this gap, by utilizing superposition theory, Gauss elimination and Stehfest numerical algorithm, the pressure-transient solution of MWPP scheme was established, as a result, type flow regimes can be identified by considering MWIP. Our results show that our proposed model demonstrates promising calculation speed and acceptable accuracy compared to numerical simulation. Part of flow regimes are significantly distorted by IWPI. In addition, well rate mainly determines the distortion of pressure curves, while fracture length, well spacing, fracture spacing mainly determine when the IWPI occurs. The smaller the gas rate, the more severely flow regimes are distorted. As the well spacing increases, fracture length decreases, fracture spacing decreases, occurrence of IWPI becomes later. Stress sensitivity coefficient approximately has no influences on distortion of pressure curves and occurrence of IWPI. This work gains some additional insights on pressure-transient response for MWPP scheme in SG reservoir, which can provide considerable guidance on fracture properties estimation as well as well pattern optimization for MWPP scheme.

Energy‐efficient routing protocol to improve energy consumption in wireless sensors networks

SummaryThe energy consumption is considered to be the major challenge in wireless sensor networks. In this paper, we shed light on a new approach to overcome the energy consumption problems. The objective of this work is to integrate an improved algorithm which is the K‐means to create a balanced energy on clusters and to use the Gaussian elimination algorithm during the election of cluster head that will guarantee the distribution of energy consumption. To address the problem of the optimal number of groups, we use Davies Bouldin index to increase the network lifetime. The simulation result shows that the proposed protocol extends the network lifetime compared to Leach, iMod‐Leach, leach‐C, and CELRP.

KAJIAN NUMERIK MASALAH SYARAT BATAS MELALUI PENYELESAIAN MATRIKS TRIDIAGONAL (Studi Kasus : Menghitung Potensial Listrik)

A numerical study of the solution of boundary value problems with the tridiagonal matrix approach has been done. The case studied is computing the electrical potential expressed by 1D Poisson’s equation with boundary conditions. The 1D Poisson’s equation are then discreted with the finite different method so as to form a system of linier equations. The system of non homogeneous linear equations that can be formed is a tridiagonal matrix. Then the matrix is solved by Gaussian elimination algorithm. On this study, the algorithm is implemented on three programming languages namely, Fortran, Java and MATLAB.