Lower-upper Research Articles

Computation of Neumann Localised Boundary Domain Integral Equations

Most integrals in Localised Boundary Domain Integral Equations (LBDIEs) comprise singularities. This paper aims to produce numerical solutions of the LBDIEs for the Partial Differential Equations with variable coefficients. The singularities of the boundary integrals in LBDIEs will be handled by using a semi-analytic for logarithmic singularity and a semi-quadratic analytic method for r−2 singularity. Whereas the singular domain integrals are handled by using the Duffy transformation. The LBDIEs that we consider are associated with the Neumann problem, which can be solved with a condition. If it can be solved, the solution is, however, unique up to an additive constant. We add a perturbation operator to the LBDIEs to convert the LBDIE to a uniquely solvable equation. The perturbed integral operator leads the perturbed LBDIEs to a dense matrix system that disable the use of methods in solving sparse matrix system. We solve the system of linear equations by Lower-Upper (LU) decomposition method. The numerical results indicate that high accuracy results can be attained. It gives the impression that the methods we use in this numerical experiment are reliable in handling the boundary and domain singular integrals.

Unique Symbolic Factorization for Fast Contingency Analysis Using Full Newton–Raphson Method

Contingency analysis plays an important role in assessing the static security of a network. Its purpose is to check whether a system can operate safely when some elements are out of service. In a real-time application, the computational time required to perform the calculation is paramount for operators to take immediate actions to prevent cascading outages. Therefore, the numerical performance of the contingency analysis is the main focus of this current research. In power flow calculation, when solving the network equations with a sparse matrix solver, most of the time is spent factorizing the Jacobian matrix. In terms of computation time, the symbolic factorization is the costliest operation in the LU (Lower-upper) factorization process. This paper proposes a novel method to perform the calculation with only one symbolic factorization using a full Newton–Raphson-based generic formulation and modular approach (GFMA). The symbolic factorization retained can be used during the iterations of any power flow contingency scenario. A computer study demonstrates that reusing the same symbolic factorization greatly reduces computation time and improves numerical performance. Power system security assessment under N-1 and N-2 contingency conditions is performed for the IEEE standard 54-bus and 108-bus to evaluate the numerical performance of the proposed method. A comparison with the conventional power flow method shows that the time required for the analysis is shortened considerably, with a minimum gain of 228%. The comparative analysis demonstrates that the proposed solution has better numerical performance for large-scale networks.

A Finite State Machine-Based Improved Cryptographic Technique

With the advent of several new means of communication, safeguarding the confidentiality of messages has become more crucial. Financial institutions, virtual currencies, and government organizations are all examples of high-risk contexts where information exchanges need particular care. The importance of data security in preventing unauthorized access to data is emphasized. Several cryptographic methods for protecting the secrecy and integrity of data were compared. In this research, the proposed work includes a new Turbo Code-based encryption algorithm. The Turbo encoder’s puncturing process is controlled by a secret key, and a typical random sequence is generated to encrypt the data and fix any mistakes. Key generation utilizing pre-existing data eliminates the requirement for sending keys over a secure channel. Using recurrence relations and the Lower–Upper (LU) decomposition method, the presented study suggests a novel approach to message encryption and decryption. The resulting encrypted grayscale image has a very high level of security, with an entropy of 7.999, a variation from perfection of 0.0245, and a correlation of 0.0092 along the diagonal, 0.0009 along the horizontal, and −0.0015 along the vertical. Directly decrypted pictures have a Peak Signal-to-Noise Ratio (PSNR) of 56.22 dB, but the suggested approach only manages an embedding capacity of 0.5 bpp (bits per pixel). This may be achieved by decreasing the size of the location map by only 0.02 bpp.

Mixed-field source localization based on robust matrix propagator and reduced-degree polynomial rooting

This paper develops two new algorithms based on multiple signal classification (MUSIC) and matrix propagator for near-field (NF) and far-field (FF) source localization, which substantially reduce the computational complexities and can be applied no matter whether the source number is known as a priori. In the developed algorithms, two special fourth-order cumulant (FOC) matrices are constructed. Then the noise subspace is obtained by incorporating lower-upper (LU) decomposition into the matrix propagator approach. This avoids the computationally burdensome eigenvalue decomposition (EVD) or singular value decomposition (SVD) as well as the dependence of source number. The direction-of-arrivals (DOAs) of FF and NF sources are distinguished by the oblique projection and then estimated based on the MUSIC spectrum. To further improve the computational efficiency, an alternating root-propagator scheme is proposed to determine DOAs, where the real lower-degree polynomial is formulated, which increases the computational speed by a factor about eight compared to the standard root-MUSIC. Finally, the NF ranges are calculated from the FOC and the DOA estimates via the least square (LS) criterion. The effectiveness and performance of the proposed approach are validated by using numerical simulation.

Energy Efficient Approximate 3D Image Reconstruction

We demonstrate an efficient and accelerated parallel, sparse depth reconstruction framework using compressed sensing (compressed sensing (CS)) and approximate computing. Employing data parallelism for rapid image formation, the depth image is reconstructed from sparsely sampled scenes using convex optimization. Coupled with faster imaging, this sparse sampling reduces significantly the projected laser power in active systems such as light detection and ranging (LiDAR) to allow eye safe operation at longer range. We also demonstrate how reduced precision is leveraged to reduce the number of logic units in field-programmable gate array (FPGA) implementations for such sparse imaging systems. It enables significant reduction in logic units, memory requirements and power consumption by over 80% with minimal impact on the quality of reconstruction. To further accelerate processing, pre-computed, important components of the lower-upper (LU) decomposition and other linear algebraic computations are used to solve the convex optimization problems. Our methodology is demonstrated by the application of the alternating direction method of multipliers (ADMM) and proximal gradient descent (PGD) algorithms. For comparison, a fully discrete least square reconstruction method ( <inline-formula><tex-math notation="LaTeX">$d$</tex-math></inline-formula> Sparse) is also presented. This demonstrates the feasibility of novel, high resolution, low power and high frame rate LiDAR depth imagers based on sparse illumination for use in applications where resources are strictly limited.

A fully coupled, fully implicit simulation method for unsteady flames using Jacobian approximation and clustering

Jacobian clustering is proposed to reduce the computational cost associated with matrix operations encountered in the Newton iteration in fully coupled, fully implicit schemes for unsteady reactive flow simulations with detailed chemistry. The iterative solver is based on the Lower-Upper Symmetric Gauss-Seidel (LUSGS) algorithm and sparse matrix technique. The evaluation and sparse Lower-Upper (LU) factorization of the diagonal block of the system Jacobian are performed within clusters rather than individual cells for Computational Fluid Dynamics (CFD) simulations. Cells that are close in the state space are clustered to provide the averaged states for calculating the Jacobians of chemical source terms and transport fluxes. The cells retrieve the factorized sparse matrices from the belonging clusters to perform the necessary iterations. For the purpose of clustering, the spatial dependency of transport Jacobian in the diagonal block is eliminated. To further reduce the computational cost, the sparsity of chemical Jacobian is augmented by removing the insignificant matrix elements. The method is tested in various one-dimensional hydrocarbon flames with both the second order Crank-Nicolson scheme and a third order implicit Runge-Kutta scheme. Various chemical mechanisms with 9 to 111 species are used to test the performance of the iterative solver. Fast convergence of Newton iteration is achieved, and the formal order of accuracy is demonstrated with Jacobian clustering. The overall costs of evaluation and factorization of the block diagonal Jacobian are negligible compared to the cost of calculating transport fluxes and chemical source terms. The averaged costs of Jacobian evaluation, LU factorization and Newton iteration, all increase only linearly with the number of chemical species. The fully coupled, fully implicit Crank-Nicolson scheme with Jacobian clustering shows 4 to 42 times speedup in computational time compared to the decoupled implicit scheme with Strang operator splitting. Jacobian clustering is promising to increase the computational efficiency of high order fully coupled, fully implicit schemes for unsteady reactive flow simulations with detailed chemistry.

Advanced Simulation-Inserted Optimization Using Combined Quasi-Newton Method With Lagrangian Method for EM-Based Design Optimization

Traditional electromagnetic (EM)-based design optimizations typically treat the EM simulation as a black-box and use optimization algorithms to externally drive EM simulations iteratively. Recently, a novel EM-based design optimization technique, named as simulation-inserted optimization (SIO) in this article, has been developed by opening the black-box and inserting the optimization algorithm into the internal of EM simulation to achieve EM field solutions and the optimal values of design variables simultaneously. This article proposes an advanced SIO algorithm using combined quasi-Newton method with Lagrangian method. In the proposed algorithm, the finite element method (FEM) is used as the EM simulation method, and the Lagrangian method is incorporated to internally integrate EM simulation with design optimization. New formulations based on quasi-Newton method are derived to approximate the computationally intensive lower upper (LU) decomposition of the FEM system matrix in the current optimization iteration by that in the previous optimization iteration to significantly reduce the number of LU decompositions. The proposed SIO technique further derives novel optimization update formulations by introducing the line search algorithm to increase the robustness of optimization. Using the proposed SIO algorithm, only a few initial LU decompositions of FEM system matrices need to be calculated instead of repetitively calculating a large number of LU decompositions of FEM system matrices during optimization, consequently speeding up the overall optimization process. The proposed technique is demonstrated by two application examples of EM-based design optimization of microwave components.

Reliability Evaluation of LU Decomposition on GPU-Accelerated System-on-Chip Under Proton Irradiation

Graphic processing units (GPUs) have become a basic accelerator both in high-performance nodes and low-power system-on-chip (SoC). They provide massive data parallelism and very high performance per watt. However, their reliability in harsh environments is an important issue to take into account, especially for safety-critical applications. In this article, we evaluate the influence of the parallelization strategy on the reliability of lower–upper (LU) decomposition on a GPU-accelerated SoC under proton irradiation. Specifically, we compare a memory bound and a compute bound implementation of the decomposition on a K20A GPU embedded on a Tegra K1 (TK1) SoC. We leverage the GPU and CPU clock frequencies both to highlight the radiation sensitivity of the GPU where we are running the benchmark and also to apply both algorithms to solve problems with the same size when exposed to the same radiation dose. Results show that more intensive use of the resources of the GPU increases the cross section. We also observed that most of the radiation-induced errors hang the operating system and even the rebooting process. Finally, we present a preliminary study of the error propagation of the LU decomposition algorithms.

A novel and efficient engine for P-/S-wave-mode vector decomposition for vertical transverse isotropic elastic reverse time migration

Wave-mode decomposition plays a very important role in elastic reverse time migration (ERTM). Improved imaging quality can be achieved due to reduced wave-mode crosstalk artifacts. The current state-of-the-art methods for anisotropic wavefield separation are based on either splitting model strategy, low-rank approximation, or lower-upper (LU) factorization. Most of these involve expensive matrix computation and Fourier transforms with strong model assumptions. Based on the anisotropic-Helmholtz (ani-Helmholtz) decomposition operator and decoupled formulations, we develop a novel and efficient P-/S-wave-mode vector decomposition method in vertical transverse isotropic (VTI) media with application in ERTM. We first review the basics of classical Helmholtz decomposition and isotropic decoupled formulations. In addition, the ani-Helmholtz decomposition operator is built from the P- and S-wave polarizations of the Christoffel equation in VTI media. We then derive novel decoupled formulations of anisotropic P-/S-waves based on the obtained ani-Helmholtz operator. Moreover, we use the first-order Taylor expansion to approximate the normalization term from the derived decoupled formulations and obtain an efficient ani-Helmholtz decomposition approach, which produces vector P- and S-wavefields with correct units, phases, and amplitudes. Compared with the previous studies, our approach mitigates model assumptions, avoids intricate calculations, such as LU factorization or low-rank approximation, and only needs three fast Fourier transforms at each time step. In addition, the graphic processing unit technique is used to dramatically accelerate various functions of ERTM, such as wavefields extrapolation, decomposition, and imaging. Three synthetic examples demonstrate the effectiveness and feasibility of our proposed approach.

A robust time delay estimation method for ultrasonic echo signals and elastography

Modern medicine cannot ignore the significance of elastography in diagnosis and treatment plans. Despite improvements in accuracy and spatial resolution of elastograms, robustness against noise remains a neglected attribute. A method that can perform in a satisfactory manner under noisy conditions may prove useful for various elastography methods. Here, we propose a method based on eigenvalue decomposition (EVD). In this method, the estimated time delay is defined as the index of the maximum element in the eigenvector that corresponds to the minimum eigenvalue in the covariance matrix of the received signal. Moreover, the implementation of the least-squares (LS) solution and the lower-upper (LU) decomposition contributes to improving the speed of computation and the accuracy of the estimation under low signal-to-noise ratio (SNR) conditions. To assess the performance of the proposed algorithm, it is evaluated along with generalized cross-correlation (GCC) and EVD. The simulation results clearly confirm that the jitter variance achieved in the proposed algorithm outperforms GCC and EVD in the proximity of the Cramer-Rau lower band. Moreover, our algorithm provides satisfactory performance in terms of variance and bias against sub-sample delay at low SNRS. According to the experimental results, the calculated values of the elastographic signal-to-noise ratio (SNRe) and the elastographic contrast-to-noise ratio (CNRe) of the proposed algorithm were 16.7 and 20.09, respectively, clearly better than the values of the other two methods. Furthermore, the proposed algorithm offers less execution time (about 30% of GCC), with a computational complexity equal to GCC and better than EVD.

Efficient extended-search space full-waveform inversion with unknown source signatures

SUMMARY Full waveform inversion (FWI) requires an accurate estimation of source signatures. Due to the coupling between the source signatures and the subsurface model, small errors in the former can translate into large errors in the latter. When direct methods are used to solve the forward problem, classical frequency-domain FWI efficiently processes multiple sources for source signature and wavefield estimations once a single lower–upper (LU) decomposition of the wave-equation operator has been performed. However, this efficient FWI formulation is based on the exact solution of the wave equation and hence is highly sensitive to the inaccuracy of the velocity model due to the cycle skipping pathology. Recent extended-space FWI variants tackle this sensitivity issue through a relaxation of the wave equation combined with data assimilation, allowing the wavefields to closely match the data from the first inversion iteration. Then, the subsurface parameters are updated by minimizing the wave-equation violations. When the wavefields and the source signatures are jointly estimated with this approach, the extended wave equation operator becomes source dependent, hence making direct methods and, to a lesser extent, block iterative methods ineffective. In this paper, we propose a simple method to bypass this issue and estimate source signatures efficiently during extended FWI. The proposed method replaces each source with a blended source during each data-assimilated wavefield reconstruction to make the extended wave equation operator source independent. Besides computational efficiency, the additional degrees of freedom introduced by spatially distributing the sources allows for a better signature estimation at the physical location when the velocity model is rough. We implement the source signature estimation with a variable projection method in the recently proposed iteratively refined wavefield reconstruction inversion (IR-WRI) method. Numerical tests on the Marmousi II and 2004 BP salt synthetic models confirm the efficiency and the robustness against velocity model errors of the new method compared to the case where source signatures are known.

On efficient frequency-domain full-waveform inversion with extended search space

Efficient frequency-domain full-waveform inversion (FD-FWI) of wide-aperture data is designed by limiting inversion to a few frequencies and by solving the Helmholtz equation with a direct solver to process multiple sources efficiently. Some variants of FD-FWI, which process the wave equation as a weak constraint, have been proposed to increase the computational efficiency or extend the search space. Among them, the contrast-source reconstruction inversion (CSRI) reparameterizes FD-FWI in terms of contrast sources (CS) and contrasts and updates them in an alternating mode. This reparameterization allows for one lower-upper (LU) decomposition of the Helmholtz operator to be performed per frequency inversion hence further improving the computational efficiency of FD-FWI. However, iteratively refined wavefield reconstruction inversion (IR-WRI) relies on the alternating-direction method of multipliers to extend the search space by matching the data from the early iterations via an aggressive relaxation of the wave equation while satisfying it at the convergence point thanks to the defect correction performed by the Lagrange multipliers. In contrast to CSRI, IR-WRI requires redoing one LU decomposition when the subsurface model is updated. In both methods, the CSs or the wavefields are computed by solving in a least-squares sense an overdetermined linear system gathering an observation equation and a wave equation. A drawback of CSRI is that CSs are estimated approximately with one iteration of a conjugate gradient method, whereas the wavefields are reconstructed exactly by IR-WRI with a Gauss-Newton method. We have combined the benefits of CSRI and IR-WRI to decrease the number of LU decomposition during IR-WRI with a fixed-point (FP) algorithm while preserving its search space extension capability. Application on the 2D complex Marmousi and the BP salt models shows that our FP-based IR-WRI manages to reconstruct these models as accurately as the classic IR-WRI while reducing the number of LU factorizations considerably. A theoretical complexity analysis and a recent application of 3D FD-FWI based upon direct solver suggest that the FP algorithm should reduce the cost of IR-WRI by a factor of approximately 2 and 10 for 3D dense ocean bottom cable and sparse ocean bottom node acquisitions, respectively.

Image hiding using upper-lower decomposition technique

Data security is one of the most important sciences nowadays. There is a huge amount of data transferred over the internet each moment and this data should be secured. Steganography is a type of data security technique that is used to hide the secret message into a cover object. Image steganography is the technique that hides an image in another image. This paper proposed a technique that depends on Lower-Upper (LU) decomposition. In the proposed technique LU decomposition is applied for both cover and secret images. The proposed method was tested using different grayscale and color images. The proposed technique achieved high results in terms of Peak Signal to Noise Ratio (PSNR), Structural Similarity Index Measure (SSIM), and Normalized Cross-Correlation (NCC). The PSNR for the cover image is ranging from 36 to 44 dB. The similarity between the secret image and the extracted image is 100% and the NCC is 1.

A new transient model to simulate and optimize liquid unloading with coiled tubing conveyed gas lift

Nowadays, unloading gas wells with coiled tubing is a common application to the field. However, it still lacks of adequate understanding of dynamic behavior of the unloading process. This paper investigates the process of liquid unloading by gas lift with coiled tubing under transient conditions. This unloading process can be divided into three stages: liquid rising in tubing, liquid slug production, and liquid production by entrainment. In each stage, the mass and conversation equations are applied as governing equations. The components of each stage include coiled tubing, coiled tubing-tubing annulus, liquid slug, gas bubble, and liquid film. Empirical correlations have been used for surface gas injection choke, check valve, friction factor, the relationship between the gas bubble and the liquid slug velocity, inflow performance relationship, and black oil fluid properties. From the above, the dynamic model coupling real-time change of inflow performance relationship is developed. The lower upper (LU) factorization and Euler's method are applied to solve the proposed dynamic model in time domain. Among all these variables, the most important ones include gas injection rates, pressures at various locations, length of the liquid slug and gas bubble, and phase velocities. Through the simulation efforts, the mechanism of liquid unloading process is revealed.Gas lift is commonly constrained by gas availability. This is a pioneering study on liquid unloading with coiled tubing. The results can be applied to design coiled tubing gas lift to optimize the usage of injected gas, choose appropriate pump, and save the energy consumed in gas lift operation.

Additional methods for the stable calculation of isotropic Gaussian filter coefficients: The case of a truncated filter kernel

The isotropic Gaussian filter is frequently used as a post-processing method for mitigating errors in Gravity Recovery and Climate Experiment (GRACE) time-variable gravity field solutions. It is known that the recurrent calculation of isotropic Gaussian filter coefficients in the spherical harmonic domain results in numerical instabilities. This issue has been recently resolved by Piretzidis & Sideris (Computers & Geosciences 133:104303) for the case of a global Gaussian filter. In this paper, we extend this work for the case of a regional Gaussian filter (i.e., a Gaussian filter with a truncated kernel). We show that instabilities also occur when the coefficients of a regional Gaussian filter are calculated using a conventional recurrence relation, and we provide methods for their proper treatment. We also briefly present some theoretical preliminaries related to recurrence relations and examine algorithms (i.e., initial value and boundary value algorithms) for their stable evaluation. Our analysis indicates that the initial value algorithms are ineffective for the calculation of regional Gaussian filter coefficients, contrary to the global Gaussian filter. The comparison of the boundary value algorithms shows that the Thomas and lower-upper (LU) decomposition algorithms are the most suitable for this case. Their main advantage is their insusceptibility to numerical limitations, such as overflow errors. The Thomas algorithm is also the most efficient in terms of computational speed.

Efficient calculation of interior scattering from cavities with partial IBC wall

The magnetic field integral equation based method of moments (MoM) with the Kirchhoff approximation can predict the interior scattering from cavities with impedance boundary condition (IBC) walls. However, if the IBC wall is a small part of the whole wall and is modified multiple times, the conventional MoM may be time-consuming due to the repeated and expensive lower–upper (LU) decompositions of impedance matrices. In this Letter, the Sherman–Morrison–Woodbury (SMW) formula is used to mitigate the problem. The authors only require to perform the LU decomposition of the cavity without IBC walls, and to efficiently obtain the solutions of the cavity with different IBC walls by reusing the LU decomposition and the SMW formula.

Explicit nonlinear predictive control algorithms for Laguerre filter and sparse least square support vector machine-based Wiener model

In this paper, three computationally proficient model predictive control (MPC) algorithms for least square support vector machine (LSSVM)-based Wiener model are described. A Wiener model with Laguerre filter as dynamic linear part and LSSVM approximator as nonlinear static part is considered. Even though having excellent approximation abilities, LSSVM suffers from lack of sparseness. A pruning algorithm for LSSVM model is proposed and its comparison is made with classical pruning algorithm. The proposed pruning algorithm is able to remove 99% of support vectors with no remarkable drop in modelling accuracy. Using pruned Wiener model, three computationally efficient MPC algorithms are described. In the first algorithm, linearization of Wiener model is performed at every sampling interval and therefore control vector is determined by carrying out a quadratic optimization task. In the second algorithm, control signal is determined by an explicit control law and parameters of this control law are computed by performing lower-upper (LU) factorization of a matrix and solving linear equations without any online optimization. In the third algorithm, the parameters of explicit control law are calculated directly by another LSSVM approximator, which is trained offline. The advantages and effectiveness of proposed methods are demonstrated on the benchmark pH neutralization reactor. The control performance and computational efficiency of proposed algorithms are compared with computationally complex nonlinear MPC, which repeats a nonlinear optimization task at every sampling interval. The impact of pruning on model accuracy, computational efficiency and control accuracy is also discussed.

Perbandingan Metode Newton-Raphson &amp; Metode Secant Untuk Mencari Akar Persamaan Dalam Sistem Persamaan Non-Linier

The numerical method is a technique used to formulate mathematical problems so that it can be solved using ordinary arithmetic operations. In general, numerical methods are used to solve mathematical problems that cannot be solved by ordinary analytic methods. In the Numerical Method, we recognize two types of systems of equations, namely the Linear Equation System and the Non-Linear Equation System. Each system of equations has several methods. In the Linear Equation System between methods is the Gauss Elimination method, the Gauss-Jordan Elimination method, the LU (Lower-Upper) Decomposition method. And for Non-Linear Equation Systems between the methods are the Bisection method, the Regula Falsi method, the Newton Raphson method, the Secant method, and the Fix Iteration method. In this study, researchers are interested in analyzing 2 methods in the Non-Linear Equation System, the Newton-Raphson method and the Secant method. And this analysis process uses the Java programming language tools, this is to facilitate the analysis of method completion algorithm, and monitoring in terms of execution time and analysis of output results. So we can clearly know the difference between what happens between the two methods.

GLU3.0: Fast GPU-based Parallel Sparse LU Factorization for Circuit Simulation

Editor's note: Many scientific computing problems, including circuit simulations, rely on efficient lower-upper (LU) decomposition of sparse matrices. Prior studies took advantage of GPUs to parallelize LU decomposition, but they suffer from nontrivial data dependencies. This article presents a new method, called GLU3.0, to accelerate GPU-based sparse LU factorization. -Umit Ogras, Arizona State University.

Efficient calculation of interior scattering from cavities with small modifications

The analysis of the interior scattering from open cavities with small modifications is an important task in designing a stealthy jet engine. Previous research has shown the magnetic field integral equation with the Kirchhoff approximation can be used to calculate the cavity interior scattering. However, it must repeat the expensive method of moments (MoM) solution even when the cavity is modified only slightly. In this Letter, the efficient method based on the partitioned-inverse formula and the Sherman–Morrison–Woodbury formula is employed to address this problem. It can avoid the repeated MoM direct solution. We only need to solve the lower-upper (LU) decomposition of the impedance matrix of the original cavity, and can efficiently derive the solution of the modified cavities via matrix identities without loss of accuracy. Numerical results are given to demonstrate the performance of the proposed approach.