Nonsingular Research Articles

3-D induction log modelling with integral equation method and domain decomposition pre-conditioning

SUMMARY The deployment of electromagnetic (EM) induction tools while drilling is one of the standard routines for assisting the geosteering decision-making process. The conductivity distribution obtained through the inversion of the EM induction log can provide important information about the geological structure around the borehole. To image the 3-D geological structure in the subsurface, 3-D inversion of the EM induction log is required. Because the inversion process is mainly dependent on forward modelling, the use of a fast and accurate forward modelling tool is essential. In this paper, we present an improved version of the integral equation (IE) based modelling technique for general anisotropic media with domain decomposition pre-conditioning. The discretized IE after domain decomposition equals a fixed-point equation that is solved iteratively with either the block Gauss–Seidel or Jacobi pre-conditioning. Within each iteration, the inverse of the block matrix is computed using a Krylov subspace method instead of a direct solver. An additional reduction in computational time is obtained by using an adaptive relative residual stopping criterion in the iterative solver. Using this domain decomposition scheme, numerical experiments show computation time reductions by factors of 1.97–2.84 compared to solving the full-domain IE with a GMRES solver and a contraction IE pre-conditioner. Additionally, the reduction of memory requirement for covering a large area of the induction tool sensitivity enables acceleration with limited GPU memory. Hence, we conclude that the domain decomposition method is improving the efficiency of the IE method by reducing the computation time and memory requirement.

IMPLEMENTASI MATRIKS ATAS GELANGGANG Z_p⨁Z_q⨁Z_r PADA KRIPTOGRAFI CIPHER HILL

Cryptography is a method for securing data or information from data leaks. In cryptography, there are two important processes, namely, the encryption and decryption processes. Encryption is the process of changing plaintext into ciphertext using certain keys and algorithms, while decryption is the process of changing ciphertext into plaintext using keys and algorithms that match the encryption. There is a cryptosystem that is well known, namely Cipher Hill. Cipher Hill uses a square matrix for the encryption and decryption process. In this research, an mxn-sized matrix will be used over the Direct Sum ring Z_p⨁Z_q⨁Z_r to expand the Cipher Hill algorithm. The author also uses the pseudoinverse concept to find the inverse of an mxn matrix. By using an mxn matrix, the encryption results allow the length of the plaintext to be different from the length of the ciphertext.Keywords: Cipher Hill, Cryptography, Decryption, Encryption, Direct Sum, Pseudoinvers

Sparse precision matrix estimation under lower polynomial moment assumption

Precision matrix (inverse covariance matrix) estimation is a rising challenge in contemporary applications while dealing with high‐dimensional data. This paper focuses on large‐scale precision matrix of the random vector that only has lower polynomial moments. We mainly investigate upper bounds of the proposed estimator under the spectral norm in terms of the probability and mean estimation respectively. It is shown that the data‐driven estimator is fully adaptive and achieves the same optimal convergence order as under Gaussian assumption on the data. Simulation studies further support our theoretical claims.

On robust estimation of the Gauss–Markov model with a singular covariance matrix

Although the robust parameter estimation has been intensively investigated in statistics and measurement science, no one considers the robust estimation within the Gauss–Markov model (GMM) with a singular covariance matrix. In this contribution, a method of finding the robust solutions in the singular linear model has been suggested based on the generalization of the unified least-squares objective function. The equivalent cofactor matrix in the robust objective function is established by the bifactor strategy based on the IGG III function. The robustness of the proposed robust algorithms is demonstrated with the examples of geodetic coordinates transformation and the free network deformation analysis.

Eigenproblem Basics and Algorithms

Some might say that the eigenproblem is one of the examples people discovered by looking at the sky and wondering. Even though it was formulated to explain the movement of the planets, today it has become the ansatz of solving many linear and nonlinear problems. Formulation in the terms of the eigenproblem is one of the key tools to solve complex problems, especially in the area of molecular geometry. However, the basic concept is difficult without proper preparation. A review paper covering basic concepts and algorithms is very useful. This review covers the basics of the topic. Definitions are provided for defective, Hermitian, Hessenberg, modal, singular, spectral, symmetric, skew-symmetric, skew-Hermitian, triangular, and Wishart matrices. Then, concepts of characteristic polynomial, eigendecomposition, eigenpair, eigenproblem, eigenspace, eigenvalue, and eigenvector are subsequently introduced. Faddeev–LeVerrier, von Mises, Gauss–Jordan, Pohlhausen, Lanczos–Arnoldi, Rayleigh–Ritz, Jacobi–Davidson, and Gauss–Seidel fundamental algorithms are given, while others (Francis–Kublanovskaya, Gram–Schmidt, Householder, Givens, Broyden–Fletcher–Goldfarb–Shanno, Davidon–Fletcher–Powell, and Saad–Schultz) are merely discussed. The eigenproblem has thus found its use in many topics. The applications discussed include solving Bessel’s, Helmholtz’s, Laplace’s, Legendre’s, Poisson’s, and Schrödinger’s equations. The algorithm extracting the first principal component is also provided.

An improved convected particle domain interpolation material point method for large deformation geotechnical problems

AbstractA novel material point method called the improved convected particle domain interpolation (ICPDI) is proposed in this paper to simulate large‐deformation problems in geotechnical engineering. The ICPDI framework is based on the existing convected particle domain interpolation (CPDI) and employs an adaptive orthogonal improved interpolating moving least square (AOIIMLS) shape functions. The AOIIMLS shape functions prevent computational instability problems caused by singular or ill‐conditioned matrices through avoiding the direct solving of inverse matrices. Additionally, ICPDI adopts the velocity gradient to calculate the velocity field in particle domains and reduces cell‐crossing errors by using AOIIMLS shape functions. To reconstruct the velocity function in the background grid, the velocities of the material points and four related corner points are employed through AOIIMLS shape functions. The bar axis‐aligned vibration, ring radial expansion, slope elasto‐plastic slumping, and post‐failure tunnel collapse behavior examples are used to verify the accuracy and effectiveness of ICPDI by comparing the results with other numerical methods and experiments. Numerical simulation results show that ICPDI outperforms the material point method (MPM) and CPDI in terms of computational accuracy and convergence while reducing cell‐crossing errors. Consequently, this study demonstrates the capability of ICPDI in simulations of large‐deformation geotechnical problems.

Big S-Boxes for Deeply Improved Hill Image Coding

This paper introduces a novel approach to encrypt color images, utilizing multiple chaotic maps and working with any size pixel blocks. The encryption procedure commences by converting the original image into a vector, which is then divided into equal dimension subblocks. These blocks undergo a new Vigenère transformation, employing two large dynamic substitution tables interconnected by an enhanced Hill operation using a large invertible matrix generated from a pseudo-random table, controlled by another binary dynamics table. Additionally, the truncated block undergoes the same encryption process using another pseudo-random invertible matrix. To evaluate the effectiveness and resilience of our algorithm, we conducted numerous simulations on randomly selected images of varying sizes and formats from diverse databases.

General strong fuzzy solutions of complex fuzzy matrix equations involving the Moore-Penrose weak group inverse

In this paper, we investigate the fuzzy solutions of the complex fuzzy matrix equation (CFME) CZ˜=W˜, in which C is a complex crisp matrix, and W˜ is a complex fuzzy matrix. The purpose of this paper is three-fold. Firstly, the necessary and sufficient conditions for the existence of strong fuzzy solutions to the CFME are obtained using the MPWG inverse of the coefficient matrix. Secondly, we obtain a necessary and sufficient condition for the existence of S†,WG≥0 (S†,WG is called the Moore-Penrose weak group inverse of the coefficient matrix S) in order to obtain a strong fuzzy matrix solution of CFME. Moreover, general strong fuzzy solutions of the CFME are derived and an algorithm for obtaining general strong fuzzy solutions of CFME by the MPWG inverse is also established. Finally, some numerical examples are given to illustrate the main results.

Higher-continuity s-version of finite element method with B-spline functions

This paper proposes a strategy to solve the problems of the conventional s-version of finite element method (SFEM) fundamentally. Because SFEM can reasonably model an analytical domain by superimposing meshes with different spatial resolutions, it has intrinsic advantages of local high accuracy, low computation time, and simple meshing procedure. However, it has disadvantages such as accuracy of numerical integration and matrix singularity. Although several additional techniques have been proposed to mitigate these limitations, they are computationally expensive or ad-hoc, and detract from the method's strengths. To solve these issues, we propose a novel strategy called B-spline based SFEM. To improve the accuracy of numerical integration, we employed cubic B-spline basis functions with C2-continuity across element boundaries as the global basis functions. To avoid matrix singularity, we applied different basis functions to different meshes. Specifically, we employed the Lagrange basis functions as local basis functions. The numerical results indicate that using the proposed method, numerical integration can be calculated with sufficient accuracy without any additional techniques used in conventional SFEM. Furthermore, the proposed method avoids matrix singularity and is superior to conventional methods in terms of convergence for solving linear equations. Therefore, the proposed method has the potential to reduce computation time while maintaining a comparable accuracy to conventional SFEM.

Evaluating Sustainable Aviation Fuel Production Technologies A Strategic Analysis Using the WASPAS Method

Over the past 25 years, international passenger air traffic has tripled, and this rapid growth is expected to continue for the next 25 years. While the expansion of the aviation sector brings significant economic benefits, it also poses substantial social and environmental challenges. To address these, a "sustainable aviation policy" has been proposed as a "balanced plan" that seeks to mitigate the environmental impacts of the sector's growth while acknowledging its economic advantages. However, this definition of "sustainable aviation" is contested by various organizations, and there is little agreement between non-governmental organizations and the aviation industry. Consequently, the concept of a "standard aviation policy" remains in dispute, with different parties attempting to modify it to align with their respective goals. To develop policies for sustainable aviation, competing environmental discourses have been identified and analyzed using a classification exercise supported by rhetorical, stylistic, and thematic analysis. The Analytic Hierarchy Process (AHP) is one approach used for multi-attribute decision-making that satisfies the consistency requirements of each reciprocal matrix. Recently, the "Weighted Aggregates Sum Product Assessment System" (WASPAS) has been introduced in the literature, combining the weighted product sum and basic aggregate weighting methods.

Fragile watermarking for image authentication using dyadic walsh ordering

A digital image is subjected to the most manipulation. This is driven by the easy manipulating process through image editing software which is growing rapidly. These problems can be solved through the watermarking model as an active authentication system for the image. One of the most popular methods is Singular Value Decomposition (SVD) which has good imperceptibility and detection capabilities. Nevertheless, SVD has high complexity and can only utilize one singular matrix S, and ignore two orthogonal matrices. This paper proposes the use of the Walsh matrix with dyadic ordering to generate a new S matrix without the orthogonal matrices. The experimental results showed that the proposed method was able to reduce computational time by 22% and 13% compared to the SVD-based method and similar methods based on the Hadamard matrix respectively. This research can be used as a reference to speed up the computing time of the watermarking methods without compromising the level of imperceptibility and authentication.

Implementation of Asynchronous Distributed Gauss-Newton Optimization Algorithms for Uncertainty Quantification by Conditioning to Production Data

Summary Previous implementation of the distributed Gauss-Newton (DGN) optimization algorithm ran multiple optimization threads in parallel, employing a synchronous running mode (S-DGN). As a result, it waits for all simulations submitted in each iteration to complete, which may significantly degrade performance because a few simulations may run much longer than others, especially for time-consuming real-field cases. To overcome this limitation and thus improve the DGN optimizer’s execution, we propose two asynchronous DGN (A-DGN) optimization algorithms in this paper. The two A-DGN optimization algorithms are (1) the local-search algorithm (A-DGN-LS) to locate multiple maximum a-posteriori (MAP) estimates and (2) the integrated global-search algorithm with the randomized maximum likelihood (RML) method (A-DGN + RML) to generate hundreds of RML samples in parallel for uncertainty quantification. We propose using batch together with a checking time interval to control the optimization process. The A-DGN optimizers check the status of all running simulations after every checking time frame. The iteration index of each optimization thread is updated dynamically according to its simulation status. Thus, different optimization threads may have different iteration indices in the same batch. A new simulation case is proposed immediately once the simulation of an optimization thread is completed, without waiting for the completion of other simulations. We modified the training data set updating algorithm using each thread’s dynamically updated iteration index to implement the asynchronous running mode. We apply the modified QR decomposition method to estimate the sensitivity matrix at the best solution of each optimization thread by linear interpolation of all or a subset of the training data to avoid the issue of solving a linear system with a singular matrix because of insufficient training data points in early batches. A new simulation case (or search point) is generated by solving the Gauss-Newton (GN) trust-region subproblem (GNTRS) using the estimated sensitivity matrix. We developed a more efficient and robust GNTRS solver using eigenvalue decomposition (EVD). The proposed A-DGN optimization methods are tested and validated on a 2D analytical toy problem and a synthetic history-matching problem and then applied to a real-field deepwater reservoir model. Numerical tests confirm that the proposed A-DGN optimization methods can converge to solutions with matching quality comparable to those obtained by the S-DGN optimizers, saving on the time required for the optimizer to converge by a factor ranging from 1.3 to 2 when compared to the S-DGN optimizer depending on the problem. The new A-DGN optimization algorithms improve efficiency and robustness in solving history-matching or inversion problems, especially for uncertainty quantification of subsurface model parameters and production forecasts of real-field reservoirs by conditioning production data.

A numerical method for inverse Helmholtz problem based on approximate inverse of a matrix

A numerical method for inverse Helmholtz problem based on approximate inverse of a matrix

Advanced structural health monitoring and damage detection using inverse methods and perturbation stiff-ness matrix analysis

Sensor technology advancements have the health monitoring task for modern intricate structures introduce many challenges, primarily centered around the augmentation of safety margins and operational dependabil-ity. The demand for diverse systems or methodologies is, therefore, increasing. These systems or methodolo-gies should be capable of detecting damage, monitoring remotely, and conducting continuous evaluations in real-time. As part of this study, inverse techniques rooted in wave propagation analysis within structural frameworks are utilized to identify damage response vectors. Additionally, mathematical formulations grounded in minimal rank perturbation theory facilitate determining a perturbed stiffness matrix for a com-promised structure. Various excitations are included in the analysis, which is based on intricate non-linear structural modeling. Using streamlined procedures, these findings confirm the inherent versatility of a range of methodologies capable of meeting both rapid monitoring needs and exhaustive analytical needs. In addition, the proposed approach promotes the use of intelligent structures embedded with a variety of sensors that are not restricted by limitations related to sensor type or spatial deployment. Monitoring structural health and detecting damage could be significantly improved by this strategic deployment.

On the independence number of regular graphs of matrix rings

Consider a graph on the non-singular matrices over a finite field, in which two distinct non-singular matrices are joined by an edge whenever their sum is singular. We prove an upper bound for the independence number of this graph. As a consequence, we obtain a lower bound for its chromatic number that significantly improves a previous result of Tomon.

Zeroing neural network approaches for computing time-varying minimal rank outer inverse

Generalized inverses are extremely effective in many areas of mathematics and engineering. The zeroing neural network (ZNN) technique, which is currently recognized as the state-of-the-art approach for calculating the time-varying Moore-Penrose matrix inverse, is investigated in this study as a solution to the problem of calculating the time-varying minimum rank outer inverse (TV-MROI) with prescribed range and/or TV-MROI with prescribed kernel. As a result, four novel ZNN models are introduced for computing the TV-MROI, and their efficiency is examined. Numerical tests examine and validate the effectiveness of the introduced ZNN models for calculating TV-MROI with prescribed range and/or prescribed kernel.

Space-time wave packets with reduced divergence and tunable group velocity generated in free space after multi-mode fiber propagation.

Previously, space-time wave packets (STWPs) have been generated in free space with reduced diffraction and a tunable group velocity by combining multiple frequency comb lines each carrying a single Bessel mode with a unique wave number. It might be potentially desirable to propagate the STWP through fiber for reconfigurable positioning. However, fiber mode coupling might degrade the output STWP and distort its propagation characteristics. In this Letter, we experimentally demonstrate STWP generation and propagation over 1-m graded-index multi-mode fiber. Fiber mode coupling is mitigated by pre-distortion according to the inverse matrix of the fiber mode coupling matrix. Measurement of the STWP at the fiber output shows that its group velocity can vary from 1.0042c to 0.9967c by tuning the wave number of the Bessel mode on each frequency. The measured time-averaged intensity profiles show that the beam radius remains similar after 150-mm free-space propagation after exiting the fiber.

Certain invertible operator-block matrices induced by C^{*}-algebras and scaled hypercomplex numbers

The main purposes of this paper are (i) to enlarge scaled hypercomplex structures to operator-valued cases, where the operators are taken from a C^{*}-subalgebra of an operator algebra on a separable Hilbert space, (ii) to characterize the invertibility conditions on the operator-valued scaled-hypercomplex structures of (i), (iii) to study relations between the invertibility of scaled hypercomplex numbers, and that of operator-valued cases of (ii), and (iv) to confirm our invertibility of (ii) and (iii) are equivalent to the general invertibility of left( 2times 2right) -block operator matrices.

Design and kinematical performance analysis of a novel reconfigurable parallel mechanism with three remote center-of-motion modes

In view of the demands for variability and safety in surgical procedures, this paper presents a novel reconfigurable parallel mechanism (RPM) capable of performing three remote center-of-motion (RCM) modes: 1R1T, 2R1T, and 3R1T. One of the RCM modes can transform into two distinct configurations from the switch configuration, in which one configuration offers a rotational degree-of-freedom (DOF) and a translational DOF in the YOZ plane, and the other configuration provides identical DOFs in the XOZ plane. A novel device is introduced for switching between the active and passive modes of the revolute joint, which serves as the basis for the RPM reconfiguration process. Utilizing screw theory, the mobility and actuation scheme of the RPM in each configuration are analyzed. Inverse kinematic analysis and Jacobian matrices are then carried out. Subsequently, kinematical performances, such as singularity, workspace, and dexterity, are studied. All results indicate that the proposed RPM has the ability to switch motion modes without requiring reassembly and can effectively meet the diverse demands of surgical procedures based on specific minimally invasive surgery (MIS) requirements.

Null ideals of sets of 3 × 3 similar matrices with irreducible characteristic polynomial

Let F be a field and M n ( F ) the ring of n × n matrices over F. Given a subset S of M n ( F ) , the null ideal of S is the set of all polynomials f with coefficients from M n ( F ) such that f ( A ) = 0 for all A ∈ S . We say that S is core if the null ideal of S is a two-sided ideal of the polynomial ring M n ( F ) [ x ] . We study sufficient conditions under which S is core in the case where S consists of 3 × 3 matrices, all of which share the same irreducible characteristic polynomial. In particular, we show that if F is finite with q elements and | S | ≥ q 3 − q 2 + 1 , then S is core. As a byproduct of our work, we obtain some results on block Vandermonde matrices, invertible matrix commutators, and graphs defined via an invertible difference relation.