Search Results (863)

Recently, a new type of derivative has been introduced, known as Caputo proportional derivatives. These are motivated by the applications of such derivatives (which are a generalization of Caputo’s standard fractional derivative) and the need to incorporate such calculus into the research on operators. The investigation therefore focuses on the equivalence of differential and integral problems for proportional calculus problems. The operators are always studied in the appropriate function spaces. Furthermore, the investigation extends these results to encompass the more general notion of Hilfer hybrid derivatives. The primary aim of this study is to preserve the maximal regularity of solutions for this class of problems. To this end, we consider such operators not only in spaces of absolutely continuous functions, but also in particular in little Hölder spaces. It is widely acknowledged that these spaces offer a natural framework for the study of classical Riemann–Liouville integral operators as inverse operators with derivatives of fractional order. This paper presents a comprehensive study of this problem for proportional derivatives and demonstrates the application of the obtained results to Langevin-type boundary problems. Full article

The grounding grid is an important piece of equipment to ensure the safety of a power system, and thus research detecting on its corrosion status is of great significance. Electrical impedance tomography (EIT) is an effective method for grounding grid corrosion imaging. However, the inverse process of image reconstruction has pathological solutions, which lead to unstable imaging results. This paper proposes a grounding grid electrical impedance imaging method based on an improved conditional generative adversarial network (CGAN), aiming to improve imaging precision and accuracy. Its generator combines a preprocessing module and a U-Net model with a convolutional block attention module (CBAM). The discriminator adopts a PatchGAN structure. First, a grounding grid forward problem model was built to calculate the boundary voltage. Then, the image was initialized through the preprocessing module, and the important features of ground grid corrosion were extracted again through the encoder module, decoder module and attention module. Finally, the generator and discriminator continuously optimized the objective function and conducted adversarial training to achieve ground grid electrical impedance imaging. Imaging was performed on grounding grids with different corrosion conditions. The results showed a final average peak signal-to-noise ratio of 20.04. The average structural similarity was 0.901. The accuracy of corrosion position judgment was 94.3%. The error of corrosion degree judgment was 9.8%. This method effectively improves the pathological problem of grounding grid imaging and improves the precision and accuracy, with certain noise resistance and universality. Full article

A major challenge in accidental or unregulated releases is the ability to identify the pollutant source, especially if the location is in a large industrial area. Usually in such cases, only a few sensors provide non-zero signal. A crucial issue is therefore the ability to use a small number of sensors in order to identify the source location and rate of emission. The general problem of characterizing source parameters based on real-time sensors is known to be a difficult task. As with many inverse problems, one of the main obstacles for an accurate estimation is the non-uniqueness of the solution, induced by the lack of sufficient information. In this study, an efficient method is proposed that aims to provide a quantitative estimation of the source of hazardous gases or breathable aerosols. The proposed solution is composed of two parts. First, the physics of the atmospheric dispersion is utilized by a well-established Lagrangian stochastic model propagated backward in time. Then, a new algorithm is formulated for the prediction of the spacial expected uncertainty reduction gained by the optimal placement of an additional sensor. These two parts together are used to construct an adaptive decision support system for the dynamical deployment of detectors, allowing for an efficient characterization of the emitting source. This method has been tested for several scenarios and is shown to significantly reduce the uncertainty that stems from the insufficient information. Full article

A number of computer experiments have investigated the effectiveness in terms of accuracy of the method for simultaneously determining the distributions of electrical conductivity and magnetic permeability in the subsurface zone of planar conductive objects when modeling the process of eddy-current measurement testing by surface probes. The method is based on the use of surrogate optimization, which involves the use of a high-performance neural network proxy-model of probe by means of a deep learning as part of the target quadratic function. The surrogate model acts as a carrier and storage of a priori information about the object and takes into account the influence of all the main factors essential in the formation of the probe output signal. The problems of the surrogate model’s cumbersomeness and mitigation of the “curse of dimensionality” effect are solved by applying techniques for reducing the dimensionality of the design space based on the PCA algorithm. We investigated options for compromise solutions regarding the dimensionality of the PCA-space and the accuracy of obtaining the desired material properties profiles by the optimization method. The results of modeling the inverse measurement problem indicate a fairly high accuracy of profile reconstruction. Full article

This paper proposes an analytical solution for the seepage field when a localized line leakage occurs in a tunnel by accurately considering the boundary conditions at the leakage site, which overcomes the problem of current methods, such as the equivalent method or methods improving on the existing analytical solution for fully drained tunnels, being unable to give an accurate analytical solution. First, the semi-infinite seepage region is converted into a rectangular seepage region using two conformal transformations. Subsequently, in order to accurately consider the boundary conditions at the leakage site, the rectangular seepage region with a discontinuous boundary is divided into three subregions with continuous boundaries, and the water head solution for each subregion is given by using the separated variable method. Finally, the principle of orthogonality of trigonometric functions is specially adopted to construct a non-homogeneous system of equations to solve the unknowns in the analytical solution, and through the inverse transformation of the conformal transformation, an analytical solution for the steady-state seepage field when localized line leakage occurs in a tunnel is obtained. The solution proposed is verified by its satisfactory agreement with the numerical simulation results and existing experimental results, and is much more accurate than the existing analytical solution. In addition, the proposed analytical solution is much less computationally demanding compared to numerical simulations. Finally, the capability of the proposed analytical solution is demonstrated by a parametric analysis of the tunnel burial depth, leakage location, and leakage width, and some meaningful conclusions are drawn. Full article

The demand for fast and accurate electromagnetic solutions to support current and emerging technologies has fueled the rapid development of various machine learning techniques for applications such as antenna design and optimization, microwave imaging, device diagnostics, and more. Multi-fidelity (MF) surrogate modeling methods have shown great promise in significantly reducing computational costs associated with surrogate modeling while maintaining high model accuracy. This work offers a comprehensive review of the available MF surrogate modeling methods in electromagnetics, focusing on specific methodologies, related challenges, and the generation of variable-fidelity datasets. The article is structured around the two main types of electromagnetic problems: forward and inverse. It begins by summarizing key machine learning concepts and limitations. This transitions to discussing multi-fidelity surrogate model architectures and low-fidelity data techniques for the forward problem. Subsequently, the unique challenges of the inverse problem are presented, along with traditional solutions and their limitations. Following this, the review examines MF surrogate modeling approaches tailored to the inverse problem. In conclusion, the review outlines promising future directions in MF modeling for electromagnetics, aiming to provide fundamental insights into understanding these developing methods. Full article

This paper studies new characterizations and expressions of the weak group (WG) inverse and its dual over the quaternion skew field. We introduce a dual to the weak group inverse for the first time in the literature and give some new characterizations for both the WG inverse and its dual, named the right and left weak group inverses for quaternion matrices. In particular, determinantal representations of the right and left WG inverses are given as direct methods for their constructions. Our other results are related to solving the two-sided constrained quaternion matrix equation AXB=C and the according approximation problem that could be expressed in terms of the right and left WG inverse solutions. Within the framework of the theory of noncommutative row–column determinants, we derive Cramer’s rules for computing these solutions based on determinantal representations of the right and left WG inverses. A numerical example is given to illustrate the gained results. Full article

This article introduces a novel modification of a Delta-type parallel robot. The robot has five degrees of freedom and provides its end-effector with a 3T2R motion pattern (three translational and two rotational degrees of freedom). The fifth degree of freedom (rotation) is kinematically decoupled from the other four motions, and it is controlled by two drives. Thus, the proposed robot has a redundant actuation. In this article, we present an algorithm to solve the inverse kinematics of this robot and apply it to a path modeling example of a spiral-like trajectory. Numerical simulations illustrate the algorithm and show how the actuated coordinates change along the considered trajectory. Forward kinematics follows next, and an approach is introduced to determine all end-effector configurations for the specified displacements in the actuated joints. A numerical example presents four assembly modes of the robot corresponding to four real solutions of the forward kinematic problem. Finally, this article demonstrates a computer-aided design and analysis of the proposed robot: we describe a procedure for analyzing inverse kinematics and calculating actuation torques. This study forms the basis for the future manufacturing and experimental analysis of a robot prototype. Full article

Seismic inversion is a process of imaging or predicting the spatial and physical properties of underground strata. The most commonly used one is sparse-spike seismic inversion with sparse regularization. There are many effective methods to solve sparse regularization, such as L0-norm, L1-norm, weighted L1-norm, etc. This paper studies the sparse-spike inversion with L0-norm. It is usually solved by the iterative hard thresholding algorithm (IHTA) or its faster variants. However, hard thresholding algorithms often lead to a sharp increase or numerical oscillation of the residual, which will affect the inversion results. In order to deal with this issue, this paper attempts the idea of the relaxed optimal thresholding algorithm (ROTA). In the solution process, due to the particularity of the sparse constraints in this convex relaxation model, this model can be considered as a L1-norm problem when dealt with the location of non-zero elements. We use a modified iterative soft thresholding algorithm (MISTA) to solve it. Hence, it forms a new algorithm called the iterative hybrid thresholding algorithm (IHyTA), which combines IHTA and MISTA. The synthetic and real seismic data tests show that, compared with IHTA, the results of IHyTA are more accurate with the same SNR. IHyTA improves the noise resistance. Full article

Non-linear least squares (NLS) methods are commonly used for quantitative magnetic resonance imaging (MRI), especially for multi-exponential T1ρ mapping, which provides precise parameter estimation for different relaxation models in tissues, such as mono-exponential (ME), bi-exponential (BE), and stretched-exponential (SE) models. However, NLS may suffer from problems like sensitivity to initial guesses, slow convergence speed, and high computational cost. While deep learning (DL)-based T1ρ fitting methods offer faster alternatives, they often face challenges such as noise sensitivity and reliance on NLS-generated reference data for training. To address these limitations of both approaches, we propose the HDNLS, a hybrid model for fast multi-component parameter mapping, particularly targeted for T1ρ mapping in the knee joint. HDNLS combines voxel-wise DL, trained with synthetic data, with a few iterations of NLS to accelerate the fitting process, thus eliminating the need for reference MRI data for training. Due to the inverse-problem nature of the parameter mapping, certain parameters in a specific model may be more sensitive to noise, such as the short component in the BE model. To address this, the number of NLS iterations in HDNLS can act as a regularization, stabilizing the estimation to obtain meaningful solutions. Thus, in this work, we conducted a comprehensive analysis of the impact of NLS iterations on HDNLS performance and proposed four variants that balance estimation accuracy and computational speed. These variants are Ultrafast-NLS, Superfast-HDNLS, HDNLS, and Relaxed-HDNLS. These methods allow users to select a suitable configuration based on their specific speed and performance requirements. Among these, HDNLS emerges as the optimal trade-off between performance and fitting time. Extensive experiments on synthetic data demonstrate that HDNLS achieves comparable performance to NLS and regularized-NLS (RNLS) with a minimum of a 13-fold improvement in speed. HDNLS is just a little slower than DL-based methods; however, it significantly improves estimation quality, offering a solution for T1ρ fitting that is fast and reliable. Full article

Distributed drive electric vehicles (DDEV), characterized by their independently drivable wheels, offer significant advantages in terms of vehicle handling, stability, and energy efficiency. These attributes collectively contribute to enhancing driving safety and extending the all-electric range for sustainable transportation. Nonetheless, the challenge persists in designing a control strategy that effectively coordinates the objectives of handling, stability, and energy efficiency under both lateral and longitudinal driving conditions. To this end, this paper proposes a cooperative control strategy for DDEVs, incorporating active rear steering (ARS) and direct yaw moment control (DYC) to enhance handling capabilities, stability, and energy efficiency. A stability boundary is delineated using an analytical expression that correlates with the front wheel steering angle, and an adjustment factor is introduced to quantify vehicle stability based on this input parameter. This factor aids in establishing a coordinated control reference for handling and stability. At the upper-level motion control layer, a model predictive control method is developed to track this reference and implement ARS and DYC for superior performance. Specifically, the rear lateral force serves as the control command for ARS, which is converted into a rear wheel steering angle using a tire inverse model. Meanwhile, the front lateral force is modeled as linear-time-varying to simplify calculations. At the lower-level torque allocation layer, the adjustment factor is utilized to balance tire workload rate and in-wheel motors’ (IWM) energy consumption, enabling efficient switching between energy consumption and driving stability targets, and the torque allocation is conducted to acquire the expected IWMs’ command. Both the upper and lower-level optimization problems are formulated as convex problems, ensuring efficient and effective solutions. Simulations verify the effectiveness of this strategy in improving handling, stability, and energy economy under DLC cases, while maintaining high computational efficiency. Full article

Summation of infinite series has played a significant role in a broad range of problems in the physical sciences and is of interest in a purely mathematical context. In a prior paper, we found that the Fourier–Legendre series of a Bessel function of the first kind JNkx and modified Bessel functions of the first kind INkx lead to an infinite set of series involving F21 hypergeometric functions (extracted therefrom) that could be summed, having values that are inverse powers of the eight primes 1/2i3j5k7l11m13n17o19p multiplying powers of the coefficient k, for the first 22 terms in each series. The present paper shows how to generate additional, doubly infinite summed series involving F21 hypergeometric functions from Chebyshev polynomial expansions of Bessel functions, and trebly infinite sets of summed series involving F21 hypergeometric functions from Gegenbauer polynomial expansions of Bessel functions. That the parameters in these new cases can be varied at will significantly expands the landscape of applications for which they could provide a solution. Full article

A novel time-dependent deterministic SEIRS model, extended with vaccination, hospitalization, and vital dynamics, is introduced. Time-varying basic and effective reproduction numbers associated with this model are defined, which are crucial metrics in understanding epidemic dynamics. Furthermore, a parameter identification approach has been used to develop a numerical method to compute these numbers for long-term epidemics. We analyze the actual COVID-19 data from the USA, Italy, and Bulgaria to solve appropriate inverse problems and gain an understanding of the time evolution behavior of the basic and effective reproduction numbers. Moreover, an insightful comparison of key coronavirus data and epidemiological parameters across these countries has been conducted. For this purpose, while the basic and effective reproduction numbers provide insights into the virus transmission potential, we propose data-driven criteria for assessing the actual realization of the transmission potential of the SARS-CoV-2 virus and the effectiveness of the applied restrictive measures. To obtain these results, we conduct a mathematical analysis to demonstrate various biological properties of the new differential model, including non-negativity, boundedness, existence, and uniqueness of the solution. The new model and the associated numerical simulation tools proposed herein could be applied to COVID-19 data in any country worldwide and hold a promising potential for the transmission capacity and impact of the virus. Full article

The structural gravimetry problem, which involves determining the geometry of a contact surface between two homogeneous layers based on observed gravity fields, is addressed in this paper. The method of local corrections is presented in a generalized form to improve its applicability to a broader range of problems. This study introduces several improvements to the local corrections method, including the use of a finite element approach for more accurate field calculations, particularly for near-surface boundaries. Additionally, the method incorporates prior knowledge of the boundary geometry, which serves as an initial approximation to enhance convergence and avoid potential divergence issues. Demonstrations on several synthetic examples are performed, which show the advantages of the generalized form of the method. For the territory of the Middle Urals, Russia, the refinement of two crustal boundaries is performed (the Moho boundary and middle crust boundary). Full article

The third-order pseudoparabolic equations represent models of filtration, the movement of moisture and salts in soils, heat and mass transfer, etc. Such non-classical equations are often referred to as Sobolev-type equations. We consider an inverse problem for identifying an unknown time-dependent boundary condition in a two-dimensional linear pseudoparabolic equation from integral-type measured output data. Using the integral measurements, we reduce the two-dimensional inverse problem to a one-dimensional problem. Then, we apply appropriate substitution to overcome the non-local nature of the problem. The inverse ill-posed problem is reformulated as a direct well-posed problem. The well-posedness of the direct and inverse problems is established. We develop a computational approach for recovering the solution and unknown boundary function. The results from numerical experiments are presented and discussed. Full article