Simple Closed-form Solution Research Articles

On network deconvolution for undirected graphs.

Network deconvolution (ND) is a method to reconstruct a direct-effect network describing direct (or conditional) effects (or associations) between any two nodes from a given network depicting total (or marginal) effects (or associations). Its key idea is that, in a directed graph, a total effect can be decomposed into the sum of a direct and an indirect effects, with the latter further decomposed as the sum of various products of direct effects. This yields a simple closed-form solution for the direct-effect network, facilitating its important applications to distinguish direct and indirect effects. Despite its application to undirected graphs, it is not well known why the method works, leaving it with skepticism. We first clarify the implicit linear model assumption underlying ND, then derive a surprisingly simple result on the equivalence between ND and use of precision matrices, offering insightful justification and interpretation for the application of ND to undirected graphs. We also establish a formal result to characterize the effect of scaling a total-effect graph. Finally, leveraging large-scale genome-wide association study data, we show a novel application of ND to contrast marginal versus conditional genetic correlations between body height and risk of coronary artery disease; the results align with an inferred causal directed graph using ND. We conclude that ND is a promising approach with its easy and wide applicability to both directed and undirected graphs.

A low-complexity ADMM-based secure beamforming for IRS-assisted uplink cognitive radio networks with NOMA

Intelligent reflecting surface (IRS) and non-orthogonal multiple access (NOMA) are both powerful means to improve the spectrum efficiency and can assist the cognitive radio network (CRN) in achieving more effective spectrum allocation. However, spectrum sharing in CRNs and the utilization of the scarce bandwidth in NOMA make it vulnerable to security threats. The secure beamforming design in IRS-assisted CRNs with NOMA is complicated and non-convex. Traditional semidefinite relaxation along with Gaussian randomization (SDR-GR) algorithm can solve the non-convex problem, however a high computational complexity. This paper proposes an alternating direction method of multipliers (ADMM)-based sum secrecy rate maximization (A-SSRM) scheme to achieve secure transmission with low computation complexity in IRS-assisted uplink CRNs with NOMA. Specifically, we derive an appropriate transformation of the original problem based on the ADMM framework and decompose it into multiple sub-problems, which are carefully designed to have closed-form solutions. Furthermore, an eavesdropper zero forcing (EZF)-based sub-optimal scheme is presented. Simulation results reveal that our proposed A-SSRM scheme outperforms other benchmarks in terms of sum secrecy rate and reduces the average running time by 6 times compared with the traditional SDR-GR algorithm. This is attributed to our effective decoupling and decomposing of the original problem, which enables each sub-problem to obtain a simple closed-form solution through derivation.

Group sparsity residual constraint model with weighted log-sum penalty for image restoration

In the field of image restoration, non-convex penalties and non-local self-similarity (NSS) prior have been found to enhance image edges and textures. However, NSS-based methods often overlook the inherent variations within patch groups by treating them as a single entity for processing. Despite the existence of numerous convex or non-convex regularizers, few of them have been able to adaptively adjust penalty strength based on NSS information in order to preserve fine details in an image. To address this issue, we propose a novel group sparsity residual constraint model with weighted log-sum penalty (GSRC-Log) for image restoration in this paper. The proposed regularization term is parameterized by an adaptive scalar parameter that relates to NSS, allowing the model to prioritize high NSS group sparse coefficients while penalizing low NSS coefficients. For image denoising, we provide detailed analysis on the optimal condition of our proposed model and prove that it has a simple closed-form global solution. For image restoration tasks, we utilize the ADMM algorithm to solve the GSRC-Log problem and empirically verify its convergence. Extensive experimental results on both image denoising and compressive sensing (CS) recovery demonstrate that our proposed GSRC-Log outperforms many popular or state-of-the-art methods.

Two-Dimensional Target Localization Approach via a Closed-Form Solution Using Range Difference Measurements Based on Pentagram Array

This paper presents a simple and fast closed-form solution approach for two-dimensional (2D) target localization using range difference (RD) measurements. The formulation of the localization problem is derived using a pentagram array. The target position is determined using passive radar measurements (RDs) between the target and the (N+1=10) receivers’ locations. The method facilitates the problem of target position and can be used as a counter-parallel method for spherical interpolation (SI) and spherical intersection (SX) methods in time difference of arrival (TDOA) and radar systems. The performance of the method is examined in 2D target localization using numerical analysis under the distribution of receivers in the pentagram array. The simulations are conducted using four different far-distance targets and comparatively large-area distributed receivers. The RD measurements were distorted by two different values of Gaussian errors based on ionosphedriec time delays of 20 and 50 nsec owing to the different receivers’ positions. The findings highly verified the validity of the method for addressing the problem of target localization. Additionally, a theoretical accuracy study of the method is given, which solely relies on the RD measurements.

A simple and accurate closed-form analytical solution to the Boussinesq equation for horizontal flow

The classical problem used to model the response of an unconfined aquifer to a sudden change in boundary head is considered in this paper. This problem is usually modeled using the nonlinear groundwater Boussinesq equation. Due to the nonlinearity of this problem, no exact solutions exist. Solutions to the Boussinesq equation are therefore obtained using numerical techniques or approximate analytical methods. In this paper, we present a novel closed-form approximate analytical solution to this problem. The Boussinesq equation is converted to a first-order ordinary differential equation (ODE) by means of the Boltzmann transformation and by introducing a new variable related to the water flow. The first-order ODE is then solved analytically after introducing an intermediate approximation involving two fitting parameters. To avoid any numerical treatment, closed-form polynomial expressions of the fitting parameters are proposed. The final-form solution is simple to use and is obtained in terms of the incomplete gamma function, which is valid for both recharge and discharge. The derived solution is tested and compared to efficient numerical solutions, as well as to two types of analytical solutions: an accurate series expansion solution and an equivalent closed-form solution. The results show excellent agreement between the proposed solution and the numerical and series solutions. The proposed solution offers a key advantage in terms of both accuracy and simplicity; notably, it can be implemented using a simple spreadsheet.

Underwater Localization of AUVs in Motion Using Two-Way Travel Time Measurements With Unknown Sound Velocity

For an autonomous underwater vehicle (AUV) underwater localization system based on acoustic two-way travel time measurements, the motion of the AUV during the interrogation-reception time interval, unknown underwater sound velocity, and beacon calibration error affect the performance of the positioning solution. Ignoring these factors significantly deteriorates the performance of the localization system. This paper focuses on the localization problem of AUV in motion with an unknown sound velocity and beacon calibration error. The Cramér–Rao lower bound of the concerned localization problem is first derived as the performance bound of an unbiased estimator. Next, through multi-stage processing, we design a closed-form solution that can attain the corresponding performance bound under small measurement noise conditions. Furthermore, considering that the AUV motion velocity is typically much lower than the sound velocity in practical applications, a simpler and more efficient simplified closed-form solution is proposed, which can guarantee localization performance under the condition that the AUV motion speed is low, as well as small measurement noise. Finally, the theoretical performance and superiority of the proposed solutions were verified through numerical simulation.

The optimal triangulation method is not really optimal

AbstractTriangulation refers to the problem of finding a 3D point from its 2D projections on multiple images taken at different camera poses. For solving this problem, it is a common practice to use the so‐called optimal triangulation method. But, the method can be optimal only if we assume no uncertainty in the camera parameters. In this paper, an extensive comparison between various existing methods for the calibrated cameras is performed, in which different poses are considered. Furthermore, uncertainty sensitivity analysis is conducted for extrinsic parameters of cameras. The results show that the optimal triangulation method is actually not the best choice when there is uncertainty in extrinsic parameters. Interestingly, it can be observed that the simple midpoint method works equally well or outperforms the optimal triangulation and many other methods. Apart from its high performance, the midpoint method has a simple closed form solution for multiple views while the optimal triangulation method is hard to be used for more than two views. Therefore, in contrast to the common practice, we argue that the simple midpoint method can be a good choice in the structure‐from‐motion process where there is uncertainty in extrinsic camera parameters.

Lateral–Torsional Buckling of Cantilever Steel Beams under 2 Types of Complex Loads

Cantilever steel beams are an essential structural element in civil engineering fields such as bridges and buildings. However, there is very little research on the critical moment (Mcr) of cantilever beams subjected to a concentrated load (CL) or a combination of concentrated load and uniformly distributed load (CUDL) when the concentrated load is not limited to the free end. Therefore, the focus of the current paper is the calculation of Mcr for cantilever steel beams under CL and CUDL. This paper proposes a program and a simple closed-form solution for Mcr that are applicable to the elastic buckling analysis of cantilever I-beams under CL and CUDL. Based on the Rayleigh–Ritz method, a matrix equation and the corresponding procedure about Mcr under CL and CUDL are derived by using infinite trigonometric series for the buckling deformation functions. The value of Mcr and the corresponding mode of buckling can be obtained efficiently by considering the symmetry of the section, the ratio of two load values and the load action position. Experimental results and finite element calculations validate the numerical solutions of the procedure. A closed-form solution for Mcr is derived according to the assumption of a small torsion angle and the specific values of each coefficient in the closed-form solution of Mcr are calculated by the proposed procedure. The results show that the procedure and closed-form solution for Mcr presented in this paper have a high degree of accuracy in calculating the Mcr of the cantilever beam under CL and CUDL. The deviations between the results calculated by the proposed procedure and data from existing literature are less than 8%. These conclusions are capable of solving the calculation problem of Mcr for cantilever beams under CL or CUDL, which are both significant load cases in engineering. The study provides a reference for the design of cantilever steel beams.

Self-Optimizing Bluetooth Low Energy Networks for Industrial IoT Applications

This letter develops an efficient self-optimizing algorithm for Bluetooth Low Energy (BLE) networks in industrial IoT (IIoT) applications. The ultra-dense nature of IIoT applications implies that BLE networks are associated with a high probability of packet collisions, increasing the data collection latency. Determining the optimal packet transmission interval that minimizes the data collection latency is very challenging due to the non-linearity of this problem. We derive an analytic solution by which to realize an optimal advertising interval in a closed-form expression and prove its optimality and uniqueness. Based on a simple closed-form solution, we develop a low-complexity self-optimizing algorithm that minimizes the data collection latency. Simulation results confirm that the analytic results are accurate and that the proposed algorithm autonomously optimizes BLE networks.

Estimating lateral groundwater inflow to rivers using heat as a tracer

Heat is a natural tracer that can be used to estimate groundwater velocity and fluxes. However, most studies, if not all, are focused on vertical flow due to the existence of a simple closed-form analytical solution that well captures the characteristics of vertical heat transport. In this study, we show that lateral groundwater inflow to rivers can also be inferred from natural heat signals by integrating theoretical development and field observation. A closed-form analytical solution for lateral heat transport in unconfined aquifers is developed and serves as the basis for the inversion of groundwater velocity. Two parameter estimation techniques, Markov Chain Monte Carlo (MCMC) and a trial-and-error method, are applied to estimate groundwater velocity using groundwater temperature observed at wells. The MCMC method requires at least one observation well and its performance increases with the number of wells. The trial-and-error method requires three observation wells and can appropriately represent the groundwater velocity. The performance of both methods is affected by the uncertainty of hydrogeological and thermal parameters, such as porosity, thermal conductivity of the overlying and underlying layers, and aquifer thickness. Porosity is the most sensitive and thermal conductivity is the least sensitive parameter to the accuracy of estimated groundwater velocity. The developed heat tracer method for estimating lateral groundwater inflow to rivers is novel (script provided) and could be a valuable alternative to the widely used environmental tracers.

Error Vector Magnitude-Based Low Complexity Adaptive Power Allocation in SM-MIMO System

In this article, we propose the error vector magnitude (EVM)-based adaptive power allocation (PA) for spatial modulation (SM) multiple-input-multiple-output (MIMO) system to optimize the minimum squared Euclidean distance and reduce the system complexity. We first investigate the typical Euclidean distance (ED)-based PA (ED-PA) algorithm. Second, an EVM-based ED-PA algorithm is proposed, where a simplified closed-form PA solution for the quadrature amplitude modulation is obtained in the case of two transmit antennas. For more transmit antennas, we further propose a low complexity-PA (LC-PA) algorithm, where all the transmit antennas are grouped with two antennas in each group and the proposed EVM-based ED-PA algorithm is utilized to assign the power within each group. Numerical results reveal that the proposed EVM-based ED-PA and LC-PA algorithms have considerably lower complexity and can achieve the optimal average bit error rate (ABER) and suboptimal ABER performance at low-and-moderate signal-to-noise ratio (SNR) region and high-SNR region, respectively, compared with the typical PA algorithms in SM-MIMO system.

Analytical solution for a focused vortex beam radiated by a Gaussian source

Current interest in focused vortex beams is motivated by the ability to trap particles axially and laterally using the resulting radiation force. A simple closed-form solution is obtained in the Fresnel approximation for a sound beam radiated by a Gaussian source with time dependence e−iωt, focal length d, amplitude distribution exp(−r2/a2), and azimuthal phase dependence einθ, where θ is the angle in the plane perpendicular to the beam axis, and the integer n is the topological charge, referred to here as the vorticity. The solution is in good agreement with the pressure field predicted in the paraxial region by numerical evaluation of the Rayleigh integral. Of interest in optics as well as acoustics is the distance from the minimum along the beam axis to the first local maximum, referred to as the vortex ring. The present solution yields rn = ηnd/ka for the vortex ring radius in the focal plane, where k is the wavenumber, and ηn = 1.69 + 0.965(n−1) for vorticities in the range 1 ≤ n < O(20). Within this range the radius rn thus increases linearly with the vorticity. Results are also presented for dependence of the focusing gain on the vorticity. [CAG was supported by the ARL:UT Chester M. McKinney Graduate Fellowship in Acoustics.]

Quasi-brittle fracture analysis of large and small wedge splitting concrete specimens with size from 150 mm to 2 m and aggregates from 10 to 100 mm

The wedge splitting (WS) test geometry, suitable for testing large concrete specimens up to a few meters in size because of the self-weight supports, will find more applications if a closed-form model is available. This study presented a linearized boundary effect model (BEM) on quasi-brittle fracture of large WS mass concrete specimens containing aggregates up to 100 mm. Two separate small WS specimens containing aggregates around 10 mm and 20 mm were also analyzed to show the versatility of the model and the necessity to include the aggregate size in modelling. Modelling concrete as a large particle composite led to a simple closed-form solution, and both the fracture toughness KIC and tensile strength ft for the formation of the crack/notch-tip fracture process zone (FPZ) were determined from the maximum splitting force Pmax-h of WS specimens. The material properties KIC and ft from the current WS model were also confirmed by three-point-bending (3-p-b) tests of the same concrete. Comparison between the linear BEM and the well-known size effect law (SEL) was provided for purposes of experimental data analysis and mathematical fitting principles.

A Comparison of General Solutions to the Non-Axisymmetric Frictionless Contact Problem with a Circular Area of Contact: When the Symmetry Does Not Matter

The non-axisymmetric problem of frictionless contact between an isotropic elastic half-space and a cylindrical punch with an arbitrarily shaped base is considered. The contact problem is formulated as a two-dimensional Fredholm integral equation of the first type in a fixed circular domain with the right-hand side being representable in the form of a Fourier series. A number of general solutions of the contact problem, which were published in the literature, are discussed. Based on the Galin–Mossakovskii general solution, new formulas are derived for the particular value of the contact pressure at the contact center and the contact stress-intensity factor at the contour of the contact area. Since the named general solution does not employ the operation of differentiation of a double integral with respect to the coordinates that enter it as parameters, the form of the general solution derived by Mossakovskii as a generalization of Galin’s solution for the special case, when the contact pressure beneath the indenter is bounded, is recommended for use as the most simple closed-form general solution of the non-axisymmetric Boussinesq contact problem.

A general expression for the maximum force in peeling a tape from a rigid substrate with an initial crack

ABSTRACT Assuming a Dugdale cohesive zone, recently a simple Euler beam model has been proposed to understand the transient peeling on an elastic film from a rigid substrate at 90° from the substrate. The model also shows with sufficiently high bending stiffness, a peak force emerges larger than the Rivlin–Kendall steady-state value, scaling with the fourth root of bending stiffness. From the maximum force during peeling, and its steady-state value which is given by the Rivlin–Kendall well-known model, a simultaneous characterisation of the most important cohesive properties (cohesive strength and toughness of the interface) is obtained from a single test. This maximum force eventually saturates at a value which depends on cohesive strength alone. The presence of possibly inevitable initial cracks at the edge of the interface obviously reduces the peak force, and here, we give a simple closed-form approximate solution for this case too.

Closed-form solution based on peak probability density function for statistical fatigue life prediction under Broad-Band random loading

In this study, an analytical method for fast spectral fatigue-life prediction under Broad-Band random loading is proposed. It is based on the power spectral density of stress in the structure and the Rice peak stress distribution of a stationary Gaussian random process. The novel simple closed-form solution employs special HyperGeometric function which analytically describes the transcendent error weighting function between the Gaussian and Rayleigh distributions inherent in the Rice model. HyperGeometric-based solution outperforms other currently existing approximate methods based on the Rice peak distribution, e.g. Chaudhury, Kam, Dover, Chow and Lee, Lü et al. etc. Straightforward computational applicability and fast engineering implementation of obtained general closed-form results is accented throughout the investigation.

Learning Sparse Graphs via Majorization-Minimization for Smooth Node Signals

In this letter, we propose an algorithm for learning a sparse weighted graph by estimating its adjacency matrix under the assumption that the observed signals vary smoothly over the nodes of the graph. The proposed algorithm is based on the principle of majorization-minimization (MM), wherein we first obtain a tight surrogate function for the graph learning objective and then solve the resultant surrogate problem which has a simple closed form solution. The proposed algorithm does not require tuning of any hyperparameter and it has the desirable feature of eliminating the inactive variables in the course of the iterations - which can help speeding up the algorithm. The numerical simulations conducted using both synthetic and real world (brain-network) data show that the proposed algorithm converges faster, in terms of the average number of iterations, than several existing methods in the literature.

Analytical Modeling, Design, and Performance Analysis of a Micromirror for Space-Based Multiobject Spectroscopy

This article presents the analytical modeling, design, and performance analysis of an electrostatic micromirror for space-based Multiobject Spectroscopy (MOS). The micromirror under investigation is a double-bridge design with hidden cantilevers architecture in which the deflection is achieved by a combination of bending and twisting cantilevers. A simple closed-form solution of pull-in voltage and deflection is obtained using the parallel-plate capacitor model that assumes micromirror deflection dominated by bending cantilevers due to electrostatic actuation. The design optimization is done to achieve a micromirror of size $200\,\,\mu \text{m}\,\,\times 200\,\,\mu \text{m}$ , deflection $2.5~\mu \text{m}$ , pull-in voltage smaller than 25 V, and a shock survival capacity of at least 10 000 g. The static and dynamic behavior of the optimized design is obtained using the analytical model and compared with the finite-element method (FEM) and characterization results and found to be in close agreement. The micromirror exhibits an analytical pull-in voltage near the FEM and measured pull-in voltage with a deviation of 5% and 0.9%, respectively. The analytical resonance frequency is also closer to simulation and measurement results with a deviation of 4.66% and 2.58%, respectively. The analytical switching time is also very close to the FEM results, with a deviation of 28%. The analytical model and the simple approach of design optimization using tuning parameters presented in the article can be used to design the micromirror according to desired specifications.

Multi-block Nonconvex Nonsmooth Proximal ADMM: Convergence and Rates Under Kurdyka–Łojasiewicz Property

We study the convergence and convergence rates of a multi-block proximal alternating direction method of multipliers (PADMM) for solving linearly constrained separable nonconvex nonsmooth optimization problems. This algorithm is an important variant of the alternating direction method of multipliers (ADMM) which includes a proximal term in each subproblem, to cancel out complicated terms in applications where subproblems are not easy to solve or do not admit a simple closed form solution. We consider an over-relaxation step size in the dual update and provide a detailed proof of the convergence for any step size \(\beta \in (0,2)\). We prove the convergence of the sequence generated by the PADMM by showing that it has a finite length and it is Cauchy. Under the powerful Kurdyka–Łojasiewicz (KŁ) property, we establish the convergence rates for the values and the iterates, and we show that various values of KŁ-exponent associated with the objective function can raise PADMM with three different convergence rates. More precisely, we show that if the (KŁ) exponent \(\theta =0\), the sequence generated by PADMM converges in a finite numbers of iterations. If \(\theta \in (0,1/2]\), then the sequential rate of convergence is \(cQ^{k}\) where \(c>0\), \(Q\in (0,1)\), and \(k\in {\mathbb {N}}\) is the iteration number. If \(\theta \in (1/2,1]\), then \({\mathcal {O}}(1/k^{r})\) rate where \(r=(1-\theta )/(2\theta -1)\) is achieved.

Accurate Frequency Estimation With Fewer DFT Interpolations Based on Padé Approximation

Frequency estimation is a fundamental problem in many areas. The well-known A&M and its variant estimators have established an estimation framework by iteratively interpolating the discrete Fourier transform (DFT) coefficients. In general, those estimators require two DFT interpolations per iteration, have uneven initial estimation performance against frequencies, and are incompetent for small sample numbers due to low-order approximations involved. Exploiting the iterative estimation framework of A&M, we unprecedentedly introduce the Padé approximation to frequency estimation, unveil some features about the updating function used for refining the estimation in each iteration, and develop a simple closed-form solution to solving the residual estimation error. Extensive simulation results are provided, validating the superiority of the new estimator over the state-the-art estimators in wide ranges of key parameters.