Bisection Method Research Articles

AnACor2.0: a GPU-accelerated open-source software package for analytical absorption corrections in X-ray crystallography

Analytical absorption corrections are employed in scaling diffraction data for highly absorbing samples, such as those used in long-wavelength crystallography, where empirical corrections pose a challenge. AnACor2.0 is an accelerated software package developed to calculate analytical absorption corrections. It accomplishes this by ray-tracing the paths of diffracted X-rays through a voxelized 3D model of the sample. Due to the computationally intensive nature of ray-tracing, the calculation of analytical absorption corrections for a given sample can be time consuming. Three experimental datasets (insulin at λ = 3.10 Å, thermolysin at λ = 3.53 Å and thaumatin at λ = 4.13 Å) were processed to investigate the effectiveness of the accelerated methods in AnACor2.0. These methods demonstrated a maximum reduction in execution time of up to 175× compared with previous methods. As a result, the absorption factor calculation for the insulin dataset can now be completed in less than 10 s. These acceleration methods combine sampling, which evaluates subsets of crystal voxels, with modifications to standard ray-tracing. The bisection method is used to find path lengths, reducing the complexity from O(n) to O(log2 n). The gridding method involves calculating a regular grid of diffraction paths and using interpolation to find an absorption correction for a specific reflection. Additionally, optimized and specifically designed CUDA implementations for NVIDIA GPUs are utilized to enhance performance. Evaluation of these methods using simulated and real datasets demonstrates that systematic sampling of the 3D model provides consistently accurate results with minimal variance across different sampling ratios. The mean difference of absorption factors from the full calculation (without sampling) is at most 2%. Additionally, the anomalous peak heights of sulfur atoms in the Fourier map show a mean difference of only 1% compared with the full calculation. This research refines and accelerates the process of analytical absorption corrections, introducing innovative sampling and computational techniques that significantly enhance efficiency while maintaining accurate results.

Joint jammer selection and power optimization in covert communications against a warden with uncertain locations

In covert communications, joint jammer selection and power optimization are important to improve performance. However, existing schemes usually assume a warden with a known location and perfect channel state information (CSI), which is difficult to achieve in practice. To be more practical, it is important to investigate covert communications against a warden with uncertain locations and imperfect CSI, which makes it difficult for legitimate transceivers to estimate the detection probability of the warden. First, the uncertainty caused by the unknown warden location must be removed, and the optimal detection position (OPTDP) of the warden is derived which can provide the best detection performance (i.e., the worst case for a covert communication). Then, to further avoid the impractical assumption of perfect CSI, the covert throughput is maximized using only the channel distribution information. Given this OPTDP based worst case for covert communications, the jammer selection, the jamming power, the transmission power, and the transmission rate are jointly optimized to maximize the covert throughput (OPTDP-JP). To solve this coupling problem, a heuristic algorithm based on maximum distance ratio (H-MAXDR) is proposed to provide a sub-optimal solution. First, according to the analysis of the covert throughput, the node with the maximum distance ratio (i.e., the ratio of the distances from the jammer to the receiver and that to the warden) is selected as the friendly jammer (MAXDR). Then, the optimal transmission and jamming power can be derived, followed by the optimal transmission rate obtained via the bisection method. In numerical and simulation results, it is shown that although the location of the warden is unknown, by assuming the OPTDP of the warden, the proposed OPTDP-JP can always satisfy the covertness constraint. In addition, with an uncertain warden and imperfect CSI, the covert throughput provided by OPTDP-JP is 80% higher than the existing schemes when the covertness constraint is 0.9, showing the effectiveness of OPTDP-JP.

An improved goal-oriented adaptive finite-element method for 3-D direct current resistivity anisotropic forward modeling using nested tetrahedra

We developed a novel adaptive finite element method (FEM) to address the problem of 3-D direct current (DC) resistivity forward modeling with complex surface topography and arbitrary conductivity anisotropy. The tetrahedra-based FEM and secondary virtual potential algorithm are first used to handle arbitrary complex geo-models. Then, to ensure the accuracy of the simulation solution, an improved goal-oriented adaptive mesh refinement (AMR) algorithm is proposed to realize an optimized mesh density distribution. To avoid the drawback of the traditional goal-oriented AMR algorithm for the DC forward modeling problem, we incorporate a volume-based weighting factor into the posterior error estimation procedure to further optimize the density distribution of the forward modeling grid. In addition, instead of traditional open source mesh generation software, we propose using the longest-edge bisection (LEB) algorithm to perform the mesh refinement process, which can preserve the topological structure between different-level meshes. Finally, the comprehensive test using a two-layered model and two complex 3-D models demonstrate the capability of our newly developed code to obtain highly accurate solutions even on relatively coarse initial grids. By incorporating the volume factor, our novel AMR algorithm achieves a more uniform and reasonable mesh density distribution during these experiments. The LEB refinement technique can generate a series of nested tetrahedral elements and provide fewer tetrahedral elements compared to the traditional Delaunay-based AMR method. The proposed 3-D DC forward modeling method has been implemented into an open source C++ code, which will contribute to the advancement of the 3-D DC resistivity imaging field.

Wave Height Estimation with a Short Return Period

Bisection and grid search methods are proposed as wave height estimation techniques for short return periods (hereafter RP; reference, one year) using long-term wave monitoring data. The proposed method is compared and evaluated with the estimation results using GP (generalized Pareto) and GEV (generalized extreme value) distribution functions, which are widely used as extreme value analysis methods. The wave height data used for the estimation were KMA Ulleungdo Ocean Buoy’s wave height data observed for 12 years from 2012 to 2023. The estimation results show that the annual frequency wave height is 4.55 m, the 90% confidence interval (±5%) is [4.18, 4.69] (m), and the confidence interval for the RP is [0.58, 1.42] (years). The difference from the GP and GEV estimation results (4.61 m and 4.53 m, respectively) was statistically ‘no significance difference.’ The method proposed in this study can estimate design variables for RPs without assumptions on candidate extreme distribution functions or parameter estimation procedures, so it can be used to estimate wave heights for short RPs of one year or less when observation data of approximately ten years or more are available.

Empowering Discovery Learning and Direct Instruction with Computer-Assisted to Enhance Mathematical Performance

In the current educational context, the low academic performance of students in mathematics indicates the need for the implementation of more effective teaching methods to support the understanding of complex concepts. The effectiveness of learning can be enhanced through computer-assisted instruction, particularly in discovery learning (DL-CA) and direct instruction (DI-CA) utilizing web-based media. This study employs a quantitative approach with statistical analysis to examine the significant impact of these teaching methods on students' mathematical performance. Discovery learning and direct instruction were applied to two groups studying numerical methods, focusing on the topic of nonlinear equation solutions using the bisection method, with each group consisting of 33 students. The statistical test results indicated that there was no significant difference in mathematical performance between the DL-CA and DI-CA groups; both methods showed comparable effectiveness. However, both methods had a significant effect on the N-Gain of the learning outcomes in both groups. The improvement in mathematical performance for the DL-CA group was better than that for the DI-CA group, with more students showing high improvement (11 students) compared to the DI-CA group (1 student). Additionally, in the low improvement category, the DL-CA group had fewer students (2 students) than the DI-CA group (8 students). This suggests that both discovery learning and direct instruction have almost equivalent reliability in achieving students' mathematical abilities, but DL-CA is more effective in enhancing performance compared to DI-CA.

Pixel-level Terrain Processing of the Martian Surface from Moderate Resolution Images of Tianwen-1

The moderate-resolution camera (MoRIC) of the Tianwen-1 orbiter, China’s first Mars exploration mission, is a frame-imaging scheme color camera. However, generating a precise digital elevation model (DEM) from these images faces challenges due to the presence of featureless and repetitive regions on the Martian surface. Traditional photogrammetric methods often struggle to achieve pixel-level resolution in these areas. To address this, we employ a shape from shading (SFS) algorithm that leverages shading patterns to reconstruct fine-scale terrain details, optimizing photogrammetric initial DEM. This paper presents a novel global bisection search algorithm, based on the Pearson correlation coefficient to determine the smoothing parameter, a crucial element in the SFS process. Comparisons with higher-resolution data from the Mars Express high-resolution stereo camera and Mars Reconnaissance Orbiter CTX images validate our approach, highlighting the potential of SFS to significantly enhance the scientific value of Tianwen-1 MoRIC data by generating high-resolution, three-dimensional maps of Mars.

A Single-Variable Method for Solving the Min–Max Programming Problem with Addition–Overlap Function Composition

Min–max programming problems with addition–min constraints have been studied in the literature to model data transfer in BitTorrent-like peer-to-peer file-sharing systems. It is well known that the class of overlap functions contains various operators, including the “min” operator. The aim of this paper is to generalize the above min–max programming problem with addition–overlap function constraints. We demonstrate that this new optimization problem can be transformed into a simplified single-variable optimization problem, which makes it easier to find an optimal solution. The bisection method will be used to find this optimal solution. In addition, when the overlap function is explicitly specified, an iterative method is given to compute the optimal objective value with a polynomial time complexity. A numerical example is provided to illustrate the procedures.

The Validity-Reliability Study of Turkish Version of Electronical Symptom Screening Tool (8-18) in SSPedi-Pediatric Patients with Cancer

Objective: The first step in symptom management is symptom screening which is necessary to keep the symptom under control. This study aimed to determine the validity and reliability of the Turkish version of the Electronic Symptom Screening Tool in Pediatrics for children aged 8-18 (SSPedi 8-18) with cancer. Methods: The research was conducted between September 2020 and June 2021. The sample in our study consists of 80 children (8-18 years) with cancer. For content validity, the scale was first translated into Turkish and then adapted using Davis’ correspondence analysis technique. The construct validity of the scale was assessed using the Barlett test and Kaiser-Mayer Olkin. For the scale’s reliability, Cronbach Alpha Reliability Coefficient, Equivalent Forms Method, bisection method and Item Total Score Correlation Coefficient correlation tests were analyzed. Before starting the study, ethics committee approval and written permits from institutions and individuals were obtained. Results: The content validity index was found between 0.93 and 1. The reliability of the scale of the equivalent form method was r:0.57, and bisection method was r:0.85 and the Cronbach internal consistency coefficient of scale=0.86. The correlation coefficient between the items and the total score was greater than r=0.20. The Kaiser-Meyer-Olkin coefficient was 0.82. The χ2 value (Bartlett test) was statistically significant (p= .001). Conclusion: The SSPedi 8-18 is a reliable and valid tool that can be used to assess the symptoms of Turkish children with cancer. Nurses’ use of the scale while providing nursing care to children diagnosed with cancer and their families will guide them in objectively determining symptoms, understanding the relationship between symptoms, and managing symptoms.

3-D parallel anisotropic inversion of controlled-source electromagnetic data using nested tetrahedral grids

SUMMARY Geophysicists today face the challenge of quickly and reliably interpreting extensive controlled-source electromagnetic (CSEM) data sets to map subsurface conductivity structures within realistic geological environments. An ideal 3-D CSEM inversion algorithm using tetrahedral grids should be capable of distinguishing different resolution requirements between forward modelling and inversion grids, have an optimal parallel strategy that fully exploits the inherent independence of CSEM data sets while also possessing the capability to handle large-scale geo-electrical models, and incorporate conductivity anisotropy which should be a common characteristic in realistic subsurface environments. However, existing tools in the geo-electromagnetic community often fall short of these three demands. Addressing this gap, our study introduces a scalable and parallel anisotropic inversion technique for CSEM data, capitalizing on the potential of unstructured tetrahedral grids. We first apply the tetrahedral longest-edge bisection method to create a refined dense, heterogeneous forward modelling grid from a coarse inversion grid. This refinement, focused on areas around transmitters and receivers, is seamlessly integrated within the coarser inversion grid’s topology, enabling precise conductivity mapping and preserving electromagnetic response accuracy during model updates. We further innovate with a source-mesh double-level parallel strategy, utilizing the message passing interface technique for parallel handling of independent CSEM data sets and large-scale geo-electrical models. Externally, we dedicate a processor for inversion model updates employing the Limited-memory Broyden–Fletcher–Goldfarb–Shanno optimization algorithm and divide other processors into groups, each associated with specific transmitting sources and frequencies. Internally, in each group, we employ a domain-decomposition-based scalable and robust iterative solvers using the Auxiliary-Space Maxwell pre-conditioner to parallel quickly calculate the electromagnetic responses from its assigned source-frequency set. Additionally, recognizing the potential for electrical conductivity anisotropy in field data, we incorporate the case of vertical transverse isotropy. We validate the effectiveness of our method through examples, including an isotropic land model with undulating topography, an anisotropic marine model and a real-field data case. Results from both synthetic and field data inversions underscore our method’s significant advancements in efficiency and practicality, particularly in addressing large-scale 3-D CSEM data sets inversion challenges in realistic geological environments.

Data-driven control of wind turbine under online power strategy via deep learning and reinforcement learning

This study proposes a data-driven wind turbine (WT) model predictive control (MPC) enhanced by a deep-learning (DL) radial basis function network (RBFN) and a reinforcement-learning (RL) deep Q-learning network (DQN). The RBFN provides comprehensive aerodynamic predictions, including thrust, torque, and power. Besides, the MPC linearization relies on the RBFN prediction to estimate force sensitivities. The DQN achieves an online power strategy (OPS) that solves the 2-degree-of-freedom (2-DOF) optimization of rotor speed and pitch angle, which can actively adjust power capture to meet different power requirements. The DQN adopts a novel bisection algorithm with a first-in-first-out (FIFO) queue for high-precision 2-DOF results. The MPC coordinates the permanent magnet synchronous generator (PMSG) and pitch servo, considering shaft rotation and tower movement. Compared with the maximum power point tracking (MPPT) and power reference point tracking (PRPT) based controls, the proposed RBFN-DQN-MPC reduces power fluctuation and ensures constant output. This study also compares the DQN with the categorical DQN (C51), which indicates that the DQN is more effective in the 2-DOF optimization. Hence, WTs enhanced by the DL-RL-MPC are intelligent and reliable for flexible wind generation.

Numerical solution for the interior electric displacement of the moving DBY‐PS model for semi‐permeable cracked piezoelectric material

AbstractIn this study, we analysed a moving crack at the interface of an infinitely long piezoelectric bilayer using the Dugdale–Barenblatt yield (DBY) model and the polarisation saturation (PS) model. To model the moving crack problem, a Yoffe‐type crack moves at a constant subsonic speed on the interface of an infinitely long piezoelectric bilayer. The crack faces are assumed to be semi‐permeable, and at the boundary of the bilayer, in‐plane electrical and out‐of‐plane mechanical stresses are applied. Due to the application of electro‐mechanical loads, cracks propagate, mechanical yielding zones and electric saturation zones are developed. To arrest the crack from further propagation, mechanical yield stress and saturation electric displacement are applied at the developed zones. To address this problem analytically and numerically, the mixed boundary value problem is transformed into a set of coupled Fredholm integral equations (FIEs) of the second kind using the Fourier transform and the Copson method. The closed‐form analytical expressions for the length of the electrical saturation zone (ESZ), whether longer, shorter or equal to the mechanical yielding zone (MYZ), show dependence on external electro‐mechanical loads under semi‐permeable crack conditions. The algorithm to solve the electric crack condition parameter (ECCP) has been defined using numerical discretization and the bisection method. Illustrative examples demonstrate the proposed technique's effectiveness and suitability for Yoffe‐type moving cracks. The numerical results show the convergence of the ECCP. Furthermore, the numerical results show how mechanical and electrical zone lengths and energy release rate (ERR) are affected by electrical and mechanical loads, strip thickness and crack velocity. In addition, the size of the mechanical yielding zone is consistently promoted by electrical load, while the promotion or prevention of the electrical saturation zone by mechanical load depends on the relative sizes of the nonlinear zones.

A new iterative fuzzy approach to the multi-objective fractional solid transportation problem with mixed constraints using a bisection algorithm

A new iterative fuzzy approach to the multi-objective fractional solid transportation problem with mixed constraints using a bisection algorithm

The Bisection Method and the Accelerated Dynamical Mean Value Optimization Method in Comparison of Different Topology Optimization Approaches

The Bisection Method and the Accelerated Dynamical Mean Value Optimization Method in Comparison of Different Topology Optimization Approaches

Learning Hidden Markov Models with Structured Transition Dynamics

The hidden Markov model (HMM) provides a natural framework for modeling the dynamic evolution of latent diseases. The unknown probability matrices of HMMs can be learned through the well-known Baum–Welch algorithm, a special case of the expectation-maximization algorithm. In many disease models, the probability matrices possess nontrivial properties that may be represented through a set of linear constraints. In these cases, the traditional Baum–Welch algorithm is no longer applicable because the maximization step cannot be solved by an explicit formula. In this paper, we propose a novel approach to efficiently solve the maximization step problem under linear constraints by providing a Lagrangian dual reformulation that we solve by an accelerated gradient method. The performance of this approach critically depends on devising a fast method to compute the gradient in each iteration. For this purpose, we employ dual decomposition and derive Karush–Kuhn–Tucker conditions to reduce our problem into a set of single variable equations, solved using a simple bisection method. We apply this method to a case study on sports-related concussion and provide an extensive numerical study using simulation. We show that our approach is in orders of magnitude computationally faster and more accurate than other alternative approaches. Moreover, compared with other methods, our approach is far less sensitive with respect to increases in problem size. Overall, our contribution lies in the advancement of accurately and efficiently handling HMM parameter estimation under linear constraints, which comprises a wide range of applications in disease modeling and beyond. History: Accepted by Paul Brooks, Area Editor for Applications in Biology, Medicine, & Healthcare. Funding: This research was funded by [Grant R01DE028283] from the National Institute of Dental and Craniofacial Research, National Institutes of Health. G.-G. Garcia was also funded by the Georgia Clinical & Translational Science Alliance National Institutes of Health award [Grant UL1-TR002378]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0342 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0342 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

PERBANDINGAN ESTIMASI VOLATILITAS HARGA OPSI BELI SAHAM APPLE INC. (AAPL) DENGAN METODE BISECTION DAN SECANT

Stock price volatility is a measure of how far a stock price moves in a given Stock price volatility is a measure of how far a stock price moves in a given time. The theory developed by Black-Scholes states that every option price with the same 'underlying asset' and the same time to maturity but with different exercise values will have the same Implied Volatility value. However, this is not always the case in the market. Therefore, it is necessary to estimate volatility known as Implied Volatility, which is considered an appropriate method in estimating volatility values. This study compares the Bisection and Secant methods to estimate the volatility of Apple Inc. (Aapl) stock. This study uses data on the closing price of the stock in the period September 29, 2022 to September 29, 2023. Volatility estimation for the Bisection and Secant methods by determining the initial approximation and limiting it to a maximum of 100 simulations and iteration stops if it has produced a relative error smaller than = . The is an error tolerance limit, the smaller the error tolerance, the more accurate it is. According to the research results, the Bisection method produces an estimated volatility value of 0.498212 at the 9th iteration, while the Secant method produces an estimated value of 0.498590 at the 10th iteration. The Secant method produces a smaller relative error value of 0.000096, indicating that the Secant method is more accurate than the Bisection method.

Low-carbon optimal scheduling of park-integrated energy system based on bidirectional Stackelberg-Nash game theory

Multi-stakeholder participation is crucial in facilitating the development of park-integrated energy systems (PIES). Balancing the diverse interests of various stakeholders, each with its distinct requirements presents a notable challenge. Concurrently, the model's complexity increases due to the engagement of various stakeholders, posing challenges to its resolution through traditional methods. In this context, this paper aims to investigate an optimal scheduling model that incorporates shared energy storage (SES) system, microgrids operator (MGO), electric vehicles station (EVS), and user aggregator (UA) with multiple prosumers. To comprehensively address the interests of all stakeholders, this study introduces a tri-level optimization model. The proposed model integrates a bidirectional Stackelberg-Nash game framework, in which the SES acts as the leader, the MGO acts as the secondary leader, and the UA-EVS acts as the followers while allocating benefits based on the asymmetric Nash bargaining theory. The Stackelberg game model between MGO and UA-EVS is analyzed using the Karush-Kuhn-Tucker (KKT) condition, while the Stackelberg game model between SES and MGO is resolved using the bisection method. Meanwhile, the Nash bargaining method among users is solved using the alternating direction method of multipliers (ADMM) technique. The analysis indicates that the proposed strategy can reduce PIES's costs and carbon emissions, yielding a win-win situation for all stakeholders.

Fuel-Optimal Trajectory Planning for Lunar Vertical Landing

In this paper we consider a trajectory planning problem arising from a lunar vertical landing with minimum fuel consumption. The vertical landing requirement is written as a final steering angle constraint, and a nonnegative regularization term is proposed to modify the cost functional. In this way, the final steering angle constraint will be inherently satisfied according to Pontryagin’s Minimum Principle. As a result, the modified optimal steering angle has to be determined by solving a transcendental equation. To this end, a transforming procedure is employed, which allows for finding the desired optimal steering angle by a simple bisection method. Consequently, the vertical landing trajectory can be found offline by the indirect shooting method. Finally, some numerical examples are presented to demonstrate and verify the developments of the paper.

Increasingly high-order hybrid multi-resolution WENO schemes in multi-dimensions

In this paper, a new type of increasingly high-order hybrid multi-resolution weighted essentially non-oscillatory (HMR-WENO) schemes is presented in the finite difference framework for solving hyperbolic conservation laws in one, two, and three dimensions. Based on the reconstruction polynomials defined on the one-point, three-point, five-point, seven-point, and nine-point spatial stencils, we reconstruct one zeroth degree reconstruction polynomial, one quadratic reconstruction polynomial, one quartic reconstruction polynomial, one sextic reconstruction polynomial, one octave reconstruction polynomial, and their derivative polynomials together with a new hierarchical bisection method to design a series of new troubled cell indicators which can precisely find all extreme points of associated unequal degree reconstruction polynomials located inside the smallest interval in one dimension. The new troubled cell indicators do not introduce any manual parameters related to different problems. The new hybrid methodology is divided into two parts: if all extreme points of the reconstruction polynomials are nonexistent or outside the smallest interval, the target cell is not a troubled cell and the simple linear upwind schemes are utilized to obtain high-order approximations. Otherwise, the target cell is a troubled cell and the MR-WENO spatial reconstruction procedures with excellent shock-capture ability are adopted. Then a series of HMR-WENO schemes are proposed by using these new troubled cell indicators, which can be easily expanded to arbitrarily high-order accuracies in multi-dimensions. The main benefits of these HMR-WENO schemes are their efficiency, since they could save about 22%-75% CPU time than that of the same order MR-WENO schemes when simulating some benchmark examples in multi-dimensions.

General Theory for Damped Beams with Elastic Supports Subjected to a Moving Damped Sprung Mass

Previously, classical boundary conditions (i.e. simple, fixed and cantilevered) have been adopted for the beams under the moving loads in deriving the dynamic response. In reality, most beams cannot be simply classified as the ones with classical supports, but are elastically restrained. For theoretical completeness, a general theory will be developed in this note for the damped beams with elastic restraints modeled by vertical and rotational springs subjected to a moving damped sprung mass. Essential to the present theory is the solution of the transcendental equation for the frequency by the bi-section method. The solution obtained can also be applied to the classical boundary conditions (i.e. simple, fixed and cantilevered) under a moving sprung mass. The reliability of the present theory, along with the bi-section method for solving the frequency and mode shape, is validated by comparison with the solution obtained by the finite element method (FEM) for various damping ratios, mass ratios and running speeds of the sprung mass.

Numerical Methods for Statistical Data Generation

A comprehensive comparison was undertaken between two numerical methods, namely bisection and secant methods, to generate random variable data for distribution. two methodologies were suggested to acquire the initial guesses for secant and interval for the bisection method. The results of 100 cases for the choices of the parameters are shown and a comparison was made according to the number of iterations between the methods that was conducted.