Ellipsoid Method Research Articles

Spatial Characterization of Single-Cracked Space Based on Microcrack Distribution in Sandstone Failure

To further understand the rock damage zone, an approach based on microcrack distribution was proposed to characterize the crack space of rock specimens in this research. Acoustic emission (AE) technology was utilized on sandstone to obtain the spatial distribution of microcracks in which uniaxial compression forms the single-cracked fracture. The proposed theoretical distribution pattern space (TDPS), 3D convex hull, and the minimum volume enclosing ellipsoid (MVEE) algorithms were adopted to analyze the geometric features of the crack space. It was found that the 3D convex hull method returned the smallest results in both area and volume of the crack space, and the largest results were provided by the proposed TDPS method. The difference between the results of the proposed TDPS method and the MVEE method became smaller after 85%. The deviation angle of the principal axis of the cracked space gradually decreased as the spatial scale decreased, while the other two major axes exhibited a tendency to increase at the 65% scale. The results indicate that a spatial scale from 65% to 85% is a reliable range for the characterization of crack space.

An ellipsoid modelling method for discrete element simulation of wheat seeds

An ellipsoid modelling method for discrete element simulation of wheat seeds

Analysis and Discussion on Ellipsoidal and virtual sphere models used for transverse arrangement INS

AbstractTransverse arrangement is one of the main methods used in the polar inertial navigation system (INS). In the traditional algorithm, the calculation of using the earth ellipsoid model is complex, while using the earth sphere model cannot satisfy a high-accuracy application. Therefore, an approach based on the virtual sphere model is proposed, which has been proved in simulation experiments to reduce the computational complexity and maintain the same accuracy as the ellipsoid algorithm, but its accuracy has not yet been proved in theory. Starting from the basic principles of the ellipsoid and virtual sphere model algorithm, this paper compares the key formulations of the two. Finally, it is proved that the two arrangements are actually the same.

A Lagrangian Algorithm for Multiple Depot Traveling Salesman Problem With Revisit Period Constraints

This work presents a Multiple Depot Traveling Salesman Problem with revisit period constraints. The revisit period constraints are relevant to persistent routing applications, where these constraints represent maximum time between successive visits to a target. This problem is first posed as a Mixed Integer Linear Program. The coupling constraints in the primal problem are then relaxed via Lagrangian relaxation. Minimizing the resulting Lagrangian over the primal variables can be separated into an individual subproblem for each salesman. An algorithm to solve the subproblems to near-optimality that scales well for larger instances is presented. The dual function is then maximized, where three different dual update methods are studied: the subgradient method, the ellipsoid algorithm, and a bundle algorithm. Further, a primal reconstruction algorithm is presented to reconstruct feasible solutions from the solution of the dual algorithm. The quality of these solutions are compared to the optimal solutions obtained from the CPLEX MILP optimizer. The results of extensive numerical testing show that the dual algorithm presented was computationally efficient, capable of finding high quality solutions, and scale well compared to CPLEX. Further, it was seen that the bundle method used showed better convergence than the other update methods within the dual algorithm. Note to Practitioners— This work was motivated by the problem of routing UAVs in the presence of timing constraints, specifically motivated by military applications. This problem has constraints on the frequency at which targets are visited, such that high priority targets must be visited more frequently. Similar prior work exists in the literature for a variety of routing problems for different timing or resource constraints. Techniques which are successful on one set of constraints may be less useful when applied to different constraints. We present a Lagrangian based approach to the multiple depot traveling salesman variant. The results show the utility of a Lagrangian based technique to this problem, as both the solution quality and computation time scale well with problem size. The Lagrangian based technique presented here is seen to provide, on average, high quality solutions with improved computational time over a branch-and-bound algorithm, which solves the problem exactly. While the solutions returned from the algorithm are on average of good quality, they do exhibit some variance. Alternative algorithms may give more consistent results. The technique presented in this paper is general and may be applied to other routing problems where the revisit period constraints are applied. Future research includes application of this method to dynamic environments, where the targets’ states are unknown to the UAVs except when being monitored. Further, alternative techniques can be developed for this problem and compared to both the branch-and-bound and the Lagrange algorithm presented here.

Solving strongly convex-concave composite saddle-point problems with low dimension of one group of variable

Algorithmic methods are developed that guarantee efficient complexity estimates for strongly convex-concave saddle-point problems in the case when one group of variables has a high dimension, while another has a rather low dimension (up to 100). These methods are based on reducing problems of this type to the minimization (maximization) problem for a convex (concave) functional with respect to one of the variables such that an approximate value of the gradient at an arbitrary point can be obtained with the required accuracy using an auxiliary optimization subproblem with respect to the other variable. It is proposed to use the ellipsoid method and Vaidya's method for low-dimensional problems and accelerated gradient methods with inexact information about the gradient or subgradient for high-dimensional problems. In the case when one group of variables, ranging over a hypercube, has a very low dimension (up to five), another proposed approach to strongly convex-concave saddle-point problems is rather efficient. This approach is based on a new version of a multidimensional analogue of Nesterov's method on a square (the multidimensional dichotomy method) with the possibility to use inexact values of the gradient of the objective functional. Bibliography: 28 titles.

LSTM-OBE based Interval Prediction of Effluent BOD for Wastewater Treatment

In this paper, a new interval prediction method of effluent BOD for wastewater treatment is proposed. Firstly, a point prediction method of effluent BOD based on a long short term memory (LSTM) neural network model is generated. Then, an improved LSTM-MTOBE algorithm is designed to predict the interval of effluent BOD, in which the minimum trace optimal boundary ellipsoid (MTOBE) algorithm is combined with LSTM to identify the real parameters of its output layer weights. Next, based on the properties of ellipsoids, the determination interval of the model uncertainty is obtained and analyzed. Finally, compared with some existing algorithms, the proposed method is illustrated to have better prediction accuracy and interval performance.

Real Objects Understanding Using 3D Haptic Virtual Reality for E-Learning Education

In the past two decades, there has been a lot of work on computer vision technology that incorporates many tasks which implement basic filtering to image classification. The major research areas of this field include object detection and object recognition. Moreover, wireless communication technologies are presently adopted and they have impacted the way of education that has been changed. There are different phases of changes in the traditional system. Perception of three-dimensional (3D) from two-dimensional (2D) image is one of the demanding tasks. Because human can easily perceive but making 3D using software will take time manually. Firstly, the blackboard has been replaced by projectors and other digital screens so such that people can understand the concept better through visualization. Secondly, the computer labs in schools are now more common than ever. Thirdly, online classes have become a reality. However, transferring to online education or e-learning is not without challenges. Therefore, we propose a method for improving the efficiency of e-learning. Our proposed system consists of two-and-a-half dimensional (2.5D) features extraction using machine learning and image processing. Then, these features are utilized to generate 3D mesh using ellipsoidal deformation method. After that, 3D bounding box estimation is applied. Our results show that there is a need to move to 3D virtual reality (VR) with haptic sensors in the field of e-learning for a better understanding of real-world objects. Thus, people will have more information as compared to the traditional or simple online education tools. We compare our result with the ShapeNet dataset to check the accuracy of our proposed method. Our proposed system achieved an accuracy of 90.77% on plane class, 85.72% on chair class, and car class have 72.14%. Mean accuracy of our method is 70.89%.

On Subgradient Methods with Polyak’s Step and Space Transformation

Introduction. Minimization of ravine convex functions, both smooth and non-smooth, arises in many problems of planning, control, stability analysis of dynamic systems, artificial intelligence, and machine learning. Therefore, the development of new and improvement of existing methods is an important task, taking into account the fact that more and more frequently the functions to be minimized depend on a large number of variables. Special attention should be paid to the methods using the operation of linear transformation of space, which allow to improve the properties of the objective function and significantly accelerate its minimization. Known methods of this type, in particular, ellipsoid methods and r-algorithms, require recalculation of the transformation matrix at each iteration. Therefore, the development of methods with a one-time transformation of space, which have a lower cost of iteration, is definitely an urgent task. The purpose of the article is to review modifications of subgradient method with Polyak’s step that use a scalar parameter m(1 and one-time space transformation operation. Supplement the justification of the convergence and convergence rate of the modifications described. Give recommendations regarding determination of parameter m and space transformation matrix B. Results. Subgradient method with Polyak’s step in transformed space and parameter m>1 is an effective method for minimizing smooth and non-smooth convex functions that have a ravine structure. Determination of an appropriate value of the parameter m(1 and space transformation matrix B allows to significantly accelerate this method and use it for minimization functions that depend on a large number of variables. Conclusions. The development of fast methods for minimization of non-smooth convex functions of many variables with a ravine structure makes it possible to effectively solve modern problems of artificial intelligence, in particular, the problems of machine learning, image recognition, big data analysis, etc. Keywords: subgradient method, space transformation, ravine functions.

A Designed Calibration Approach for the Measurement-While-Drilling Instrument

The Measurement-While-Drilling (MWD) system, composed of a tri-axial magnetometer and a tri-axial accelerometer, is widely used in the Horizontal Directional Drilling machine in coal mines. This system can provide attitude information of each measuring point in the borehole, which will eventually allow the trajectory of the borehole to be drawn. The attitude information, however, showed a low-level accuracy, due to the sensor’s imperfection and mounting errors. The accuracy worsened when low-cost sensors were employed, as they had higher random noise. Therefore, an exploration of ways to eliminate the sensor imperfection and mounting tolerance as well as to suppress the noise is needed. In this paper, a feasible calibration approach was designed to address these issues. This new approach combined three foundational calibration algorithms, including the ellipsoidal fitting method, the planar fitting method, and the inner product invariance method. The traditional ellipsoidal fitting method and planar fitting method were optimized by using the recursive least square criterion and omitting the steps of sample data acquisition, respectively. In addition, the noise suppression method was involved in our approach to improve the calibration accuracy. The numerical simulation results showed that the number of sampling points decreased significantly, but the accuracy of the azimuthal angle and the pitch angle fully met the engineering requirements. The experimental results showed that the pitch angle error was reduced by less than 0.5°, and the azimuth error was also reduced by less than 2.5°. It should be noted that this new approach could be implemented without the help of other expensive auxiliary equipment.

Minimizing Convex Functions with Rational Minimizers

Given a separation oracle SO for a convex function f defined on ℝ n that has an integral minimizer inside a box with radius R , we show how to find an exact minimizer of f using at most O(n (n log log (n)/ log (n) + log ( R ))) calls to SO and poly ( n , log ( R )) arithmetic operations, or O(n log (nR) calls to SO and exp ( O(n) ) ⋅ poly (log (R) ) arithmetic operations. When the set of minimizers of f has integral extreme points, our algorithm outputs an integral minimizer of f . This improves upon the previously best oracle complexity of O(n 2 ( n + log ( R ))) for polynomial time algorithms and O(n 2 log ( nR ) for exponential time algorithms obtained by [Grötschel, Lovász and Schrijver, Prog. Comb. Opt. 1984, Springer 1988] over thirty years ago. Our improvement on Grötschel, Lovász and Schrijver’s result generalizes to the setting where the set of minimizers of f is a rational polyhedron with bounded vertex complexity. For the Submodular Function Minimization problem, our result immediately implies a strongly polynomial algorithm that makes at most O(n 3 log log ( n )/log ( n )) calls to an evaluation oracle, and an exponential time algorithm that makes at most O(n 2 log ( n )) calls to an evaluation oracle. These improve upon the previously best O(n 3 log 2 ( n )) oracle complexity for strongly polynomial algorithms given in [Lee, Sidford and Wong, FOCS 2015] and [Dadush, Végh and Zambelli, SODA 2018], and an exponential time algorithm with oracle complexity O(n 3 log ( n )) given in the former work. Our result is achieved via a reduction to the Shortest Vector Problem in lattices. We show how an approximately shortest vector of an auxiliary lattice can be used to effectively reduce the dimension of the problem. Our analysis of the oracle complexity is based on a potential function that simultaneously captures the size of the search set and the density of the lattice, which we analyze via tools from convex geometry and lattice theory.

The influence exerted by active cooling on the permanent connection formed as a result of electric arc welding

The problem of temperature stresses is solved in the work. The process of electric arc welding with guaranteed penetration of steel plates of various thicknesses is simulated. The possibility of reducing residual stresses formed as a result of welding by applying heat removal from the surface of the near-weld region of a steel plate is being investigated. The temperature distribution is determined by solving a non-linear heat equation, in which the specific heat and thermal conductivity depend on temperature. The heat source is modeled using the double ellipsoid method proposed by John A. Goldak. The horseshoe-shaped heat sink follows immediately after the anode spot and is set by the heat flux in such a way that the main temperature of the plate does not decrease below its initial temperature. This limitation reduces the level of the temperature gradient, thereby reducing the stresses in the metal plate. To illustrate the effect of active heat sink, the temperature distribution in the top layer at the moment of termination of welding is compared with and without active heat sink. The material is assumed to be elastoviscoplastic, the deformations are small and consist of reversible and irreversible ones. Reversible deformations are related to stresses by the Duhamel - Neumann law. Irreversible strains increase when the von Mises plastic flow condition is satisfied, in which there is a plastic strain rate component responsible for the plastic flow viscosity. The elastic moduli (Young's modulus, Poisson's ratio) and the yield strength are assumed to be temperature dependent. The solution of the mechanical part of the problem is found by the method of simple iterations. For plates with different thicknesses, diagrams of residual stresses are given, located in the center of the plate from the seam to the periphery. As a result of the work, according to the obtained distributions of residual stresses in the plate material, it is concluded that the use of an active horseshoe-shaped heat sink from the near-weld zone when welding thin steel plates reduces residual stresses, therefore it is recommended to use in the welding process. The use of an active heat sink on the reverse side of the plate leads to an increase in the stress level, therefore it is not recommended for use.

A scenario-based robust distribution expansion planning under ellipsoidal uncertainty set using second-order cone programming

• A robust trilevel distribution expansion planning model is presented. • The model considers short-term demand and wind uncertainty. • The uncertainties are modelled by ellipsoidal uncertainty set. • Second-order cone programming is used to model the AC distribution network. • The robust trilevel model is solved by classic column-and-constraint generation. This paper intends to propose a new mathematical framework for distribution expansion planning (DEP) based on which the uncertainties associated with electric demand and wind production are modeled through several plausible ellipsoidal uncertainty sets. In this regard, a hybrid model combining stochastic programming and robust optimization is constructed. Data related to the demand-wind scenarios are first divided into clusters of similar elements using an existing clustering technique. Then, an ellipsoidal uncertainty set is created corresponding to each of these generated clusters based on the theory of minimum volume covering ellipsoid (MVCE) using the Khachiyan algorithm (KA). The hybrid robust/stochastic scheme is formulated as a two-stage tri-level min-max-min optimization problem in which the convex conic relaxation of AC power flow is used to represent the electric distribution network. The multi-uncertainty-set-based model is then solved by employing the classic column-and-constraint-generation (C&CG) technique which guarantees the convergence to the optimal solution in a limited number of iterations. In this regard, a master problem and several subproblems related to each scenario will be solved, both of which are second-order cone programs. Numerical simulations reveal the superiority of the hybrid model compared to the existing ones.

Influence of Sociodemographic and Anthropometric Factors on Gallbladder Volume in Pregnancy in a Tertiary Hospital in Nigeria.

Our study sought to determine the effect of age, parity, body mass index (BMI), and previous oral contraceptive use on gallbladder volume and ejection fraction in pregnancy to enable obstetricians to identify patients at risk of gallbladder disease in pregnancy. We conducted a prospective cross-sectional study involving 190 pregnant women who were evaluated within 32 and 40 weeks gestation at the Fetal Assessment Unit of the Department of Radiology, University of Ilorin Teaching Hospital, Nigeria. Patients were scanned with a commercially available Siemens Ultrasound scanner using a curvilinear probe and a transducer frequency of 3.5 MHz. The fasting gallbladder volume (FGBV) and 30 minutes postprandial gallbladder volume (PGBV) were calculated using the prolate ellipsoid method. Gallbladder ejection fraction was determined using FGBV and PGBV. Each patient's biodata, medical history, and somatometric parameters were documented. A total of 190 volunteers were included in the study. BMI showed statistically significant positive, though weak correlations with FGBV (r = 0.179, p =0.015) and PGBV (r = 0.216, p =0.003). Maternal age, parity, and previous oral contraceptive use did not show any statistically significant correlation with gallbladder parameters. BMI showed a statistically significant but weakly positive correlation with FGBV and PGBV in pregnancy. Our study demonstrated that gallbladder volume in pregnancy might be dependent on BMI. Thus, high BMI may predispose to large gallbladder volume, increasing the risk of biliary stasis and gallstone formation in pregnancy. This is relevant for early detection and prevention of gallbladder disease and follow-up, including expectant management of subjects that may develop gallbladder disease in pregnancy.

Integral sliding‐mode‐based robust output‐regulation for constrained and uncertain linear systems

AbstractThis article proposes a solution to the output regulation problem for uncertain linear systems with input saturation, state constraints, and external disturbances. The proposed robust control is composed of a linear and a nonlinear part. Each part of the control can be designed independently due to the use of an integral sliding‐mode control approach. The linear control, whose design is based on the barrier Lyapunov function and the attractive ellipsoid method, considers the input saturation, the state constraints, and the parameter uncertainties. The nonlinear part is based on an integral sliding‐mode control approach and it can compensate the effect of some matched disturbances. Also, a safe set where the system trajectories do not transgress the state constraints and input saturation, is provided. The proposed scheme guarantees asymptotic convergence to zero of the output regulation error coping with the system constraints and disturbances. A constructive and simple method, based on linear matrix inequalities, is proposed to compute the controller gains. Some simulation results illustrate the feasibility of the proposed scheme.

A piecewise ellipsoidal reachable set estimation method for continuous bimodal piecewise affine systems

In this work, the issue of estimation of reachable sets in continuous bimodal piecewise affine systems is studied. A new method is proposed, in the framework of ellipsoidal bounding, using piecewise quadratic Lyapunov functions. Although bimodal piecewise affine systems can be seen as a special class of affine hybrid systems, reachability methods developed for affine hybrid systems might be inappropriately complex for bimodal dynamics. This work goes in the direction of exploiting the dynamical structure of the system to propose a simpler approach. More specifically, because of the piecewise nature of the Lyapunov function, we first derive conditions to ensure that a given quadratic function is positive on half spaces. Then, we exploit the property of bimodal piecewise quadratic functions being continuous on a given hyperplane. Finally, linear matrix characterizations of the estimate of the reachable set are derived.

A simple optimal contention resolution scheme for uniform matroids

Contention resolution schemes (or CR schemes), introduced by Chekuri, Vondrak and Zenklusen, are a class of randomized rounding algorithms for converting a fractional solution to a relaxation for a down-closed constraint family into an integer solution. A CR scheme takes a fractional point x in a relaxation polytope, rounds each coordinate xi independently to get a possibly non-feasible set, and then drops some elements in order to satisfy the constraints. Intuitively, a CR scheme is c-balanced if every element i is selected with probability at least c⋅xi.It is known that general matroids admit a (1−1/e)-balanced CR scheme, and that this is (asymptotically) optimal. This is in particular true for the special case of uniform matroids of rank one. In this work, we provide a simple and explicit monotone CR scheme for uniform matroids of rank k on n elements with a balancedness of 1−(nk)(1−kn)n+1−k(kn)k, and show that this is optimal. As n grows, this expression converges from above to 1−e−kkk/k!. While this asymptotic bound can be obtained by combining previously known results, these require defining an exponential-sized linear program, as well as using random sampling and the ellipsoid algorithm. Our procedure, on the other hand, has the advantage of being simple and explicit. This scheme extends naturally into an optimal CR scheme for partition matroids.

Comparison of PSA density and lesion volume strategies for selecting men with equivocal PI-RADS 3 lesions on bpMRI for biopsies

PurposeTo compare two strategies: Prostate-specific antigen density (PSAd) and lesion volume measurement in ruling out significant prostate cancer (sPCa) in men with equivocal Prostate Imaging Reporting and Data System (PI-RADS) category 3 index lesions on biparametric magnetic resonance imaging.MethodsIn total, 130 men from our database had index lesions with PI-RADS scores of 3. Prostate volume was measured using the ellipsoid method, in accordance with PI-RADS version 2.1 criteria. Index lesion volumes were also measured using the ellipsoidal formula on the diffusion-weighted imaging sequence with the highest b-value and sagittal T2 sequences.ResultsAmong 130 men with PI-RADS category 3 index lesions, 23 (18%) had sPCa. In total, 6 of the 89 men with PSAd < 0.15 ng/mL2 (7%) had sPCa, whereas 8 of the 49 men with index lesion volumes < 0.5 mL (16%) had sPCa. The difference was statistically significant (McNemar, p < 0.0001).ConclusionThe PSAd strategy performed better than the lesion volume strategy in ruling out sPCa in men with equivocal PI-RADS category 3 index lesions.

On the Computability of Continuous Maximum Entropy Distributions with Applications

We initiate a study of the following problem: Given a continuous domain $\Omega$ along with its convex hull $\mathcal{K}$, a point $A \in \mathcal{K}$ and a prior measure $\mu$ on $\Omega$, find the probability density over $\Omega$ whose marginal is $A$ and that minimizes the KL-divergence to $\mu$. This framework gives rise to several extremal distributions that arise in mathematics, quantum mechanics, statistics, and theoretical computer science. Our technical contributions include a polynomial bound on the norm of the optimizer of the dual problem that holds in a very general setting and relies on a balance property of the measure $\mu$ on $\Omega$, and exact algorithms for evaluating the dual and its gradient for several interesting settings of $\Omega$ and $\mu$. Together, along with the ellipsoid method, these results imply polynomial-time algorithms to compute such KL-divergence minimizing distributions in several cases. Applications of our results include: 1) an optimization characterization of the Goemans-Williamson measure that is used to round a positive semidefinite matrix to a vector, 2) the computability of the entropic barrier for polytopes studied by Bubeck and Eldan, and 3) a polynomial-time algorithm to compute the barycentric quantum entropy of a density matrix that was proposed as an alternative to von Neumann entropy in the 1970s: this corresponds to the case when $\Omega$ is the set of rank one projections matrices and $\mu$ corresponds to the Haar measure on the unit sphere. Our techniques generalize to the setting of Hermitian rank $k$ projections using the Harish-Chandra-Itzykson-Zuber formula, and are applicable even beyond, to adjoint orbits of compact Lie groups.

Robust Output Regulation for a Constrained Omnidirectional Mobile Robot

This paper aims to design a robust output control to regulate the output for an uncertain omnidirectional mobile robot with input saturation and state constraints, in the presence of external disturbances. The proposed robust control approach is composed of a linear control part, that is designed based on the attractive ellipsoid method and a barrier Lyapunov function approach, taking into account the input and state constraints, and a nonlinear control part that can compensate the effect of matched disturbances. The synthesis of the robust control is given in terms of linear matrix inequalities. Simulation results show the workability of the proposed robust control approach.

Ellipsoidal Design of Robust Stabilization for Markov Jump Power Systems under Normal and Contingency Conditions

The essential prerequisites for secure customer service are power system stability and reliability. This work shows how to construct a robust switching control for studying power system load changes using an invariant ellipsoid method. Furthermore, the suggested control ensures stability when the system is subjected to random stochastic external disturbances, and functions randomly in two conditions: normal and contingency. The extreme (least) reliability state is chosen as the most severe scenario (corresponding to a transmission line outage). As a two-state Markov random chain, the transition probabilities are utilized to simulate the switching between normal and contingency modes (or processes). To characterize the dynamics of the studied system, a stochastic mathematical model is developed. The effect of stochastic disturbances and random normal/contingency operations is taken into account during the design stage. For a stochastic power system, a novel excitation control is designed. The attractive ellipsoid approach and linear matrix inequalities (LMIs) optimization are used to build the best two-controller gains. Therefore, the proposed modeling/design technique can be employed for the power system under load changes, stochastic topological changes, and random disturbances. Finally, the system’s random dynamics simulation indicates the effectiveness of the designed control law.