Step Length Calculation Research Articles

Error analyses and evaluation for solving burnup equations with MMPA method

The Mini-Max Polynomial Approximation (MMPA) is a new method to solve burnup equations, which has attracted much attention due to its advantages of good computational accuracy and efficiency. Understanding its error mechanism is significant for its application in practical burnup calculations. In this research, a mathematical model is proposed to clarify the error mechanism of the MMPA method for the burnup calculation. Then the coefficients of the MMPA method with different orders are given by the Remez algorithm. The errors of the MMPA method with different orders in burnup calculation are compared and analyzed, especially in single-step burnup calculation and infinite-step burnup calculation. Three cases are selected to verify the correctness of the mathematical error model, including fresh fuel burnup, depleted fuel burnup and decay calculation. The results show that when the order is greater than 36, the calculation accuracy of the MMPA method is less than the numerical noise of double precision calculation. As the order increases, the step length sensitivity of the MMPA method decreases. Finally, the computational accuracy and computational efficiency of the MMPA method with different orders are compared with those of the 16th order Chebyshev Rational Approximation Method (CRAM). The accuracy of the 16th order CRAM is between the 32nd and 36th order MMPA method. And the computation time of MMPA methods below order 76 is less than that of the 16th order CRAM. The theoretical analysis results of the mathematical error model are consistent with the numerical results. Then several best practices for solving burnup equations using the MMPA method are presented: (1) It is recommended to use the MMPA method with orders above 36 and below 76, whose accuracy and efficiency are better than 16th order CRAM in different degrees. (2) In multi-physics coupled computation which may require a strategy of large step length calculation, increasing the order of the MMPA method can not only improve the computational accuracy but also improve the computational efficiency.

A Benchmark Study on Steepest Descent and Conjugate Gradient Methods-Line Search Conditions Combinations in Unconstrained Optimization

In this paper, it is aimed to computationally conduct a performance benchmarking for the steepest descent and the three well-known conjugate gradient methods (i.e., Fletcher-Reeves, Polak-Ribiere and Hestenes-Stiefel) along with six different step length calculation techniques/conditions, namely Backtracking, Armijo-Backtracking, Goldstein, weakWolfe, strongWolfe, Exact local minimizer in the unconstrained optimization. To this end, a series of computational experiments on a test function set is completed using the combinations of those optimization methods and line search conditions. During these experiments, the number of function evaluations for every iteration are monitored and recorded for all the optimization method-line search condition combinations. The total number of function evaluations are then set a performance measure when the combination in question converges to the functions minimums within the given convergence tolerance. Through those data, the performance and data profiles are created for all the optimization method-line search condition combinations with the purpose of a reliable and an efficient benchmarking. It has been determined that, for this test function set, the steepest descent-Goldstein combination is the fastest one whereas the steepest descent-exact local minimizer is the most robust one with a high convergence accuracy. By making a trade-off between convergence speed and robustness, it has been identified that the steepest descent-weak Wolfe combination is the optimal choice for this test function set.

A Novel 3-D Indoor Localization Algorithm Based on BLE and Multiple Sensors

Indoor wireless localization using Bluetooth low energy (BLE) beacons has attracted considerable attention due to its extensive distribution and low cost properties. This article proposes a novel 3-D indoor localization algorithm which uses the combination of BLE and multiple sensors (3D-LBMS). The inertial navigation system (INS) and pedestrian dead reckoning (PDR) mechanizations are combined for accurate heading and speed estimation, which contains a multilevel constraints-based quasistatic magnetic field (QSMF) detection algorithm. In addition, dynamic-time-warping (DTW)-based BLE landmark detection algorithm is proposed to provide absolute 3-D location reference to multiple sensors-based positioning method, and the detected BLE landmark points are also used to calibrate the parameter of step-length calculation. Finally, the adaptive unscented Kalman filter (AUKF) is applied to fuse the results of INS/PDR mechanizations, QSMF and locations of detected BLE landmarks to achieve accurate and concrete multisource-based 3-D indoor localization performance. The experimental results show that the proposed 3-D-LBMS is proved to achieve meterlevel 2-D positioning accuracy and submeter level 3-D altitude estimation accuracy in typical indoor environments.

A Hybrid Dead Reckon System Based on 3-Dimensional Dynamic Time Warping

In recent years, using smartphones for indoor positioning has become increasingly popular with consumers. This paper presents an integrated localization technique for inertial and magnetic field sensors to challenge indoor positioning without Wi-Fi signals. For dead-reckoning (DR), attitude angle estimation, step length calculation, and step counting estimation are introduced. Dynamic time warping (DTW) usually calculates the distance between the measured magnetic field and magnetic fingerprint in the database. For DR/Magnetic matching (MM), we creatively propose 3-dimensional dynamic time warping (3DDTW) to calculate the distance. Unlike traditional DTW, 3DDTW extends the original one-dimensional signal to a two-dimensional signal. Finally, the weighted least squares further improves indoor positioning accuracy. In the three different experimental scenarios—teaching building, study room, office building—DR/MM hybrid positioning accuracy is about 3.34 m.

A Bluetooth/PDR Integration Algorithm for an Indoor Positioning System.

This paper proposes two schemes for indoor positioning by fusing Bluetooth beacons and a pedestrian dead reckoning (PDR) technique to provide meter-level positioning without additional infrastructure. As to the PDR approach, a more effective multi-threshold step detection algorithm is used to improve the positioning accuracy. According to pedestrians’ different walking patterns such as walking or running, this paper makes a comparative analysis of multiple step length calculation models to determine a linear computation model and the relevant parameters. In consideration of the deviation between the real heading and the value of the orientation sensor, a heading estimation method with real-time compensation is proposed, which is based on a Kalman filter with map geometry information. The corrected heading can inhibit the positioning error accumulation and improve the positioning accuracy of PDR. Moreover, this paper has implemented two positioning approaches integrated with Bluetooth and PDR. One is the PDR-based positioning method based on map matching and position correction through Bluetooth. There will not be too much calculation work or too high maintenance costs using this method. The other method is a fusion calculation method based on the pedestrians’ moving status (direct movement or making a turn) to determine adaptively the noise parameters in an Extended Kalman Filter (EKF) system. This method has worked very well in the elimination of various phenomena, including the “go and back” phenomenon caused by the instability of the Bluetooth-based positioning system and the “cross-wall” phenomenon due to the accumulative errors caused by the PDR algorithm. Experiments performed on the fourth floor of the School of Environmental Science and Spatial Informatics (SESSI) building in the China University of Mining and Technology (CUMT) campus showed that the proposed scheme can reliably achieve a 2-meter precision.

Optimal reactive power dispatch based on the improved seeker optimisation algorithm and statistical analysis

Optimal reactive power dispatch (ORPD) is one of the important non-linear mixed-variable optimisation problems in power system which includes both continuous and discrete control variables satisfying both equality and inequality constraints. This paper presents a new evolutionary optimisation algorithm to solve the optimal reactive power dispatch problem with operational constraints using the improved seeker optimisation algorithm (ISOA). In this paper, we improvise the step length determination strategy in the conventional seeker optimisation method by considering an optimistically contracting step length calculation. The feasibility of the proposed approach was tested on IEEE 30-bus system with three different single objectives like minimise the total real power loss, improve the load bus voltage profile and improve the voltage stability index. Simulation result obtained reveals the robustness and ability of the proposed methodology over other existing techniques recently reported in the literature. An innovative statistical analysis based on central tendency measures and dispersion measures were carried out on the three different single objectives.

Optimal short-term hydrothermal generation scheduling using modified seeker optimisation algorithm

This paper presents a new evolutionary optimisation algorithm to solve the short-term hydrothermal generation problem with operational constraints using the modified seeker optimisation algorithm. Seeker optimisation algorithm is a recently developed empirical gradient based, meta-heuristic optimisation algorithm, which draws inspiration from the random process of human search strategy. In this paper, we improvise the step length determination strategy in the classical seeker optimisation method by considering an optimistically contracting step length calculation. The proposed methodology easily takes care of solving non-linear hydrothermal generation problem along with different constraints like power balance, capacity limits, valve-point loading and prohibited operating zones. Simulations were performed over various standard test cases and a comparative study is carried out with other existing relevant approaches. The result obtained reveals the robustness and ability of the proposed methodology over other existing techniques.

Scalar Correction Method for Solving Large Scale Unconstrained Minimization Problems

We introduce a gradient descent algorithm for solving large scale unconstrained nonlinear optimization problems. The computation of the initial trial steplength is based on the usage of both the quasi-Newton property and the Hessian inverse approximation by an appropriate scalar matrix. The nonmonotone line search technique for the steplength calculation is applied later. The computational and storage complexity of the new method is equal to the computational and storage complexity of the Barzilai and Borwein method. On the other hand, the reported numerical results indicate improvements in favor of the new method with respect to the well known global Barzilai and Borwein method.

Accelerated gradient descent methods with line search

We introduced an algorithm for unconstrained optimization based on the transformation of the Newton method with the line search into a gradient descent method. Main idea used in the algorithm construction is approximation of the Hessian by an appropriate diagonal matrix. The steplength calculation algorithm is based on the Taylor’s development in two successive iterative points and the backtracking line search procedure. The linear convergence of the algorithm is proved for uniformly convex functions and strictly convex quadratic functions satisfying specified conditions.

Multiplicative parameters in gradient descent methods

We introduced an algorithm for unconstrained optimization based on the reduction of the modified Newton method with line search into a gradient descent method. Main idea used in the algorithm construction is approximation of Hessian by a diagonal matrix. The step length calculation algorithm is based on the Taylor's development in two successive iterative points and the backtracking line search procedure.

On the Implementation of Interior Point Decomposition Algorithms for Two-Stage Stochastic Conic Programs

In this paper we develop a practical primal interior decomposition algorithm for two-stage stochastic programming problems. The framework of this algorithm is similar to the framework in [S. Mehrotra and M. G. Özevin, “Decomposition based interior point methods for two-stage stochastic convex quadratic programs with recourse,” to appear in Oper. Res.; S. Mehrotra and M. G. Özevin, SIAM J. Optim., 18 (2007), pp. 206–222] and [G. Zhao, Math. Program., 90 (2001), pp. 507–536], however, their algorithm is altered in a simple yet fundamental way to achieve practical performance. In particular, this new algorithm weighs the log-barrier terms in the second stage problems differently from the theoretical algorithms analyzed in [S. Mehrotra and M. G. Özevin, Oper. Res., to appear], [S. Mehrotra and M. G. Özevin, SIAM J. Optim., 18 (2007), pp. 206–222], and [G. Zhao, Math. Program., 90 (2001), pp. 507–536]. We give a method for generating a suitable starting point; a method for selecting a good starting barrier parameter; a heuristic for first stage step-length calculation without performing line searches; and a method for adaptive addition of new scenarios over the course of the algorithm. The decomposition algorithm is implemented to solve two-stage stochastic conic programs with recourse whose underlying cones are Cartesian products of linear, second order, and semidefinite cones. The performance of primal decomposition method is studied on a set of randomly generated test problems as well as a two-stage stochastic programming extension of the Markowitz portfolio selection model. The computational results show that an efficient and stable implementation of the primal decomposition method is possible. These results also show that in problems with a large number of scenarios, the adaptive addition of scenarios can yield computational savings of up to 80%.

Investigation of minimal forms with conjugate gradient method

Amongst the numerous calculation methods available to investigate minimal forms, an approach based on the mechanical consideration of uniform tension in the domain leads to the writing of innovative relationships. Indeed, by establishing the equivalence between the vector of nodal internal forces due to the prestressed domain and the gradient of the function to be minimized, this study proposes to use the conjugate gradient method as a minimal area shape form-finding tool. The determination of descent directions may refer to Fletcher–Reeves suggestion or to an optimized value based on the Polak–Ribiere formula. Moreover, the steplength calculation is envisaged in accordance with the More and Thuente's line search algorithm or with a modified approach especially adapted to the studied configurations. Numerical experiments illustrate the use of the conjugate gradient method and thus point out the efficiency of the suggested modifications related to required computation times. In a second time, the conjugate gradient method is compared with density methods and a mixed formulation is therefore put forward. Numerical tests enable the comparison between these different approaches.

Partial BFGS update and efficient step-length calculation for three-layer neural networks.

Second-order learning algorithms based on quasi-Newton methods have two problems. First, standard quasi-Newton methods are impractical for large-scale problems because they require N2 storage space to maintain an approximation to an inverse Hessian matrix (N is the number of weights). Second, a line search to calculate a reasonably accurate step length is indispensable for these algorithms. In order to provide desirable performance, an efficient and reasonably accurate line search is needed. To overcome these problems, we propose a new second-order learning algorithm. Descent direction is calculated on the basis of a partial Broydon-Fletcher-Goldfarb-Shanno (BFGS) update with 2Ns memory space (s < < N), and a reasonably accurate step length is efficiently calculated as the minimal point of a second-order approximation to the objective function with respect to the step length. Our experiments, which use a parity problem and a speech synthesis problem, have shown that the proposed algorithm outperformed major learning algorithms. Moreover, it turned out that an efficient and accurate step-length calculation plays an important role for the convergence of quasi-Newton algorithms, and a partial BFGS update greatly saves storage space without losing the convergence performance.

On the solution of non-linear finite element equations

The paper deals with the basic requirements in the construction of a reliable continuation procedure. Adaptive step length determination and calculation of critical equilibrium states are discussed. For simple critical points an algorithm, which does not need classification between different types of bifurcations or even distinction between limit vs bifurcation point, is described. Situations where the extension of the parameter space could reveal vital information concerning the behaviour of the structure being analysed, are addressed.