Fixed Step Length Research Articles

A New Method Based on Locally Optimal Step Length in Accelerated Gradient Descent for Quantum State Tomography.

Quantum state tomography (QST) is one of the key steps in determining the state of the quantum system, which is essential for understanding and controlling it. With statistical data from measurements and Positive Operator-Valued Measures (POVMs), the goal of QST is to find a density operator that best fits the measurement data. Several optimization-based methods have been proposed for QST, and one of the most successful approaches is based on Accelerated Gradient Descent (AGD) with fixed step length. While AGD with fixed step size is easy to implement, it is computationally inefficient when the computational time required to calculate the gradient is high. In this paper, we propose a new optimal method for step-length adaptation, which results in a much faster version of AGD for QST. Numerical results confirm that the proposed method is much more time-efficient than other similar methods due to the optimized step size.

Fast high quality highlight removal using a simplified matrix iteration method

Recently, a global non-smooth optimization method was developed for efficiently eliminating specular highlights from single color image. Although this method was reported to be effective on both natural and medical images, its algorithm’s speed is slow due to the non-smoothness of the objective function. This paper proposes a global smooth optimization method, where two gradient regularization terms are used to describe diffuse and specular component features for texture details. A simplified optimality condition is established and thus a new iteration algorithm for specular highlight removal is proposed. By employing the gradient regularization of diffuse reflection, image texture detail can be preserved during specular highlight removal. Due to the smoothness of the objective function, the proposed smooth optimization algorithm has lower computational complexity and less storage requirement than the global non-smooth optimization algorithm. Furthermore, the proposed iteration algorithm is guaranteed to converge to a global optimal solution under a fixed step length. Experimental results demonstrate that the proposed smooth optimization algorithm is much faster than the non-smooth optimization algorithm. Moreover, the proposed highlight removal algorithm is more effective on benchmark natural images and real laparoscopic images than conventional highlight removal algorithms.

Short Step Length Estimation for Parkinson's Disease Patients by Using Fusion Data From Camera-IMU in Smart Glasses.

Shortened step length is a prominent motor abnormality in Parkinson's disease (PD) patients. Current methods for estimating short step length have the limitation of relying on laboratory scenarios, wearing multiple sensors, and inaccurate estimation results from a single sensor. In this paper, we proposed a novel method for estimating short step length for PD patients by fusing data from camera and inertial measurement units in smart glasses. A simultaneous localization and mapping technique and acceleration thresholding-based step detection technique were combined to realize the step length estimation. Two sets of experiments were conducted to demonstrate the performance of our method. In the first set of experiments with 12 healthy subjects, the proposed method demonstrated an average error of 8.44% across all experiments including six fixed step lengths below 30 cm. The second set of straightly walking experiments were implemented with 12 PD patients, the proposed method exhibited an average error of 4.27% compared to a standard gait evaluation technique in total walking distance. Notably, among the results of step lengths below 40 cm, our method agreed with the standard technique (R 2=0.8659). This study offers a promising approach for estimating short step length for PD patients during smart glasses-based gait training.

Iteration Complexity of Fixed-Step Methods by Nesterov and Polyak for Convex Quadratic Functions

This note considers the momentum method by Polyak and the accelerated gradient method by Nesterov, both without line search but with fixed step length applied to strongly convex quadratic functions assuming that exact gradients are used and appropriate upper and lower bounds for the extreme eigenvalues of the Hessian matrix are known. Simple 2-d-examples show that the Euclidean distance of the iterates to the optimal solution is non-monotone. In this context, an explicit bound is derived on the number of iterations needed to guarantee a reduction of the Euclidean distance to the optimal solution by a factor ϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon $$\\end{document}. For both methods, the bound is optimal up to a constant factor, it complements earlier asymptotically optimal results for the momentum method, and it establishes another link of the momentum method and Nesterov’s accelerated gradient method.

An Improved Perturbation Observation Method with Variable

Aiming at the problems of oscillation and misjudgment in traditional MPPT technology, the influence of voltage disturbance step size on the tracking of the maximum power point is studied. On this basis, an improved method based on the slope and space division of output characteristic curve is proposed. The voltage disturbance is carried out by using a fixed step length outside the set search interval, so that the photovoltaic cells can quickly work near the maximum power point. Reduce the early search time; When the photovoltaic cell is working near the maximum power point, the voltage disturbance is carried out with small steps to avoid the oscillation caused by repeated pulsation of the operating point around the maximum power point. Meanwhile, the search space is further divided to reduce the search interval. Static and dynamic lighting modes were simulated by Simulink.

Conditions for linear convergence of the gradient method for non-convex optimization

In this paper, we derive a new linear convergence rate for the gradient method with fixed step lengths for non-convex smooth optimization problems satisfying the Polyak-Łojasiewicz (PŁ) inequality. We establish that the PŁ inequality is a necessary and sufficient condition for linear convergence to the optimal value for this class of problems. We list some related classes of functions for which the gradient method may enjoy linear convergence rate. Moreover, we investigate their relationship with the PŁ inequality.

Numerical simulation of the general rate model of chromatography using orthogonal collocation

Process chromatography is a critical manufacturing step in the purification of biopharmaceutical therapeutics and monoclonal Antibody (mAb) capture using Protein A-chromatography is the most expensive step in the entire production chain. Protein A-chromatography performance is hard to predict due to variations in chromatography column characteristics causing wastage and reduced efficiency. Additionally, it is not yet established how to use Protein A-chromatography optimally in continuous manufacturing. Mechanistic modeling of the chromatography process enables simulation-based optimization of operations to both reduce waste and increase productivity in continuous manufacturing. We present an easy-to-use, yet computationally efficient Python package called Simulation Testbed for Liquid Chromatography (STLC). STLC is a chromatography simulator implementing a solver for the General rate model, a comprehensive mathematical model in the field, and the simplified Lumped kinetic model. STLC uses orthogonal collocation to discretize the spatial domain turning the equations into an algebraic system of ordinary differential equations. The equations are integrated in time with a fixed step length backward Euler method. We evaluate STLC in numerical trials and benchmark against high-resolution references showing high-order convergence. This allows for reduced computation system size making the package a good choice for use in the rapid prototyping of simulated chromatography processes.

IMPROVEMENT OF PEDESTRIAN DEAD RECKONING ALGORITHM FOR INDOOR POSITIONING BY USING STEP LENGTH ESTIMATION

Abstract. Pedestrian dead reckoning (PDR) can solve the position information by its inertial measurement unit (IMU), which is a method to achieve indoor autonomous positioning without deploying positioning base stations in advance. However, affected by the accumulated error, the positioning accuracy will decrease with the increase of the moving distance. To improve the indoor positioning accuracy, the step length estimation of PDR is improved. A PDR method combining the information on walking frequency, acceleration variance, and fixed step is proposed. Firstly, the pedestrian's walking steps are detected and estimated by the threshold peak method; secondly, the step frequency and acceleration variance are calculated, and the step length estimation model is fitted with the fixed step length of the pedestrian to estimate the step length of each step; then, the fusion algorithm of magnetometer, accelerometer, and gyroscope are used to estimate the heading; finally, combined with the step number, step length, and heading angle, the position information of walking is calculated by dead reckoning. The experimental results show that in the practical distance of 42.6 m, the accuracy of dead reckoning distance based on the combination of step frequency, acceleration variance, and fixed step length can reach 98.47%, which is better than the fixed step length and linear step length version, 5.09% higher than the linear step length and 0.35% higher than the fixed step length version.

The exact worst-case convergence rate of the gradient method with fixed step lengths for L-smooth functions

In this paper, we study the convergence rate of the gradient (or steepest descent) method with fixed step lengths for finding a stationary point of an L-smooth function. We establish a new convergence rate, and show that the bound may be exact in some cases, in particular when all step lengths lie in the interval (0, 1/L]. In addition, we derive an optimal step length with respect to the new bound.

Three-Dimensional Surface Reconstruction of Space Targets Using a Terahertz MIMO Linear Array Based on Multilayer Wideband Frequency Interferometry Techniques

In this article, the three-dimensional (3-D) surface reconstruction techniques of space targets were studied with a multiple-input-multiple-output (MIMO) linear array in the terahertz band. Conventional wavenumber domain integration-based reconstruction algorithm is unable to accurately reconstruct the continuous surfaces owing to the fixed step length in the range direction. To overcome this shortcoming, a phase unwrapping-free multilayer wideband frequency interferometry algorithm (MWFIA) was proposed. The MWFIA divides the targets into several layers based on the range resolution corresponding to the bandwidth of interference phase images, which can achieve surface reconstruction of targets with arbitrary varying range of distance. Besides, to ensure a satisfactory reconstruction accuracy under low signal-to-noise ratio environments, the interference phase errors introduced by noise are greatly suppressed by performing interference phase extraction between wideband two-dimensional imaging results of adjacent center frequency points. Both simulation and experimental results in the 0.22 THz band were provided to demonstrate the performance of the proposed method in comparison with those of the wavenumber domain integration-based and the previous frequency interferometry-based surface reconstruction techniques.

A Collision Avoidance Method for Intelligent Ship Based on the Improved Bacterial Foraging Optimization Algorithm

The bacterial foraging optimization algorithm (BFOA) is an intelligent population optimization algorithm widely used in collision avoidance problems; however, the BFOA is inappropriate for the intelligent ship collision avoidance planning with high safety requirements because BFOA converges slowly, optimizes inaccurately, and has low stability. To fix the above shortcomings of BFOA, an autonomous collision avoidance algorithm based on the improved bacterial foraging optimization algorithm (IBFOA) is demonstrated in this paper. An adaptive diminishing fractal dimension chemotactic step length is designed to replace the fixed step length to achieve the adaptive step length adjustment, an optimal swimming search method is proposed to solve the invalid searching and repeated searching problems of the traditional BFOA, and the adaptive migration probability is developed to take the place of the fixed migration probability to prevent elite individuals from being lost in BOFA. The simulation of benchmark tests shows that the IBFOA has a better convergence speed, optimized accuracy, and higher stability; according to a collision avoidance simulation of intelligent ships which applies the IBFOA, it can realize the autonomous collision avoidance of intelligent ships in dynamic obstacles environment is quick and safe. This research can also be used for intelligent collision avoidance of automatic driving ships.

Adapted block hybrid method for the numerical solution of Duffing equations and related problems

<abstract><p>Problems of non-linear equations to model real-life phenomena have a long history in science and engineering. One of the popular of such non-linear equations is the Duffing equation. An adapted block hybrid numerical integrator that is dependent on a fixed frequency and fixed step length is proposed for the integration of Duffing equations. The stability and convergence of the method are demonstrated; its accuracy and efficiency are also established.</p></abstract>

Active set expansion strategies in MPRGP algorithm

The paper investigates strategies for expansion of active set that can be employed by the MPRGP algorithm. The standard MPRGP expansion uses a projected line search in the free gradient direction with a fixed step length. Such a scheme is often too slow to identify the active set, requiring a large number of expansions. We propose to use adaptive step lengths based on the current gradient, which guarantees the decrease of the unconstrained cost function with different gradient-based search directions. Moreover, we also propose expanding the active set by projecting the optimal step for the unconstrained minimization. Numerical experiments demonstrate the benefits (up to 78% decrease in the number of Hessian multiplications) of our expansion step modifications on two benchmarks – contact problem of linear elasticity solved by TFETI and machine learning problems of SVM type, both implemented in PERMON toolbox.

A New Filter Nonmonotone Adaptive Trust Region Method for Unconstrained Optimization

In this paper, a new filter nonmonotone adaptive trust region with fixed step length for unconstrained optimization is proposed. The trust region radius adopts a new adaptive strategy to overcome additional computational costs at each iteration. A new nonmonotone trust region ratio is introduced. When a trial step is not successful, a multidimensional filter is employed to increase the possibility of the trial step being accepted. If the trial step is still not accepted by the filter set, it is possible to find a new iteration point along the trial step and the step length is computed by a fixed formula. The positive definite symmetric matrix of the approximate Hessian matrix is updated using the MBFGS method. The global convergence and superlinear convergence of the proposed algorithm is proven by some classical assumptions. The efficiency of the algorithm is tested by numerical results.

A Fast and Robust Algorithm for Orientation Estimation Using Inertial Sensors

We present a novel algorithm for online, real-time orientation estimation. Our algorithm integrates gyroscope data and corrects the resulting orientation estimate for integration drift using accelerometer and magnetometer data. This correction is computed, at each time instance, using a single gradient descent step with fixed step length. This fixed step length results in robustness against model errors, e.g. caused by large accelerations or by short-term magnetic field disturbances, which we numerically illustrate using Monte Carlo simulations. Our algorithm estimates a three-dimensional update to the orientation rather than the entire orientation itself. This reduces the computational complexity by approximately 1/3 with respect to the state of the art. It also improves the quality of the resulting estimates, specifically when the orientation corrections are large. We illustrate the efficacy of the algorithm using experimental data.

An enhanced Bacterial Foraging Optimization and its application for training kernel extreme learning machine

The Bacterial Foraging Optimization (BFO) algorithm is a swarm intelligent algorithm widely used in various optimization problems. However, BFO suffers from multiple drawbacks, including slow convergence speed, inability to jump out of local optima and fixed step length. In this study, an enhanced BFO with chaotic chemotaxis step length, Gaussian mutation and chaotic local search (CCGBFO) is proposed for overcoming the existing weakness of original BFO. First, a chaotic chemotaxis step length operation is used to produce adaptive chemotaxis step length. Then, by combining the optimal position in the current bacteria with the Gaussian mutation operation to make full use of the information of the optimal position. Finally, a chaotic local search is introduced into the chemotaxis step to ensure that the algorithm can explore a large search space in the early stage. The performance of CCGBFO was evaluated on a comprehensive set of numerical benchmark functions including IEEE CEC2014 and CEC2017 problems. In addition, CCGBFO was also used to tune the key parameters of kernel extreme learning machine for dealing with the real-world problems. The experimental results show that the proposed CCGBFO significantly outperforms the original BFO in terms of both convergence speed and solution accuracy.

A collocation method for differential equations with one‐point Taylor initial conditions

This paper presents a new multistep collocation method for nth order differential equations. The interval of interest is first divided into N subintervals. By determining interval conditions in Taylor interpolation in each subinterval, Taylor polynomials are calculated with different step lengths. Then the obtained solutions in each subinterval are used as initial conditions in the next step. Optimal error is assessed using Peano lemma, which shows that the errors decay by decreasing the step length. In particular, for fixed step length h, the error is of O(m−m), where m is the number of collocation points in each subinterval. Meanwhile, for fixed m, the error is of O(hm). Numerical examples demonstrate the validity and high accuracy of the proposed method even for stiff problems.

Laplace-domain waveform inversion using the l-BFGS method

ABSTRACTWe examine the performance of the limited-memory Broyden–Fletcher–Goldfarb–Shanno (l-BFGS) optimization method for full waveform inversions in the Laplace domain. Full waveform inversion methods generally adopt gradient-based optimization schemes for efficiency. Most Laplace-domain full waveform inversion methods published to date use the preconditioned steepest descent (PSD) method with the diagonal elements of the pseudo-Hessian as the preconditioner. These methods use a small fixed step length for stable convergence. The l-BFGS method is a quasi-Newton optimization method that efficiently approximates the inverse of the Hessian. We compare Laplace-domain inversion results from the PSD method with those from the l-BFGS method. Numerical examples using the SEG/EAGE salt model and the Pluto model demonstrate that the l-BFGS method with the Wolfe line search yields smaller cost function values than the PSD method.

Depth from defocus (DFD) based on VFISTA optimization algorithm in micro/nanometer vision

In the three-dimensional (3D) morphological reconstruction of micro/nano-scale vision, the global depth from defocus algorithm (DFD) transforms the depth information of the scene into a dynamic optimization problem to solve. In order to improve the problem of dynamic optimization in the recovery process of global DFD, a variable-step-size fast iterative shrinkage-thresholding algorithm (VFISTA) is proposed. The traditional iterative shrinkage-thresholding algorithm (ISTA) is often used to solve this dynamic optimization problem in the global DFD method. The ISTA algorithm is an extension of the gradient descent method, which is close to the minimal value point of the optimization process, and the convergence speed is slow. What is more, the ISTA algorithm uses fixed step length in the iterative process, The search direction tend to be “orthogonal”, prone to “saw tooth” phenomenon, so close to the minimum point when the convergence rate is slower. First, the VFISTA algorithm joins the acceleration operator on the basis of the ISTA algorithm. Further, it combines linear search method to find the optimal iteration length, and changes the limit of the ISTA algorithm step fixed. Finally, it is applied to the depth measurement of defocus scene in micro/nanometer scale vision. The experimental results show that the proposed fast depth from defocus algorithm based on VFISTA has faster convergent speed. Moreover, the precision of the measurement is obviously improved in micro/nanometer scale vision.

Influence of the random walk finite step on the first-passage probability

A well known connection between first-passage probability of random walk and distribution of electrical potential described by Laplace equation is studied. We simulate random walk in the plane numerically as a discrete time process with fixed step length. We measure first-passage probability to touch the absorbing sphere of radius R in 2D. We found a regular deviation of the first-passage probability from the exact function, which we attribute to the finiteness of the random walk step.