Iteration Step Research Articles

A Modified Dai–Liao Conjugate Gradient Method Based on a Scalar Matrix Approximation of Hessian and Its Application

We introduce and investigate proper accelerations of the Dai–Liao (DL) conjugate gradient (CG) family of iterations for solving large-scale unconstrained optimization problems. The improvements are based on appropriate modifications of the CG update parameter in DL conjugate gradient methods. The leading idea is to combine search directions in accelerated gradient descent methods, defined based on the Hessian approximation by an appropriate diagonal matrix in quasi-Newton methods, with search directions in DL-type CG methods. The global convergence of the modified Dai–Liao conjugate gradient method has been proved on the set of uniformly convex functions. The efficiency and robustness of the newly presented methods are confirmed in comparison with similar methods, analyzing numerical results concerning the CPU time, a number of function evaluations, and the number of iterative steps. The proposed method is successfully applied to deal with an optimization problem arising in 2D robotic motion control.

Variational Principles for the Trajectory Tracking Control of Underactuated Mechanical Systems

Abstract Robotics is undergoing dynamic progression with the spread of soft robots and compliant mechanisms. The mechanical models describing these systems are underactuated, having more degrees-of-freedom than inputs. The trajectory tracking control of underactuated systems is not straightforward. The solution of the inverse dynamics is not stable in all cases, as it only considers the actual state of the system. Therefore employing the advances of optimal control theory is a reasonable choice. However, the real-time application of these is challenging as the solution to the discretized optimization problems is numerically expensive. This paper presents a novel iterative approach to solving nonlinear optimal control problems. The authors first define the iteration formula after which the obtained equations are discretized to prepare the numerical solution, contrarily to the accessible works in the literature having reverse order. The main idea is to approximate the cost functional with a second-order expansion in each iteration step, which is then extremized to get the subsequent approximation of the optimum. In the case of nonlinear optimal control problems, the process leads to a sequence of time-variant linear–quadratic regulator (LQR) problems. The proposed technique was effectively applied to the trajectory tracking control of a flexible rotational-rotational joint (RR) manipulator. The case study showed that the initialization of the iteration is simple, and the convergence is rapid.

An alternative switching filtered-x LMS algorithm for road impact noise control in vehicle

An adaptive switching FxLMS algorithm is proposed to deal with the problem that the performance of the active control system of interior noise drops sharply when come across road impact. According to the state of the system, the algorithm updates the weight coefficient of the control filter in two different ways. Firstly, the normalization algorithm is adopted to improve the convergence speed. After convergence, the relationship between iteration step size and residual error is established by using deformed Gaussian function. The adverse effects of road impact will be dealt with by a new iteration step. On the premise of increasing a small amount of calculation, it improves the convergence speed of the system and ensures the steady-state performance of the system. Numerical simulation and real vehicle test verify the effectiveness of the proposed algorithm in the face of road impact.

Generalized Inexact Newton-Landweber Iteration for Possibly Non-Smooth Inverse Problems in Banach Spaces

In this paper, we consider a generalized inexact Newton-Landweber iteration to solve nonlinear ill-posed inverse problems in Banach spaces, where the forward operator might not be Gâteaux differentiable. The method is designed with non-smooth convex penalty terms, including L1-like and total variation-like penalty functionals, to capture special features of solutions such as sparsity and piecewise constancy. Furthermore, the inaccurate inner solver is incorporated into the minimization problem in each iteration step. Under some assumptions, based on ε-subdifferential, we establish the convergence analysis of the proposed method. Finally, some numerical simulations are provided to illustrate the effectiveness of the method for solving both smooth and non-smooth nonlinear inverse problems.

Large-Dimensional Multibody Dynamics Simulation Using Contact Nodalization and Diagonalization

In this article, we propose a novel multibody dynamics simulation framework that can efficiently deal with large-dimensionality and complementarity multicontact conditions. Typical contact simulation approaches require performing contact impulse fixed-point iteration, which has high time-complexity from large-size matrix factorization and multiplication, as well as susceptibility to ill-conditioned contact situations. To circumvent this, we propose a novel framework based on velocity fixed-point iteration (V-FPI), which, by utilizing a certain surrogate dynamics and contact nodalization (with virtual nodes), we achieve not only intercontact decoupling but also their interaxes decoupling (i.e., contact diagonalization) at each iteration step. This then enables us to one-shot/parallel-solve the contact problem during each V-FPI iteration-loop, while avoiding large-size/dense matrix inversion/multiplication, thereby, significantly speeding up the simulation time with improved convergence property. We theoretically show that the solution of our framework is consistent with that of the original problem and, further, elucidate mathematical conditions for the convergence of our proposed solver. Performance and properties of our proposed simulation framework are also demonstrated and experimentally validated for various large-dimensional/multicontact scenarios including deformable objects.

PowerSynth 2: Physical Design Automation for High-Density 3-D Multichip Power Modules

Moving towards an electrified world requires ultra-high-density power converters. With the adoption of wide-bandgap semiconductors (e.g., SiC, GaN), the next-generation power converters are on the horizon. However, the complexity of compact and high-speed converters is beyond the current industry-standard design flow based on manual and iterative steps. Therefore, research on design automation and optimization has been identified as an emerging topic in the power electronics society. Among different components of the converter, the physical design of power modules has proven critical for achieving high power density and energy efficiency. We present the latest design automation flow for high-density (2D/2.5D/3D) and heterogeneous multi-chip power modules (MCPM) through PowerSynth 2 framework. In this work, we further demonstrate electro-thermal optimization for State-of-the-Art (SOTA) 3D packaging technologies. Using the latest PowerSynth 2 framework, electro-thermal design optimization is carried out on a sample 3D MCPM layout using both exhaustive and evolutionary search methods. The optimized design is hardware-validated against physical measurements. The measurement result has shown an order of magnitude productivity improvement within 10% accuracy compared to the SOTA industry design flow.

InDuDoNet+: A deep unfolding dual domain network for metal artifact reduction in CT images.

During the computed tomography (CT) imaging process, metallic implants within patients often cause harmful artifacts, which adversely degrade the visual quality of reconstructed CT images and negatively affect the subsequent clinical diagnosis. For the metal artifact reduction (MAR) task, current deep learning based methods have achieved promising performance. However, most of them share two main common limitations: (1) the CT physical imaging geometry constraint is not comprehensively incorporated into deep network structures; (2) the entire framework has weak interpretability for the specific MAR task; hence, the role of each network module is difficult to be evaluated. To alleviate these issues, in the paper, we construct a novel deep unfolding dual domain network, termed InDuDoNet+, into which CT imaging process is finely embedded. Concretely, we derive a joint spatial and Radon domain reconstruction model and propose an optimization algorithm with only simple operators for solving it. By unfolding the iterative steps involved in the proposed algorithm into the corresponding network modules, we easily build the InDuDoNet+ with clear interpretability. Furthermore, we analyze the CT values among different tissues, and merge the prior observations into a prior network for our InDuDoNet+, which significantly improve its generalization performance. Comprehensive experiments on synthesized data and clinical data substantiate the superiority of the proposed methods as well as the superior generalization performance beyond the current state-of-the-art (SOTA) MAR methods. Code is available at https://github.com/hongwang01/InDuDoNet_plus.

Corrosion test and three-dimensional cellular machine simulation of Q235 steel by ion diffusion in different soil environments

In the actual soil environment, due to the unpredictable characteristics of metal pitting corrosion. Therefore, in this study, soil composed of different concentrations of corrosion medium was used for the mental corrosion test, and the experimental results of single ion and multiple ions were compared. At the same time, the corrosion behavior of metals is introduced in combination with the mathematical model of unsteady diffusion. In addition, the cellular automata model verification method is used to study the comparison and verification of the diffusion growth model of the model corrosion pit in different iteration steps under the discrete mathematical model. The results show that the growth and diffusion trend of corrosion pits controlled by the concentration of SO42-, HCO3–, and Cl- is consistent basically with the law reflected by cellular automata.

An efficient fourth-order accurate conservative scheme for Riesz space fractional Schrödinger equation with wave operator

In this study, we investigate a high-order accurate conservative finite difference scheme by utilizing a fourth-order fractional central finite difference method for the two-dimensional Riesz space-fractional nonlinear Schrödinger wave equation. The conservation laws of the discrete difference scheme are shown. Meanwhile, the exactness, uniqueness, and prior estimate of the numerical solution are rigorously established. Then, it is proved that the proposed scheme is unconditionally convergent in the discrete L2 and Hγ/2 norm, where γ is a fractional order. Furthermore, we demonstrate that when the fractional order γ and the spatial grid number J increase, the block-Toeplitz coefficient matrix generated by the spatial discretization becomes ill-conditioned. As a result, we adopt an effective linearized iteration method for the nonlinear system, allowing it to be solved efficiently by the Krylov subspace solver with an appropriate circulant preconditioner, in which the fast Fourier transform is applied to speed up the computational cost at each iterative step. Finally, numerical experiments are presented to validate the theoretical findings and the efficiency of the fast algorithm.

Influence factors on illuminance distribution uniformity and energy saving of the indoor illumination control method.

This study investigated the influence factors of illuminance distribution uniformity and the energy-saving ability of the indoor illumination control system, which consists of a white light-emitting diodes (WLED) matrix and a tabletop matrix. In the proposed illumination control method, comprehensive effects of time-invariant and time-variant sunlight of the outdoor environment, arrangement of the WLED matrix, iterative functions selected for illuminance distribution optimization, and compositions of WLED optical spectra are considered. The non-symmetrical spatial distribution of WLED-tabletop matrices, the optical spectrum selection of WLEDs, and the shifting sunlight intensity introduce obvious influence on (a) emission intensity and distribution uniformity of the WLED matrix and (b) receiving illuminance intensity and distribution uniformity of the tabletop matrix. Additionally, the selection of iterative functions, the WLED matrix dimension, the target error coefficient in the iteration, and the optical spectra of the WLEDs cause non-negligible influence on the energy saving percentage and iteration steps of the proposed algorithm, influencing the effectiveness and accuracy of the proposed method. Our investigation provides guidelines for the improvement of optimization speed and accuracy of indoor illumination control systems, and the method is hopefully to be widely applied in manufacturing industry and intelligent office buildings.

Hybrid Alternated Inertial Projection and Contraction Algorithm for Solving Bilevel Variational Inequality Problems

This paper considers an alternated inertial-type extrapolation algorithm for solving bilevel pseudomonotone variational inequality problem in the framework of real Hilbert spaces with split variational inequality and fixed-point constraints of demimetric mapping. The algorithm which involves alternated inertial uses self-adjustment stepsize condition that depends solely on the information from previous iterative step. The bilevel problem considered in the work consists of upper-level problem with an underlying operator which is pseudomonotone, while the lower problem is associated with strongly monotone mapping and the fixed-point constraint of demimetric mapping. Our algorithm is anchored on modified projection and contraction techniques, fused with alternated inertial and relaxation. Under some suitable conditions on the algorithm control parameters, we obtain a strong convergence result of the proposed method without the prior knowledge of the operator norm or the coefficients of the underlying operators within the scope of real Hilbert spaces. Finally, some numerical examples are presented to illustrate the gain of our method in comparison to some related algorithms in the literature.

Ten strategies towards successful calibration of environmental models

Model calibration is the procedure of finding model settings such that simulated model outputs best match the observed data. Model calibration is necessary when the model parameters cannot directly be measured as is the case with a wide range of environmental models where parameters are conceptually describing upscaled and effective physical processes. Model calibration is therefore an important step of environmental modeling as the model might otherwise provide random outputs if never compared to a ground truth. Model calibration itself is often referred to be an art due to its plenitude of intertwined steps and necessary decisions along the way before a calibration can be carried out or can be regarded successful.This work provides a general guide specifying which steps a modeler needs to undertake, how to diagnose the success of each step, and how to identify the right action to revise steps that were not successful. The procedure is formalized into ten iterative steps generally appearing in calibration experiments. Each step of this “calibration life cycle” is either illustrated with an exemplary calibration experiment or providing an explicit checklist the modeler can follow. These ten strategies are: (1) using sensitivity information to guide the calibration, (2) handling of parameters with constraints, (3) handling of data ranging orders of magnitude, (4) choosing the data to base the calibration on, (5) presenting various methods to sample model parameters, (6) finding appropriate parameter ranges, (7) choosing objective functions, (8) selecting a calibration algorithm, (9) determining the success and quality of a multi-objective calibration, and (10) providing a checklist to diagnose calibration performance using ideas introduced in the previous steps. The formal definition of strategies through the calibration process is providing an overview while shedding a light on connections between these main ingredients to calibrate an environmental model and will therefore enable especially novice modelers to succeed.

An effective nonlinear interval sequential quadratic programming method for uncertain inverse problems

An effective nonlinear interval sequential quadratic programming method is proposed to provide an efficient tool for uncertain inverse problems. Assisted by the ideology of sequential quadratic programming and dimension-reduction analysis theory, the interval inverse problem is transformed into several interval arithmetic and deterministic optimizations, which could enhance computational efficiency without losing much accuracy. The novelty of the proposed method lies in two main aspects. First, an alternate updating strategy is proposed to identify the radii and midpoints of the interval inputs in each cycle, which could reduce the number of iterative steps. Second, the standard quadratic models are constructed based on the dimension-reduction analysis results, rather than the second-order Taylor expansion. Therefore, the interval arithmetic can be applied to efficiently calculate the interval response, which avoids the inner optimization. Moreover, a novel iterative mechanism is developed to accelerate the convergence rate of the proposed method. Finally, two numerical examples and an engineering application are adopted to verify its feasibility, accuracy and efficiency.

Combining machine learning and physics modelling to determine the natural cave property with fracturing curves

Currently, it is still challenging to determine the geological parameters of natural caves in deep reservoirs accurately. By inheriting the advantages of Machine Learning (ML) method and physics modelling, a novel ML-Physics method is developed to determine the geological parameters of natural caves based on the data mining of fracturing curves obtained during Hydraulic Fracturing (HF) operation. The computational time of ML-Physics method is divided into two stages, preparation-stage and operation-stage. The preparation-stage happens before HF operation, therefore there is no limitation to the computational time. During this preparation-stage, the implicit relationship between cave property and fracturing curve is generated using ML, which usually fails to ensure the accuracy under different geological and operational conditions. The operation-stage happens during HF operation, in which the computational time is limited because the geological parameters of natural caves are required to be determined in real time. During this operation-stage, the physical modelling based inverse analysis method is carried out, in which the initial value is chosen based on the ML results obtained in preparation-stage. Results show that, with the same target error, the iteration step of ML-Physics method (1 iteration) is much less than that of traditional inverse analysis method (5 iterations). After the same iterations, the error of fracturing curve using the traditional inverse analysis method is 0.40%, while the error using ML-Physics method is 0.02%. Meanwhile, the error of permeability using the traditional ML is up to 10.33%, while the error of ML-Physics method is 0.29%. The present ML-Physics method is potentially useful to optimize the HF design based on the data mining of fracturing curves in real time.

A sequential deep learning algorithm for sampled mixed-integer optimisation problems

Mixed-integer optimisation problems can be computationally challenging. Here, we introduce and analyse two efficient algorithms with a specific sequential design that are aimed at dealing with sampled problems within this class. At each iteration step of both algorithms, we first test the feasibility of a given test solution for each and every constraint associated with the sampled optimisation at hand, while also identifying those constraints that are violated. Subsequently, an optimisation problem is constructed with a constraint set consisting of the current basis—namely, the smallest set of constraints that fully specifies the current test solution—as well as constraints related to a limited number of the identified violating samples. We show that both algorithms exhibit finite-time convergence towards the optimal solution. Algorithm 2 features a neural network classifier that notably improves the computational performance compared to Algorithm 1. We quantitatively establish these algorithms' efficacy through three numerical tests: robust optimal power flow, robust unit commitment, and robust random mixed-integer linear program.

An Iterative Ray Tracing Algorithm to Increase Simulation Speed While Maintaining Overall Precision

Ray tracing is a frequently used method for acoustic simulations, valued for its calculation speed and ease of use. Although it is fast, there are no fully ray tracing-based real-time simulation methods or engines. Under real-time restrictions, ray tracing simulations lose precision and the variance inherent in the random simulation method has too much impact on the outcome. In this paper, an algorithm called iterative ray tracing is presented that reduces the negative effects of real-time restrictions by iteratively improving the initial calculation and increasing the precision over time. In addition, new estimates of the expected value and variance of ray tracing simulations are presented and used to show the iteration steps in the new algorithm reduce variance, while maintaining the expected value. Simulations using iterative ray tracing are compared to measurements and simulations using the classical ray tracing method, and it is shown that iterative ray tracing can be used to improve precision over time. Although more testing is needed, iterative ray tracing can be used to extend most ray tracing algorithms, in order to decrease the adverse effects of real-time restrictions.

Markov clustering regularized multi-hop graph neural network

Graph Neural Networks (GNNs) have shown great potential for graph data analysis. In this paper, we focus on multi-hop graph neural networks and aim to extend existing models to a high-order multi-hop form for graph-level representation learning. However, such a directly extending method suffers from two limitations, i.e., computational inefficiency and limited representation ability of the multi-hop neighbor. For the former limitation, we utilize an iteration approach to approximate the power of a complex adjacency matrix to achieve linear computational complexity. For the latter limitation, we introduce the Regularized Markov Clustering (R-MCL) to regularize the flow matrix, i.e., the adjacency matrix, in each iteration step. With these two strategies, we construct Markov Clustering Regularized Multi-hop Graph Neural Network (MCMGN) for graph-level representation learning tasks. Specifically, MCMGN consists of a multi-hop message passing phase and a readout phase, where the multi-hop message passing phase aims to learn multi-hop node embedding, and then the readout phase aggregates multi-hop node representations to generate graph embedding for graph-level representation learning tasks. Extensive experiments on eight graph benchmark datasets strongly demonstrate the effectiveness of Markov Clustering Regularized Multi-hop Graph Neural Network, leading to superior performance on graph classification.

Adaptive learning rate optimization algorithms with dynamic bound based on Barzilai-Borwein method

The training effect of the neural network model is directly influenced by optimization algorithms. The Barzilai-Borwein(BB) method is used in the stochastic gradient descent (SGD) and other deep learning optimization algorithms because of its outstanding performance in terms of convergence speed. In order to improve the stability of BB step size and the output quality of network model in deep learning, this paper presents two improved optimization algorithms based on the BB method: BBbound and AdaBBbound. BBbound reduces the floating BB step size by generating the upper bound of the step in the current iteration. It avoids the occasional occurrence of long steps; AdaBBbound is an adaptive gradient method based on BB method with dynamic bound. The algorithm has a fast convergence effect in the early stage, then calculates learning rate and smoothly transitions to SGD by setting the better conditions of the BB method. We analyzed the performance of the two algorithms based on the initial conditions and the learning rate change curve. Meanwhile, we tested our algorithms on popular network models such as ResNet and DenseNet. The results showed that the new optimization algorithms have achieved high stability and significantly improved the performance of the network model.

On Computational Efficiency of Nine Iterative Algorithms for Hammerstein Equations

Many fixed point iterative methods have been proposed. Their authors provide, theoretically, the rate of convergence which indicates how much the error changes after one iteration step. Of practical importance are the CPU time required by each method and their accuracy. This has never been investigated. In this work we investigate nine iterative processes and compare their CPU time and accuracy. The schemes are the Picard, Mann, Ishikawa, Noor, Argawal, Abbas-Nazir, Thakur, Ullah, and S* methods. By applying a recently proposed fourth order quadrature rule to the integral equation and replacing the analytical operator with the discretized one, the iterative schemes are derived. They are tested on five Hammerstein equations. The results show that (i) all the methods converge to the exact solution at the correct discretization order of convergence, (ii) the Picard scheme is the fastest and also has good accuracy, and (iii) the S* and Abbas-Nazir schemes are the least efficient. It is concluded that for a contractive problem the Picard scheme is sufficient but if the operator is non-expansive, then the Mann scheme is sufficient.

Stochastic Geometric Iterative Method for Loop Subdivision Surface Fitting

In this paper, we propose a stochastic geometric iterative method (S-GIM) to approximate the high-resolution 3D models by finite loop subdivision surfaces. Given an input mesh as the fitting target, the initial control mesh is generated using the mesh simplification algorithm. Then, our method adjusts the control mesh iteratively to make its finite loop subdivision surface approximate the input mesh. In each geometric iteration, we randomly select part of points on the subdivision surface to calculate the difference vectors and distribute the vectors to the control points. Finally, the control points are updated by adding the weighted average of these difference vectors. We prove the convergence of S-GIM and verify it by demonstrating error curves in the experiment. In addition, compared with existing geometric iterative methods, S-GIM has a shorter running time under the same number of iteration steps.