Optimal Smoothing Research Articles

Smoothing analysis of two-color distributive relaxation for solving 2D Stokes flow by multigrid method

Smoothing properties of two-color distributive relaxation for solving a two-dimensional (2D) Stokes flow by multigrid method are theoretically investigated by using the local Fourier analysis (LFA) method. The governing equation of the 2D Stokes flow in consideration is discretized with the non-staggered grid and an added pressure stabilization term with stabilized parameters to be determined is introduced into the discretization system in order to enhance the smoothing effectiveness in the analysis. So, an important problem caused by the added pressure stabilization term is how to determine a suitable zone of parameters in the added term. To that end, theoretically, a two-color distributive relaxation, developed on the two-color Jacobi point relaxation, is established for the 2D Stokes flow. Firstly, a mathematical constitution based on the Fourier modes with various frequency components is constructed as a base of the two-color smoothing analysis, in which the related Fourier representation is presented by the form of two-color Jacobi point relaxation. Then, an optimal one-stage relaxation parameter and related smoothing factor for the two-color distributive relaxation are applied to the discretization system, and an analytical expression of the parameter zone on the added pressure stabilization term is established by LFA. The obtained analytical results show that numerical schemes for solving 2D Stokes flow by multigrid method on the two-color distributive relaxation have a specific convergence zone on the parameters of the added pressure stabilization term, and the property of convergence is independent of mesh size, but depends on the parameters of the pressure stabilization term.

On first and second order multiobjective programming with interval-valued objective functions

The growing use of optimization models to help decision making has created a demand for such tools that allow formulating and solving more models of real-world processes and systems related to human activity in which hypotheses are not verify in a way specific for classical optimization. One of the approaches for real-world extremum problems under uncertainty is interval-valued optimization. In this paper, a twice differentiable vector optimization problem with multiple interval-valued objective function and both inequality and equality constraints is considered. In this paper, the first order necessary optimality conditions of Karush-Kuhn-Tucker type are proved for differentiable interval-valued vector optimization problems under the first order constraint qualification. If the interval-valued objective function is assumed to be twice weakly differentiable and constraints functions are assumed to be twice differentiable, then two types of second order necessary optimality conditions under two various constraint qualifications are proved for such smooth interval-valued vector optimization problems. Finally, in order to illustrate the Karush-Kuhn-Tucker type necessary optimality conditions established in the paper, an example of an interval-valued optimization is given.

Robust Multi-Criteria Traffic Network Equilibrium Problems with Path Capacity Constraints

With the progress of society and the diversification of transportation modes, people are faced with more and more complicated travel choices, and thus, multi-criteria route choosing optimization problems have drawn increased attention in recent years. A number of multi-criteria traffic network equilibrium problems have been proposed, but most of them do not involve data uncertainty nor computational methods. This paper focuses on the methods for solving robust multi-criteria traffic network equilibrium problems with path capacity constraints. The concepts of the robust vector equilibrium and the robust vector equilibrium with respect to the worst case are introduced, respectively. For the robust vector equilibrium, an equivalent min–max optimization problem is constructed. A direct search algorithm, in which the step size without derivatives and redundant parameters, is proposed for solving this min–max problem. In addition, we construct a smoothing optimization problem based on a variant version of ReLU activation function to compute the robust weak vector equilibrium flows with respect to the worst case and then find robust vector equilibrium flows with respect to the worst case by using the heaviside step function. Finally, extensive numerical examples are given to illustrate the excellence of our algorithms compared with existing algorithms. It is shown that the proposed min–max algorithm may take less time to find the robust vector equilibrium flows and the smoothing method can more effectively generate a subset of the robust vector equilibrium with respect to the worst case.

Reliable strategies for implementing model-based navigation on fixed-wing drones

AbstractA relatively novel approach of autonomous navigation employing platform dynamics as the primary process model raises new implementational challenges. These are related to: (i) potential numerical instabilities during longer flights; (ii) the quality of model self-calibration and its applicability to different flights; (iii) the establishment of a global estimation methodology when handling different initialisation flight phases; and (iv) the possibility of reducing computational load through model simplification. We propose a unified strategy for handling different flight phases with a combination of factorisation and a partial Schmidt–Kalman approach. We then investigate the stability of the in-air initialisation and the suitability of reusing pre-calibrated model parameters with their correlations. Without GNSS updates, we suggest setting a subset of the state vector as ‘considered’ states within the filter to remove their estimation from the remaining observations. We support all propositions with new empirical evidence: first in model-parameter self-calibration via optimal smoothing and second through applying our methods on three test flights with dissimilar durations and geometries. Our experiments demonstrate a significant improvement in autonomous navigation quality for twelve different scenarios.

Convergence of Random Reshuffling under the Kurdyka–Łojasiewicz Inequality

We study the random reshuffling method for smooth nonconvex optimization problems with a finite-sum structure. Though this method is widely utilized in practice, e.g., in the training of neural networks, its convergence behavior is only understood in several limited settings. In this paper, under the well-known Kurdyka–Łojasiewicz (KL) inequality, we establish strong limit-point convergence results for with appropriate diminishing step sizes; namely, the whole sequence of iterates generated by is convergent and converges to a single stationary point in an almost sure sense. In addition, we derive the corresponding rate of convergence, depending on the KL exponent and suitably selected diminishing step sizes. When the KL exponent lies in , the convergence is at a rate of with counting the number of iterations. When the KL exponent belongs to , our derived convergence rate is of the form with depending on the KL exponent. The standard KL inequality-based convergence analysis framework only applies to algorithms with a certain descent property. We conduct a novel convergence analysis for the nondescent method with diminishing step sizes based on the KL inequality, which generalizes the standard KL framework. We summarize our main steps and core ideas in an informal analysis framework, which is of independent interest. As a direct application of this framework, we establish similar strong limit-point convergence results for the reshuffled proximal point method.

Artificial-neural-network-based optimal Smoother design for oscillation suppression control of underactuated overhead cranes with distributed mass beams

As a kind of convenient and practical oscillation suppression method, the open-loop controllers are widely utilized in the control of actual cranes, yet most of them need to be designed for each mode frequency when targeting multimodal systems, which will degrade the robustness of the controller. Moreover, most open-loop controllers require linearization of the model in the design process, which will inevitably reduce the performance of the controller in terms of oscillation suppression. To solve the above prevalent problems, an optimal command-smoothing method is proposed in this paper. Specifically, the dynamic analysis of a 2-D overhead crane with distributed-mass beams (DMB) is first performed. Then, the particle swarm optimization algorithm (PSO) is used to solve the optimal values of the parameters of the Smoother under different system parameters, and a data set for ANN training is generated, together with a series of indices to verify the effectiveness of the designed ANN. Finally, in combination with the PD controller, the proposed controller is experimentally proven to accomplish the positioning requirements while improving the oscillation suppression efficiency by an average of 29% compared to the basic Smoother.

Accelerated gradient methods with absolute and relative noise in the gradient

In this paper, we investigate accelerated first-order methods for smooth convex optimization problems under inexact information on the gradient of the objective. The noise in the gradient is considered to be additive with two possibilities: absolute noise bounded by a constant, and relative noise proportional to the norm of the gradient. We investigate the accumulation of the errors in the convex and strongly convex settings with the main difference with most of the previous works being that the feasible set can be unbounded. The key to the latter is to prove a bound on the trajectory of the algorithm. We also give a stopping criterion for the algorithm and consider extensions to the cases of stochastic optimization and composite nonsmooth problems.

A Vanka‐based parameter‐robust multigrid relaxation for the Stokes–Darcy Brinkman problems

AbstractWe consider a block‐structured multigrid method based on Braess–Sarazin relaxation for solving the Stokes–Darcy Brinkman equations discretized by the marker and cell scheme. In the relaxation scheme, an element‐based additive Vanka operator is used to approximate the inverse of the corresponding shifted Laplacian operator involved in the discrete Stokes–Darcy Brinkman system. Using local Fourier analysis, we present the stencil for the additive Vanka smoother and derive an optimal smoothing factor for Vanka‐based Braess–Sarazin relaxation for the Stokes–Darcy Brinkman equations. Although the optimal damping parameter is dependent on meshsize and physical parameter, it is very close to one. In practice, we find that using three sweeps of Jacobi relaxation on the Schur complement system is sufficient. Numerical results of two‐grid and V(1,1)‐cycle are presented, which show high efficiency of the proposed relaxation scheme and its robustness to physical parameters and the meshsize. Using a damping parameter equal to one gives almost the same convergence results as these for the optimal damping parameter.

Forecasting the Indonesian Coffee Production and Consumption Using the Modified Golden Section Search to Estimate the Smoothing Parameters

The Double Exponential Smoothing (DES) and Triple Exponential Smoothing (TES) are forecasting methods that require two and three smoothing parameters, respectively. Smoothing parameters are often determined through a trial and error process that is not really efficient since many experiments need to be done. Therefore, in this study, a smoothing parameter estimation algorithm is conducted in the form of the modified Golden Section Search (GSS) to obtain the optimal smoothing parameters from the DES and TES methods. Forecasting is carried out on production, domestic consumption, and export consumption data of Indonesian coffee, which is one of the leading agricultural sub-sector commodities. The data is obtained from the Ministry of Agriculture of the Republic of Indonesia. The smoothing parameters obtained by applying the modified GSS are used to forecast production and domestic consumption data using the DES method, while the forecasting of the export consumption data is done with the TES method. All of the MAPE values are less than 20% which indicates that the smoothing parameters obtained by using the modified GSS are able to perform good forecasting. The results show that coffee production in Indonesia cannot meet its demand until 2024 since the total coffee consumption exceeds the production.

Forecasting the Indonesian Coffee Production and Consumption Using the Modified Golden Section Search to Estimate the Smoothing Parameters

The Double Exponential Smoothing (DES) and Triple Exponential Smoothing (TES) are forecasting methods that require two and three smoothing parameters, respectively. Smoothing parameters are often determined through a trial and error process that is not really efficient since many experiments need to be done. Therefore, in this study, a smoothing parameter estimation algorithm is conducted in the form of the modified Golden Section Search (GSS) to obtain the optimal smoothing parameters from the DES and TES methods. Forecasting is carried out on production, domestic consumption, and export consumption data of Indonesian coffee, which is one of the leading agricultural sub-sector commodities. The data is obtained from the Ministry of Agriculture of the Republic of Indonesia. The smoothing parameters obtained by applying the modified GSS are used to forecast production and domestic consumption data using the DES method, while the forecasting of the export consumption data is done with the TES method. All of the MAPE values are less than 20% which indicates that the smoothing parameters obtained by using the modified GSS are able to perform good forecasting. The results show that coffee production in Indonesia cannot meet its demand until 2024 since the total coffee consumption exceeds the production.

Research on Market Share Prediction of Highway Passenger Transport Based on Exponential Smoothing Method

Highway transportation is an important part of our country traffic transportation system, taking on most of the capacity demand in passenger transport markets. But with the rapid development of domestic rail transportation technology, people's travel mode has gradually changed from highway transportation to rail transportation. This makes the highway passenger transport management department must consider the impact of the change of people's travel mode on the highway passenger transport market demand when making production plans. The market share of highway passenger transport is an important index to measure the development level of highway passenger transport, which can directly reflect the supply-demand relationship between highway passenger transport and passenger transport market. Therefore, this paper selects the market share of highway passenger transport as the prediction evaluation index, and predicts and analyzes the market share data of highway passenger transport in Henan Province from 2010 to 2021 based on the exponential smoothing method. It mainly calculates the single exponential smoothing results under several different smoothing coefficients by using the smoothing analysis tool in EXCEL, and determines the optimal smoothing coefficient by taking the minimum root mean square error (RMSE) as the criterion. Then, it solves the lag defect of the single exponential smoothing prediction results by using the quadratic exponential smoothing. Finally, the trend prediction model of highway passenger transport market share in Henan Province is obtained. The prediction results of this model can guide the rational allocation of highway transportation resources and the formulation of passenger transport production plan in Henan Province.

The Forecast of Jute Export in Bangladesh for Optimal Smoothing Constants

The Forecast of Jute Export in Bangladesh for Optimal Smoothing Constants

Fault diagnosis model of rolling bearing based on parameter adaptive VMD algorithm and Sparrow Search Algorithm-Based PNN

Fault diagnosis of rolling bearings is essential to ensure the proper functioning of the entire machinery and equipment. Variational mode decomposition (VMD) and neural networks have gained widespread attention in the field of bearing fault diagnosis due to their powerful feature extraction and feature learning capacity. However, past methods usually utilize experiential knowledge to determine the key parameters in the VMD and neural networks, such as the penalty factor, the smooth factor, and so on, so that generates a poor diagnostic result. To address this problem, an Adaptive Variational Mode Decomposition (AVMD) is proposed to obtain better features to construct the fault feature matrix and Sparrow probabilistic neural network (SPNN) is constructed for rolling bearing fault diagnosis. Firstly, the unknown parameters of VMD are estimated by using the genetic algorithm (GA), then the suitable features such as kurtosis and singular value entropy are extracted by automatically adjusting the parameters of VMD. Furthermore, a probabilistic neural network (PNN) is used for bearing fault diagnosis. Meanwhile, embedding the sparrow search algorithm (SSA) into PNN to obtain the optimal smoothing factor. Finally, the proposed method is tested and evaluated on a public bearing dataset and bearing tests. The results demonstrate that the proposed method can extract suitable features and achieve high diagnostic accuracy.

A closed‐form multigrid smoothing factor for an additive Vanka‐type smoother applied to the Poisson equation

AbstractWe consider an additive Vanka‐type smoother for the Poisson equation discretized by the standard finite difference centered scheme. Using local Fourier analysis, we derive analytical formulas for the optimal smoothing factors for vertex‐wise and element‐wise Vanka smoothers. In one dimension the element‐wise Vanka smoother is equivalent to the scaled mass operator obtained from the linear finite element method and in two dimensions the element‐wise Vanka smoother is equivalent to the scaled mass operator discretized by bilinear finite element method plus a scaled identity operator. Based on these findings, the mass matrix obtained from finite element method can be used as a smoother for the Poisson equation, and the resulting mass‐based relaxation scheme yields small smoothing factors in one, two, and three dimensions, while avoiding the need to compute an inverse of a matrix. Our analysis may help better understand the smoothing properties of additive Vanka approaches and develop fast solvers for numerical solutions of other partial differential equations.

Determination of the power smoothing effect in a photovoltaic-hydrokinetic system by experimental analysis and pattern search

Sizing optimization and power smoothing methodology in highly fluctuating grid-connected renewable systems represent important challenges nowadays, the total cost and energy quality depend on the effective balance between source and load with certain restrictions to increase reliability and enhance the use of renewable sources at all scales. This article presents a novel power smoothing effect and sizing optimization methodology for a grid-connected photovoltaic-hydrokinetic system by pattern search method comparatively with three optimization algorithms encompassing a multi-objective optimization. In this context, the method considers the reduction of power fluctuations using supercapacitors, where the objective function considers technical and economic indexes. The main contribution of the research is that the objective function decreases the fluctuation suppression ratio to sizing optimization of renewable system through three pattern search size optimization algorithms. Experiments have shown that pattern search method can find the global optimum by reducing the computational effort; the Latin hypercube algorithm reduces the photovoltaic capacity by approximately 5 % and 2.3 % with respect to the Nelder Mead and genetic algorithm respectively. The influence of the supercapacitor successfully reduces power fluctuations and causes an increase in self-consumption of 32.32 %, this result represents an annual cost savings of 42 % on energy purchased from the grid. Furthermore, this paper goes further by calculating the optimal location of a hydrokinetic turbine in a river using the HEC-RAS software to determine with high precision the HKT power fluctuations as input signal to novel proposed method.

Are Regionally Calibrated Seismicity Models More Informative than Global Models? Insights from California, New Zealand, and Italy

Abstract Earthquake forecasting models express hypotheses about seismogenesis that underpin global and regional probabilistic seismic hazard assessments (PSHAs). An implicit assumption is that the comparatively higher spatiotemporal resolution datasets from which regional models are generated lead to more informative seismicity forecasts than global models, which are however calibrated on greater datasets of large earthquakes. Here, we prospectively assess the ability of the Global Earthquake Activity Rate (GEAR1) model and 19 time-independent regional models to forecast M 4.95+ seismicity in California, New Zealand, and Italy from 2014 through 2021, using metrics developed by the Collaboratory for the Study of Earthquake Predictability (CSEP). Our results show that regional models that adaptively smooth small earthquake locations perform best in California and Italy during the evaluation period; however, GEAR1, based on global seismicity and geodesy datasets, performs surprisingly well across all testing regions, ranking first in New Zealand, second in California, and third in Italy. Furthermore, the performance of the models is highly sensitive to spatial smoothing, and the optimal smoothing likely depends on the regional tectonic setting. Acknowledging the limited prospective test data, these results provide preliminary support for using GEAR1 as a global reference M 4.95+ seismicity model that could inform eight-year regional and global PSHAs.

Worst-case evaluation complexity of a quadratic penalty method for nonconvex optimization

This paper addresses the worst-case evaluation complexity of a version of the standard quadratic penalty method for smooth nonconvex optimization problems with constraints. The method analysed allows inexact solution of the subproblems and do not require prior knowledge of the Lipschitz constants related with the problem. When an approximate feasible point is used as starting point, it is shown that the referred method takes at most outer iterations to generate an ϵ-approximate KKT point, where is the first penalty parameter. For equality constrained problems, this bound yields to an evaluation complexity bound of , when and suitable first-order methods are used as inner solvers. For problems having only linear equality constraints, an evaluation complexity bound of is established when appropriate p-order methods () are used as inner solvers. Illustrative numerical results are also presented and corroborate the theoretical predictions.

Automatic performance tuning for Albany Land Ice

Accurate simulation of the evolution of polar ice-sheets requires a massive amount of computational power. In order to take advantage of the newest generation of supercomputing clusters, the Albany Land Ice code has been modernized for performance portability across a variety of parallel architectures, with a focus on enabling end-to-end GPU capability. Albany uses a multigrid preconditioning approach for solving linear systems via performance portable smoothers from the Trilinos package Ifpack2. Since the Albany Land Ice code is constantly evolving and both Albany and Trilinos are in constant development, it is likely that the optimal choice of solver parameters will change over time. It is therefore critical to have an automatic performance tuning framework to ensure that the best possible performance is maintained. Toward this effect, we have developed an automatic performance tuning framework to determine the best fine- and coarse-grid smoothing algorithms and parameters. We treat the underlying performance model of the linear solve as a black box and use the python-based GPTune Bayesian optimization library to determine the optimal smoother choice and parameters. Using this approach, we have found smoothers and their corresponding parameters that result in, on average, 1.2 times faster, and up to 1.5 times faster solve-times than our manually-tuned parameters. We also show that the proposed auto-tuning approach produces reliably better parameters than naive black box optimization techniques like random search for a given function evaluation budget. By implementing our tuning framework in the Python-based workflow management tool parsl, we also ensure that we efficiently use available computing resources during the tuning process and avoid unnecessary long wait times in computing cluster job queues.

Transient Growth of Accelerated Optimization Algorithms

Optimization algorithms are increasingly being used in applications with limited time budgets. In many real-time and embedded scenarios, only a few iterations can be performed and traditional convergence metrics cannot be used to evaluate performance in these nonasymptotic regimes. In this article, we examine the transient behavior of accelerated first-order optimization algorithms. For convex quadratic problems, we employ tools from linear systems theory to show that transient growth arises from the presence of nonnormal dynamics. We identify the existence of modes that yield an algebraic growth in early iterations and quantify the transient excursion from the optimal solution caused by these modes. For strongly convex smooth optimization problems, we utilize the theory of integral quadratic constraints to establish an upper bound on the magnitude of the transient response of Nesterov’s accelerated algorithm. We show that both the Euclidean distance between the optimization variable and the global minimizer and the rise time to the transient peak are proportional to the square root of the condition number of the problem. Finally, for problems with large condition numbers, we demonstrate tightness of the bounds that we derive up to constant factors.

General Hölder Smooth Convergence Rates Follow from Specialized Rates Assuming Growth Bounds

Often in the analysis of first-order methods for both smooth and nonsmooth optimization, assuming the existence of a growth/error bound or KL condition facilitates much stronger convergence analysis. Hence, separate analysis is typically needed for the general case and for the growth bounded cases. We give meta-theorems for deriving general convergence rates from those assuming a growth lower bound. Applying this simple but conceptually powerful tool to the proximal point, subgradient, bundle, dual averaging, gradient descent, Frank–Wolfe and universal accelerated methods immediately recovers their known convergence rates for general convex optimization problems from their specialized rates. New convergence results follow for bundle methods, dual averaging and Frank–Wolfe. Our results can lift any rate based on Hölder continuous gradients and Hölder growth bounds. Moreover, our theory provides simple proofs of optimal convergence lower bounds under Hölder growth from textbook examples without growth bounds.