Sum Of Squares Method Research Articles

Theory and applications of the sum-of-squares technique

Abstract The sum-of-squares (SOS) approximation method is a technique used in optimization problems to derive lower bounds on the optimal value of an objective function. By representing the objective function as a sum of squares in a feature space, the SOS method transforms non-convex global optimization problems into solvable semidefinite programs. This note presents an overview of the SOS method. We start with its application in finite-dimensional feature spaces and, subsequently, we extend it to infinite-dimensional feature spaces using reproducing kernels (k-SOS). Additionally, we highlight the utilization of SOS for estimating some relevant quantities in information theory, including the log-partition function.

Ex-vivo parametric study of laser ablation-based drilling of cortical bone.

Frequently orthopedic surgeries require mechanical drilling processes especially for inserted biodegradable screws or removing small bone lesions. However mechanical drilling techniques induce large number of forces as well as have substantially lower material removal rates resulting in prolong healing times. This study focuses on analyzing the impact of quasi-continuous laser drilling on the bone's surface as well as optimizing the drilling conditions to achieve high material removal rates. An ex-vivo study was conducted on the cortical region of desiccated bovine bone. The laser-based drilling on the bovine bine specimens was conducted in an argon atmosphere using a number of laser pulses ranging from 100 to 15,000. The morphology of the resulting laser drilled cavities was characterized using Energy dispersive Spectroscopy (EDS) and the width and depth of the drills were measured using a laser based Profilometer. Data from the profilometer was then used to calculate material removal rates. At last, the material removal rates and laser processing parameters were used to develop a statistical model based on Design of Experiments (DOE) approach to predict the optimal laser drilling parameters. The main outcome of the study based on the laser drilled cavities was that as the number of laser pulses increases, the depth and diameter of the cavities progressively increase. However, the material removal rates revealed a decrease in value at a point between 4000 and 6000 laser pulses. Therefore, based on the sequential sum of square method, a polynomial curve to the 6th power was fit to the experimental data. The predicted equation of the curve had a p-value of 0.0010 indicating statistical significance and predicted the maximum material removal rate to be 32.10 mm3/s with 95%CI [28.3,35.9] which was associated with the optimum number of laser pulses of 4820. Whereas the experimental verification of bone drilling with 4820 laser pulses yielded a material removal rate of 33.37 mm3/s. Therefore, this study found that the carbonized layer formed due to laser processing had a decreased carbon content and helped in increasing the material removal rate. Then using the experimental data, a polymetric equation to the sixth power was developed which predicted the optimized material removal rate to occur at 4820 pulses.

An analytical approach for shock response of fractional viscoelastic beams with tip mass

This paper specifically analyzes the behavior of a beam with fractional viscoelastic properties and a tip mass under shock. The shock is applied longitudinally to the beam, and a fractional Kelvin–Voigt model is employed to describe the viscoelastic properties. Additionally, the paper examines the fractional viscoelastic power as a function of time during the shock. The equation of motion for the fractional viscoelastic beam with a tip mass is derived using Hamilton’s principle. In order to ensure accurate analysis of the results, the graphs are carefully analyzed in both the time and frequency domains. To solve the equations, the special technique provided for differential equations of variable order (VO) is used. Two distinct methods, namely, absolute method and square root of the sum of squares method, are thoroughly examined to calculate the absolute acceleration. Subsequently, the most suitable method is selected based on the evaluation results. The study’s results demonstrate that the damping parameters of the viscoelastic beam have a significant impact on shock transmissibility. Additionally, the findings demonstrate that the ability to transfer shocks can be manipulated by modifying the system’s geometry and mass.

Optimized wake-superposition approach for multiturbine wind farms

Optimizing the wind farm layout requires accurately quantifying the wind-turbine wake distribution to minimize interference between wakes. Thus, the accuracy of wind turbine wake superposition models is critical. The sum of squares (SS) model is currently touted as the most accurate, but its application in engineering is hampered by its overestimation of the velocity deficit of the mixed wake. Therefore, previous work relied on approximate power calculations for performing optimization. The physical meaning of the SS model is unclear, which makes optimization difficult. In this study, a univariate linear correction idea is proposed based on the linear increase phenomenon of the SS method error. The unknown coefficients are obtained by fitting experimental data. The results demonstrate that the proposed method can accurately quantify the full-wake two-dimensional distribution of the mixed wake.

Determination of The Best Box-Jenkins’ Seasonal Model to Inflows of Reservoir of Bekhme Dam

Prediction of water inflow to a reservoir is of great interest in the policy of the reservoir operation throughout the year. When significant amounts of inflow series entering to the reservoir are nondeterministic events, the utilization of stochastic models to check the reliability of the recorded data and forecast the future events become preferable. One of the most powerful and widely used methodology for forecasting time series is the class of models called the Box-Jenkins models. In this study, time series analysis was applied to records of 69 monthly mean inflows to Bekhme reservoir, in the northern part of Iraq, for the water year period from 1933 to 2006. Nine multiplicative seasonal models were fitted to this series; these were the seasonal autoregressive integrated (SARI) (1, 1, 0) × (1, 1, 0)12, (2, 1, 0) × (1, 1, 0)12, and (1, 1, 0) × (2, 1, 0)12 models, the seasonal autoregressive integrated moving average (SARIMA) (1, 1, 1) × (1, 1, 1)12, (2, 1, 2) × (1, 1, 1)12, and (1, 1, 1) × (2, 1, 2)12 models, and the seasonal integrated moving average (SIMA) (0, 1, 1) × (0, 1, 1)12, (0, 1, 2) × (0, 1, 1)12, and (0, 1, 1) × (0, 1, 2)12 models. The unconditional sum of squares method was used to estimate the parameters of the models and to compute the sum of squared errors for each one. It was found that the best model which corresponded to the minimum sum of squared errors was the SIMA (0, 1, 1) × (0, 1, 1)12 model. The estimated moving average parameters of this model were 0.378 and 0.953 for both θ and Θ respectively. The adequacy of this model was checked by plotting the normalized cumulative periodogram which does not indicate nonrandomness of the residuals. Forecasts of monthly inflow for the period from October, 2002, to September, 2006 were graphically compared with observed inflow for the same period and since agreement was very precise, adequacy of the selected model was confirmed.

Deep parallel MRI reconstruction based on a complex-valued loss function

To propose a new method for fast MRI reconstruction based on deep learning in parallel MRI data using a new loss function defined as the summation of the mean squared errors of the magnitude and phase. The multicoil image data were combined into single-coil image data to eliminate the correlation between noises and used as a label in the training process. Considering the importance of the phase information in some applications, where the phase information was lost when combining multicoil data using sum of square method, a new loss function was introduced, defined as the weighted sum of the mean squared error (MSE) of the magnitude and phase. The single weight in the loss function was used to balance the importance of the magnitude and phase in different applications. To validate the proposed method, real brain and knee data in FastMRI dataset were used for training and testing. We also compared this proposed method with two other methods that used MSE or mean absolute error (MAE) as a loss function. The experimental results showed that the proposed method was capable of accurate reconstruction of multicoil MR images with significantly reduced artifacts compared with the other two methods. Quantitative analysis showed that the propose method increased the peak signal-to-noise ratio (PSNR) of the reconstructed images by about 1 dB. The proposed deep MRI reconstruction method using a new loss function to fit the noise in parallel MRI data can accelerate MRI reconstruction and significantly improve the quality of the reconstructed images.

Engineering uncertainty analysis of the heat transfer of fuel plates

Uranium-containing aluminum-cladding dispersion plate fuels are widely used in various research reactors owing to their excellent heat-transfer capability. In the nuclear design stage, both the maximum temperature of the cladding surface and the minimum departure from nucleate boiling ratio (MDNBR) must satisfy safety acceptance criteria to ensure the performance of the fuel plates. Many engineering-related uncertainties lead to deviations between the actual and design values. For safety considerations, these engineering-related uncertainties must be considered when analyzing the maximum temperature of the cladding surface and MDNBR. In this study, the various uncertainties of the engineering hot channel and hot spot factors, three methods of uncertainty analysis, and sensitivity analysis of these uncertainty factors were investigated. The results show that the square root of the sum of squares method is both conservative and economical. The sensitivity analysis results show that the flow channel gap deviation, produced by fuel plate manufacturing, has the greatest influence on the maximum temperature of the fuel cladding surface.

Ecoregional Variation of Crimean Pine (Pinus nigrasubspeciespallasiana[Lamb.] Holmboe) Stand Growth

AbstractInvestigation and numerical determination of the effects of ecologically different regions on forest growth are important issues to be considered for sustainable forest management. In this study, differences in growth patterns of Crimean pine (Pinus nigra subsp. pallasiana [Lamb.] Holmboe) stands among the ecoregions located in the Kastamonu region of northwestern Turkey were investigated. For this purpose, stand-level growth models for dominant height, number of trees, basal area, and volume were fitted using the data from 274 sample plots and 293 stem analysis in even-aged, pure, and productive Crimean pine stands. The nonlinear extra sum of squares method and graphical examinations revealed that growth characteristics of these Crimean pine stands were affected by ecoregional differences. Ecoregion-based growth models, which are more flexible and have the ability to reflect the ecoregional differences in growth patterns, were found to be superior to regional models. Applying general models without any ecoregional classification resulted in biases of 5.3%, 23.1%, 5.5%, and 6.2% in the estimates of dominant height, number of trees, basal area, and volume of Crimean pine stands, respectively. The results showed that ecological differences must be considered in studies on forest growth modeling and forest management.

Adaptive Optimal Control for Unknown Constrained Nonlinear Systems With a Novel Quasi-Model Network.

A policy-iteration-based algorithm is presented in this article for optimal control of unknown continuous-time nonlinear systems subject to bounded inputs by utilizing the adaptive dynamic programming (ADP). Three neural networks (NNs), called critic network, actor network, and quasi-model network, are utilized in the proposed algorithm to give approximations of the control law, the cost function, and the function constituted by partial derivatives of value functions with respect to states and unknown input gain dynamics, respectively. At each iteration, based on the least sum of squares method, the parameters of critic and quasi-model networks will be tuned simultaneously, which eliminates the necessity of separately learning the system model in advance. Then, the control law is improved by satisfying the necessary optimality condition. Then, the proposed algorithm's optimality and convergence properties are exhibited. Finally, the simulation results demonstrate the availability of the proposed algorithm.

CFD and Machine Learning based Simulation of Flow and Heat Transfer Characteristics of Micro Lattice Structures

In this paper, systematic CFD analysis using ANSYS Fluent was carried out to generate the dataset for developing the Machine Learning model, which predicts the average final temperature of water and the pressure drop from the set of input parameters considered for applications. There are six micro lattice structures, kagome, tetrahedral, pyramidal, hexagonal, windward bent and hexagonal-windward bent, modelled for this study using FUSION 360 by Autodesk. The study of heat transfer between liquid water and the micro lattice structures realized with the independent variables, initial fluid flow velocity, lattice temperature, and fluid temperature as well as lattice materials and its different structures. About 2146 output data of average final fluid temperature and the pressure drop were collected from the CFD simulations by varying input parameters. To predict the output parameter against the set of input parameters, Machine Learning model with regression based classification algorithm was adopted while training the ML model. The quality metric of the ML model was calculated using residual sum of squares method. The final average temperature of the fluid and pressure drop as predicted by the ML model is closer to simulated data.

Ecoregion-based height-diameter models for Crimean pine

ABSTRACT Generalized height (h)–diameter (d) models were developed for Crimean pine stands in the Kastamonu Regional Directorate of Forestry, northwestern Turkey. Nine generalized h-d models were fitted and evaluated based on data set consisting of 2697 h-d measurements obtained from 274 sample plots located in three ecoregions of the studied region. The best results were obtained by modified Richards h-d model accounting for the 93% of the total variance in height–diameter relationship. It is a known fact that h-d relationship can vary in ecological conditions, especially for regions with large areas and various ecological conditions such as Kastamonu. To investigate the differences of height–diameter relationship among the ecoregions located in Kastamonu region, nonlinear extra sum of squares method was performed. The test results showed that there were significant differences, especially between coastal and the inland ecoregions. When compared to reduced model with the same set of parameters for the whole region, full model with a different set of parameters for each ecoregion achieved better accuracy in estimating tree height. Besides, applying h-d model fitted for the coastal ecoregion to the others, or vice versa, could lead under- or over-estimates of tree heights by about 1.8%.

Uncertainty Propagation Through An Empirical Model of Cutting Forces in End Milling

Abstract Empirical mathematical models of cutting forces in machining processes use experimentally determined input parameters to make predictions. A general method for propagation of input parameter uncertainties through such predictive models is developed. Sources of uncertainty are identified and classified. First, a classical uncertainty procedure is employed to estimate uncertainties associated with the data reduction equation using a first-order Taylor series expansion. Small values of input parameter uncertainties justify this local linearization. Coverage factors required to estimate confidence intervals are computed based on appropriate underlying statistical distributions. A root sum of squares method yields the overall expanded uncertainty in force predictions. A popular model used for predicting cutting forces in end milling is selected to demonstrate the procedure, but the demonstrated approach is general. The analysis is applied to experimental data. Force predictions are quoted along with a confidence interval attached to them. An alternative analysis based on Monte Carlo simulations is also presented. This procedure yields different insights compared with the classical uncertainty analysis and complements it. Monte Carlo simulation provides combined uncertainties directly without sensitivity calculations. Classical uncertainty analysis reveals the impacts of random effects and systematic effects separately. This information can prompt the user to improve the experimental setup if the impact of systematic effects is observed to be comparatively large. The method of quoting an estimate of the uncertainty in force predictions presented in this paper will permit users to assess the suitability of given empirical force prediction models in specific applications.

Development of Eco-Friendly Basalt Fibre Reinforced Cementitious Composites Using Taguchi Method

In order to improve the sustainability of cementitious composites used for repair and construction applications, environmentally friendly materials can be incorporated as components in the composites. As materials with a higher ductility are preferred for the repair of structures such as pavement, short fibres can be incorporated into cementitious composites to improve the flexural and tensile strength. In order to develop an eco-friendly composite, this study utilized fly ash to replace a high proportion of Portland cement alongside basalt fibre which is an eco-friendly binder to reinforce to developed composites. Taguchi method was used to design the mixtures in order to investigate the effect of various parameters on the performance of the developed composites. A total of nine mixtures was developed according to the L9 array. Analysis of variance of the experimental results was carried out to evaluate the effect of each of the investigated parameters, and the sum of squares method used to calculate the percentage contribution of each parameter. Results obtained from this study showed that the use of fly ash to Portland cement ratio of 2 to 4 alongside a basalt fibre dosage of 2% resulted in acceptable mechanical properties for repair applications.

Biogeography of Medically Important Insects using Quantitative Analysis

We summarized distributional information of medically important insects from 76 families and 4531 genera occurring worldwide. The continents were divided into 67 basic geographical units. Using a new similarity formula and a new clustering method for quantitative analysis, 67 basic geographical units were clustered into 7 large unit groups and 20 small unit groups. The results were superior to the traditional single linkage method, average group linkage method, or sum of squares method. The cluster results were similar with the result of mainly phytophagous insects 104,344 genera in the world, but were different from the Wallace’s mammal geographical division scheme. Based on these seemingly contradictory results, we infer that animals, insect and plants may have the same distribution pattern and that it is necessary to conduct precise quantitative analysis for animals and plants worldwide.

Binder-free torrefied biomass pellets: significance of torrefaction temperature and pelletization parameters by multivariate analysis

Torrefaction is a promising technology to improve fuel properties of biomass. However, torrefied char does not show densified energy content compared with coal and typically requires densification such as pelletization. Several key parameters including pelletization and torrefaction conditions were investigated in order to determine the effects on energy density of torrefied char pellets. In this study, multivariate analyses were performed using three independent variables, torrefaction temperature, pelletization temperature, and pelletization pressure; and five dependent variables: mass density, durability, volume expansion, higher heating value (HHV), and energy density (ED). The variation and correlation of the dependent variables were evaluated using principal component analysis (PCA). Additionally, multivariate linear regression was performed to correlate independent variables and dependent variable responses using an adjusted sum of squares method and a two-sided, 90% confidence interval for the mean response; the model was fitted with statistically significant linear, quadratic, and interaction terms. The statistical analysis showed that all independent parameters are statistically significant for ED with the largest model variation from experimental data being 1.44%. Response curve of the regression analysis showed that mass density, HHV, and energy density model have the highest output predictability with higher R2 values of 95.3%, 98.1%, and 74.1%, respectively.

Synchronizing Stability Analysis and Region of Attraction Estimation of Grid-Feeding VSCs Using Sum-of-Squares Programming

Grid-synchronising Stability (GSS) is an emerging issue that exists in grid-feeding voltage-source converters (VSCs). Its occurrence is primarily related to the nonlinear dynamics of a type of vastly applied synchronization units - Phase-locked Loops (PLL). Dynamic characterisation and modelling for the GSS analysis can be achieved by using a simplified system model, which is a second-order and autonomous nonlinear equation; however, with the presence of an indefinite damping term. As revealed and demonstrated in this work, this indefinite damping effect can result in an inaccurate Region-of-Attraction (ROA) estimation of the traditional Equal Area Principle (EAP)-based method. To overcome this issue and achieve a valid ROA estimation, this paper adopts the sum-of-squares (SOS) programming technique, which is a numeric optimisation method with SOS relaxations. The development and implementation of the SOS program for ROA estimation are presented. Numerical case studies and time-domain verifications demonstrate that this method is valid for GSS analysis, and an almost precise estimation is achieved in the first quadrant. This evidence makes the SOS method a promising tool for GSS analysis because the GSS problem is most concerned with the stability within the first swing cycle, i.e., in the first quadrant.

Workers\u2019 physical activity data contribute to estimating maximal oxygen consumption: a questionnaire study to concurrently assess workers\u2019 sedentary behavior and cardiorespiratory fitness

BackgroundSedentary behavior (SB) and cardiorespiratory fitness (CRF) are important issues in occupational health. Developing a questionnaire to concurrently assess workers’ SB and CRF could fundamentally improve epidemiological research. The Worker’s Living Activity-time Questionnaire (WLAQ) was developed previously to assess workers’ sitting time. WLAQ can be modified to evaluate workers’ CRF if additional physical activity (PA) data such as PA frequency, duration, and intensity are collected.MethodsA total of 198 working adults (93 women and 105 men; age, 30–60 years) completed anthropometric measurements, a treadmill exercise test for measuring maximal oxygen consumption (VO2max), and modified WLAQ (m-WLAQ, which included questions about PA data additional to the original questions). Multiple regression analyses were performed to develop prediction equations for VO2max. The generated models were cross-validated using the predicted residual error sum of squares method. Among the participants, the data of 97 participants who completed m-WLAQ twice after a 1-week interval were used to calculate intraclass correlation coefficient (ICC) for the test–retest reliability analyses.ResultsAge (r = − 0.29), sex (r = 0.48), body mass index (BMI, r = − 0.20), total sitting time (r = − 0.15), and PA score (total points for PA data, r = 0.47) were significantly correlated with VO2max. The models that included age, sex, and BMI accounted for 43% of the variance in measured VO2max [standard error of the estimate (SEE) = 5.04 ml·kg− 1·min− 1]. These percentages increased to 59% when the PA score was included in the models (SEE = 4.29 ml·kg− 1·min− 1). Cross-validation analyses demonstrated good stability of the VO2max prediction models, while systematic underestimation and overestimation of VO2max were observed in individuals with high and low fitness, respectively. The ICC of the PA score was 0.87 (0.82–0.91), indicating excellent reliability.ConclusionsThe PA score obtained using m-WLAQ, rather than sitting time, correlated well with measured VO2max. The equation model that included the PA score as well as age, sex, and BMI had a favorable validity for estimating VO2max. Thus, m-WLAQ can be a useful questionnaire to concurrently assess workers’ SB and CRF, which makes it a reasonable resource for future epidemiological surveys on occupational health.

Application of Sum-of-Squares Method in Estimation of Region of Attraction for Nonlinear Polynomial Systems

We present a sum of squares (SOS) method for the synthesis of nonlinear polynomial control systems. As an emerging numerical solution method in recent years, SOS targets polynomials as the research object. It guarantees that the polynomial we solve for is always nonnegative. In this paper, we give a generalized S-procedure to solve the SOS problem. As an illustration of how the SOS method can be used, the region of attraction (ROA) in a nonlinear polynomial system is analyzed in detail. The method of determining decision variables is given in the SOS problem. We discuss the determination and solution of set-containment constraints and the conservatism problem in solving the SOS problem. SOS provides a convenient numerical method to solve nonlinear problems that are not easy to solve analytically.

Non-invasive assessment of ventriculo-arterial coupling using aortic wave intensity analysis combining central blood pressure and phase-contrast cardiovascular magnetic resonance.

Wave intensity analysis (WIA) in the aorta offers important clinical and mechanistic insight into ventriculo-arterial coupling, but is difficult to measure non-invasively. We performed WIA by combining standard cardiovascular magnetic resonance (CMR) flow-velocity and non-invasive central blood pressure (cBP) waveforms. Two hundred and six healthy volunteers (age range 21-73 years, 47% male) underwent sequential phase contrast CMR (Siemens Aera 1.5 T, 1.97 × 1.77 mm2, 9.2 ms temporal resolution) and supra-systolic oscillometric cBP measurement (200 Hz). Velocity (U) and central pressure (P) waveforms were aligned using the waveform foot, and local wave speed was calculated both from the PU-loop (c) and the sum of squares method (cSS). These were compared with CMR transit time derived aortic arch pulse wave velocity (PWVtt). Associations were examined using multivariable regression. The peak intensity of the initial compression wave, backward compression wave, and forward decompression wave were 69.5 ± 28, -6.6 ± 4.2, and 6.2 ± 2.5 × 104 W/m2/cycle2, respectively; reflection index was 0.10 ± 0.06. PWVtt correlated with c or cSS (r = 0.60 and 0.68, respectively, P < 0.01 for both). Increasing age decade and female sex were independently associated with decreased forward compression wave (-8.6 and -20.7 W/m2/cycle2, respectively, P < 0.01) and greater wave reflection index (0.02 and 0.03, respectively, P < 0.001). This novel non-invasive technique permits straightforward measurement of wave intensity at scale. Local wave speed showed good agreement with PWVtt, and correlation was stronger using the cSS than the PU-loop. Ageing and female sex were associated with poorer ventriculo-arterial coupling in healthy individuals.

Coil combination methods for multi-channel hyperpolarized 13C imaging data from human studies.

Effective coil combination methods for human hyperpolarized 13C spectroscopy multi-channel data had been relatively unexplored. This study implemented and tested several coil combination methods, including (1) the sum-of-squares (SOS), (2) singular value decomposition (SVD), (3) Roemer method by using reference peak area as a sensitivity map (RefPeak), and (4) Roemer method by using ESPIRiT-derived sensitivity map (ESPIRiT). These methods were evaluated by numerical simulation, thermal phantom experiments, and human cancer patient studies. Overall, the SVD, RefPeak, and ESPIRiT methods demonstrated better accuracy and robustness than the SOS method. Extracting complex pyruvate signal provides an easy and excellent approximation of the coil sensitivity map while maintaining valuable phase information of the coil-combined data.