Merit Function Research Articles

Regularized Normalization Methods for Solving Linear and Nonlinear Eigenvalue Problems

To solve linear and nonlinear eigenvalue problems, we develop a simple method by directly solving a nonhomogeneous system obtained by supplementing a normalization condition on the eigen-equation for the uniqueness of the eigenvector. The novelty of the present paper is that we transform the original homogeneous eigen-equation to a nonhomogeneous eigen-equation by a normalization technique and the introduction of a simple merit function, the minimum of which leads to a precise eigenvalue. For complex eigenvalue problems, two normalization equations are derived utilizing two different normalization conditions. The golden section search algorithms are employed to minimize the merit functions to locate real and complex eigenvalues, and simultaneously, we can obtain precise eigenvectors to satisfy the eigen-equation. Two regularized normalization methods can accelerate the convergence speed for two extensions of the simple method, and a derivative-free fixed-point Newton iterative scheme is developed to compute real eigenvalues, the convergence speed of which is ten times faster than the golden section search algorithm. Newton methods are developed for solving two systems of nonlinear regularized equations, and the efficiency and accuracy are significantly improved. Over ten examples demonstrate the high performance of the proposed methods. Among them, the two regularization methods are better than the simple method.

Two global quasi-Newton algorithms for solving matrix polynomial equations

In this article, we globalize the quasi-Newton algorithm proposed in Macías et al. (Appl Math Comput 441:127678, 2023) introducing an exact line search, we propose a polynomial approximation to the merit function and we deduce a sufficient condition for its minimization interval. We use the exact merit function and its approximation to propose two global quasi-Newton algorithms for solving matrix polynomial equations. For each algorithm, we prove that the exact line search does not affect the convergence of quasi-Newton method. In addition, we present comparative numerical tests of the algorithmic proposals in which we also compare with global Newton’s method.

Patch seriation to visualize data and model parameters

We developed a new seriation merit function for enhancing the visual information of data matrices. A local similarity matrix is calculated, where the average similarity of neighbouring objects is calculated in a limited variable space and a global function is constructed to maximize the local similarities and cluster them into patches by simple row and column ordering. The method identifies data clusters in a powerful way, if the similarity of objects is caused by some variables and these variables differ for the distinct clusters. The method can be used in the presence of missing data and also on more than two-dimensional data arrays. We show the feasibility of the method on different data sets: on QSAR, chemical, material science, food science, cheminformatics and environmental data in two- and three-dimensional cases. The method can be used during the development and the interpretation of artificial neural network models by seriating different features of the models. It helps to identify interpretable models by elucidating clusters of objects, variables and hidden layer neurons.Graphical

Dynamical Optimal Values of Parameters in the SSOR, AOR, and SAOR Testing Using Poisson Linear Equations

This paper proposes a dynamical approach to determine the optimal values of the parameters used in each iteration of the symmetric successive over-relaxation (SSOR), accelerated over-relaxation (AOR), and symmetric accelerated over-relaxation (SAOR) methods for solving linear equation systems. When the optimal values of the parameters in the SSOR, AOR, and SAOR are hard to determine as some fixed values, they are obtained by minimizing the merit functions, which are based on the maximal projection technique between the left- and right-hand-side vectors, which involves the input vector, the previous step values of the variables, and the parameters. The novelty is a new concept of the dynamical optimal values of the parameters, instead of the fixed values and the maximal projection technique. In a preferred range, the optimal values of the parameters can be quickly determined by using the golden section search algorithm with a loose convergence criterion. Without knowing and having the theoretical optimal values in general, the new methods might provide an alternative and proper choice of the values of the parameters for accelerating the convergence speed. Numerical testings of the linear Poisson equation discretized to a matrix–vector form and a Lyapunov equation form were used to assess the performance of the DOSSOR, DOAOR, and DOSAOR dynamical optimal methods.

The meshless solutions of Laplacian non-harmonic and Cauchy problems by developing novel hybrid methods

In the first part of the paper a non-harmonic boundary value problem of the Laplace equation is solved by new hybrid methods. We propose four hybrid methods, namely the Trefftz-Trefftz method (TTM), the Trefftz-MFS method (TMM), the MFS-Trefftz method (MTM), and the MFS-MFS method (MMM), by supplementing extra linear equations with the Trefftz functions or the fundamental solutions as the testers in Green’s second identity. In doing so the range of the coefficient matrix can be enlarged such as more close to the input vector, and meanwhile the accuracy is improved. In the second part we improve the accuracy of the MFS, which deteriorates due to the ill-posed property of the Cauchy problem. We derive two simple merit functions and minimize them to determine the source points in the MFS, which can further enhance the accuracy obtained by two optimal MTMs. These new hybrid methods apparently improve the accuracy than the MFS and the Trefftz method, when we solve the non-harmonic and Cauchy problems of the Laplace equation.

Improving the lean burn performance of a high compression ratio methanol engine by multiple-injection

Methanol is a promising fuel for achieving carbon neutrality in sectors requiring internal combustion engines. The multiple-injection approach is effective in reducing cycle-to-cycle variations and improving the performance of methanol engines under lean burn conditions. This study proposes a concept for transforming diesel engines into methanol engines with flexible injection and lean burning capabilities. This study fills the research gap in the study of lean methanol combustion in swirl combustion chambers. The effects of 22 different multiple-injection strategies on combustion characteristics, emission performance, and combustion stability are experimentally assessed in a direct-injection spark ignition engine with an excess air ratio (λ) of 1.35. Each multiple-injection strategy is comprehensively evaluated using a merit function and compared with the baseline case that uses single injection. The results show that the double- and triple-injection strategies substantially shorten the durations of combustion and flame development. Therefore, a higher effective expansion ratio is obtained. Meanwhile, the cycle-to-cycle variation in the IMEP decreases from 4.2% of the baseline to 1.49% at T8. CO and HC emissions deteriorate; however, NOx and CO2 emissions under triple injection are lower than those under the baseline. The T7 case with triple injection achieves the highest thermal efficiency, which is 6.8% higher than that for the baseline case. Overall, triple injection achieves higher merit function values, which indicates a better capability to improve lean-burn performance than double injection.

Correlation between figure of merit function and deviation in the assessment of kinetic parameters in realistic TLD-100 behavior using fully-connected neural networks

In this study, a realistic model was employed to research the LiF:Mg:Ti behavior (commercially known as TLD-100), a widely employed as thermoluminescent (TL) material in dosimetry services. The research was focused on exploring the association between the Figure of Merit (FOM) function and deviations from the true kinetic parameter values, that were obtained through fitting experimental measures. FOM function is the most used magnitude to measure the quality of a fit achieved between a mathematical model and experimental data. Indeed, FOM function is particularly relevant when the TL glow curve deconvolution method is implemented to obtain the kinetic parameters: Activation Energy E (eV), Frequency Factor s (s−1), value of kinetic order b, Temperature at which the intensity maximum TMax(oC), Intensity Maximum IMax (a.u.) and Distribution Width σ (eV) of every peak. FOM function is subject to degeneracy under certain conditions, this means that different sets of kinetic parameters can lead to similar FOM values. Furthermore, as the obtained kinetic parameters are deemed acceptable when the FOM value is below 5%, then the physical degeneracy is greater than the mathematical ones. This contribution provides a rigorous analysis of how the FOM values can be impacted by deviations between experimental and estimated data of kinetic parameters from the analysis process. This studio was carried out using a mathematical method based on Artificial Intelligence with a high capability to manage a large volume of data. Another advantage of this method is that the analysis to be extended to a wide range of experimental data.

Simplified analysis and suppression of polarization aberration in planar symmetric optical systems.

Polarized remote sensing imaging has attracted more attention in recent years due to its wider detection information dimension compared to traditional imaging methods. However, the inherent instrument errors in optical systems can lead to errors in the polarization state of the incident and outgoing light, which is the polarization aberration of the optical system, resulting in a decrease in polarization detection accuracy. We propose a polarization aberration simplification calculation method for planar symmetric optical systems, by what only three ray samples are needed to obtain the distribution of polarization aberrations within the pupil. This method has a calculation accuracy close to traditional methods, and the sampling rate is 0.003 times that of traditional methods. Based on this, we designed a merit function that optimizes both wavefront and polarization aberrations simultaneously. It is found that diattenuation and retardance of the optical system are 62% and 58% of the original, and the polarization crosstalk term is reduced by 37% when the polarization weight factor takes an appropriate value. And at the same time, the wavefront aberration has also been well optimized.

Pupil aberrations correction of the afocal telescope for the TianQin project

TianQin is a planned Chinese space-based gravitational wave (GW) observatory with a frequency band of 10−4–1 Hz. Optical telescopes are essential for the delivery of the measurement beam to support a precise distance measurement between pairs of proof masses. As the design is driven by the interferometric displacement sensitivity requirements, the stability control of optical path length (OPL) is extremely important beyond the traditional requirement of diffraction-limited imaging quality. The recurring tilt-to-length (TTL) coupling noise arises from the OPL variation due to the wavefront deformation and angular misalignment. Reducing the residual chief ray aberration in the optical design helps suppress TTL coupling noise. To correct the pupil aberrations, we derive primary pupil aberrations in a series expansion form, and then refine the formulation of merit function by combining the pupil aberration theory and traditional image aberration theory. The automatic correction of pupil aberrations is carried out by using the macro programming in the commercial optical software Zemax, leading to a high performance telescope design. The design results show that on one side the pupil aberrations have been corrected, and on the other side, its optical performance meets the requirements for TianQin project. The RMS wavefront error over the science field of view (FOV) is less than λ/200 and the maximum TTL coupling noise over the entire ±300 μrad FOV is 0.0152 nm µrad−1 . We believe that our design approach can be a good guide for the space telescope design in any other space-based GW detection project, as well as other similar optical systems.

Extracting product design guidance from online reviews: An explainable neural network-based approach

With the development of data mining techniques, user-generated data has become a valuable resource in diverse research areas. In product design research, many studies have been utilizing user data to discover implications for new product design. However, previous works focused on analyzing existing features, whereas companies also need strategies for new features. Some studies discovered new feature ideas from user data but did not provide design implications. This paper addresses the above limitation by extracting comprehensive design implications for both features from user data. The method first defines the lists of existing/new features and collects spec data for these features. Then, it constructs customer choice sets based on the online review and spec data. Regarding spec values, this study presents a newness merit function that reflects the changing value of new features and applies it to the choice sets. The final stage trains a neural network model based on choice sets and conducts SHAP (SHapley Additive exPlanations) on the model. The method draws design implications by further analyzing the resultant SHAP values. The suggested methodology was tested on real-world datasets. The result provides design guidance, including strategies for new features and recommended spec ranges for existing features. This article validates the result by showing that the obtained design implications are consistent with previous market research for product features.

CQD-ATO hybrid nanofluid with good stability in the application of spectral beam splitters

In photovoltaic power systems, nanofluids are promising candidates for spectral beam splitters (SBSs) owing to their excellent spectral absorption regulation characteristics. The drawbacks of high cost and instability restrain the practical application of nanofluid SBSs. Here, we proposed a hybrid nanofluid SBS containing carbon quantum dots (CQDs) and antimony tin oxide (ATO) nanoparticles, which has been proven to be an excellent SBS from multiple perspectives. Both CQDs and ATO nanoparticles have low cost and super stability, which ensures the long-term practical application of hybrid nanofluid SBSs. Then, the light absorption characteristics of hybrid nanofluids can be effectively regulated because of the complementary optical properties of CQDs and ATO nanoparticles. Furthermore, the PV/T conversion model and indoor test were used to assess the filtering performance of hybrid nanofluid SBSs. The results indicate that CQD-ATO hybrid nanofluids can achieve a high merit function value of 1.815, which is significantly higher than pure CQD nanofluids and outperforms other nanofluid SBSs reported in the literature. This paper may promote the practical utilization of nanofluid SBSs in photovoltaic power systems.

Automatic Method of Exploring the Landscape of Freeform Dioptric Optical Problems, Working in the Infrared Region

We present an automated method of finding different freeform dioptric starting systems, working in the infrared region, for further optimization in commercial optical design software. Our developed method couples the simultaneous multiple surface (SMS) method, introduced by Benítez and Miñano, with automatic optimization in Zemax OpticStudio. The method allows an optical designer to explore the merit function (MF) landscape of freeform optical problems. In this article, we apply our method to a size, weight, and power (SWaP) problem, and we compare our designed system with a system found in the literature that has the same aperture of F/1.2. Then, we increase the aperture of the system up to F/0.9, taking advantage of the use of freeform surfaces.

Inverse-Dynamics MPC via Nullspace Resolution

Optimal control (OC) using inverse dynamics provides numerical benefits, such as coarse optimization, cheaper computation of derivatives, and a high convergence rate. However, to take advantage of these benefits in model predictive control (MPC) for legged robots, it is crucial to handle efficiently its large number of equality constraints. To accomplish this, we first propose a novel approach to handle equality constraints based on nullspace parameterization. Our approach balances optimality, and both dynamics and equality-constraint feasibility appropriately, which increases the basin of attraction to high-quality local minima. To do so, we modify our feasibility-driven search by incorporating a merit function. Furthermore, we introduce a condensed formulation of inverse dynamics that considers arbitrary actuator models. We also propose a novel MPC based on inverse dynamics within a perceptive locomotion framework. Finally, we present a theoretical comparison of OC with forward and inverse dynamics and evaluate both numerically. Our approach enables the first application of inverse-dynamics MPC on hardware, resulting in the state-of-the-art dynamic climbing on the ANYmal robot. We benchmark it over a wide range of robotics problems and generate agile and complex maneuvers. We show the computational reduction of our nullspace resolution and condensed formulation (up to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${47.3}\boldsymbol{\%}$</tex-math></inline-formula> ). We provide evidence of the benefits of our approach by solving coarse optimization problems with a high convergence rate (up to 10 Hz of discretization). Our algorithm is publicly available inside <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Crocoddyl</small> .

Error Bounds and Merit Functions for Exponentially General Variational Inequalities

In this paper, some new classes of classes of exponentially general variational inequalities are introduced. It is shown that the odd-order and nonsymmetric exponentially boundary value problems can be studied in the framework of exponentially general variational inequalities. We consider some classes of merit functions for exponentially general variational inequalities. Using these functions, we derive error bounds for the solution of exponentially general variational inequalities under some mild conditions. Since the exponentially general variational inequalities include general variational inequalities, quasi-variational inequalities and complementarity problems as special cases, results proved in this paper hold for these problems. Results obtained in this paper represent a refinement of previously known results for several classes of variational inequalities and related optimization problems.

A neuro-structural framework for bankruptcy prediction

We develop a framework to simultaneously compute the unobservable parameters underlying the structural-parametric models for bankruptcy prediction. More specifically, we compute the unobservable parameters such as, asset value and asset volatility, through learning by embedding in the structural models a neural network that maps the neural network’s input space (e.g. companies’ observable financial and market data) to the unobservable parameter space. Within such a ‘neuro-structural’ framework, the neural network and the structural model work together as a one unit during the learning phase by providing to each other forward and backward information, respectively, until the weights of the neural network are optimized according to a merit function. Empirical results show that structural models, like the Black-Scholes-Merton and the Down-and-Out option models, with parameters computed with our approach, perform better than alternative specifications of the structural models, out of sample, in terms of discriminatory power, information content and economic impact. Importantly, they also perform better than a standard neural network, suggesting that the co-joint dynamics between the neural network and the structural model are useful during the learning phase and can improve the prediction performance (and the training efficiency) of neural networks. Finally, our approach provides methodological (and empirical) enhancements over logit specifications such as, Campbell et al. [In search of distress risk. J Finance, 2008, 63, 2899–2939]. There, financial and market data are the inputs, and the output is the probability of bankruptcy whereas our approach includes an intermediary step to obtain the unobservable parameters and subsequently the probability of bankruptcy.

Minimization of exergy destruction with discrete metal foam filling in a pipe under turbulent flow condition

PurposeThis study aims to present the numerical analysis of exergy transfer and irreversibility through the discrete filling of high-porosity aluminum metal foams inside the horizontal pipe.Design/methodology/approachIn this study, the heater is embedded on the pipe’s circumference and is assigned with known heat input. To enhance the heat transfer, metal foam of 10 pores per inch with porosity 0.95 is filled into the pipe. In filling, two kinds of arrangements are made, in the first arrangement, the metal foam is filled adjacent to the inner wall of the pipe [Model (1)–(3)], and in the second arrangement, the foam is located at the center of the pipe [Models (4)–(6)]. So, six different models are examined in this research for a fluid velocity ranging from 0.7 to7 m/s under turbulent flow conditions. Darcy Extended Forchheimer is combined with local thermal non-equilibrium models for forecasting the flow and heat transfer features via metal foams.FindingsThe numerical methodology implemented in this study is confirmed by comparing the outcomes with the experimental outcomes accessible in the literature and found a fairly good agreement between them. The application of the second law of thermodynamics via metal foams is the novelty of current investigation. The evaluation of thermodynamic performance includes the parameters such as mean exergy-based Nusselt number (Nue), rate of irreversibility, irreversibility distribution ratio (IDR), merit function (MF) and non-dimensional exergy destruction (I*). In all the phases, Models (1)–(3) exhibit better performance than Models (4)–(6).Practical implicationsThe present study helps to enhance the heat transfer performance with the introduction of metal foams and reveals the importance of available energy (exergy) in the system which helps in arriving at optimum design criteria for the thermal system.Originality/valueThe uniqueness of this study is to analyze the impact of discrete metal foam filling on exergy and irreversibility in a pipe under turbulent flow conditions.

New merit functions for multiobjective optimization and their properties

A merit (gap) function is a map that returns zero at the solutions of problems and strictly positive values otherwise. Its minimization is equivalent to the original problem by definition, and it can estimate the distance between a given point and the solution set. Ideally, this function should have some properties, including the ease of computation, continuity, differentiability, boundedness of the level set, and error boundedness. In this work, we propose new merit functions for multiobjective optimization with lower semicontinuous objectives, convex objectives, and composite objectives, and we show that they have such desirable properties under reasonable assumptions.

Clean hydrogen energy production via purification of hydrogen sulfide thermolysis products employing supersonic separator

Catalytic thermolysis of hydrogen sulfide rich acid gases at elevated temperatures is a novel process to produce clean hydrogen fuel. Separation of the hydrogen gas from furnace effluents is rather a difficult task. Several approaches such as membranes, adsorption and absorption processes are traditionally used to achieve this goal. Each of these conventional processes has its own difficulties such as high capital and operation investments, large equipment sizes and elevated corrosion rates. Employment of supersonic separators for the above purpose strongly alleviates the above problems. The present work investigates the use of supersonic separator units both as a final separation device and also as a hybrid pre-treatment method via numerical simulation using MATLAB-Simulink toolbox. It was noticed that a single supersonic separator unit is unable to accomplish the target specification as a final separation device. Hence, consecutive combination of supersonic separator units is used to obtain the desired hydrogen purity. The performance of the entire supersonic separator network strongly depends on the inlet pressure of each individual supersonic separator unit. Detailed optimization of the entire process revealed that for the case study at hand, the optimal inlet pressure for all eleven required supersonic separator units would be around 7 MPa. The number of required 3S units and the overall power of compressors required between the 3S stages are used as the optimization merit functions. Furthermore, in a hybrid approach, a combination of supersonic separator units is used prior to absorption process to reduce the hydrogen sulfide concentration of the hydrogen gas product to less than 4 ppm. Use of four consecutive supersonic separator units along with the absorption process can lead to more than 95% saving both in DEA circulation rate and reboiler duty. The entire above issues have not been addressed previously in the literature.

A Fast Temporal Decomposition Procedure for Long-Horizon Nonlinear Dynamic Programming

We propose a fast temporal decomposition procedure for solving long-horizon nonlinear dynamic programs. The core of the procedure is sequential quadratic programming (SQP) that utilizes a differentiable exact augmented Lagrangian as the merit function. Within each SQP iteration, we approximately solve the Newton system using an overlapping temporal decomposition strategy. We show that the approximate search direction is still a descent direction of the augmented Lagrangian provided the overlap size and penalty parameters are suitably chosen, which allows us to establish the global convergence. Moreover, we show that a unit step size is accepted locally for the approximate search direction and further establish a uniform, local linear convergence over stages. This local convergence rate matches the rate of the recent Schwarz scheme (Na et al. 2022). However, the Schwarz scheme has to solve nonlinear subproblems to optimality in each iteration, whereas we only perform a single Newton step instead. Numerical experiments validate our theories and demonstrate the superiority of our method. Funding: This work was supported by the National Science Foundation [Grant CNS-1545046] and the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research [Grant DE-AC02-06CH11347].

Dynamic programming network for point target detection

To improve the efficiency of the dim point target detection based on dynamic programming (DP), this paper proposes a multi-frame target detection method based on a DP ring network (DPRN). In the proposed method, first, the target trajectory is approximated using the piecewise linear function. The velocity space partition DP (VSP-DP) is used to accumulate the merit functions of a target on each piecewise linear trajectory segment to avoid the merit function diffusion in different velocity spaces. In addition, the velocity space matching DP (VSM-DP) is employed to realize the state transition of a target between adjacent piecewise linear trajectory segments. Then, the VSP-DP and VSM-DP are used to construct a DP network (DPN). Second, to suppress the merit function diffusion further, the sequential and reverse DPNs are connected in a head-to-tail manner to form a DPRN, and the merit function of the DPRN is obtained by averaging the merit functions of the sequential and reverse DPNs. Finally, the target trajectory is obtained by tracking the extreme points of the merit functions of the DPRN. The simulation and analysis results show that the proposed DPRN combines the advantages of high detection probability of the high-order DP and high execution efficiency of the first-order DP. The proposed DPRN is suitable for radars and infrared searching and tracking systems.