Test Problems Research Articles

A-4 Step Chebyshev Based Multiderivative Direct Solver For Third Order Ordinary Differential Equations

This paper develops and examines a uniform order, 4 step block method with Chebyshev series as bases function. Collocation and interpolation technique was used to modeled implicit discrete schemes from continuous scheme to derive our block. The block method obtained was of order 4. It is consistent, zero stable and consequently zero stable. The results obtained from four test problems shown that the method converges to exact solutions and perform better than some existing methods in the literatures.

Preventing mass loss in the standard level set method: New insights from variational analyses

For decades, the computational multiphase flow community has grappled with mass loss in the level set method. Numerous solutions have been proposed, from fixing the reinitialization step to combining the level set method with other conservative schemes. However, our work reveals a more fundamental culprit: the smooth Heaviside and delta functions inherent to the standard formulation. Even if reinitialization is done exactly, i.e., the zero contour interface remains stationary, the use of smooth functions lead to violation of mass conservation. We propose a novel approach using variational analysis to incorporate a mass conservation constraint. This introduces a Lagrange multiplier that enforces overall mass balance. Notably, as the delta function sharpens, i.e., approaches the Dirac delta limit, the Lagrange multiplier approaches zero. However, the exact Lagrange multiplier method disrupts the signed distance property of the level set function. This motivates us to develop an approximate version of the Lagrange multiplier that preserves both overall mass and signed distance property of the level set function. Our framework even recovers existing mass-conserving level set methods, revealing some inconsistencies in prior analyses. We extend this approach to three-phase flows for fluid-structure interaction (FSI) simulations. We present variational equations in both immersed and non-immersed forms, demonstrating the convergence of the former formulation to the latter when the body delta function sharpens. Rigorous test problems confirm that the FSI dynamics produced by our simple, easy-to-implement immersed formulation with the approximate Lagrange multiplier method are accurate and match state-of-the-art solvers.

Significance-Based Decision Tree for Interpretable Categorical Data Clustering

Numerous clustering algorithms prioritize accuracy, but in high-risk domains, the interpretability of clustering methods is crucial as well. The inherent heterogeneity of categorical data makes it particularly challenging for users to comprehend clustering outcomes. Currently, the majority of interpretable clustering methods are tailored for numerical data and utilize decision tree models, leaving interpretable clustering for categorical data as a less explored domain. Additionally, existing interpretable clustering algorithms often depend on external, potentially non-interpretable algorithms and lack transparency in the decision-making process during tree construction. In this paper, we tackle the problem of interpretable categorical data clustering by growing a decision tree in a statistically meaningful manner. We formulate the evaluation of candidate splits as a multivariate two-sample testing problem, where a single p-value is derived by combining significance evidence from all individual categories. This approach provides a reliable and controllable method for selecting the optimal split while determining its statistical significance. Extensive experimental results on real-world data sets demonstrate that our algorithm achieves comparable performance in terms of cluster quality, running efficiency, and explainability relative to its counterparts.

Numerical Study of the Reaction Diffusion Prey–Predator Model Having Holling II Increasing Function in the Predator Under Noisy Environment

The current study deals with the stochastic diffusive prey–predator model with Holling II increasing function in the predator. As it is a population dynamics its population can be affected by various factors such as disease in prey–predator species, environmental fluctuation, etc. So, the underlying model is considered under the impact of a temporal multiplicative noise. The numerical solution of the governing model is obtained by finite difference schemes. To check the accuracy of the numerical solution, the stability and consistency of the schemes in the mean square sense are derived. All the schemes are consistent with the model and von Neumann stable. Several simulations are drawn to check the efficacy of the scheme. A test problem is investigated with numerical schemes and for the different values of the parameters, 3D and 2D plots are drawn. A graphical representation of the given system is drawn which shows the behavior of both species and the efficiency of the novel schemes. Hope so, these results will motivate the researcher to consider the stochastic prey–predator model for a better understanding of the ecosystem.

Near-optimal March Tests for Three-Cell and Four-Cell Coupling Fault Models in Random-Access Memories

This research work addresses the problem of testing n×1 RAMs in which complex models of unlinked static three or four-cell coupling faults are considered. As in other works, it is assumed that only physically neighboring memory cells could be involved in a three or four-cell coupling fault. For this reason, these fault models can also be considered to be of the neighborhood pattern sensitive type. As extensions of the well-known model of all unlinked static two-cell coupling faults, the fault models we address are complex including faults sensitized by a transition write operation as well as faults sensitized by a non-transition write or a read operation. For these complex models, near-optimal multirun march tests are proposed. This optimality assessment is based on the fact that, for any group of cells corresponding to the considered fault model, the state graph is completely covered, and each arc is traversed only once, which means that the graph is of the Eulerian type. Additional write operations are only required for data background changes.

CT-Based method for determining the elastic properties of an inhomogeneous medium element

The determination of mechanical parameters is one of the relevant directions in the calculation of inhomogeneous media elements. The need to build numerical models is primarily due to the development of modern approaches to the reconstruction of materials of various composite structures. The use of computational models in clinical practice also makes it possible to study the strength characteristics of objects of biological origin, the features of which include the uniqueness of the material structural distribution. The most widespread approaches in this case are based on the use of image data. Within the framework of this work, a method for constructing an elasticity matrix based on digital twin data is presented. During the work, images with a given material structure are constructed, as well as data extraction from CT scans of full-scale samples. The averaging of mechanical parameters was performed on the basis of the finite element method for the area of square or cubic geometry defining the unit cell. The calculations performed by the proposed method showed the convergence of the method with the analytical calculation data on the example of solving test problems, and also determined the qualitative correspondence of the results to the features of the structural distribution of the material in full-scale samples. The presented homogenization method makes it possible to determine the elastic characteristics of a medium element, taking into account the inhomogeneous properties of the forming material.

Analysis of the Effectiveness of Using a High-Accuracy Numerical Algorithm in the Simulation of Two-Phase Flow Problems

The continuous technical development of weapons and ammunition, as well as the growing requirements placed on new equipment, require the use of modern tools for their designing and testing. The analysis of the operation of powder propellant systems, which is the focus of interior ballistics, requires particularly specialised tools due to the complexity of the phenomena occurring during a shot. The development of numerical methods and the increase in the computing power of modern computers have enabled the modelling and simulation of highly complex physical problems and the development of numerical schemes with a theoretically arbitrary high order of accuracy. In this study, an attempt was made to develop a one-dimensional model of two-phase flows with a high order of solution accuracy in time, based on the finite volume method. The use of a two-phase flow model in the context of interior ballistics allows, among other things, the analysis of wave phenomena occurring during firing, which is crucial for safety reasons, especially in tank guns. The developed algorithm for solving two-phase systems was checked using recognised test problems for this type of model. The results of the work will be used to develop a numerical model to solve the main problem of interior ballistics.

Column generation approach for 1.5-dimensional cutting stock problem with technical constraints

In this study, the 1.5-dimensional cutting stock problem with technical constraints is considered. In the literature, this problem is also defined as a strip packing or open dimension problem. When given a strip of infinite length and bounded width, the problem is to define a packing of rectangular objects into a strip that minimizes its final length. Technical constraints, such as the order type and the number of strips, are indispensable in real life; however, they are often neglected in the literature because they make the problem difficult to solve. Only one study was reached in the literature that took into account technical constraints, but in that mentioned study, only a mathematical model was proposed for the problem. In this context, our aim is to solve the problem with a more effective approach. The research question in this study is the usability of the column generation technique to solve the 1.5-dimensional cutting stock problem. In this study, the column generation approach was proposed for the first time for the considered problem. To demonstrate the performance of the proposed solution method, randomly generated test problems were solved with GAMS/Cplex. As we report the results, proposed column generation approach (CG) reaches very close (such as %1 and %2 error) solutions to integrated mathematical model (IM) for small sized problems in a second. On the other hand, while CG solved all the problems in a reasonable time, IM could not produce a feasible solution to some problems. Numerical experiments showed that the column generation algorithm outperforms the integrated mathematical model for the problem.

Adaptive Constraint Relaxation-Based Evolutionary Algorithm for Constrained Multi-Objective Optimization

The key problem to solving constrained multi-objective optimization problems (CMOPs) is how to achieve a balance between objectives and constraints. Unfortunately, most existing methods for CMOPs still cannot achieve the above balance. To this end, this paper proposes an adaptive constraint relaxation-based evolutionary algorithm (ACREA) for CMOPs. ACREA adaptively relaxes the constraints according to the iteration information of population, whose purpose is to induce infeasible solutions to transform into feasible ones and thus improve the ability to explore the unknown regions. Completely ignoring constraints can cause the population to waste significant resources searching for infeasible solutions, while excessively satisfying constraints can trap the population in local optima. Therefore, balancing constraints and objectives is a crucial approach to improving algorithm performance. By appropriately relaxing the constraints, it induces infeasible solutions to be transformed into feasible ones, thus obtaining more information from infeasible solutions. At the same time, it also establishes an archive for the storage and update of solutions. In the archive update process, a diversity-based ranking is proposed to improve the convergence speed of the algorithm. In the selection process of the mating pool, common density selection metrics are incorporated to enable the algorithm to obtain higher-quality solutions. The experimental results show that the proposed ACREA algorithm not only achieved the best Inverse Generation Distance (IGD) value in 54.6% of the 44 benchmark test problems and the best Hyper Volume (HV) value in 50% of them, but also obtained the best results in seven out of nine real-world problems. Clearly, CP-TSEA outperforms its competitors.

Застосування методу двобічних наближень до розв’язання першої крайової задачі для одновимірного рівняння теплопровідності з експоненціально нелінійним коефіцієнтом теплопровідності

The problem of heat conduction in nonlinear media reduces to solving boundary value problems for nonlinear heat conduction equations, where the coefficients of the equation or the function of the power of heat sources depend on temperature according to some law. Among the numerical methods for solving problems for nonlinear equations of mathematical physics, one can distinguish finite difference methods, finite element methods, variational and projection methods, as well as iterative methods. Among the latter group, the two-sided approximation method is particularly attractive due to its ability to provide a convenient estimate for the error of the approximate solution and to prove the existence of a solution to the original problem. The theory of linear partially ordered spaces was developed by L. V. Kantorovich in the second half of the 1930s. Further development of this theory is associated with the works of M. A. Krasnosel’skii, H. Amann, V. I. Opojtsev, N. S. Kurpel, B. A. Shuvar, A. I. Kolosov etc. The aim of this article is to develop a two-sided approximation method based on the use of Green's functions for solving the first boundary value problem for a one-dimensional nonlinear heat conduction equation and to investigate its performance when solving test problems. To achieve this goal, the unknown function was replaced, and the boundary value problem was reduced to a Hammerstein integral equation, which was considered as a nonlinear operator equation in a partially ordered Banach space. Conditions for the existence of a unique positive solution to the problem and conditions for two-sided convergence of successive approximations to it were obtained. The developed method was implemented in software and tested on solving test problems. The results of the computational experiment are illustrated with graphical and tabular information

Fractional logarithmic Chebyshev cardinal functions: Application for Caputo–Hadamard fractional optimal control problems

This study investigates a specific type of fractional optimization problems, subject to a dynamical system including the Caputo–Hadamard fractional differentiation. In this way, a novel class of basis functions known as the fractional logarithmic Chebyshev cardinal functions is introduced. An operational matrix regarding the Hadamard fractional integral of these fractional functions is derived and employed to develop an efficient numerical method for the provided problem. The established algorithm converts the solution of the primary fractional optimization problem into the solution of an algebraic system of equations by representing the state and control variables via the introduced fractional functions. Two test problems are considered to confirm the effectiveness of the suggested method. The derived outcomes of solving these examples highlight the satisfactory accuracy and efficiency of the designed method.

An integer mathematical model proposal for an agile external milkrun system

A milkrun system is a cyclical material delivery system that enables frequent small-batch deliveries with short lead times and low inventories. The literature classifies milkrun systems into external and internal categories. External milkrun systems in agile production environments, on the other hand, have not received much attention in the literature thus far. This study presents an integer mathematical model for the cost minimisation of external milkrun tours and discusses agile external milkrun systems. In addition to minimising costs, the proposed mathematical model can determine the types and number of milkrun vehicles. This is an important advantage for logistics professionals. The proposed model was coded using the Lingo software, and its effectiveness was shown using both a real-world problem and randomly generated test problems. The proposed model found the best solution in a few seconds for some real-world and most test problems. The results have shown that the proposed mathematical model for external milkrun systems in agile environments is useful.

Notes on the Practical Application of Nested Sampling: MultiNest, (Non)convergence, and Rectification

Nested sampling is a promising tool for Bayesian statistical analysis because it simultaneously performs parameter estimation and facilitates model comparison. MultiNest is one of the most popular nested sampling implementations, and has been applied to a wide variety of problems in the physical sciences. However, MultiNest results, like those of any sampling tool, can be unreliable, and accompanying convergence tests are a component of any analysis. Using analytically tractable test problems, I illustrate how MultiNest, when applied without rigorously chosen hyperparameters, (1) can produce systematically erroneous estimates of the Bayesian evidence, which are more significantly biased for problems of higher dimensionality; (2) can derive posterior estimates with errors on the order of ~100%; (3) can, particularly when sampling noisy likelihood functions, systematically underestimate posterior widths. Furthermore, I show how MultiNest, thanks to the advantageous speed at which it explores parameter space, can also be used to jump-start Markov chain Monte Carlo sampling or more rigorous nested sampling techniques, potentially accelerating more robust measurements of posterior distributions and Bayesian evidences, and overcoming the challenge of Markov chain Monte Carlo initialization.

Fuzzy-stochastic distributional robust approach for microalgae-based bio-plastic supply chain with new sustainability aspects

The current study offers a sustainable supply chain model that can be used to identify opportunities and constraints for the future of the microalgae-based bio-diesel and bio-plastic industry. The network’s carbon emissions are restricted by an uncertain constraint that allows for flexibility in controlling emissions as needed. The social constraints of the model provides a transparent depiction of labour needs in dynamic business landscape and foster women’s empowerment through a preference coefficient. The proposed model introduces a new upside risk measures under fuzzy-stochastic setting to quantify the upward financial risk. To characterize and manage the model uncertainty, a fuzzy-stochastic distributional robust model is presented. Credibility distribution is utilized to handle fuzzy uncertainty, while box and polyhedral uncertainty sets are included to handle stochastic uncertainty. After that, tractable deterministic formulations of the uncertain model for two ambiguity sets are presented to solve the model. Finally, a number of insights are obtained by means of five test problems that are implemented in Python and resolved by one exact and five meta-heuristic algorithms. Experiments indicate that, when faced with uncertainty, two robust models have produced more stable solutions (16.23% and 19.67% higher stability, respectively) than nominal stochastic model. Further, a 4.68% decline in total cost and 3.27% increment in bio-diesel production can be achieved by increasing bio-diesel conversion rate to 10%. In addition, a 9.09% downturn in total cost and 10.44% rise in bio-plastic production can be obtained by 10% increment in bio-plastic conversion rate. Moreover, 45.38% more woman labours can be recruited by increasing the preference coefficient from −0.2 to +0.2.

Two modified conjugate gradient methods for unconstrained optimization

Conjugate gradient (CG) methods are a popular class of iterative methods for solving linear systems of equations and nonlinear optimization problems. Motivated by the construction of some modern CG methods, we propose two modified CG methods, named DHS* and DPRP*. Under the strong Wolfe line search (SWLS), the two presented methods are proven to be sufficient descent and globally convergent. Preliminary numerical results show that the DHS* and DPRP* methods are effective for the given test problems.

Global Inference and Test for Eigensystems of Imaging Data Over Complicated Domains

A nonparametric approach for analyzing eigensystems of image data over a complex domain is novelly developed. The proposed estimators, which are based on bivariate splines, have both oracle efficiency of O p ( n − 1 / 2 ) , meaning they are asymptotically indistinguishable from estimators computed with true trajectories, and computational efficiency with more convenient spectrum decomposition forms. Under mild conditions, we derive consistency and weak convergence of our proposed estimators, which can be used to construct confidence intervals and simultaneous confidence corridors for any individual eigenvalue and eigenfunction, as well as design uniform inference procedures for eigensystems with a diverging number of components. In addition, we extend our method to two-sample test problems. Extensive simulation results provide strong support for the asymptotic theory, and the procedures are applied to two potential seawater temperature data sets to demonstrate its validity.

Beyond Neyman–Pearson: E-values enable hypothesis testing with a data-driven alpha

A standard practice in statistical hypothesis testing is to mention the P-value alongside the accept/reject decision. We show the advantages of mentioning an e-value instead. With P-values, it is not clear how to use an extreme observation (e.g. [Formula: see text]) for getting better frequentist decisions. With e-values it is straightforward, since they provide Type-I risk control in a generalized Neyman-Pearson setting with the decision task (a general loss function) determined post hoc, after observation of the data-thereby providing a handle on "roving [Formula: see text]'s." When Type-II risks are taken into consideration, the only admissible decision rules in the post hoc setting turn out to be e-value-based. Similarly, if the loss incurred when specifying a faulty confidence interval is not fixed in advance, standard confidence intervals and distributions may fail, whereas e-confidence sets and e-posteriors still provide valid risk guarantees. Sufficiently powerful e-values have by now been developed for a range of classical testing problems. We discuss the main challenges for wider development and deployment.

Drift Control of High-Dimensional Reflected Brownian Motion: A Computational Method Based on Neural Networks

Motivated by applications in queueing theory, we consider a stochastic control problem whose state space is the d-dimensional positive orthant. The controlled process Z evolves as a reflected Brownian motion whose covariance matrix is exogenously specified, as are its directions of reflection from the orthant’s boundary surfaces. A system manager chooses a drift vector [Formula: see text] at each time t based on the history of Z, and the cost rate at time t depends on both [Formula: see text] and [Formula: see text]. In our initial problem formulation, the objective is to minimize expected discounted cost over an infinite planning horizon, after which we treat the corresponding ergodic control problem. Extending the earlier work by Han et al. [Han J, Jentzen A, Weinan E (2018) Solving high-dimensional partial differential equations using deep learning. Proc. Natl. Acad. Sci. USA 115(34):8505–8510], we develop and illustrate a simulation-based computational method that relies heavily on deep neural network technology. For the test problems studied thus far, our method is accurate to within a fraction of 1% and is computationally feasible in dimensions up to at least [Formula: see text].

Universally consistent K-sample tests via dependence measures

The K-sample testing problem involves determining whether K groups of data points are each drawn from the same distribution. Analysis of variance is arguably the most classical method to test mean differences, along with several recent methods to test distributional differences. In this paper, we demonstrate the existence of a transformation that allows K-sample testing to be carried out using any dependence measure. Consequently, universally consistent K-sample testing can be achieved using a universally consistent dependence measure, such as distance correlation and the Hilbert–Schmidt independence criterion. This enables a wide range of dependence measures to be easily applied to K-sample testing.

Rank-transformed subsampling: inference for multiple data splitting and exchangeable p-values

Abstract Many testing problems are readily amenable to randomized tests, such as those employing data splitting. However, despite their usefulness in principle, randomized tests have obvious drawbacks. Firstly, two analyses of the same dataset may lead to different results. Secondly, the test typically loses power because it does not fully utilize the entire sample. As a remedy to these drawbacks, we study how to combine the test statistics or p-values resulting from multiple random realizations, such as through random data splits. We develop rank-transformed subsampling as a general method for delivering large-sample inference about the combined statistic or p-value under mild assumptions. We apply our methodology to a wide range of problems, including testing unimodality in high-dimensional data, testing goodness-of-fit of parametric quantile regression models, testing no direct effect in a sequentially randomized trial and calibrating cross-fit double machine learning confidence intervals. In contrast to existing p-value aggregation schemes that can be highly conservative, our method enjoys Type I error control that asymptotically approaches the nominal level. Moreover, compared to using the ordinary subsampling, we show that our rank transform can remove the first-order bias in approximating the null under alternatives and greatly improve power.