In this paper we present two approaches for estimating matrix-inverse quadratic forms , where A is a symmetric positive definite matrix of order n, and . Using the first, analytic approach, we establish two families of estimates which are convenient for matrices with small condition number. Based on the second, heuristic approach, we derive two families of estimates which are suitable for matrices when vector x is close enough to an eigenvector. The low complexity and stability of the estimates is proved. Several numerical results illustrating the effectiveness of the methods are presented.
本文提出了估计矩阵逆二次型 xTA-1x 的两种方法，其中 A 是阶数为 n 的对称正定矩阵，x∈Rn。利用第一种分析方法，我们建立了两个估计族，这对条件数较小的矩阵很方便。基于第二种启发式方法，我们得出了两个估计族，当向量 x 与特征向量足够接近时，这两个估计族适用于矩阵。我们证明了这些估计值的低复杂性和稳定性。我们还给出了一些数值结果，以说明这些方法的有效性。
{"title":"Efficient estimates for matrix-inverse quadratic forms","authors":"Emmanouil Bizas , Marilena Mitrouli , Ondřej Turek","doi":"10.1016/j.apnum.2024.01.013","DOIUrl":"10.1016/j.apnum.2024.01.013","url":null,"abstract":"<div><div>In this paper we present two approaches for estimating matrix-inverse quadratic forms <span><math><msup><mrow><mi>x</mi></mrow><mrow><mi>T</mi></mrow></msup><msup><mrow><mi>A</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>x</mi></math></span>, where <em>A</em><span> is a symmetric positive definite matrix of order </span><em>n</em>, and <span><math><mi>x</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span><span>. Using the first, analytic approach, we establish two families of estimates which are convenient for matrices with small condition number. Based on the second, heuristic approach, we derive two families of estimates which are suitable for matrices when vector </span><em>x</em><span> is close enough to an eigenvector. The low complexity and stability of the estimates is proved. Several numerical results illustrating the effectiveness of the methods are presented.</span></div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"208 ","pages":"Pages 76-91"},"PeriodicalIF":2.2,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139558043","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-01DOI: 10.1016/j.apnum.2024.03.002
B. Ali Ibrahimoglu
Interpolation by polynomials on equispaced points is not always convergent due to the Runge phenomenon, and also, the interpolation process is exponentially ill-conditioned. By taking advantage of the optimality of the interpolation processes on the Chebyshev-Lobatto nodes, one of the best strategies to defeat the Runge phenomenon is to use the mock-Chebyshev nodes for polynomial interpolation. Mock-Chebyshev nodes asymptotically follow the Chebyshev distribution, and they are selected from a sufficiently large set of equispaced nodes. However, there are few studies in the literature regarding the computation of these points.
In a recent paper [1], we have introduced a fast algorithm for computing the mock-Chebyshev nodes for a given set of Chebyshev-Lobatto points using the distance between each pair of consecutive points. In this study, we propose a modification of the algorithm by changing the function to compute the quotient of the distance and show that this modified algorithm is also fast and stable; and gives a more accurate grid satisfying the conditions of a mock-Chebyshev grid with the complexity being . Some numerical experiments using the points obtained by this modified algorithm are given to show its effectiveness and numerical results are also provided. A bivariate generalization of the mock-Chebyshev nodes to the Padua interpolation points is discussed.
{"title":"A new fast algorithm for computing the mock-Chebyshev nodes","authors":"B. Ali Ibrahimoglu","doi":"10.1016/j.apnum.2024.03.002","DOIUrl":"10.1016/j.apnum.2024.03.002","url":null,"abstract":"<div><div>Interpolation by polynomials on equispaced points is not always convergent due to the Runge phenomenon, and also, the interpolation process is exponentially ill-conditioned. By taking advantage of the optimality of the interpolation processes on the Chebyshev-Lobatto nodes, one of the best strategies to defeat the Runge phenomenon is to use the mock-Chebyshev nodes for polynomial interpolation. Mock-Chebyshev nodes asymptotically follow the Chebyshev distribution, and they are selected from a sufficiently large set of equispaced nodes. However, there are few studies in the literature regarding the computation of these points.</div><div>In a recent paper <span><span>[1]</span></span>, we have introduced a fast algorithm for computing the mock-Chebyshev nodes for a given set of <span><math><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span> Chebyshev-Lobatto points using the distance between each pair of consecutive points. In this study, we propose a modification of the algorithm by changing the function to compute the quotient of the distance and show that this modified algorithm is also fast and stable; and gives a more accurate grid satisfying the conditions of a mock-Chebyshev grid with the complexity being <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span><span>. Some numerical experiments using the points obtained by this modified algorithm are given to show its effectiveness and numerical results are also provided. A bivariate generalization of the mock-Chebyshev nodes to the Padua interpolation points is discussed.</span></div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"208 ","pages":"Pages 246-255"},"PeriodicalIF":2.2,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140125652","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-01DOI: 10.1016/j.apnum.2024.10.010
Mariano De Leo , Juan Pablo Borgna , Cristian Huenchul
This article addresses both the existence and properties of non-trivial solutions for a system of coupled Ginzburg-Landau equations derived from nematic-superconducting models. Its main goal is to provide a thorough numerical description of the region in the parameter space containing solutions that behave as a mixed (non trivial) nematic-superconducting state along with a rigorous proof for the existence of this region. More precisely, the rigorous approach establishes that the parameter space is divided into two regions with qualitatively different properties, according to the magnitude of the coupling constant: for small values (weak coupling), there is a unique non-trivial solution, and for large values (strong coupling), only trivial solutions exist. In addition, using a shooting method-based numerical approach, the profiles for the nematic and superconducting components of the non trivial solution are given, together with an algorithm computing the transition values representing the boundaries for the weak coupling region: from superconducting to mixed, and from mixed to nematic. Finally, numerical evidence is given for the existence of a third region, related to neither a small nor a strong coupling parameter (medium coupling) for which multiple non trivial solutions exist.
{"title":"Non trivial solutions for a system of coupled Ginzburg-Landau equations","authors":"Mariano De Leo , Juan Pablo Borgna , Cristian Huenchul","doi":"10.1016/j.apnum.2024.10.010","DOIUrl":"10.1016/j.apnum.2024.10.010","url":null,"abstract":"<div><div>This article addresses both the existence and properties of non-trivial solutions for a system of coupled Ginzburg-Landau equations derived from nematic-superconducting models. Its main goal is to provide a thorough numerical description of the region in the parameter space containing solutions that behave as a mixed (non trivial) nematic-superconducting state along with a rigorous proof for the existence of this region. More precisely, the rigorous approach establishes that the parameter space is divided into two regions with qualitatively different properties, according to the magnitude of the coupling constant: for small values (weak coupling), there is a unique non-trivial solution, and for large values (strong coupling), only trivial solutions exist. In addition, using a shooting method-based numerical approach, the profiles for the nematic and superconducting components of the non trivial solution are given, together with an algorithm computing the transition values representing the boundaries for the weak coupling region: from superconducting to mixed, and from mixed to nematic. Finally, numerical evidence is given for the existence of a third region, related to neither a small nor a strong coupling parameter (medium coupling) for which multiple non trivial solutions exist.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"208 ","pages":"Pages 271-289"},"PeriodicalIF":2.2,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143141234","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-01DOI: 10.1016/j.apnum.2024.05.014
Protopapas Eleftherios , Vafeas Panayiotis , Hadjinicolaou Maria
The susceptibility of the human Red Blood Cells (RBCs) under the action of magnetic fields, either serves as a biomarker in medical tests, e.g.. Magnetic Resonance Imaging, Nuclear Magnetic Resonance, Magnetoencephalography, or it is used in diagnostic and therapeutical processes, e.g.. magnetophoresis for cell sorting. In the present manuscript we provide analytical expressions for the magnetic potential and the magnetic field strength vector, when a magnetic field is applied to a RBC, modeled as a two-layered inverted spheroid. We introduce this way in the model the biconcave shape of the RBC and its structure (membrane and cytocol) in a more realistic representation, as until now, the RBC's shape was considered either as a sphere or a spheroid. The solution inside the RBC is obtained in R-separable form in terms of Legendre functions of the first and of the second kind and cyclic trigonometric functions, by applying appropriate boundary conditions on each layer. Our results reveal a non-uniform magnetic field inside the RBC. Parametric study of the solution, for various values of the physical properties of the RBC, is also provided, demonstrating the diamagnetic or the paramagnetic property of the RBC, which is strongly related to the health condition of the blood. The obtained solution may also serve for the justification of experimental results.
{"title":"A mathematical model for studying the Red Blood Cell magnetic susceptibility","authors":"Protopapas Eleftherios , Vafeas Panayiotis , Hadjinicolaou Maria","doi":"10.1016/j.apnum.2024.05.014","DOIUrl":"10.1016/j.apnum.2024.05.014","url":null,"abstract":"<div><div><span><span>The susceptibility of the human Red Blood Cells (RBCs) under the action of magnetic fields, either serves as a biomarker in medical tests, e.g.. Magnetic Resonance Imaging, Nuclear Magnetic Resonance, </span>Magnetoencephalography, or it is used in diagnostic and therapeutical processes, e.g.. magnetophoresis for cell sorting. In the present manuscript we provide analytical expressions for the magnetic potential and the magnetic field </span>strength vector, when a magnetic field is applied to a RBC, modeled as a two-layered inverted spheroid. We introduce this way in the model the biconcave shape of the RBC and its structure (membrane and cytocol) in a more realistic representation, as until now, the RBC's shape was considered either as a sphere or a spheroid. The solution inside the RBC is obtained in R-separable form in terms of Legendre functions of the first and of the second kind and cyclic trigonometric functions, by applying appropriate boundary conditions on each layer. Our results reveal a non-uniform magnetic field inside the RBC. Parametric study of the solution, for various values of the physical properties of the RBC, is also provided, demonstrating the diamagnetic or the paramagnetic property of the RBC, which is strongly related to the health condition of the blood. The obtained solution may also serve for the justification of experimental results.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"208 ","pages":"Pages 356-365"},"PeriodicalIF":2.2,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141138058","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-01DOI: 10.1016/j.apnum.2024.09.030
Savio B. Rodrigues , Giovanni Belloni Fernandes Braga , Marcello Augusto Faraco de Medeiros
In numerical time-integration with implicit-explicit (IMEX) methods, a within-step adaptable decomposition called residual balanced (RB) decomposition is introduced. This new decomposition maintains time-stepping accuracy even when the implicit equation is only roughly approximated. This novel property is possible because a suitable modification of the traditional IMEX algorithm allows the remaining residual to be seamlessly transferred to the explicit part of the decomposition. The RB decomposition allows an early termination of iterations while preserving time-step accuracy. It can gain computational efficiency by exploring the trade-off between the computational effort placed in the iterative solver and the numerically stable step size. We develop a rigorous theory showing that RB maintains the order of singly diagonally implicit schemes. In computational experiments we show that, in many cases, RB-IMEX reduces the number of iterations when compared with the traditional IMEX method. It is often more stable also. The stability of RB-IMEX is studied using a model containing diffusion and dispersion; in this way, one can visualize how the stability region changes as a function of the number of iterations. Here, computational experiments use ESDIRK schemes for a stiff reaction-advection-diffusion equation, for a Navier-Stokes simulation with acoustic stiffness, and for a semi-implicit implementation of Burguers equation.
{"title":"The residual balanced IMEX decomposition for singly-diagonally-implicit schemes","authors":"Savio B. Rodrigues , Giovanni Belloni Fernandes Braga , Marcello Augusto Faraco de Medeiros","doi":"10.1016/j.apnum.2024.09.030","DOIUrl":"10.1016/j.apnum.2024.09.030","url":null,"abstract":"<div><div>In numerical time-integration with implicit-explicit (IMEX) methods, a within-step adaptable decomposition called residual balanced (RB) decomposition is introduced. This new decomposition maintains time-stepping accuracy even when the implicit equation is only roughly approximated. This novel property is possible because a suitable modification of the traditional IMEX algorithm allows the remaining residual to be seamlessly transferred to the explicit part of the decomposition. The RB decomposition allows an early termination of iterations while preserving time-step accuracy. It can gain computational efficiency by exploring the trade-off between the computational effort placed in the iterative solver and the numerically stable step size. We develop a rigorous theory showing that RB maintains the order of singly diagonally implicit schemes. In computational experiments we show that, in many cases, RB-IMEX reduces the number of iterations when compared with the traditional IMEX method. It is often more stable also. The stability of RB-IMEX is studied using a model containing diffusion and dispersion; in this way, one can visualize how the stability region changes as a function of the number of iterations. Here, computational experiments use ESDIRK schemes for a stiff reaction-advection-diffusion equation, for a Navier-Stokes simulation with acoustic stiffness, and for a semi-implicit implementation of Burguers equation.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"208 ","pages":"Pages 58-78"},"PeriodicalIF":2.2,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143097229","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-01DOI: 10.1016/j.apnum.2024.01.019
Emmanouil Lardas, Marilena Mitrouli
The aim of this work is to provide a complete list of all the possible values that the first six pivots of an Hadamard matrix of order 20 can take. This is accomplished by determining the possible values of certain minors of such matrices, in combination with the fact that the pivots can be computed in terms of these minors. We extend known results, by giving a different proof for the complete list of possible values for each of the first five pivots, as well as determining the complete list of possible values for the sixth pivot. By using a computational approach to search for new pivot patterns, we have also found at least 1246 different pivot patterns of Hadamard matrices of order 20.
{"title":"On the growth factor of Hadamard matrices of order 20","authors":"Emmanouil Lardas, Marilena Mitrouli","doi":"10.1016/j.apnum.2024.01.019","DOIUrl":"10.1016/j.apnum.2024.01.019","url":null,"abstract":"<div><div>The aim of this work is to provide a complete list of all the possible values that the first six pivots of an Hadamard matrix of order 20 can take. This is accomplished by determining the possible values of certain minors of such matrices, in combination with the fact that the pivots can be computed in terms of these minors. We extend known results, by giving a different proof for the complete list of possible values for each of the first five pivots, as well as determining the complete list of possible values for the sixth pivot. By using a computational approach to search for new pivot patterns, we have also found at least 1246 different pivot patterns of Hadamard matrices of order 20.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"208 ","pages":"Pages 310-316"},"PeriodicalIF":2.2,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139666010","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-01DOI: 10.1016/j.apnum.2024.10.004
Lin Yang, Qilong Zhai, Ran Zhang
In this paper, we develop a weak Galerkin (WG) finite element scheme for the Stokes interface problems with curved interface. The conventional numerical schemes rely on the use of straight segments to approximate the curved interface and the accuracy is limited by geometric errors. Hence in our method, we directly construct the weak Galerkin finite element space on the curved cells to avoid geometric errors. For the integral calculation on curved cells, we employ non-affine transformations to map curved cells onto the reference element. The optimal error estimates are obtained in both the energy norm and the norm. A series of numerical experiments are provided to validate the efficiency of the proposed WG method.
{"title":"The weak Galerkin finite element method for Stokes interface problems with curved interface","authors":"Lin Yang, Qilong Zhai, Ran Zhang","doi":"10.1016/j.apnum.2024.10.004","DOIUrl":"10.1016/j.apnum.2024.10.004","url":null,"abstract":"<div><div>In this paper, we develop a weak Galerkin (WG) finite element scheme for the Stokes interface problems with curved interface. The conventional numerical schemes rely on the use of straight segments to approximate the curved interface and the accuracy is limited by geometric errors. Hence in our method, we directly construct the weak Galerkin finite element space on the curved cells to avoid geometric errors. For the integral calculation on curved cells, we employ non-affine transformations to map curved cells onto the reference element. The optimal error estimates are obtained in both the energy norm and the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> norm. A series of numerical experiments are provided to validate the efficiency of the proposed WG method.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"208 ","pages":"Pages 98-122"},"PeriodicalIF":2.2,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143141566","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-01DOI: 10.1016/j.apnum.2023.12.006
N. Krejić , E.H.M. Krulikovski , M. Raydan
To solve convex constrained minimization problems, that also include a cardinality constraint, we propose an augmented Lagrangian scheme combined with alternating projection ideas. Optimization problems that involve a cardinality constraint are NP-hard mathematical programs and typically very hard to solve approximately. Our approach takes advantage of a recently developed and analyzed continuous formulation that relaxes the cardinality constraint. Based on that formulation, we solve a sequence of smooth convex constrained minimization problems, for which we use projected gradient-type methods. In our setting, the convex constraint region can be written as the intersection of a finite collection of convex sets that are easy and inexpensive to project. We apply our approach to a variety of over and under determined constrained linear least-squares problems, with both synthetic and real data that arise in variable selection, and demonstrate its effectiveness.
{"title":"An augmented Lagrangian approach for cardinality constrained minimization applied to variable selection problems","authors":"N. Krejić , E.H.M. Krulikovski , M. Raydan","doi":"10.1016/j.apnum.2023.12.006","DOIUrl":"10.1016/j.apnum.2023.12.006","url":null,"abstract":"<div><div>To solve convex constrained minimization problems, that also include a cardinality constraint, we propose an augmented Lagrangian scheme combined with alternating projection ideas. Optimization problems that involve a cardinality constraint are NP-hard mathematical programs and typically very hard to solve approximately. Our approach takes advantage of a recently developed and analyzed continuous formulation that relaxes the cardinality constraint. Based on that formulation, we solve a sequence of smooth convex constrained minimization problems, for which we use projected gradient-type methods. In our setting, the convex constraint region can be written as the intersection of a finite collection of convex sets that are easy and inexpensive to project. We apply our approach to a variety of over and under determined constrained linear least-squares problems, with both synthetic and real data that arise in variable selection, and demonstrate its effectiveness.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"208 ","pages":"Pages 284-296"},"PeriodicalIF":2.2,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138820449","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-01DOI: 10.1016/j.apnum.2023.12.016
Ioannis Iordanis, Christos Koukouvinos, Iliana Silou
In this paper, Latin Hypercube Sampling (LHS) method is compared as per its effectiveness in supervised machine learning procedures. Employing LHS saves computer's processing time and in conjunction with Latin hypercube design properties and space filling ability, is considered as one of the most advanced mechanisms in terms of sampling. Although more data usually deliver better results, when using LHS techniques, same quality outputs can be produced with less data and, as a result, storage cost and training time are reduced. Conditioned Latin Hypercube Sampling (cLHS) is one of those techniques, successfully performing in supervised machine learning tasks. Unfortunately, the minimum sufficient training dataset size cannot be known in advance. In this case, progressive sampling is recommended since it begins with a small sample and progressively increases its size until model accuracy no longer improves. Combining Latin hypercube sampling and the idea of sequentially incrementing sampling, we test Progressive Latin Hypercube Sampling (PLHS) while monitoring the performance of the sampling-based training as the sample size grows. PLHS and cLHS algorithms are applied in datasets with discrete variables securing that each sample is provided with the Latin hypercube design properties and preserves the principal ability of LHS for space filling, as illustrated in respective sample projecting diagrams. The performance of the above LHS methods in supervised machine learning is evaluated by the degree of training of the model, which is certified through the accuracy of the produced confusion matrices in test files. The results from the use of the above Latin Hypercube Sampling techniques compared against benchmark sampling method empirically prove that machine learning training process becomes less costfull, while remaining reliable.
{"title":"On the efficacy of conditioned and progressive Latin hypercube sampling in supervised machine learning","authors":"Ioannis Iordanis, Christos Koukouvinos, Iliana Silou","doi":"10.1016/j.apnum.2023.12.016","DOIUrl":"10.1016/j.apnum.2023.12.016","url":null,"abstract":"<div><div><span>In this paper, Latin Hypercube Sampling<span> (LHS) method is compared as per its effectiveness in supervised machine learning procedures. Employing LHS saves computer's processing time and in conjunction with Latin hypercube design properties and space filling ability, is considered as one of the most advanced mechanisms in terms of sampling. Although more data usually deliver better results, when using LHS techniques, same quality outputs can be produced with less data and, as a result, </span></span>storage cost<span> and training time are reduced. Conditioned Latin Hypercube Sampling (cLHS) is one of those techniques, successfully performing in supervised machine learning tasks. Unfortunately, the minimum sufficient training dataset size cannot be known in advance. In this case, progressive sampling is recommended since it begins with a small sample and progressively increases its size until model accuracy no longer improves. Combining Latin hypercube sampling and the idea of sequentially incrementing sampling, we test Progressive Latin Hypercube Sampling (PLHS) while monitoring the performance of the sampling-based training as the sample size grows. PLHS and cLHS algorithms are applied in datasets with discrete variables securing that each sample is provided with the Latin hypercube design properties and preserves the principal ability of LHS for space filling, as illustrated in respective sample projecting diagrams. The performance of the above LHS methods in supervised machine learning is evaluated by the degree of training of the model, which is certified through the accuracy of the produced confusion matrices in test files. The results from the use of the above Latin Hypercube Sampling techniques compared against benchmark sampling method empirically prove that machine learning training process becomes less costfull, while remaining reliable.</span></div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"208 ","pages":"Pages 256-270"},"PeriodicalIF":2.2,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139094504","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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