Many problems arise from science and engineering which can be expressed as an unconstrained minimization problem. Therefore, developing numerical methods to obtain their approximate solutions has become necessary, as their exact solutions cannot be obtained. Several such numerical methods have been proposed, with the conjugate gradient (CG) method stands out to be more efficient in handling this type of problem, due to its nice theoretical structure and promising numerical result. In this article, we consider a CG algorithm based on a generalized conjugacy condition. The new CG parameter is selected to ensure a convex combination of modified version of the Polak, Ribière-Polyak (PRP) and Fletcher-Revees (FR) CG algorithms. The numerical implementation adopts inexact line search which revealed that the scheme is robust when compared with some known efficient algorithms in literature. Furthermore, the theoretical analysis shows that the proposed method converge globally. The method is also applicable to solve three degree of freedom motion control robotic model.
科学和工程中出现的许多问题都可以表示为无约束最小化问题。因此，发展数值方法来获得它们的近似解是必要的，因为它们的精确解不能得到。目前已经提出了几种这样的数值方法，其中共轭梯度法（CG）由于其良好的理论结构和令人满意的数值结果，在处理这类问题时更为有效。在本文中，我们考虑了一种基于广义共轭条件的CG算法。选择新的CG参数是为了确保Polak、ribire - polyak （PRP）和Fletcher-Revees (FR) CG算法的改进版本的凸组合。数值实现采用非精确直线搜索，与文献中已知的一些高效算法相比，该算法具有较强的鲁棒性。理论分析表明，该方法具有全局收敛性。该方法同样适用于求解三自由度运动控制机器人模型。
{"title":"An efficient matrix free optimization algorithm combining a revised PRP and FR-CG type methods with application to robotics","authors":"Nasiru Salihu , Poom Kumam , Aliyu Muhammed Awwal , Mathew Remilekun Odekunle , Thidaporn Seangwattana","doi":"10.1016/j.cam.2026.117378","DOIUrl":"10.1016/j.cam.2026.117378","url":null,"abstract":"<div><div>Many problems arise from science and engineering which can be expressed as an unconstrained minimization problem. Therefore, developing numerical methods to obtain their approximate solutions has become necessary, as their exact solutions cannot be obtained. Several such numerical methods have been proposed, with the conjugate gradient (CG) method stands out to be more efficient in handling this type of problem, due to its nice theoretical structure and promising numerical result. In this article, we consider a CG algorithm based on a generalized conjugacy condition. The new CG parameter is selected to ensure a convex combination of modified version of the Polak, Ribière-Polyak (PRP) and Fletcher-Revees (FR) CG algorithms. The numerical implementation adopts inexact line search which revealed that the scheme is robust when compared with some known efficient algorithms in literature. Furthermore, the theoretical analysis shows that the proposed method converge globally. The method is also applicable to solve three degree of freedom motion control robotic model.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"483 ","pages":"Article 117378"},"PeriodicalIF":2.6,"publicationDate":"2026-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146079949","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 : 2026-01-22DOI: 10.1016/j.cam.2026.117377
Ruyun Chen, Hong Du
This work develops two matrix-based quadrature rules to compute the integrals containing products of two Bessel functions. By reformulating these integrals into a matrix framework and employing low-order derivatives of Bessel functions of the first kind in combination with integration by parts, we construct both a matrix-based asymptotic rule and a matrix-based Filon-type rule. Numerical experiments confirm the theoretical analysis and highlight the efficiency of the proposed rules.
{"title":"Efficient matrix-based quadrature rules for oscillatory integrals with products of two Bessel functions","authors":"Ruyun Chen, Hong Du","doi":"10.1016/j.cam.2026.117377","DOIUrl":"10.1016/j.cam.2026.117377","url":null,"abstract":"<div><div>This work develops two matrix-based quadrature rules to compute the integrals containing products of two Bessel functions. By reformulating these integrals into a matrix framework and employing low-order derivatives of Bessel functions of the first kind in combination with integration by parts, we construct both a matrix-based asymptotic rule and a matrix-based Filon-type rule. Numerical experiments confirm the theoretical analysis and highlight the efficiency of the proposed rules.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"483 ","pages":"Article 117377"},"PeriodicalIF":2.6,"publicationDate":"2026-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146079935","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}
This paper develops a Frobenius series framework for the stochastic analysis of second–order random differential equations of the formwhere the damping coefficient A(t) is a positive stochastic process and the initial conditions are square–integrable random variables. Assuming mean–square analyticity of A(t) in a neighborhood of the initial time, we establish existence and uniqueness of the solution in L2(Ω) and derive exponentially convergent truncation error bounds for the associated Frobenius expansion. The resulting series representation enables the numerical approximation of the probability density function of Y(t) via Monte Carlo simulation. To improve computational efficiency, a control variates strategy is incorporated for variance reduction.
A comprehensive numerical study is conducted for a broad family of positive, right–skewed damping distributions, including the Lindley, XLindley, New XLindley (NXLD), Gamma–Lindley, Inverse–Lindley, Truncated–Lindley, Log–Lindley, and a newly proposed Mixed Lindley–Uniform model. The simulations illustrate how different tail behaviors and boundedness properties of the damping coefficient influence the stochastic dynamics and the accuracy of density estimation. Finally, stylized applications to option pricing and Value–at–Risk estimation are presented to illustrate how the Frobenius–based framework and control variates methodology can be embedded within standard uncertainty quantification workflows. Overall, the proposed approach provides a flexible and computationally efficient tool for the analysis of randomly damped dynamical systems.
{"title":"Fröbenius expansions for second-order random differential equations: Stochastic analysis and applications to Lindley-type damping models","authors":"Halim Zeghdoudi , Mohamed Amine Kerker , Elif Boduroglu","doi":"10.1016/j.cam.2026.117379","DOIUrl":"10.1016/j.cam.2026.117379","url":null,"abstract":"<div><div>This paper develops a Frobenius series framework for the stochastic analysis of second–order random differential equations of the form<span><span><span><math><mrow><mover><mi>Y</mi><mo>¨</mo></mover><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>+</mo><mi>A</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mover><mi>Y</mi><mo>˙</mo></mover><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn><mo>,</mo></mrow></math></span></span></span>where the damping coefficient <em>A</em>(<em>t</em>) is a positive stochastic process and the initial conditions are square–integrable random variables. Assuming mean–square analyticity of <em>A</em>(<em>t</em>) in a neighborhood of the initial time, we establish existence and uniqueness of the solution in L<sup>2</sup>(Ω) and derive exponentially convergent truncation error bounds for the associated Frobenius expansion. The resulting series representation enables the numerical approximation of the probability density function of <em>Y</em>(<em>t</em>) via Monte Carlo simulation. To improve computational efficiency, a control variates strategy is incorporated for variance reduction.</div><div>A comprehensive numerical study is conducted for a broad family of positive, right–skewed damping distributions, including the Lindley, XLindley, New XLindley (NXLD), Gamma–Lindley, Inverse–Lindley, Truncated–Lindley, Log–Lindley, and a newly proposed Mixed Lindley–Uniform model. The simulations illustrate how different tail behaviors and boundedness properties of the damping coefficient influence the stochastic dynamics and the accuracy of density estimation. Finally, stylized applications to option pricing and Value–at–Risk estimation are presented to illustrate how the Frobenius–based framework and control variates methodology can be embedded within standard uncertainty quantification workflows. Overall, the proposed approach provides a flexible and computationally efficient tool for the analysis of randomly damped dynamical systems.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"483 ","pages":"Article 117379"},"PeriodicalIF":2.6,"publicationDate":"2026-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146079947","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 : 2026-01-21DOI: 10.1016/j.cam.2026.117376
Alessandra Aimi , Mattia Alex Leoni , Sara Remogna
The paper deals with the numerical solution of Cauchy singular integral equations, by means of spline quasi interpolating projectors and their variant quasi2-interpolating projectors, within a collocation approach which takes into account the particular features of the problem at hand. Several numerical results, including those related to the application of the presented approach to an extended model problem, validate the proposed error estimates.
{"title":"Spline quasi-interpolating and quasi2-interpolating projectors for the numerical solution of Cauchy singular integral equations","authors":"Alessandra Aimi , Mattia Alex Leoni , Sara Remogna","doi":"10.1016/j.cam.2026.117376","DOIUrl":"10.1016/j.cam.2026.117376","url":null,"abstract":"<div><div>The paper deals with the numerical solution of Cauchy singular integral equations, by means of spline quasi interpolating projectors and their variant quasi<sup>2</sup>-interpolating projectors, within a collocation approach which takes into account the particular features of the problem at hand. Several numerical results, including those related to the application of the presented approach to an extended model problem, validate the proposed error estimates.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"483 ","pages":"Article 117376"},"PeriodicalIF":2.6,"publicationDate":"2026-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146079948","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 : 2026-01-21DOI: 10.1016/j.cam.2026.117375
Yonglei Fang , Changying Liu , Xiong You
This paper is devoted to the effective integration of general second-order highly oscillatory systems. By approximating the nonlinear integrals appeared in the matrix-variation-of-constants formula with the Birkhoff-Hermite interpolating polynomial, new ERKN integrators (BHERKN) are obtained. The symmetry and nonlinear stability of the BHERKN integrators are analyzed. By energy analysis, the BHERKN integrators are shown to converge with an arbitrary high-order. Finally, numerical experiments are reported to show the high efficiency, accuracy and robustness of our new methods.
{"title":"Novel Birkhoff-hermite ERKN methods for solving general second-order highly oscillatory systems","authors":"Yonglei Fang , Changying Liu , Xiong You","doi":"10.1016/j.cam.2026.117375","DOIUrl":"10.1016/j.cam.2026.117375","url":null,"abstract":"<div><div>This paper is devoted to the effective integration of general second-order highly oscillatory systems. By approximating the nonlinear integrals appeared in the matrix-variation-of-constants formula with the Birkhoff-Hermite interpolating polynomial, new ERKN integrators (BHERKN) are obtained. The symmetry and nonlinear stability of the BHERKN integrators are analyzed. By energy analysis, the BHERKN integrators are shown to converge with an arbitrary high-order. Finally, numerical experiments are reported to show the high efficiency, accuracy and robustness of our new methods.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"483 ","pages":"Article 117375"},"PeriodicalIF":2.6,"publicationDate":"2026-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146079950","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 : 2026-01-20DOI: 10.1016/j.cam.2026.117373
Wenfei Cao , Xicui Peng , Yang Chen , Jiahui Ji
In this paper, we study the tensor recovery problem from linear measurements corrupted by the ℓ1-bounded noise plus the adversarial noise with sparsity ratio ω. To handle this problem, we propose a novel least-absolute-deviation (LAD) loss minimization model based on low tubal-rank tensor decomposition. For the requirement of theoretical studies, we extend the mixed ℓ1/ℓ2-RIP and the ω-robustness to the tensor case, i.e., ℓ1/ℓ2-t-RIP and ω-t-robustness. Then leveraging these tools, we establish a reliable recovery guarantee for the proposed model, showing that when the sampling complexity reaches , the model’s optimal solution can robustly recover the original low tubal-rank tensor for any ω < 0.239. Moreover, we develop a subgradient descent algorithm to solve the proposed model and prove that it achieves geometrical convergence under appropriate initialization conditions. Finally, extensive experiments on the synthetic tensors and real video datasets are conducted to validate the exactness of the established theories and demonstrate the effectiveness of the proposed approach.
{"title":"Low tubal-rank tensor recovery with adversarial sparse noises","authors":"Wenfei Cao , Xicui Peng , Yang Chen , Jiahui Ji","doi":"10.1016/j.cam.2026.117373","DOIUrl":"10.1016/j.cam.2026.117373","url":null,"abstract":"<div><div>In this paper, we study the tensor recovery problem from linear measurements corrupted by the ℓ<sub>1</sub>-bounded noise plus the adversarial noise with sparsity ratio <em>ω</em>. To handle this problem, we propose a novel least-absolute-deviation (LAD) loss minimization model based on low tubal-rank tensor decomposition. For the requirement of theoretical studies, we extend the mixed ℓ<sub>1</sub>/ℓ<sub>2</sub>-RIP and the <em>ω</em>-<em>robustness</em> to the tensor case, i.e., ℓ<sub>1</sub>/ℓ<sub>2</sub>-<em>t</em>-RIP and <em>ω</em>-<em>t</em>-<em>robustness</em>. Then leveraging these tools, we establish a reliable recovery guarantee for the proposed model, showing that when the sampling complexity reaches <span><math><mrow><mi>O</mi><mo>(</mo><mrow><mo>(</mo><msub><mi>n</mi><mn>1</mn></msub><mo>+</mo><msub><mi>n</mi><mn>2</mn></msub><mo>+</mo><mn>1</mn><mo>)</mo></mrow><msub><mi>n</mi><mn>3</mn></msub><mi>r</mi><mo>)</mo></mrow></math></span>, the model’s optimal solution can robustly recover the original low tubal-rank tensor for any <em>ω</em> < 0.239. Moreover, we develop a subgradient descent algorithm to solve the proposed model and prove that it achieves geometrical convergence under appropriate initialization conditions. Finally, extensive experiments on the synthetic tensors and real video datasets are conducted to validate the exactness of the established theories and demonstrate the effectiveness of the proposed approach.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"483 ","pages":"Article 117373"},"PeriodicalIF":2.6,"publicationDate":"2026-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146079951","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 : 2026-01-20DOI: 10.1016/j.cam.2026.117374
Bohdan Datsko , Vasyl Gafiychuk
Different types of instability and resulting pattern formation in a two-component incommensurate fractional reaction-diffusion system are studied. Considered system sets the possibility of continuous transitions between classical systems with integer derivatives. As a result, the presented investigations provide a better understanding of the instability conditions and nonlinear solutions not only in systems with fractional-order derivatives but also in classical two-component elliptic, parabolic, and hyperbolic systems, as well as those of a mixed type. Based on the linear stability analysis, the computer simulation of nonlinear dynamics in the fractional two-component system with cubic-like nonlinearity has been performed, demonstrating the rich diversity of pattern formation phenomena.
{"title":"Instabilities and pattern formation in fractional incommensurate activator-inhibitor reaction-diffusion systems","authors":"Bohdan Datsko , Vasyl Gafiychuk","doi":"10.1016/j.cam.2026.117374","DOIUrl":"10.1016/j.cam.2026.117374","url":null,"abstract":"<div><div>Different types of instability and resulting pattern formation in a two-component incommensurate fractional reaction-diffusion system are studied. Considered system sets the possibility of continuous transitions between classical systems with integer derivatives. As a result, the presented investigations provide a better understanding of the instability conditions and nonlinear solutions not only in systems with fractional-order derivatives but also in classical two-component elliptic, parabolic, and hyperbolic systems, as well as those of a mixed type. Based on the linear stability analysis, the computer simulation of nonlinear dynamics in the fractional two-component system with cubic-like nonlinearity has been performed, demonstrating the rich diversity of pattern formation phenomena.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"483 ","pages":"Article 117374"},"PeriodicalIF":2.6,"publicationDate":"2026-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039577","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 : 2026-01-19DOI: 10.1016/j.cam.2026.117370
E. Aourir , H. Laeli Dastjerdi , M. Oudani
This work presents a new algorithm for solving a kind of Volterra delay integral equations of the third kind (VDIEs). Using the Tau method and generalized polynomial bases, our developed method is a robust approach for solving these equations. Specifically, we employ simple matrix operations to enhance the Tau approach. The underlying strategy leverages orthogonal polynomial bases to change the original equation into a matrix-vector form. Such a transformation makes the third-kind VDIEs easier to handle by turning them into a set of algebraic equations. Importantly, this method exhibits good stability, reduces memory usage, and is computationally cost-effective. The paper details the algorithm’s formulation and shows its capability to provide approximate polynomial solutions. We perform a thorough error estimation to check the method’s accuracy. To demonstrate its practical effectiveness, we use several numerical examples. The obtained results highlight the performance of the method and prove its alignment with theoretical error predictions. Furthermore, a comparative analysis with analytical solutions and alternative methods reaffirms the efficiency of the developed approach.
{"title":"An efficient Tau approach for solving a class of third-kind Volterra integral equations with proportional delays","authors":"E. Aourir , H. Laeli Dastjerdi , M. Oudani","doi":"10.1016/j.cam.2026.117370","DOIUrl":"10.1016/j.cam.2026.117370","url":null,"abstract":"<div><div>This work presents a new algorithm for solving a kind of Volterra delay integral equations of the third kind (VDIEs). Using the Tau method and generalized polynomial bases, our developed method is a robust approach for solving these equations. Specifically, we employ simple matrix operations to enhance the Tau approach. The underlying strategy leverages orthogonal polynomial bases to change the original equation into a matrix-vector form. Such a transformation makes the third-kind VDIEs easier to handle by turning them into a set of algebraic equations. Importantly, this method exhibits good stability, reduces memory usage, and is computationally cost-effective. The paper details the algorithm’s formulation and shows its capability to provide approximate polynomial solutions. We perform a thorough error estimation to check the method’s accuracy. To demonstrate its practical effectiveness, we use several numerical examples. The obtained results highlight the performance of the method and prove its alignment with theoretical error predictions. Furthermore, a comparative analysis with analytical solutions and alternative methods reaffirms the efficiency of the developed approach.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"483 ","pages":"Article 117370"},"PeriodicalIF":2.6,"publicationDate":"2026-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039554","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 : 2026-01-18DOI: 10.1016/j.cam.2026.117372
Jana Vráblíková, Bert Jüttler
Computing the envelope of deforming planar domains is a significant and challenging problem with a wide range of potential applications. We approximate the envelope using circular arc splines, curves that balance geometric flexibility and computational simplicity. Our approach combines two concepts to achieve these benefits.
First, we represent a planar domain by its medial axis transform (MAT), which is a geometric graph in Minkowski space (possibly with degenerate branches). We observe that circular arcs in the Minkowski space correspond to MATs of arc spline domains. Furthermore, as a planar domain evolves over time, each branch of its MAT evolves and forms a surface in the Minkowski space. This allows us to reformulate the problem of envelope computation as a problem of computing cyclographic images of finite sets of curves on these surfaces. We propose and compare two pairs of methods for approximating the curves and boundaries of their cyclographic images. All of these methods result in an arc spline approximation of the envelope of the evolving domain.
Second, we exploit the geometric flexibility of circular arcs in both the plane and Minkowski space to achieve a high approximation rate. The computational simplicity ensures the efficient trimming of redundant branches of the generated envelope using a sweep line algorithm with optimal computational complexity.
{"title":"Arc spline approximation of envelopes of evolving planar domains","authors":"Jana Vráblíková, Bert Jüttler","doi":"10.1016/j.cam.2026.117372","DOIUrl":"10.1016/j.cam.2026.117372","url":null,"abstract":"<div><div>Computing the envelope of deforming planar domains is a significant and challenging problem with a wide range of potential applications. We approximate the envelope using circular arc splines, curves that balance geometric flexibility and computational simplicity. Our approach combines two concepts to achieve these benefits.</div><div>First, we represent a planar domain by its medial axis transform (MAT), which is a geometric graph in Minkowski space <span><math><msup><mi>R</mi><mrow><mn>2</mn><mo>,</mo><mn>1</mn></mrow></msup></math></span> (possibly with degenerate branches). We observe that circular arcs in the Minkowski space correspond to MATs of arc spline domains. Furthermore, as a planar domain evolves over time, each branch of its MAT evolves and forms a surface in the Minkowski space. This allows us to reformulate the problem of envelope computation as a problem of computing cyclographic images of finite sets of curves on these surfaces. We propose and compare two pairs of methods for approximating the curves and boundaries of their cyclographic images. All of these methods result in an arc spline approximation of the envelope of the evolving domain.</div><div>Second, we exploit the geometric flexibility of circular arcs in both the plane and Minkowski space to achieve a high approximation rate. The computational simplicity ensures the efficient trimming of redundant branches of the generated envelope using a sweep line algorithm with optimal computational complexity.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"483 ","pages":"Article 117372"},"PeriodicalIF":2.6,"publicationDate":"2026-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039576","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 : 2026-01-18DOI: 10.1016/j.cam.2026.117367
Paweł Przybyłowicz, Michał Sobieraj
In this paper, we investigate the problem of strong approximation of the solutions of stochastic differential equations (SDEs) when the drift coefficient is given in integral form. We investigate its upper error bounds, in terms of the discretization parameter n and the size M of the random sample drawn at each step of the algorithm, in different subclasses of coefficients of the underlying SDE presenting various rates of convergence. Integral-form drift often appears when analyzing stochastic dynamics of optimization procedures in machine learning (ML) problems. Hence, we additionally discuss connections of the defined randomized Euler approximation scheme with the perturbed version of the stochastic gradient descent (SGD) algorithm. Finally, the results of numerical experiments performed using GPU architecture are also reported, including a comparison with other popular optimizers used in ML.
{"title":"On the randomized Euler scheme for stochastic differential equations with integral-form drift","authors":"Paweł Przybyłowicz, Michał Sobieraj","doi":"10.1016/j.cam.2026.117367","DOIUrl":"10.1016/j.cam.2026.117367","url":null,"abstract":"<div><div>In this paper, we investigate the problem of strong approximation of the solutions of stochastic differential equations (SDEs) when the drift coefficient is given in integral form. We investigate its upper error bounds, in terms of the discretization parameter <em>n</em> and the size <em>M</em> of the random sample drawn at each step of the algorithm, in different subclasses of coefficients of the underlying SDE presenting various rates of convergence. Integral-form drift often appears when analyzing stochastic dynamics of optimization procedures in machine learning (ML) problems. Hence, we additionally discuss connections of the defined randomized Euler approximation scheme with the perturbed version of the stochastic gradient descent (SGD) algorithm. Finally, the results of numerical experiments performed using GPU architecture are also reported, including a comparison with other popular optimizers used in ML.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"483 ","pages":"Article 117367"},"PeriodicalIF":2.6,"publicationDate":"2026-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039582","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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