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Optimal complexity solution of space-time finite element systems for state-based parabolic distributed optimal control problems 基于状态的抛物型分布最优控制问题的时空有限元系统最优复杂性解
IF 1.8 2区 数学
Journal of Complexity Pub Date : 2026-02-01 Epub Date: 2025-07-10 DOI: 10.1016/j.jco.2025.101976
Richard Löscher, Michael Reichelt, Olaf Steinbach
{"title":"Optimal complexity solution of space-time finite element systems for state-based parabolic distributed optimal control problems","authors":"Richard Löscher,&nbsp;Michael Reichelt,&nbsp;Olaf Steinbach","doi":"10.1016/j.jco.2025.101976","DOIUrl":"10.1016/j.jco.2025.101976","url":null,"abstract":"<div><div>In this paper we consider a distributed optimal control problem subject to a parabolic evolution equation as constraint. The approach presented here is based on the variational formulation of the parabolic evolution equation in anisotropic Sobolev spaces, considering the control in <span><math><msup><mrow><mo>[</mo><msubsup><mrow><mi>H</mi></mrow><mrow><mn>0</mn><mo>;</mo><mo>,</mo><mn>0</mn></mrow><mrow><mn>1</mn><mo>,</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msubsup><mo>(</mo><mi>Q</mi><mo>)</mo><mo>]</mo></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>. Since the state equation defines an isomorphism from <span><math><msubsup><mrow><mi>H</mi></mrow><mrow><mn>0</mn><mo>;</mo><mn>0</mn><mo>,</mo></mrow><mrow><mn>1</mn><mo>,</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msubsup><mo>(</mo><mi>Q</mi><mo>)</mo></math></span> onto <span><math><msup><mrow><mo>[</mo><msubsup><mrow><mi>H</mi></mrow><mrow><mn>0</mn><mo>;</mo><mo>,</mo><mn>0</mn></mrow><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msubsup><mo>(</mo><mi>Q</mi><mo>)</mo><mo>]</mo></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>, we can eliminate the control to end up with a minimization problem in <span><math><msubsup><mrow><mi>H</mi></mrow><mrow><mn>0</mn><mo>;</mo><mn>0</mn><mo>,</mo></mrow><mrow><mn>1</mn><mo>,</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msubsup><mo>(</mo><mi>Q</mi><mo>)</mo></math></span> where the anisotropic Sobolev norm can be realized using a modified Hilbert transformation. In the unconstrained case, the minimizer is the unique solution of a singularly perturbed elliptic equation. In the case of a space-time tensor-product mesh, we can use sparse factorization techniques to construct a solver of almost linear complexity. Numerical examples also include additional state constraints, and a nonlinear state equation.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"92 ","pages":"Article 101976"},"PeriodicalIF":1.8,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144632611","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}
引用次数: 0
Changes of the Editorial Board 编辑委员会的变动
IF 1.8 2区 数学
Journal of Complexity Pub Date : 2026-02-01 Epub Date: 2025-10-13 DOI: 10.1016/j.jco.2025.101997
Josef Dick, Erich Novak, Friedrich Pillichshammer, Klaus Ritter, Jan Vybíral, Henryk Woźniakowski
{"title":"Changes of the Editorial Board","authors":"Josef Dick,&nbsp;Erich Novak,&nbsp;Friedrich Pillichshammer,&nbsp;Klaus Ritter,&nbsp;Jan Vybíral,&nbsp;Henryk Woźniakowski","doi":"10.1016/j.jco.2025.101997","DOIUrl":"10.1016/j.jco.2025.101997","url":null,"abstract":"","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"92 ","pages":"Article 101997"},"PeriodicalIF":1.8,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145332349","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}
引用次数: 0
A high-efficiency fourth-order iterative method for nonlinear equations: Convergence and computational gains 非线性方程的高效四阶迭代法：收敛性与计算增益
IF 1.8 2区 数学
Journal of Complexity Pub Date : 2026-02-01 Epub Date: 2025-09-23 DOI: 10.1016/j.jco.2025.101994
Amir Naseem , Krzysztof Gdawiec , Sania Qureshi , Ioannis K. Argyros , Muhammad Aziz ur Rehman , Amanullah Soomro , Evren Hincal , Kamyar Hosseini , Ausif Padder
{"title":"A high-efficiency fourth-order iterative method for nonlinear equations: Convergence and computational gains","authors":"Amir Naseem ,&nbsp;Krzysztof Gdawiec ,&nbsp;Sania Qureshi ,&nbsp;Ioannis K. Argyros ,&nbsp;Muhammad Aziz ur Rehman ,&nbsp;Amanullah Soomro ,&nbsp;Evren Hincal ,&nbsp;Kamyar Hosseini ,&nbsp;Ausif Padder","doi":"10.1016/j.jco.2025.101994","DOIUrl":"10.1016/j.jco.2025.101994","url":null,"abstract":"<div><div>This study introduces an optimal fourth-order iterative method derived by combining two established methods, resulting in enhanced convergence when solving nonlinear equations. Through rigorous convergence analysis using both Taylor expansion and the Banach space framework, the fourth-order optimality condition is verified. We demonstrate the superior efficiency and stability of this new method compared to traditional alternatives. Numerical experiments confirm its effectiveness, showing a reduction in the average number of iterations and computational time. Visual analysis with polynomiographs confirms the method's robustness, focusing on convergence area index, iteration count, computational time, fractal dimension, and Wada measure of basins. These findings underscore the potential of this optimal method for tackling complex nonlinear problems in various scientific and engineering fields.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"92 ","pages":"Article 101994"},"PeriodicalIF":1.8,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145227117","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}
引用次数: 0
On the packing functions of some linear sets of Lebesgue measure zero 若干Lebesgue测度为0的线性集的填充函数
IF 1.8 2区 数学
Journal of Complexity Pub Date : 2026-02-01 Epub Date: 2025-09-01 DOI: 10.1016/j.jco.2025.101990
Austin Anderson , Steven Damelin
{"title":"On the packing functions of some linear sets of Lebesgue measure zero","authors":"Austin Anderson ,&nbsp;Steven Damelin","doi":"10.1016/j.jco.2025.101990","DOIUrl":"10.1016/j.jco.2025.101990","url":null,"abstract":"<div><div>We use a characterization of Minkowski measurability to study the asymptotics of best packing on cut-out subsets of the real line with Minkowski dimension <span><math><mi>d</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>. Our main result is a proof that Minkowski measurability is a sufficient condition for the existence of best packing asymptotics on monotone rearrangements of these sets. For each such set, the main result provides an explicit constant of proportionality <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span>, depending only on the Minkowski dimension <em>d</em>, that relates its packing limit and Minkowski content. We later use the Digamma function to study the limiting value of <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span> as <span><math><mi>d</mi><mo>→</mo><msup><mrow><mn>1</mn></mrow><mrow><mo>−</mo></mrow></msup></math></span>. This study further provides two sharpness results illustrating the necessity of the hypotheses of the main result. Finally, the aforementioned characterization of Minkowski measurability motivates the asymptotic study of an infinite multiple subset sum problem.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"92 ","pages":"Article 101990"},"PeriodicalIF":1.8,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144996569","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}
引用次数: 0
On optimal recovery and information complexity in numerical differentiation and summation 数值微分与求和的最优恢复与信息复杂度
IF 1.8 2区 数学
Journal of Complexity Pub Date : 2026-02-01 Epub Date: 2025-07-04 DOI: 10.1016/j.jco.2025.101975
Y.V. Semenova , S.G. Solodky
{"title":"On optimal recovery and information complexity in numerical differentiation and summation","authors":"Y.V. Semenova ,&nbsp;S.G. Solodky","doi":"10.1016/j.jco.2025.101975","DOIUrl":"10.1016/j.jco.2025.101975","url":null,"abstract":"<div><div>In this paper, we study the problems of numerical differentiation and summation of univariate functions from the weighted Wiener classes. To solve these problems, we propose an approach based on the truncation method. The essence of this method is to replace the infinite Fourier series with a finite sum. It is only necessary to properly select the order of this sum, which plays the role of a regularization parameter here. The results show that the proposed approach not only ensures a stability of approximations and does not require cumbersome computational procedures, but also constructs algorithms that achieve the optimal order of accuracy using the minimal amount of perturbed values of Fourier-Chebyshev coefficients. Moreover, we establish under what conditions the summation problem is well-posed on the considered function classes.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"92 ","pages":"Article 101975"},"PeriodicalIF":1.8,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144570627","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}
引用次数: 0
Regularized reduced-rank regression for structured output prediction 结构化输出预测的正则化降阶回归
IF 1.8 2区 数学
Journal of Complexity Pub Date : 2026-02-01 Epub Date: 2025-07-18 DOI: 10.1016/j.jco.2025.101977
Heng Chen , Di-Rong Chen , Kun Cheng , Yang Zhou
{"title":"Regularized reduced-rank regression for structured output prediction","authors":"Heng Chen ,&nbsp;Di-Rong Chen ,&nbsp;Kun Cheng ,&nbsp;Yang Zhou","doi":"10.1016/j.jco.2025.101977","DOIUrl":"10.1016/j.jco.2025.101977","url":null,"abstract":"<div><div>Reduced-rank regression (RRR) has been widely used to strength the dependency among multiple outputs. This paper develops a regularized vector-valued RRR approach, which plays an important role in predicting multiple outputs with structures. The estimator of vector-valued RRR is obtained by minimizing the empirically squared reproducing kernel Hilbert space (RKHS) distances between output feature kernel and all <em>r</em> dimensional subspaces in vector-valued RKHS. The algorithm is implemented easily with kernel tricks. We establish the learning rate of vector-valued RRR estimator under mild assumptions. Moreover, as a reduced-dimensional approximation of output kernel regression function, the estimator converges to the output regression function in probability when the rank <em>r</em> tends to infinity appropriately. It implies the consistency of structured predictor in general settings, especially in a misspecified case where the true regression function is not contained in the hypothesis space. Numerical experiments are provided to illustrate the efficiency of our method.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"92 ","pages":"Article 101977"},"PeriodicalIF":1.8,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144679194","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}
引用次数: 0
A sixth-order bi-parametric iterative method for nonlinear systems: Theory, stability and computational complexity 非线性系统的六阶双参数迭代法：理论、稳定性和计算复杂度
IF 1.8 2区 数学
Journal of Complexity Pub Date : 2025-12-01 Epub Date: 2025-06-06 DOI: 10.1016/j.jco.2025.101960
G Thangkhenpau, Sunil Panday
{"title":"A sixth-order bi-parametric iterative method for nonlinear systems: Theory, stability and computational complexity","authors":"G Thangkhenpau,&nbsp;Sunil Panday","doi":"10.1016/j.jco.2025.101960","DOIUrl":"10.1016/j.jco.2025.101960","url":null,"abstract":"<div><div>In this paper, we propose a new bi-parametric three-step iterative method with sixth-order convergence for solving systems of nonlinear equations. The method is formulated using the composition technique, which we uniquely apply twice within a single method - an approach that, to our knowledge, is the first of its kind in the literature. This allows us to incorporate two free disposable parameters, offering flexibility with enhanced performance, stability and adaptability. The local convergence behaviour is rigorously analysed within the framework of Banach spaces, where we establish theoretical bounds for the convergence radius and demonstrate uniqueness conditions under Lipschitz-continuous Fréchet derivative assumptions. The theoretical outcomes are supported by numerical experiments. Lastly, we evaluate the method's efficiency by exploring its basins of attraction and applying it to systems of nonlinear equations and boundary value problems.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"91 ","pages":"Article 101960"},"PeriodicalIF":1.8,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144255436","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}
引用次数: 0
Rademacher learning rates for iterated random functions 迭代随机函数的Rademacher学习率
IF 1.8 2区 数学
Journal of Complexity Pub Date : 2025-12-01 Epub Date: 2025-06-19 DOI: 10.1016/j.jco.2025.101971
Nikola Sandrić
{"title":"Rademacher learning rates for iterated random functions","authors":"Nikola Sandrić","doi":"10.1016/j.jco.2025.101971","DOIUrl":"10.1016/j.jco.2025.101971","url":null,"abstract":"<div><div>Most supervised learning methods assume training data is drawn from an i.i.d. sample. However, real-world problems often exhibit temporal dependence and strong correlations between marginals of the data-generating process, rendering the i.i.d. assumption unrealistic. Such cases naturally involve time-series processes and Markov chains. The learning rates typically obtained in these settings remain independent of the data distribution, potentially leading to restrictive hypothesis classes and suboptimal sample complexities. We consider training data generated by an iterated random function that need not be irreducible or aperiodic. Assuming the governing function is contractive in its first argument and subject to certain regularity conditions on the hypothesis class, we first establish uniform convergence for the sample error. We then prove learnability of approximate empirical risk minimization and derive its learning rate bound. Both bounds depend explicitly on the data distribution through the Rademacher complexities of the hypothesis class, thereby better capturing properties of the data-generating distribution.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"91 ","pages":"Article 101971"},"PeriodicalIF":1.8,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144321580","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}
引用次数: 0
Convergence analysis of a regularized iterative scheme for solving nonlinear problems 求解非线性问题的正则迭代格式的收敛性分析
IF 1.8 2区 数学
Journal of Complexity Pub Date : 2025-12-01 Epub Date: 2025-06-23 DOI: 10.1016/j.jco.2025.101972
M.P. Rajan, Niloopher Salam
{"title":"Convergence analysis of a regularized iterative scheme for solving nonlinear problems","authors":"M.P. Rajan,&nbsp;Niloopher Salam","doi":"10.1016/j.jco.2025.101972","DOIUrl":"10.1016/j.jco.2025.101972","url":null,"abstract":"<div><div>Nonlinear inverse and ill-posed problems occur in many practical applications and the regularization techniques are employed to get a stable approximate solution for the same. Although many schemes are available in literature, iterative regularization techniques are the most commonly used approaches. One such important method is the Levenberg-Marquardt scheme. However, the scheme involves computation of the Fréchet derivative at every iterate which makes it tedious and the restrictive assumptions on it often difficult to verify for practical scenarios. In this paper, we propose a simplified Levenberg-Marquardt scheme that has two benefits. Firstly, computation of the Fréchet derivative is required only once at the initial point and secondly, the convergence and optimal convergence rate of the method is established with weaker assumptions as compared to the standard method. We also provide numerical examples to illustrate the theory and, results clearly illustrate the advantages of the proposed scheme over the standard method.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"91 ","pages":"Article 101972"},"PeriodicalIF":1.8,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144490696","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}
引用次数: 0
Lower bounds on the minimal dispersion of point sets via cover-free families 无复盖族的点集最小色散的下界
IF 1.8 2区 数学
Journal of Complexity Pub Date : 2025-12-01 Epub Date: 2025-06-30 DOI: 10.1016/j.jco.2025.101974
M. Trödler , J. Volec , J. Vybíral
{"title":"Lower bounds on the minimal dispersion of point sets via cover-free families","authors":"M. Trödler ,&nbsp;J. Volec ,&nbsp;J. Vybíral","doi":"10.1016/j.jco.2025.101974","DOIUrl":"10.1016/j.jco.2025.101974","url":null,"abstract":"<div><div>We elaborate on the intimate connection between the largest volume of an empty axis-parallel box in a set of <em>n</em> points from <span><math><msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mrow><mi>d</mi></mrow></msup></math></span> and cover-free families from the extremal set theory. This connection was discovered in a recent paper of the authors. In this work, we apply a very recent result of Michel and Scott to obtain a whole range of new lower bounds on the number of points needed so that the largest volume of such a box is bounded by a given <em>ε</em>. Surprisingly, it turns out that for each of the new bounds, there is a choice of the parameters <em>d</em> and <em>ε</em> such that the bound outperforms the others.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"91 ","pages":"Article 101974"},"PeriodicalIF":1.8,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144522696","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}
引用次数: 0
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