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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 : 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":"2025-07-10","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
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 : 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":"2025-07-04","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
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 : 2025-06-30 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":"2025-06-30","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
Upper and lower error bounds for a compact fourth-order finite-difference scheme for the wave equation with nonsmooth data 非光滑波动方程紧致四阶有限差分格式的误差上限和下限
IF 1.8 2区 数学
Journal of Complexity Pub Date : 2025-06-30 DOI: 10.1016/j.jco.2025.101973
A. Zlotnik
{"title":"Upper and lower error bounds for a compact fourth-order finite-difference scheme for the wave equation with nonsmooth data","authors":"A. Zlotnik","doi":"10.1016/j.jco.2025.101973","DOIUrl":"10.1016/j.jco.2025.101973","url":null,"abstract":"<div><div>A compact three-level fourth-order finite-difference scheme for solving the 1d wave equation is studied. New error bounds of the fractional order <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>h</mi></mrow><mrow><mn>4</mn><mo>(</mo><mi>λ</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>5</mn></mrow></msup><mo>)</mo></math></span> are proved in the mesh energy norm in terms of data, for two initial functions from the Sobolev and Nikolskii spaces with the smoothness orders <em>λ</em> and <span><math><mi>λ</mi><mo>−</mo><mn>1</mn></math></span> and the free term with a dominated mixed smoothness of order <span><math><mi>λ</mi><mo>−</mo><mn>1</mn></math></span>, for <span><math><mn>1</mn><mo>⩽</mo><mi>λ</mi><mo>⩽</mo><mn>6</mn></math></span>. The corresponding lower error bounds are proved as well to ensure the sharpness in order of the above error bounds with respect to each of the initial functions and the free term for any <em>λ</em>. Moreover, they demonstrate that the upper error bounds cannot be improved if the Lebesgue summability indices in the error norm are weakened down to 1 both in <em>x</em> and <em>t</em> and simultaneously the summability indices in the norms of data are strengthened up to ∞ both in <em>x</em> and <em>t</em>. Numerical experiments confirming the sharpness of the mentioned orders for half-integer <em>λ</em> and piecewise polynomial data have already been carried out previously.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"91 ","pages":"Article 101973"},"PeriodicalIF":1.8,"publicationDate":"2025-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144549672","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 : 2025-06-23 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":"2025-06-23","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
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 : 2025-06-19 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":"2025-06-19","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
Geo-indistinguishable location obfuscation with inference error bounds 具有推理误差界的地理不可区分位置混淆
IF 1.8 2区 数学
Journal of Complexity Pub Date : 2025-06-11 DOI: 10.1016/j.jco.2025.101970
Shun Zhang , Benfei Duan , Zhili Chen , Hong Zhong
{"title":"Geo-indistinguishable location obfuscation with inference error bounds","authors":"Shun Zhang ,&nbsp;Benfei Duan ,&nbsp;Zhili Chen ,&nbsp;Hong Zhong","doi":"10.1016/j.jco.2025.101970","DOIUrl":"10.1016/j.jco.2025.101970","url":null,"abstract":"<div><div>Geo-indistinguishability and expected inference error are two complementary statistical notions for location privacy. The joint guarantee of differential privacy (indistinguishability) and distortion privacy (inference error) limits the information leakage. This paper analyzes the dynamic location obfuscation mechanism called PIVE by Yu, Liu and Pu (NDSS 2017), and shows that PIVE fails to offer either of the privacy guarantees on adaptive Protection Location Sets (PLSs) as claimed. Specifically, we demonstrate that different PLSs could intersect with one another due to the defined search algorithm, and different apriori locations in the same PLS could have different protection diameters which causes the problematic proof of local differential privacy for PIVE. Besides, the condition introduced in PIVE is confirmed to be not sufficient for bounding expected inference errors against Bayesian attacks. To address these issues, we introduce a relaxed definition of geo-indistinguishability, propose a couple of correction approaches, and analyze their satisfied privacy characteristics.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"91 ","pages":"Article 101970"},"PeriodicalIF":1.8,"publicationDate":"2025-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144364448","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 : 2025-06-06 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":"2025-06-06","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
On upper and lower bounds for pathwise approximation of scalar SDEs with reflection 带反射的标量SDEs路径逼近的上界和下界
IF 1.8 2区 数学
Journal of Complexity Pub Date : 2025-05-16 DOI: 10.1016/j.jco.2025.101959
Mario Hefter , André Herzwurm , Klaus Ritter
{"title":"On upper and lower bounds for pathwise approximation of scalar SDEs with reflection","authors":"Mario Hefter ,&nbsp;André Herzwurm ,&nbsp;Klaus Ritter","doi":"10.1016/j.jco.2025.101959","DOIUrl":"10.1016/j.jco.2025.101959","url":null,"abstract":"<div><div>For scalar SDEs with a one-sided reflection we study pathwise approximation, globally on a compact time interval or at a single time point. We consider algorithms based on sequential evaluations of the driving Brownian motion and establish upper and lower bounds for the minimal errors. Exploiting the relation to a reflected Ornstein-Uhlenbeck process, we also provide a new upper bound for a Cox-Ingersoll-Ross process.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"90 ","pages":"Article 101959"},"PeriodicalIF":1.8,"publicationDate":"2025-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144115566","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
Skewness of a randomized quasi-Monte Carlo estimate 随机拟蒙特卡罗估计的偏度
IF 1.8 2区 数学
Journal of Complexity Pub Date : 2025-05-05 DOI: 10.1016/j.jco.2025.101956
Zexin Pan, Art B. Owen
{"title":"Skewness of a randomized quasi-Monte Carlo estimate","authors":"Zexin Pan,&nbsp;Art B. Owen","doi":"10.1016/j.jco.2025.101956","DOIUrl":"10.1016/j.jco.2025.101956","url":null,"abstract":"<div><div>Some recent work on confidence intervals for randomized quasi-Monte Carlo (RQMC) sampling found a surprising result: ordinary Student's <em>t</em> 95% confidence intervals based on a modest number of replicates were seen to be very effective and even more reliable than some bootstrap <em>t</em> intervals that were expected to be best. One potential explanation is that those RQMC estimates have small skewness. In this paper we give conditions under which the skewness is <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>ϵ</mi></mrow></msup><mo>)</mo></math></span> for any <span><math><mi>ϵ</mi><mo>&gt;</mo><mn>0</mn></math></span>, so ‘almost <span><math><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span>’. Under a random generator matrix model, we can improve this rate to <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>+</mo><mi>ϵ</mi></mrow></msup><mo>)</mo></math></span> with very high probability. We also improve some probabilistic bounds on the distribution of the quality parameter <em>t</em> for a digital net in a prime base under random sampling of generator matrices.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"90 ","pages":"Article 101956"},"PeriodicalIF":1.8,"publicationDate":"2025-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143908058","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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