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No existence of a linear algorithm for the one-dimensional Fourier phase retrieval 不存在一维傅立叶相位检索的线性算法
IF 1.8 2区 数学
Journal of Complexity Pub Date : 2025-02-01 Epub Date: 2024-07-04 DOI: 10.1016/j.jco.2024.101886
Meng Huang , Zhiqiang Xu
{"title":"No existence of a linear algorithm for the one-dimensional Fourier phase retrieval","authors":"Meng Huang ,&nbsp;Zhiqiang Xu","doi":"10.1016/j.jco.2024.101886","DOIUrl":"10.1016/j.jco.2024.101886","url":null,"abstract":"<div><p>Fourier phase retrieval, which aims to reconstruct a signal from its Fourier magnitude, is of fundamental importance in fields of engineering and science. In this paper, we provide a theoretical understanding of algorithms for the one-dimensional Fourier phase retrieval problem. Specifically, we demonstrate that if an algorithm exists which can reconstruct an arbitrary signal <span><math><mi>x</mi><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span> in <span><math><mtext>Poly</mtext><mo>(</mo><mi>N</mi><mo>)</mo><mi>log</mi><mo>⁡</mo><mo>(</mo><mn>1</mn><mo>/</mo><mi>ϵ</mi><mo>)</mo></math></span> time to reach <em>ϵ</em>-precision from its magnitude of discrete Fourier transform and its initial value <span><math><mi>x</mi><mo>(</mo><mn>0</mn><mo>)</mo></math></span>, then <span><math><mi>P</mi><mo>=</mo><mrow><mi>NP</mi></mrow></math></span>. This partially elucidates the phenomenon that, despite the fact that almost all signals are uniquely determined by their Fourier magnitude and the absolute value of their initial value <span><math><mo>|</mo><mi>x</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>|</mo></math></span>, no algorithm with theoretical guarantees has been proposed in the last few decades. Our proofs employ the result in computational complexity theory that the Product Partition problem is NP-complete in the strong sense.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"86 ","pages":"Article 101886"},"PeriodicalIF":1.8,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0885064X24000633/pdfft?md5=306cc05c455de6efb9f908455c6f3128&pid=1-s2.0-S0885064X24000633-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141637397","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}
引用次数: 0
High-efficiency parametric iterative schemes for solving nonlinear equations with and without memory 求解有内存和无内存非线性方程的高效参数迭代方案
IF 1.8 2区 数学
Journal of Complexity Pub Date : 2025-02-01 Epub Date: 2024-09-11 DOI: 10.1016/j.jco.2024.101896
Raziyeh Erfanifar, Masoud Hajarian
{"title":"High-efficiency parametric iterative schemes for solving nonlinear equations with and without memory","authors":"Raziyeh Erfanifar,&nbsp;Masoud Hajarian","doi":"10.1016/j.jco.2024.101896","DOIUrl":"10.1016/j.jco.2024.101896","url":null,"abstract":"<div><p>Many practical problems, such as the Malthusian population growth model, eigenvalue computations for matrices, and solving the Van der Waals' ideal gas equation, inherently involve nonlinearities. This paper initially introduces a two-parameter iterative scheme with a convergence order of two. Building on this, a three-parameter scheme with a convergence order of four is proposed. Then we extend these schemes into higher-order schemes with memory using Newton's interpolation, achieving an upper bound for the efficiency index of <span><math><msup><mrow><mn>7.88748</mn></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup><mo>≈</mo><mn>1.99057</mn></math></span>. Finally, we validate the new schemes by solving various numerical and practical examples, demonstrating their superior efficiency in terms of computational cost, CPU time, and accuracy compared to existing methods.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"86 ","pages":"Article 101896"},"PeriodicalIF":1.8,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0885064X24000736/pdfft?md5=6dc221bf6c1e2ffc085c2b830768b4e4&pid=1-s2.0-S0885064X24000736-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142240453","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}
引用次数: 0
Best Paper Award of the Journal of Complexity 复杂性期刊》最佳论文奖
IF 1.8 2区 数学
Journal of Complexity Pub Date : 2025-02-01 Epub Date: 2024-11-05 DOI: 10.1016/j.jco.2024.101904
{"title":"Best Paper Award of the Journal of Complexity","authors":"","doi":"10.1016/j.jco.2024.101904","DOIUrl":"10.1016/j.jco.2024.101904","url":null,"abstract":"","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"86 ","pages":"Article 101904"},"PeriodicalIF":1.8,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142586491","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
Matthieu Dolbeault is the winner of the 2024 Joseph F. Traub Information-Based Complexity Young Researcher Award 马蒂厄-多尔贝奥（Matthieu Dolbeault）是 2024 年约瑟夫-特劳布基于信息的复杂性青年研究员奖（Joseph F. Traub Information-Based Complexity Young Researcher Award）的获得者。
IF 1.8 2区 数学
Journal of Complexity Pub Date : 2025-02-01 Epub Date: 2024-10-22 DOI: 10.1016/j.jco.2024.101902
Erich Novak, Kateryna Pozharska, Mathias Sonnleitner, Michaela Szölgyenyi, Henryk Woźniakowski
{"title":"Matthieu Dolbeault is the winner of the 2024 Joseph F. Traub Information-Based Complexity Young Researcher Award","authors":"Erich Novak,&nbsp;Kateryna Pozharska,&nbsp;Mathias Sonnleitner,&nbsp;Michaela Szölgyenyi,&nbsp;Henryk Woźniakowski","doi":"10.1016/j.jco.2024.101902","DOIUrl":"10.1016/j.jco.2024.101902","url":null,"abstract":"","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"86 ","pages":"Article 101902"},"PeriodicalIF":1.8,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142525769","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 two-point Newton-like method of optimal fourth order convergence for systems of nonlinear equations 非线性方程系统最优四阶收敛的类似牛顿的两点法
IF 1.8 2区 数学
Journal of Complexity Pub Date : 2025-02-01 Epub Date: 2024-11-05 DOI: 10.1016/j.jco.2024.101907
Harmandeep Singh , Janak Raj Sharma
{"title":"A two-point Newton-like method of optimal fourth order convergence for systems of nonlinear equations","authors":"Harmandeep Singh ,&nbsp;Janak Raj Sharma","doi":"10.1016/j.jco.2024.101907","DOIUrl":"10.1016/j.jco.2024.101907","url":null,"abstract":"<div><div>A two-step Newton-like method is proposed to efficiently solve the systems of nonlinear equations. Extending Newton scheme to a next step as weighted-Newton iteration, the proposed iteration scheme shows optimal fourth order of convergence. The primary objective in formulating the method is to keep the computational efficiency as high as possible. In this context, the efficiency analysis is thoroughly examined using a systematic approach, wherein the efficiency index of the new method is compared with those of existing methods of comparable complexity. Numerical experimentation is performed to investigate the computational efficacy of the developed method. Results indicate higher efficiency and numerical precision in comparison to the existing counterparts.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"86 ","pages":"Article 101907"},"PeriodicalIF":1.8,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142656796","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
High probability bounds on AdaGrad for constrained weakly convex optimization 受约束弱凸优化的 AdaGrad 高概率边界
IF 1.8 2区 数学
Journal of Complexity Pub Date : 2025-02-01 Epub Date: 2024-07-31 DOI: 10.1016/j.jco.2024.101889
Yusu Hong , Junhong Lin
{"title":"High probability bounds on AdaGrad for constrained weakly convex optimization","authors":"Yusu Hong ,&nbsp;Junhong Lin","doi":"10.1016/j.jco.2024.101889","DOIUrl":"10.1016/j.jco.2024.101889","url":null,"abstract":"<div><p>In this paper, we study the high probability convergence of AdaGrad-Norm for constrained, non-smooth, weakly convex optimization with bounded noise and sub-Gaussian noise cases. We also investigate a more general accelerated gradient descent (AGD) template (Ghadimi and Lan, 2016) encompassing the AdaGrad-Norm, the Nesterov's accelerated gradient descent, and the RSAG (Ghadimi and Lan, 2016) with different parameter choices. We provide a high probability convergence rate <span><math><mover><mrow><mi>O</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><mn>1</mn><mo>/</mo><msqrt><mrow><mi>T</mi></mrow></msqrt><mo>)</mo></math></span> without knowing the information of the weak convexity parameter and the gradient bound to tune the step-sizes.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"86 ","pages":"Article 101889"},"PeriodicalIF":1.8,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0885064X24000669/pdfft?md5=7c5c4999e38fd8c865761fe3213f35cf&pid=1-s2.0-S0885064X24000669-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935984","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}
引用次数: 0
Optimal recovery of linear operators from information of random functions 从随机函数信息中优化恢复线性算子
IF 1.8 2区 数学
Journal of Complexity Pub Date : 2025-02-01 Epub Date: 2024-10-21 DOI: 10.1016/j.jco.2024.101903
K.Yu. Osipenko
{"title":"Optimal recovery of linear operators from information of random functions","authors":"K.Yu. Osipenko","doi":"10.1016/j.jco.2024.101903","DOIUrl":"10.1016/j.jco.2024.101903","url":null,"abstract":"<div><div>The paper concerns problems of the recovery of linear operators defined on sets of functions from information of these functions given with stochastic errors. The constructed optimal recovery methods, in general, do not use all the available information. As a consequence, optimal methods are obtained for recovering derivatives of functions from Sobolev classes by the information of their Fourier transforms given with stochastic errors. A similar problem is considered for solutions of the heat equation.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"86 ","pages":"Article 101903"},"PeriodicalIF":1.8,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142527379","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
Intractability results for integration in tensor product spaces 张量乘空间积分的难解性结果
IF 1.8 2区 数学
Journal of Complexity Pub Date : 2025-02-01 Epub Date: 2024-10-16 DOI: 10.1016/j.jco.2024.101901
Erich Novak , Friedrich Pillichshammer
{"title":"Intractability results for integration in tensor product spaces","authors":"Erich Novak ,&nbsp;Friedrich Pillichshammer","doi":"10.1016/j.jco.2024.101901","DOIUrl":"10.1016/j.jco.2024.101901","url":null,"abstract":"<div><div>We prove lower bounds on the worst-case error of numerical integration in tensor product spaces. The information complexity is the minimal number <em>N</em> of function evaluations that is necessary such that the <em>N</em>-th minimal error is less than a factor <em>ε</em> times the initial error, i.e., the error for <span><math><mi>N</mi><mo>=</mo><mn>0</mn></math></span>, where <em>ε</em> belongs to <span><math><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>. We are interested to which extent the information complexity depends on the number <em>d</em> of variables of the integrands. If the information complexity grows exponentially fast in <em>d</em>, then the integration problem is said to suffer from the curse of dimensionality.</div><div>Under the assumption of the existence of a worst-case function for the uni-variate problem, we present two methods for providing lower bounds on the information complexity. The first method is based on a suitable decomposition of the worst-case function and can be seen as a generalization of the method of decomposable reproducing kernels. The second method, although only applicable for positive quadrature rules, does not require a suitable decomposition of the worst-case function. Rather, it is based on a spline approximation of the worst-case function and can be used for analytic functions. Several applications of both methods are presented.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"86 ","pages":"Article 101901"},"PeriodicalIF":1.8,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142445798","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}
引用次数: 0
Kernel multigrid on manifolds 流形上的核多网格
IF 1.8 2区 数学
Journal of Complexity Pub Date : 2025-02-01 Epub Date: 2024-10-09 DOI: 10.1016/j.jco.2024.101900
Thomas Hangelbroek , Christian Rieger
{"title":"Kernel multigrid on manifolds","authors":"Thomas Hangelbroek ,&nbsp;Christian Rieger","doi":"10.1016/j.jco.2024.101900","DOIUrl":"10.1016/j.jco.2024.101900","url":null,"abstract":"<div><div>Kernel methods for solving partial differential equations work coordinate-free on the surface and yield high approximation rates for smooth solutions. Localized Lagrange bases have proven to alleviate the computational complexity of usual kernel methods for data fitting problems, but the efficient numerical solution of the ill-conditioned linear systems of equations arising from kernel-based Galerkin solutions to PDEs is a challenging problem which has not been addressed in the literature so far. In this article we apply the framework of the geometric multigrid method with a <span><math><mi>τ</mi><mo>≥</mo><mn>2</mn></math></span>-cycle to scattered, quasi-uniform point clouds on the surface. We show that the resulting solver can be accelerated by using the Lagrange function decay and obtain satisfying convergence rates by a rigorous analysis. In particular, we show that the computational cost of the linear solver scales log-linear in the degrees of freedom.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"86 ","pages":"Article 101900"},"PeriodicalIF":1.8,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142527392","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 iteratively regularized Landweber iteration with uniformly convex constraints in Banach spaces 巴拿赫空间中具有均匀凸约束条件的迭代正则化 Landweber 迭代的收敛性分析
IF 1.8 2区 数学
Journal of Complexity Pub Date : 2025-02-01 Epub Date: 2024-09-13 DOI: 10.1016/j.jco.2024.101897
Gaurav Mittal , Harshit Bajpai , Ankik Kumar Giri
{"title":"Convergence analysis of iteratively regularized Landweber iteration with uniformly convex constraints in Banach spaces","authors":"Gaurav Mittal ,&nbsp;Harshit Bajpai ,&nbsp;Ankik Kumar Giri","doi":"10.1016/j.jco.2024.101897","DOIUrl":"10.1016/j.jco.2024.101897","url":null,"abstract":"<div><p>In Banach spaces, the convergence analysis of iteratively regularized Landweber iteration (IRLI) is recently studied via conditional stability estimates. But the formulation of IRLI does not include general non-smooth convex penalty functionals, which is essential to capture special characteristics of the sought solution. In this paper, we formulate a generalized form of IRLI so that its formulation includes general non-smooth uniformly convex penalty functionals. We study the convergence analysis and derive the convergence rates of the generalized method solely via conditional stability estimates in Banach spaces for both the perturbed and unperturbed data. We also discuss few examples of inverse problems on which our method is applicable.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"86 ","pages":"Article 101897"},"PeriodicalIF":1.8,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0885064X24000748/pdfft?md5=5ae8eeac0a143f493ee150c18db69cf1&pid=1-s2.0-S0885064X24000748-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142240454","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}
引用次数: 0
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