Journal of Complexity最新文献

{"title":"Upper bounds for generalized Lp-discrepancy of random points","authors":"Erich Novak ,&nbsp;Friedrich Pillichshammer","doi":"10.1016/j.jco.2026.102028","DOIUrl":"10.1016/j.jco.2026.102028","url":null,"abstract":"<div><div>We study the <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-discrepancy of random point sets in high dimensions, with emphasis on small values of <em>p</em>. Although the classical <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-discrepancy suffers from the curse of dimensionality for all <span><math><mi>p</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span>, the gap between known upper and lower bounds remains substantial, in particular for small <span><math><mi>p</mi><mo>≥</mo><mn>1</mn></math></span>. To clarify this picture, we review the existing results for i.i.d. uniformly distributed points and derive new upper bounds for <em>generalized</em> <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-discrepancies, obtained by allowing non-uniform sampling densities and corresponding non-negative quadrature weights. Using the probabilistic method, we show that random points drawn from optimally chosen product densities lead to significantly improved upper bounds. For <span><math><mi>p</mi><mo>=</mo><mn>2</mn></math></span> these bounds are explicit and optimal; for general <span><math><mi>p</mi><mo>∈</mo><mo>[</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span> we obtain sharp asymptotic estimates. The improvement can be interpreted as a form of importance sampling for the underlying Sobolev space <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>d</mi><mo>,</mo><mi>q</mi></mrow></msub></math></span>. Our results also reveal that, even with optimal densities, the curse of dimensionality persists for random points when <span><math><mi>p</mi><mo>≥</mo><mn>1</mn></math></span>, and it becomes most pronounced for small <em>p</em>. This suggests that the curse should also hold for the classical <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-discrepancy for deterministic point sets.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"95 ","pages":"Article 102028"},"PeriodicalIF":1.8,"publicationDate":"2026-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146191461","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}

{"title":"Computable type and computably categorical spaces","authors":"Zvonko Iljazović ,&nbsp;Patrik Vasung","doi":"10.1016/j.jco.2026.102026","DOIUrl":"10.1016/j.jco.2026.102026","url":null,"abstract":"<div><div>We examine effective separating sequences on a metric space and, in particular, conditions under which on a metric space every two such sequences are equivalent up to an isometry. Such a metric space is called computably categorical. We prove that an effectively compact metric space <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>)</mo></math></span> is computably categorical if the space <span><math><mrow><mi>Iso</mi></mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>)</mo></math></span> of all isometries of <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>)</mo></math></span> has computable type (which in particular holds if <span><math><mrow><mi>Iso</mi></mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>)</mo></math></span> is a manifold). Using this, we prove that each effectively compact subspace of Euclidean space is computably categorical.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"95 ","pages":"Article 102026"},"PeriodicalIF":1.8,"publicationDate":"2026-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146116728","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}

{"title":"Average case tractability of additive random fields with Korobov kernels","authors":"Jia Chen ,&nbsp;Heping Wang","doi":"10.1016/j.jco.2025.102015","DOIUrl":"10.1016/j.jco.2025.102015","url":null,"abstract":"<div><div>We investigate average case tractability of approximation of additive random fields with marginal random processes corresponding to the Korobov kernels for the non-homogeneous case. We use the absolute error criterion (ABS) or the normalized error criterion (NOR). We show that the approximation problem is always polynomially tractable for ABS or NOR, and give sufficient and necessary conditions for strong polynomial tractability for ABS or NOR.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"94 ","pages":"Article 102015"},"PeriodicalIF":1.8,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145927151","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}

{"title":"Landweber iteration for inverse problems using multiple repeated measurements data in Banach spaces","authors":"Yuxin Xia ,&nbsp;Wei Wang ,&nbsp;Yong Chen","doi":"10.1016/j.jco.2025.102014","DOIUrl":"10.1016/j.jco.2025.102014","url":null,"abstract":"<div><div>In this work we consider Landweber iteration for solving generic linear inverse problems in the Banach spaces setting. Landweber iteration, along with its variants, is widely recognized as one of the most prominent iterative regularization methods due to its ease of implementation. Unlike classical theoretical analyses, this work considers the absence of noise level information, making it more relevant to real-world applications. We assume that multiple repeated independent identically distributed unbiased measurements of the exact data are available. The average of these repeated measurements is then utilized to update the iterative process. Under a statistical variant of the discrepancy principle, we establish rigorous regularizing property in the sense of expectation. Furthermore, a series of numerical experiments are conducted to evaluate and validate the performance of the approach.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"94 ","pages":"Article 102014"},"PeriodicalIF":1.8,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145884923","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}

{"title":"Weighted approximate sampling recovery and integration based on B-spline interpolation and quasi-interpolation","authors":"Dinh Dũng","doi":"10.1016/j.jco.2026.102016","DOIUrl":"10.1016/j.jco.2026.102016","url":null,"abstract":"<div><div>We propose novel methods for approximate sampling recovery and integration of functions in the Freud-weighted Sobolev space <span><math><msubsup><mrow><mi>W</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>w</mi></mrow><mrow><mi>r</mi></mrow></msubsup><mo>(</mo><mi>R</mi><mo>)</mo></math></span>. The approximation error of sampling recovery is measured in the norm of the Freud-weighted Lebesgue space <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>q</mi><mo>,</mo><mi>w</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span>. Namely, we construct equidistant, compact-supported B-spline quasi-interpolation and interpolation sampling algorithms <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>ρ</mi><mo>,</mo><mi>m</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>ρ</mi><mo>,</mo><mi>m</mi></mrow></msub></math></span> which are asymptotically optimal in terms of the sampling <em>n</em>-widths <span><math><msub><mrow><mi>ϱ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msubsup><mrow><mi>W</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>w</mi></mrow><mrow><mi>r</mi></mrow></msubsup><mo>(</mo><mi>R</mi><mo>)</mo><mo>,</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>q</mi><mo>,</mo><mi>w</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo><mo>)</mo></math></span> for every pair <span><math><mi>p</mi><mo>,</mo><mi>q</mi><mo>∈</mo><mo>[</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo>]</mo></math></span>, and prove the exact convergence rate of these sampling <em>n</em>-widths, where <span><math><msubsup><mrow><mi>W</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>w</mi></mrow><mrow><mi>r</mi></mrow></msubsup><mo>(</mo><mi>R</mi><mo>)</mo></math></span> denotes the unit ball in <span><math><msubsup><mrow><mi>W</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>w</mi></mrow><mrow><mi>r</mi></mrow></msubsup><mo>(</mo><mi>R</mi><mo>)</mo></math></span>. The algorithms <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>ρ</mi><mo>,</mo><mi>m</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>ρ</mi><mo>,</mo><mi>m</mi></mrow></msub></math></span> are based on truncated scaled B-spline quasi-interpolation and interpolation, respectively. We also prove the asymptotical optimality and exact convergence rate of the equidistant quadratures generated from <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>ρ</mi><mo>,</mo><mi>m</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>ρ</mi><mo>,</mo><mi>m</mi></mrow></msub></math></span>, for Freud-weighted numerical integration of functions in <span><math><msubsup><mrow><mi>W</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>w</mi></mrow><mrow><mi>r</mi></mrow></msubsup><mo>(</mo><mi>R</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"94 ","pages":"Article 102016"},"PeriodicalIF":1.8,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145978012","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}

{"title":"Regularization operators for identifying the unknown source in the time-fractional convection-diffusion-reaction equation","authors":"Guillermo Federico Umbricht ,&nbsp;Diana Rubio","doi":"10.1016/j.jco.2025.102013","DOIUrl":"10.1016/j.jco.2025.102013","url":null,"abstract":"<div><div>This article presents a mathematical study of the problem of identifying a time-dependent source term in transport processes described by a time-fractional parabolic equation, based on noisy time-dependent measurements taken at an arbitrary position. The problem is analytically solved using Fourier techniques, and it is shown that the solution is unstable. To address this instability, three one-parameter families of regularization operators are proposed, each designed to counteract the factors responsible for the instability of the inverse operator. Additionally, a new rule for selecting the regularization parameter is introduced, and an error bound is derived for each estimate. Numerical examples with varying characteristics are provided to illustrate the advantages of the proposed strategies.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"94 ","pages":"Article 102013"},"PeriodicalIF":1.8,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145753765","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}

{"title":"On recovering the Radon-Nikodym derivative under the big data assumption","authors":"Hanna L. Myleiko ,&nbsp;Sergei G. Solodky","doi":"10.1016/j.jco.2025.102001","DOIUrl":"10.1016/j.jco.2025.102001","url":null,"abstract":"<div><div>The present paper is focused on recovering the Radon-Nikodym derivative under the big data assumption. To address the above problem, we design an algorithm that is a combination of the Nyström subsampling and the standard Tikhonov regularization. The convergence rate of the corresponding algorithm is established both in the case when the Radon-Nikodym derivative belongs to RKHS and in the case when it does not. We prove that the proposed approach not only ensures the order of accuracy as algorithms based on the whole sample size, but also allows to achieve subquadratic computational costs in the number of observations.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"93 ","pages":"Article 102001"},"PeriodicalIF":1.8,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145571800","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}

{"title":"Global optimality of 3- and 5-point Fibonacci lattices for quasi-Monte Carlo integration and general energies","authors":"Nicolas Nagel","doi":"10.1016/j.jco.2025.102012","DOIUrl":"10.1016/j.jco.2025.102012","url":null,"abstract":"<div><div>We use linear programming bounds to analyze point sets in the torus with respect to their optimality for problems in discrepancy theory and quasi-Monte Carlo methods. These concepts will be unified by introducing tensor product energies.</div><div>We show that the canonical 3-point lattice in any dimension is globally optimal among all 3-point sets in the torus with respect to a large class of such energies. This is a new instance of universal optimality, a special phenomenon that is only known for a small class of highly structured point sets.</div><div>In the case of <span><math><mi>d</mi><mo>=</mo><mn>2</mn></math></span> dimensions it is conjectured that so-called Fibonacci lattices should also be optimal with respect to a large class of potentials. To this end we show that the 5-point Fibonacci lattice is globally optimal for a continuously parametrized class of potentials relevant to the analysis of the quasi-Monte Carlo method.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"93 ","pages":"Article 102012"},"PeriodicalIF":1.8,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145736523","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}

{"title":"Weighted sampling recovery of functions with mixed smoothness","authors":"Dinh Dũng","doi":"10.1016/j.jco.2025.102000","DOIUrl":"10.1016/j.jco.2025.102000","url":null,"abstract":"<div><div>We studied linear weighted sampling algorithms and their optimality for approximate recovery of functions with mixed smoothness on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> from a set of <em>n</em> their sampled values. Functions to be recovered are in weighted Sobolev spaces <span><math><msubsup><mrow><mi>W</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>w</mi></mrow><mrow><mi>r</mi></mrow></msubsup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> of mixed smoothness, and the approximation error is measured by the norm of the weighted Lebesgue space <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>q</mi><mo>,</mo><mi>w</mi></mrow></msub><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span>. Here, the weight <em>w</em> is a tensor-product Freud-type weight. The optimality of linear sampling algorithms is investigated in terms of sampling <em>n</em>-widths. We constructed linear sampling algorithms on sparse grids of sampled points which form a step hyperbolic cross in the function domain, and which give upper bounds for the corresponding sampling <em>n</em>-widths. We proved that in the one-dimensional case, these algorithms realize the exact convergence rate of the <em>n</em>-sampling widths.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"93 ","pages":"Article 102000"},"PeriodicalIF":1.8,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145521239","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}

{"title":"Hammersley point sets and inverse of star-discrepancy","authors":"Christian Weiß","doi":"10.1016/j.jco.2025.101998","DOIUrl":"10.1016/j.jco.2025.101998","url":null,"abstract":"<div><div>We establish the existence of <em>N</em>-point sets in dimension <em>d</em> whose star-discrepancy is bounded above by <span><math><mn>2.4631832</mn><msqrt><mrow><mfrac><mrow><mi>d</mi></mrow><mrow><mi>N</mi></mrow></mfrac></mrow></msqrt></math></span>, where the numerical constant improves upon all previously known bounds. This improvement is obtained by combining a recent result by Gnewuch on bracketing numbers in high dimensions with discrepancy bounds for Hammersley point sets due to Atanassov in dimensions <span><math><mn>1</mn><mo>≤</mo><mi>d</mi><mo>≤</mo><mn>4</mn></math></span>.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"93 ","pages":"Article 101998"},"PeriodicalIF":1.8,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145442663","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}