Foundations of Computational Mathematics最新文献

{"title":"Optimal Polynomial Meshes Exist on any Multivariate Convex Domain","authors":"Feng Dai, Andriy Prymak","doi":"10.1007/s10208-023-09606-x","DOIUrl":"https://doi.org/10.1007/s10208-023-09606-x","url":null,"abstract":"We show that optimal polynomial meshes exist for every convex body in $${mathbb {R}}^d$$ , confirming a conjecture by A. Kroó.","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"248 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136297270","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Scattering and Uniform in Time Error Estimates for Splitting Method in NLS","authors":"R. Carles, C. Su","doi":"10.1007/s10208-022-09600-9","DOIUrl":"https://doi.org/10.1007/s10208-022-09600-9","url":null,"abstract":"","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":" ","pages":""},"PeriodicalIF":3.0,"publicationDate":"2022-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47665434","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Sharp Bounds on the Approximation Rates, Metric Entropy, and n-Widths of Shallow Neural Networks","authors":"Jonathan W. Siegel, Jinchao Xu","doi":"10.1007/s10208-022-09595-3","DOIUrl":"https://doi.org/10.1007/s10208-022-09595-3","url":null,"abstract":"","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":" ","pages":""},"PeriodicalIF":3.0,"publicationDate":"2022-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44959269","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Pseudospectral Shattering, the Sign Function, and Diagonalization in Nearly Matrix Multiplication Time","authors":"Jess Banks, Jorge Garza-Vargas, Archit Kulkarni, Nikhil Srivastava","doi":"10.1007/s10208-022-09577-5","DOIUrl":"https://doi.org/10.1007/s10208-022-09577-5","url":null,"abstract":"<p>We exhibit a randomized algorithm which, given a square matrix <span>(Ain mathbb {C}^{ntimes n})</span> with <span>(Vert AVert le 1)</span> and <span>(delta &gt;0)</span>, computes with high probability an invertible <i>V</i> and diagonal <i>D</i> such that <span>( Vert A-VDV^{-1}Vert le delta )</span> using <span>(O(T_mathsf {MM}(n)log ^2(n/delta )))</span> arithmetic operations, in finite arithmetic with <span>(O(log ^4(n/delta )log n))</span> bits of precision. The computed similarity <i>V</i> additionally satisfies <span>(Vert VVert Vert V^{-1}Vert le O(n^{2.5}/delta ))</span>. Here <span>(T_mathsf {MM}(n))</span> is the number of arithmetic operations required to multiply two <span>(ntimes n)</span> complex matrices numerically stably, known to satisfy <span>(T_mathsf {MM}(n)=O(n^{omega +eta }))</span> for every <span>(eta &gt;0)</span> where <span>(omega )</span> is the exponent of matrix multiplication (Demmel et al. in Numer Math 108(1):59–91, 2007). The algorithm is a variant of the spectral bisection algorithm in numerical linear algebra (Beavers Jr. and Denman in Numer Math 21(1-2):143–169, 1974) with a crucial Gaussian perturbation preprocessing step. Our result significantly improves the previously best-known provable running times of <span>(O(n^{10}/delta ^2))</span> arithmetic operations for diagonalization of general matrices (Armentano et al. in J Eur Math Soc 20(6):1375–1437, 2018) and (with regard to the dependence on <i>n</i>) <span>(O(n^3))</span> arithmetic operations for Hermitian matrices (Dekker and Traub in Linear Algebra Appl 4:137–154, 1971). It is the first algorithm to achieve nearly matrix multiplication time for diagonalization in any model of computation (real arithmetic, rational arithmetic, or finite arithmetic), thereby matching the complexity of other dense linear algebra operations such as inversion and <i>QR</i> factorization up to polylogarithmic factors. The proof rests on two new ingredients. (1) We show that adding a small complex Gaussian perturbation to <i>any</i> matrix splits its pseudospectrum into <i>n</i> small well-separated components. In particular, this implies that the eigenvalues of the perturbed matrix have a large minimum gap, a property of independent interest in random matrix theory. (2) We give a rigorous analysis of Roberts’ Newton iteration method (Roberts in Int J Control 32(4):677–687, 1980) for computing the sign function of a matrix in finite arithmetic, itself an open problem in numerical analysis since at least 1986.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"6 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2022-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138534306","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Bad and Good News for Strassen’s Laser Method: Border Rank of $$mathrm{Perm}_3$$ and Strict Submultiplicativity","authors":"Austin Conner, Hang Huang, J. Landsberg","doi":"10.1007/s10208-022-09579-3","DOIUrl":"https://doi.org/10.1007/s10208-022-09579-3","url":null,"abstract":"","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"1 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2022-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43274289","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Kähler Geometry of Framed Quiver Moduli and Machine Learning","authors":"G. Jeffreys, Siu-Cheong Lau","doi":"10.1007/s10208-022-09587-3","DOIUrl":"https://doi.org/10.1007/s10208-022-09587-3","url":null,"abstract":"","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"23 1","pages":"1899-1957"},"PeriodicalIF":3.0,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48055726","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Stirling-Type Formula for the Distribution of the Length of Longest Increasing Subsequences","authors":"F. Bornemann","doi":"10.1007/s10208-023-09604-z","DOIUrl":"https://doi.org/10.1007/s10208-023-09604-z","url":null,"abstract":"","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"1 1","pages":"1-39"},"PeriodicalIF":3.0,"publicationDate":"2022-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47944057","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Spectral Graph Matching and Regularized Quadratic Relaxations I Algorithm and Gaussian Analysis","authors":"Z. Fan, Cheng Mao, Yihong Wu, Jiaming Xu","doi":"10.1007/s10208-022-09570-y","DOIUrl":"https://doi.org/10.1007/s10208-022-09570-y","url":null,"abstract":"","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"23 1","pages":"1511 - 1565"},"PeriodicalIF":3.0,"publicationDate":"2022-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49612197","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Implicitisation and Parameterisation in Polynomial Functors","authors":"A. Blatter, J. Draisma, Emanuele Ventura","doi":"10.1007/s10208-023-09619-6","DOIUrl":"https://doi.org/10.1007/s10208-023-09619-6","url":null,"abstract":"","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":" ","pages":""},"PeriodicalIF":3.0,"publicationDate":"2022-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44023418","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Optimal Polynomial Meshes Exist on any Multivariate Convex Domain","authors":"Feng Dai, A. Prymak","doi":"10.48550/arXiv.2205.14111","DOIUrl":"https://doi.org/10.48550/arXiv.2205.14111","url":null,"abstract":"We show that optimal polynomial meshes exist for every convex body in $${mathbb {R}}^d$$ R d , confirming a conjecture by A. Kroó.","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"1 1","pages":"1-30"},"PeriodicalIF":3.0,"publicationDate":"2022-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47356223","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}