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Foundations of Computational Mathematics最新文献

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Convergence of Numerical Methods for the Navier–Stokes–Fourier System Driven by Uncertain Initial/Boundary Data 不确定初始/边界数据驱动的纳维-斯托克斯-傅里叶系统数值方法的收敛性
IF 3 1区 数学
Foundations of Computational Mathematics Pub Date : 2024-08-06 DOI: 10.1007/s10208-024-09666-7
Eduard Feireisl, Mária Lukáčová-Medvid’ová, Bangwei She, Yuhuan Yuan
{"title":"Convergence of Numerical Methods for the Navier–Stokes–Fourier System Driven by Uncertain Initial/Boundary Data","authors":"Eduard Feireisl, Mária Lukáčová-Medvid’ová, Bangwei She, Yuhuan Yuan","doi":"10.1007/s10208-024-09666-7","DOIUrl":"https://doi.org/10.1007/s10208-024-09666-7","url":null,"abstract":"<p>We consider the Navier–Stokes–Fourier system governing the motion of a general compressible, heat conducting, Newtonian fluid driven by random initial/boundary data. Convergence of the stochastic collocation and Monte Carlo numerical methods is shown under the hypothesis that approximate solutions are bounded in probability. Abstract results are illustrated by numerical experiments for the Rayleigh–Bénard convection problem.\u0000</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"156 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141899473","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}
引用次数: 0
Polynomial and Rational Measure Modifications of Orthogonal Polynomials via Infinite-Dimensional Banded Matrix Factorizations 通过无限维带状矩阵因式分解对正交多项式进行多项式和有理测度修正
IF 3 1区 数学
Foundations of Computational Mathematics Pub Date : 2024-08-05 DOI: 10.1007/s10208-024-09671-w
Timon S. Gutleb, Sheehan Olver, Richard Mikaël Slevinsky
{"title":"Polynomial and Rational Measure Modifications of Orthogonal Polynomials via Infinite-Dimensional Banded Matrix Factorizations","authors":"Timon S. Gutleb, Sheehan Olver, Richard Mikaël Slevinsky","doi":"10.1007/s10208-024-09671-w","DOIUrl":"https://doi.org/10.1007/s10208-024-09671-w","url":null,"abstract":"<p>We describe fast algorithms for approximating the connection coefficients between a family of orthogonal polynomials and another family with a polynomially or rationally modified measure. The connection coefficients are computed via infinite-dimensional banded matrix factorizations and may be used to compute the modified Jacobi matrices all in linear complexity with respect to the truncation degree. A family of orthogonal polynomials with modified classical weights is constructed that support banded differentiation matrices, enabling sparse spectral methods with modified classical orthogonal polynomials. We present several applications and numerical experiments using an open source implementation which make direct use of these results.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"23 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141895228","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}
引用次数: 0
Stable Liftings of Polynomial Traces on Tetrahedra 多项式轨迹在四面体上的稳定提升
IF 3 1区 数学
Foundations of Computational Mathematics Pub Date : 2024-07-29 DOI: 10.1007/s10208-024-09670-x
Charles Parker, Endre Süli
{"title":"Stable Liftings of Polynomial Traces on Tetrahedra","authors":"Charles Parker, Endre Süli","doi":"10.1007/s10208-024-09670-x","DOIUrl":"https://doi.org/10.1007/s10208-024-09670-x","url":null,"abstract":"<p>On the reference tetrahedron <span>(K)</span>, we construct, for each <span>(k in {mathbb {N}}_0)</span>, a right inverse for the trace operator <span>(u mapsto (u, partial _{textbf{n}} u, ldots , partial _{textbf{n}}^k u)|_{partial K})</span>. The operator is stable as a mapping from the trace space of <span>(W^{s, p}(K))</span> to <span>(W^{s, p}(K))</span> for all <span>(p in (1, infty ))</span> and <span>(s in (k+1/p, infty ))</span>. Moreover, if the data is the trace of a polynomial of degree <span>(N in {mathbb {N}}_0)</span>, then the resulting lifting is a polynomial of degree <i>N</i>. One consequence of the analysis is a novel characterization for the range of the trace operator.\u0000</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"19 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141836767","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}
引用次数: 0
Analysis of Langevin Monte Carlo from Poincaré to Log-Sobolev 从 Poincaré 到 Log-Sobolev 的 Langevin 蒙特卡洛分析
IF 3 1区 数学
Foundations of Computational Mathematics Pub Date : 2024-07-26 DOI: 10.1007/s10208-024-09667-6
Sinho Chewi, Murat A. Erdogdu, Mufan Li, Ruoqi Shen, Matthew S. Zhang
{"title":"Analysis of Langevin Monte Carlo from Poincaré to Log-Sobolev","authors":"Sinho Chewi, Murat A. Erdogdu, Mufan Li, Ruoqi Shen, Matthew S. Zhang","doi":"10.1007/s10208-024-09667-6","DOIUrl":"https://doi.org/10.1007/s10208-024-09667-6","url":null,"abstract":"<p>Classically, the continuous-time Langevin diffusion converges exponentially fast to its stationary distribution <span>(pi )</span> under the sole assumption that <span>(pi )</span> satisfies a Poincaré inequality. Using this fact to provide guarantees for the discrete-time Langevin Monte Carlo (LMC) algorithm, however, is considerably more challenging due to the need for working with chi-squared or Rényi divergences, and prior works have largely focused on strongly log-concave targets. In this work, we provide the first convergence guarantees for LMC assuming that <span>(pi )</span> satisfies either a Latała–Oleszkiewicz or modified log-Sobolev inequality, which interpolates between the Poincaré and log-Sobolev settings. Unlike prior works, our results allow for weak smoothness and do not require convexity or dissipativity conditions.\u0000</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"57 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141768486","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}
引用次数: 0
Resonances as a Computational Tool 作为计算工具的共振
IF 3 1区 数学
Foundations of Computational Mathematics Pub Date : 2024-07-26 DOI: 10.1007/s10208-024-09665-8
Frédéric Rousset, Katharina Schratz
{"title":"Resonances as a Computational Tool","authors":"Frédéric Rousset, Katharina Schratz","doi":"10.1007/s10208-024-09665-8","DOIUrl":"https://doi.org/10.1007/s10208-024-09665-8","url":null,"abstract":"<p>A large toolbox of numerical schemes for dispersive equations has been established, based on different discretization techniques such as discretizing the variation-of-constants formula (e.g., exponential integrators) or splitting the full equation into a series of simpler subproblems (e.g., splitting methods). In many situations these classical schemes allow a precise and efficient approximation. This, however, drastically changes whenever non-smooth phenomena enter the scene such as for problems at low regularity and high oscillations. Classical schemes fail to capture the oscillatory nature of the solution, and this may lead to severe instabilities and loss of convergence. In this article we review a new class of resonance-based schemes. The key idea in the construction of the new schemes is to tackle and deeply embed the underlying nonlinear structure of resonances into the numerical discretization. As in the continuous case, these terms are central to structure preservation and offer the new schemes strong properties at low regularity.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"32 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141768455","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}
引用次数: 0
A Polynomial Time Iterative Algorithm for Matching Gaussian Matrices with Non-vanishing Correlation 匹配非相关性高斯矩阵的多项式时间迭代算法
IF 3 1区 数学
Foundations of Computational Mathematics Pub Date : 2024-07-22 DOI: 10.1007/s10208-024-09662-x
Jian Ding, Zhangsong Li
{"title":"A Polynomial Time Iterative Algorithm for Matching Gaussian Matrices with Non-vanishing Correlation","authors":"Jian Ding, Zhangsong Li","doi":"10.1007/s10208-024-09662-x","DOIUrl":"https://doi.org/10.1007/s10208-024-09662-x","url":null,"abstract":"<p>Motivated by the problem of matching vertices in two correlated Erdős-Rényi graphs, we study the problem of matching two correlated Gaussian Wigner matrices. We propose an iterative matching algorithm, which succeeds in polynomial time as long as the correlation between the two Gaussian matrices does not vanish. Our result is the first polynomial time algorithm that solves a graph matching type of problem when the correlation is an arbitrarily small constant.\u0000</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"60 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141737076","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}
引用次数: 0
Quantitative Stability of the Pushforward Operation by an Optimal Transport Map 通过最优传输图实现前推操作的定量稳定性
IF 3 1区 数学
Foundations of Computational Mathematics Pub Date : 2024-07-19 DOI: 10.1007/s10208-024-09669-4
Guillaume Carlier, Alex Delalande, Quentin Mérigot
{"title":"Quantitative Stability of the Pushforward Operation by an Optimal Transport Map","authors":"Guillaume Carlier, Alex Delalande, Quentin Mérigot","doi":"10.1007/s10208-024-09669-4","DOIUrl":"https://doi.org/10.1007/s10208-024-09669-4","url":null,"abstract":"<p>We study the quantitative stability of the mapping that to a measure associates its pushforward measure by a fixed (non-smooth) optimal transport map. We exhibit a tight Hölder-behavior for this operation under minimal assumptions. Our proof essentially relies on a new bound that quantifies the size of the singular sets of a convex and Lipschitz continuous function on a bounded domain.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"25 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141730631","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}
引用次数: 0
Koszul Complexes and Relative Homological Algebra of Functors Over Posets Koszul 复数和 Posets 上函数的相对同调代数
IF 3 1区 数学
Foundations of Computational Mathematics Pub Date : 2024-06-18 DOI: 10.1007/s10208-024-09660-z
Wojciech Chachólski, Andrea Guidolin, Isaac Ren, Martina Scolamiero, Francesca Tombari
{"title":"Koszul Complexes and Relative Homological Algebra of Functors Over Posets","authors":"Wojciech Chachólski, Andrea Guidolin, Isaac Ren, Martina Scolamiero, Francesca Tombari","doi":"10.1007/s10208-024-09660-z","DOIUrl":"https://doi.org/10.1007/s10208-024-09660-z","url":null,"abstract":"<p>Under certain conditions, Koszul complexes can be used to calculate relative Betti diagrams of vector space-valued functors indexed by a poset, without the explicit computation of global minimal relative resolutions. In relative homological algebra of such functors, free functors are replaced by an arbitrary family of functors. Relative Betti diagrams encode the multiplicities of these functors in minimal relative resolutions. In this article we provide conditions under which grading the chosen family of functors leads to explicit Koszul complexes whose homology dimensions are the relative Betti diagrams, thus giving a scheme for the computation of these numerical descriptors.\u0000</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"14 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141425513","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}
引用次数: 0
A Local Nearly Linearly Convergent First-Order Method for Nonsmooth Functions with Quadratic Growth 具有二次增长的非光滑函数的局部近线性收敛一阶方法
IF 3 1区 数学
Foundations of Computational Mathematics Pub Date : 2024-06-14 DOI: 10.1007/s10208-024-09653-y
Damek Davis, Liwei Jiang
{"title":"A Local Nearly Linearly Convergent First-Order Method for Nonsmooth Functions with Quadratic Growth","authors":"Damek Davis, Liwei Jiang","doi":"10.1007/s10208-024-09653-y","DOIUrl":"https://doi.org/10.1007/s10208-024-09653-y","url":null,"abstract":"<p>Classical results show that gradient descent converges linearly to minimizers of smooth strongly convex functions. A natural question is whether there exists a locally nearly linearly convergent method for nonsmooth functions with quadratic growth. This work designs such a method for a wide class of nonsmooth and nonconvex locally Lipschitz functions, including max-of-smooth, Shapiro’s decomposable class, and generic semialgebraic functions. The algorithm is parameter-free and derives from Goldstein’s conceptual subgradient method.\u0000</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"29 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141326869","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}
引用次数: 0
Convergent Regularization in Inverse Problems and Linear Plug-and-Play Denoisers 逆问题中的收敛正则化和线性即插即用去噪器
IF 3 1区 数学
Foundations of Computational Mathematics Pub Date : 2024-06-03 DOI: 10.1007/s10208-024-09654-x
Andreas Hauptmann, Subhadip Mukherjee, Carola-Bibiane Schönlieb, Ferdia Sherry
{"title":"Convergent Regularization in Inverse Problems and Linear Plug-and-Play Denoisers","authors":"Andreas Hauptmann, Subhadip Mukherjee, Carola-Bibiane Schönlieb, Ferdia Sherry","doi":"10.1007/s10208-024-09654-x","DOIUrl":"https://doi.org/10.1007/s10208-024-09654-x","url":null,"abstract":"<p>Regularization is necessary when solving inverse problems to ensure the well-posedness of the solution map. Additionally, it is desired that the chosen regularization strategy is convergent in the sense that the solution map converges to a solution of the noise-free operator equation. This provides an important guarantee that stable solutions can be computed for all noise levels and that solutions satisfy the operator equation in the limit of vanishing noise. In recent years, reconstructions in inverse problems are increasingly approached from a data-driven perspective. Despite empirical success, the majority of data-driven approaches do not provide a convergent regularization strategy. One such popular example is given by iterative plug-and-play (PnP) denoising using off-the-shelf image denoisers. These usually provide only convergence of the PnP iterates to a fixed point, under suitable regularity assumptions on the denoiser, rather than convergence of the method as a regularization technique, thatis under vanishing noise and regularization strength. This paper serves two purposes: first, we provide an overview of the classical regularization theory in inverse problems and survey a few notable recent data-driven methods that are provably convergent regularization schemes. We then continue to discuss PnP algorithms and their established convergence guarantees. Subsequently, we consider PnP algorithms with learned linear denoisers and propose a novel spectral filtering technique of the denoiser to control the strength of regularization. Further, by relating the implicit regularization of the denoiser to an explicit regularization functional, we are the first to rigorously show that PnP with a learned linear denoiser leads to a convergent regularization scheme. The theoretical analysis is corroborated by numerical experiments for the classical inverse problem of tomographic image reconstruction.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"47 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141246310","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}
引用次数: 0
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