Foundations of Computational Mathematics最新文献

{"title":"Adaptive Mesh Refinement for Arbitrary Initial Triangulations","authors":"Lars Diening, Lukas Gehring, Johannes Storn","doi":"10.1007/s10208-025-09698-7","DOIUrl":"https://doi.org/10.1007/s10208-025-09698-7","url":null,"abstract":"<p>We introduce a simple initialization of the Maubach bisection routine for adaptive mesh refinement which applies to any conforming initial triangulation and terminates in linear time with respect to the number of initial vertices. We show that Maubach’s routine with this initialization always terminates and generates meshes that preserve shape regularity and satisfy the closure estimate needed for optimal convergence of adaptive schemes. Our ansatz allows for the intrinsic use of existing implementations.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"21 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143695241","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":"Local Space-Preserving Decompositions for the Bubble Transform","authors":"Richard Falk, Ragnar Winther","doi":"10.1007/s10208-025-09700-2","DOIUrl":"https://doi.org/10.1007/s10208-025-09700-2","url":null,"abstract":"<p>The bubble transform is a procedure to decompose differential forms, which are piecewise smooth with respect to a given triangulation of the domain, into a sum of local bubbles. In this paper, an improved version of a construction in the setting of the de Rham complex previously proposed by the authors is presented. The major improvement in the decomposition is that unlike the previous results, in which the individual bubbles were rational functions with the property that groups of local bubbles summed up to preserve piecewise smoothness, the new decomposition is strictly space-preserving in the sense that each local bubble preserves piecewise smoothness. An important property of the transform is that the construction only depends on the given triangulation of the domain and is independent of any finite element space. On the other hand, all the standard piecewise polynomial spaces are invariant under the transform. Other key properties of the transform are that it commutes with the exterior derivative, is bounded in <span>(L^2)</span>, and satisfies the <i>stable decomposition property</i>.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"61 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143641098","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":"Towards a Fluid Computer","authors":"Robert Cardona, Eva Miranda, Daniel Peralta-Salas","doi":"10.1007/s10208-025-09699-6","DOIUrl":"https://doi.org/10.1007/s10208-025-09699-6","url":null,"abstract":"<p>In 1991, Moore (Nonlinearity 4:199–230, 1991) raised a question about whether hydrodynamics is capable of performing computations. Similarly, in 2016, Tao (J Am Math Soc 29(3):601–674, 2016) asked whether a mechanical system, including a fluid flow, can simulate a universal Turing machine. In this expository article, we review the construction in Cardona et al. (Proc Natl Acad Sci 118(19):e2026818118, 2021) of a “Fluid computer” in dimension 3 that combines techniques in symbolic dynamics with the connection between steady Euler flows and contact geometry unveiled by Etnyre and Ghrist. In addition, we argue that the metric that renders the vector field Beltrami cannot be critical in the Chern-Hamilton sense (Chern and Hamilton in On Riemannian metrics adapted to three-dimensional contact manifolds, Springer, Berlin, 1985). We also sketch the completely different construction for the Euclidean metric in <span>(mathbb {R}^3)</span> as given in Cardona et al. (J Math Pures Appl 169:50–81, 2023). These results reveal the existence of undecidable fluid particle paths. We conclude the article with a list of open problems.\u0000</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"16 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143618496","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":"Non-parametric Learning of Stochastic Differential Equations with Non-asymptotic Fast Rates of Convergence","authors":"Riccardo Bonalli, Alessandro Rudi","doi":"10.1007/s10208-025-09705-x","DOIUrl":"https://doi.org/10.1007/s10208-025-09705-x","url":null,"abstract":"<p>We propose a novel non-parametric learning paradigm for the identification of drift and diffusion coefficients of multi-dimensional non-linear stochastic differential equations, which relies upon discrete-time observations of the state. The key idea essentially consists of fitting a RKHS-based approximation of the corresponding Fokker–Planck equation to such observations, yielding theoretical estimates of non-asymptotic learning rates which, unlike previous works, become increasingly tighter when the regularity of the unknown drift and diffusion coefficients becomes higher. Our method being kernel-based, offline pre-processing may be profitably leveraged to enable efficient numerical implementation, offering excellent balance between precision and computational complexity.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"11 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143569769","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":"Sums of Squares Certificates for Polynomial Moment Inequalities","authors":"Igor Klep, Victor Magron, Jurij Volčič","doi":"10.1007/s10208-025-09703-z","DOIUrl":"https://doi.org/10.1007/s10208-025-09703-z","url":null,"abstract":"<p>This paper introduces and develops the algebraic framework of moment polynomials, which are polynomial expressions in commuting variables and their formal mixed moments. Their positivity and optimization over probability measures supported on semialgebraic sets and subject to moment polynomial constraints is investigated. On the one hand, a positive solution to Hilbert’s 17th problem for pseudo-moments is given. On the other hand, moment polynomials positive on actual measures are shown to be sums of squares and formal moments of squares up to arbitrarily small perturbation of their coefficients. When only measures supported on a bounded semialgebraic set are considered, a stronger algebraic certificate for moment polynomial positivity is derived. This result gives rise to a converging hierarchy of semidefinite programs for moment polynomial optimization. Finally, as an application, two open nonlinear Bell inequalities from quantum physics are settled.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"35 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143546173","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":"An Unfiltered Low-Regularity Integrator for the KdV Equation with Solutions Below $$mathbf{H^1}$$","authors":"Buyang Li, Yifei Wu","doi":"10.1007/s10208-025-09702-0","DOIUrl":"https://doi.org/10.1007/s10208-025-09702-0","url":null,"abstract":"<p>This article is concerned with the construction and analysis of new time discretizations for the KdV equation on a torus for low-regularity solutions below <span>(H^1)</span>. New harmonic analysis tools, including averaging approximations to the exponential phase functions and trilinear estimates of the KdV operator, are established for the construction and analysis of time discretizations with higher convergence orders under low-regularity conditions. In addition, new perturbation techniques are introduced to establish stability estimates of time discretizations under low-regularity conditions without using filters when the energy techniques fail. The proposed method is proved to be convergent with order <span>(gamma )</span> (up to a logarithmic factor) in <span>(L^2)</span> under the regularity condition <span>(uin C([0,T];H^gamma ))</span> for <span>(gamma in (0,1])</span>.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"59 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143546174","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":"Restarts Subject to Approximate Sharpness: A Parameter-Free and Optimal Scheme For First-Order Methods","authors":"Ben Adcock, Matthew J. Colbrook, Maksym Neyra-Nesterenko","doi":"10.1007/s10208-024-09673-8","DOIUrl":"https://doi.org/10.1007/s10208-024-09673-8","url":null,"abstract":"<p>Sharpness is an almost generic assumption in continuous optimization that bounds the distance from minima by objective function suboptimality. It facilitates the acceleration of first-order methods through <i>restarts</i>. However, sharpness involves problem-specific constants that are typically unknown, and restart schemes typically reduce convergence rates. Moreover, these schemes are challenging to apply in the presence of noise or with approximate model classes (e.g., in compressive imaging or learning problems), and they generally assume that the first-order method used produces feasible iterates. We consider the assumption of <i>approximate sharpness</i>, a generalization of sharpness that incorporates an unknown constant perturbation to the objective function error. This constant offers greater robustness (e.g., with respect to noise or relaxation of model classes) for finding approximate minimizers. By employing a new type of search over the unknown constants, we design a restart scheme that applies to general first-order methods and does not require the first-order method to produce feasible iterates. Our scheme maintains the same convergence rate as when the constants are known. The convergence rates we achieve for various first-order methods match the optimal rates or improve on previously established rates for a wide range of problems. We showcase our restart scheme in several examples and highlight potential future applications and developments of our framework and theory.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"13 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143443283","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":"Multilinear Hyperquiver Representations","authors":"Tommi Muller, Vidit Nanda, Anna Seigal","doi":"10.1007/s10208-025-09692-z","DOIUrl":"https://doi.org/10.1007/s10208-025-09692-z","url":null,"abstract":"<p>We count singular vector tuples of a system of tensors assigned to the edges of a directed hypergraph. To do so, we study the generalisation of quivers to directed hypergraphs. Assigning vector spaces to the nodes of a hypergraph and multilinear maps to its hyperedges gives a hyperquiver representation. Hyperquiver representations generalise quiver representations (where all hyperedges are edges) and tensors (where there is only one multilinear map). The singular vectors of a hyperquiver representation are a compatible assignment of vectors to the nodes. We compute the dimension and degree of the variety of singular vectors of a sufficiently generic hyperquiver representation. Our formula specialises to known results that count the singular vectors and eigenvectors of a generic tensor. Lastly, we study a hypergraph generalisation of the inverse tensor eigenvalue problem and solve it algorithmically.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"9 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143417487","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":"Representations of the Symmetric Group are Decomposable in Polynomial Time","authors":"Sheehan Olver","doi":"10.1007/s10208-025-09697-8","DOIUrl":"https://doi.org/10.1007/s10208-025-09697-8","url":null,"abstract":"<p>We introduce an algorithm to decompose matrix representations of the symmetric group over the reals into irreducible representations, which as a by-product also computes the multiplicities of the irreducible representations. The algorithm applied to a <i>d</i>-dimensional representation of <span>(S_n)</span> is shown to have a complexity of <span>({mathcal {O}}(n^2 d^3))</span> operations for determining which irreducible representations are present and their corresponding multiplicities and a further <span>({mathcal {O}}(n d^4))</span> operations to fully decompose representations with non-trivial multiplicities. These complexity bounds are pessimistic and in a practical implementation using floating point arithmetic and exploiting sparsity we observe better complexity. We demonstrate this algorithm on the problem of computing multiplicities of two tensor products of irreducible representations (the Kronecker coefficients problem) as well as higher order tensor products. For hook and hook-like irreducible representations the algorithm has polynomial complexity as <i>n</i> increases. We also demonstrate an application to constructing a basis of homogeneous polynomials so that applying a permutation of variables induces an irreducible representation.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"62 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143385014","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":"Safely Learning Dynamical Systems","authors":"Amir Ali Ahmadi, Abraar Chaudhry, Vikas Sindhwani, Stephen Tu","doi":"10.1007/s10208-025-09689-8","DOIUrl":"https://doi.org/10.1007/s10208-025-09689-8","url":null,"abstract":"<p>A fundamental challenge in learning an unknown dynamical system is to reduce model uncertainty by making measurements while maintaining safety. In this work, we formulate a mathematical definition of what it means to safely learn a dynamical system by sequentially deciding where to initialize the next trajectory. In our framework, the state of the system is required to stay within a safety region for a horizon of <i>T</i> time steps under the action of all dynamical systems that (i) belong to a given initial uncertainty set, and (ii) are consistent with the information gathered so far. For our first set of results, we consider the setting of safely learning a linear dynamical system involving <i>n</i> states. For the case <span>(T=1)</span>, we present a linear programming-based algorithm that either safely recovers the true dynamics from at most <i>n</i> trajectories, or certifies that safe learning is impossible. For <span>(T=2)</span>, we give a semidefinite representation of the set of safe initial conditions and show that <span>(lceil n/2 rceil )</span> trajectories generically suffice for safe learning. For <span>(T = infty )</span>, we provide semidefinite representable inner approximations of the set of safe initial conditions and show that one trajectory generically suffices for safe learning. Finally, we extend a number of our results to the cases where the initial uncertainty set contains sparse, low-rank, or permutation matrices, or when the dynamical system involves a control input. Our second set of results concerns the problem of safely learning a general class of nonlinear dynamical systems. For the case <span>(T=1)</span>, we give a second-order cone programming based representation of the set of safe initial conditions. For <span>(T=infty )</span>, we provide semidefinite representable inner approximations to the set of safe initial conditions. We show how one can safely collect trajectories and fit a polynomial model of the nonlinear dynamics that is consistent with the initial uncertainty set and best agrees with the observations. We also present extensions of some of our results to the cases where the measurements are noisy or the dynamical system involves disturbances.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"28 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143125227","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}