Mathematics of Computation最新文献

{"title":"Polynomial preserving recovery for the finite volume element methods under simplex meshes","authors":"Yonghai Li, Peng Yang, Zhimin Zhang","doi":"10.1090/mcom/3980","DOIUrl":"https://doi.org/10.1090/mcom/3980","url":null,"abstract":"<p>The recovered gradient, using the polynomial preserving recovery (PPR), is constructed for the finite volume element method (FVEM) under simplex meshes. Regarding the main results of this paper, there are two aspects. Firstly, we investigate the supercloseness property of the FVEM, specifically examining the quadratic FVEM under tetrahedral meshes. Secondly, we present several guidelines for selecting computing nodes such that the least-squares fitting procedure of the PPR admits a unique solution. Numerical experiments demonstrate that the recovered gradient by the PPR exhibits superconvergence.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"181 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511725","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":"Convergence of the numerical approximations and well-posedness: Nonlocal conservation laws with rough flux","authors":"Aekta Aggarwal, Ganesh Vaidya","doi":"10.1090/mcom/3976","DOIUrl":"https://doi.org/10.1090/mcom/3976","url":null,"abstract":"<p>We study a class of nonlinear nonlocal conservation laws with discontinuous flux, modeling crowd dynamics and traffic flow. The discontinuous coefficient of the flux function is assumed to be of bounded variation (BV) and bounded away from zero, and hence the spatial discontinuities of the flux function can be infinitely many with possible accumulation points. Strong compactness of the Godunov and Lax-Friedrichs type approximations is proved, providing the existence of entropy solutions. A proof of the uniqueness of the adapted entropy solutions is provided, establishing the convergence of the entire sequence of finite volume approximations to the adapted entropy solution. As per the current literature, this is the first well-posedness result for the aforesaid class and connects the theory of nonlocal conservation laws (with discontinuous flux), with its local counterpart in a generic setup. Some numerical examples are presented to display the performance of the schemes and explore the limiting behavior of these nonlocal conservation laws to their local counterparts.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"35 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140935469","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":"Wavenumber-explicit stability and convergence analysis of ℎ𝑝 finite element discretizations of Helmholtz problems in piecewise smooth media","authors":"M. Bernkopf, T. Chaumont-Frelet, J. Melenk","doi":"10.1090/mcom/3958","DOIUrl":"https://doi.org/10.1090/mcom/3958","url":null,"abstract":"<p>We present a wavenumber-explicit convergence analysis of the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"h p\"> <mml:semantics> <mml:mrow> <mml:mi>h</mml:mi> <mml:mi>p</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">hp</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Finite Element Method applied to a class of heterogeneous Helmholtz problems with piecewise analytic coefficients at large wavenumber <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=\"application/x-tex\">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Our analysis covers the heterogeneous Helmholtz equation with Robin, exact Dirichlet-to-Neumann, and second order absorbing boundary conditions, as well as perfectly matched layers.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"33 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141195350","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":"Convergence proof for the GenCol algorithm in the case of two-marginal optimal transport","authors":"Gero Friesecke, Maximilian Penka","doi":"10.1090/mcom/3968","DOIUrl":"https://doi.org/10.1090/mcom/3968","url":null,"abstract":"<p>The recently introduced Genetic Column Generation (GenCol) algorithm has been numerically observed to efficiently and accurately compute high-dimensional optimal transport (OT) plans for general multi-marginal problems, but theoretical results on the algorithm have hitherto been lacking. The algorithm solves the OT linear program on a dynamically updated low-dimensional submanifold consisting of sparse plans. The submanifold dimension exceeds the sparse support of optimal plans only by a fixed factor <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"beta\"> <mml:semantics> <mml:mi>β<!-- β --></mml:mi> <mml:annotation encoding=\"application/x-tex\">beta</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Here we prove that for <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"beta greater-than-or-equal-to 2\"> <mml:semantics> <mml:mrow> <mml:mi>β<!-- β --></mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">beta geq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and in the two-marginal case, GenCol always converges to an exact solution, for arbitrary costs and marginals. The proof relies on the concept of c-cyclical monotonicity. As an offshoot, GenCol rigorously reduces the data complexity of numerically solving two-marginal OT problems from <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper O left-parenthesis script l squared right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">O(ell ^2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper O left-parenthesis script l right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">O(ell )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> without any loss in accuracy, where <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script l\"> <mml:semantics> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:annotation encoding=\"application/x-tex\">ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the number of discretization points for a single marginal. At the end of the paper we also present some insights into the convergence behavior in the multi-marginal case.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"65 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140935884","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":"Six-dimensional sphere packing and linear programming","authors":"Matthew de Courcy-Ireland, Maria Dostert, Maryna Viazovska","doi":"10.1090/mcom/3959","DOIUrl":"https://doi.org/10.1090/mcom/3959","url":null,"abstract":"<p>We prove that the Cohn–Elkies linear programming bound for sphere packing is not sharp in dimension 6. The proof uses duality and optimization over a space of modular forms, generalizing a construction of Cohn–Triantafillou [Math. Comp. 91 (2021), pp. 491–508] to the case of odd weight and non-trivial character.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"133 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140935684","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":"Convergence of Langevin-simulated annealing algorithms with multiplicative noise","authors":"Pierre Bras, Gilles Pagès","doi":"10.1090/mcom/3899","DOIUrl":"https://doi.org/10.1090/mcom/3899","url":null,"abstract":"&lt;p&gt;We study the convergence of Langevin-Simulated Annealing type algorithms with multiplicative noise, i.e. for &lt;inline-formula content-type=\"math/mathml\"&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper V colon double-struck upper R Superscript d Baseline right-arrow double-struck upper R\"&gt; &lt;mml:semantics&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;V&lt;/mml:mi&gt; &lt;mml:mo&gt;:&lt;/mml:mo&gt; &lt;mml:msup&gt; &lt;mml:mrow&gt; &lt;mml:mi mathvariant=\"double-struck\"&gt;R&lt;/mml:mi&gt; &lt;/mml:mrow&gt; &lt;mml:mi&gt;d&lt;/mml:mi&gt; &lt;/mml:msup&gt; &lt;mml:mo stretchy=\"false\"&gt;→&lt;!-- → --&gt;&lt;/mml:mo&gt; &lt;mml:mrow&gt; &lt;mml:mi mathvariant=\"double-struck\"&gt;R&lt;/mml:mi&gt; &lt;/mml:mrow&gt; &lt;/mml:mrow&gt; &lt;mml:annotation encoding=\"application/x-tex\"&gt;V : mathbb {R}^d to mathbb {R}&lt;/mml:annotation&gt; &lt;/mml:semantics&gt; &lt;/mml:math&gt; &lt;/inline-formula&gt; a potential function to minimize, we consider the stochastic differential equation &lt;inline-formula content-type=\"math/mathml\"&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d upper Y Subscript t Baseline equals minus sigma sigma Superscript down-tack Baseline nabla upper V left-parenthesis upper Y Subscript t Baseline right-parenthesis\"&gt; &lt;mml:semantics&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;d&lt;/mml:mi&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;Y&lt;/mml:mi&gt; &lt;mml:mi&gt;t&lt;/mml:mi&gt; &lt;/mml:msub&gt; &lt;mml:mo&gt;=&lt;/mml:mo&gt; &lt;mml:mo&gt;−&lt;!-- − --&gt;&lt;/mml:mo&gt; &lt;mml:mi&gt;σ&lt;!-- σ --&gt;&lt;/mml:mi&gt; &lt;mml:msup&gt; &lt;mml:mi&gt;σ&lt;!-- σ --&gt;&lt;/mml:mi&gt; &lt;mml:mi mathvariant=\"normal\"&gt;⊤&lt;!-- ⊤ --&gt;&lt;/mml:mi&gt; &lt;/mml:msup&gt; &lt;mml:mi mathvariant=\"normal\"&gt;∇&lt;!-- ∇ --&gt;&lt;/mml:mi&gt; &lt;mml:mi&gt;V&lt;/mml:mi&gt; &lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;Y&lt;/mml:mi&gt; &lt;mml:mi&gt;t&lt;/mml:mi&gt; &lt;/mml:msub&gt; &lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;mml:annotation encoding=\"application/x-tex\"&gt;dY_t = - sigma sigma ^top nabla V(Y_t)&lt;/mml:annotation&gt; &lt;/mml:semantics&gt; &lt;/mml:math&gt; &lt;/inline-formula&gt; &lt;inline-formula content-type=\"math/mathml\"&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d t plus a left-parenthesis t right-parenthesis sigma left-parenthesis upper Y Subscript t Baseline right-parenthesis d upper W Subscript t plus a left-parenthesis t right-parenthesis squared normal upper Upsilon left-parenthesis upper Y Subscript t Baseline right-parenthesis d t\"&gt; &lt;mml:semantics&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;d&lt;/mml:mi&gt; &lt;mml:mi&gt;t&lt;/mml:mi&gt; &lt;mml:mo&gt;+&lt;/mml:mo&gt; &lt;mml:mi&gt;a&lt;/mml:mi&gt; &lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt; &lt;mml:mi&gt;t&lt;/mml:mi&gt; &lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt; &lt;mml:mi&gt;σ&lt;!-- σ --&gt;&lt;/mml:mi&gt; &lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;Y&lt;/mml:mi&gt; &lt;mml:mi&gt;t&lt;/mml:mi&gt; &lt;/mml:msub&gt; &lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt; &lt;mml:mi&gt;d&lt;/mml:mi&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;W&lt;/mml:mi&gt; &lt;mml:mi&gt;t&lt;/mml:mi&gt; &lt;/mml:msub&gt; &lt;mml:mo&gt;+&lt;/mml:mo&gt; &lt;mml:mi&gt;a&lt;/mml:mi&gt; &lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt; &lt;mml:mi&gt;t&lt;/mml:mi&gt; &lt;mml:msup&gt; &lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt; &lt;mml:mn&gt;2&lt;/mml:mn&gt; &lt;/mml:msup&gt; &lt;mml:mi mathvariant=\"normal\"&gt;Υ&lt;!-- Υ --&gt;&lt;/mml:mi&gt; &lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;Y&lt;/mml:mi&gt; &lt;mml:mi&gt;t&lt;/mml:mi&gt; &lt;/mml:msub&gt; &lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt; &lt;mml:mi&gt;d&lt;/mml:mi&gt; &lt;mml:mi&gt;t&lt;/mml:mi&gt; &lt;/mml:mro","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"156 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931328","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":"Stochastic nested primal-dual method for nonconvex constrained composition optimization","authors":"Lingzi Jin, Xiao Wang","doi":"10.1090/mcom/3965","DOIUrl":"https://doi.org/10.1090/mcom/3965","url":null,"abstract":"<p>In this paper we study the nonconvex constrained composition optimization, in which the objective contains a composition of two expected-value functions whose accurate information is normally expensive to calculate. We propose a STochastic nEsted Primal-dual (STEP) method for such problems. In each iteration, with an auxiliary variable introduced to track the inner layer function values we compute stochastic gradients of the nested function using a subsampling strategy. To alleviate difficulties caused by possibly nonconvex constraints, we construct a stochastic approximation to the linearized augmented Lagrangian function to update the primal variable, which further motivates to update the dual variable in a weighted-average way. Moreover, to better understand the asymptotic dynamics of the update schemes we consider a deterministic continuous-time system from the perspective of ordinary differential equation (ODE). We analyze the Karush-Kuhn-Tucker measure at the output by the STEP method with constant parameters and establish its iteration and sample complexities to find an <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"epsilon\"> <mml:semantics> <mml:mi>ϵ<!-- ϵ --></mml:mi> <mml:annotation encoding=\"application/x-tex\">epsilon</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-stationary point, ensuring that expected stationarity, feasibility as well as complementary slackness are below accuracy <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"epsilon\"> <mml:semantics> <mml:mi>ϵ<!-- ϵ --></mml:mi> <mml:annotation encoding=\"application/x-tex\">epsilon</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. To leverage the benefit of the (near) initial feasibility in the STEP method, we propose a two-stage framework incorporating a feasibility-seeking phase, aiming to locate a nearly feasible initial point. Moreover, to enhance the adaptivity of the STEP algorithm, we propose an adaptive variant by adaptively adjusting its parameters, along with a complexity analysis. Numerical results on a risk-averse portfolio optimization problem and an orthogonal nonnegative matrix decomposition reveal the effectiveness of the proposed algorithms.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"38 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931201","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":"Uniform accuracy of implicit-explicit Runge-Kutta (IMEX-RK) schemes for hyperbolic systems with relaxation","authors":"Jingwei Hu, Ruiwen Shu","doi":"10.1090/mcom/3967","DOIUrl":"https://doi.org/10.1090/mcom/3967","url":null,"abstract":"<p>Implicit-explicit Runge-Kutta (IMEX-RK) schemes are popular methods to treat multiscale equations that contain a stiff part and a non-stiff part, where the stiff part is characterized by a small parameter <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"epsilon\"> <mml:semantics> <mml:mi>ε<!-- ε --></mml:mi> <mml:annotation encoding=\"application/x-tex\">varepsilon</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In this work, we prove rigorously the uniform stability and uniform accuracy of a class of IMEX-RK schemes for a linear hyperbolic system with stiff relaxation. The result we obtain is optimal in the sense that it holds regardless of the value of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"epsilon\"> <mml:semantics> <mml:mi>ε<!-- ε --></mml:mi> <mml:annotation encoding=\"application/x-tex\">varepsilon</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the order of accuracy is the same as the design order of the original scheme, i.e., there is no order reduction.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"32 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140942540","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":"Extending error bounds for radial basis function interpolation to measuring the error in higher order Sobolev norms","authors":"T. Hangelbroek, C. Rieger","doi":"10.1090/mcom/3960","DOIUrl":"https://doi.org/10.1090/mcom/3960","url":null,"abstract":"<p>Radial basis functions (RBFs) are prominent examples for reproducing kernels with associated reproducing kernel Hilbert spaces (RKHSs). The convergence theory for the kernel-based interpolation in that space is well understood and optimal rates for the whole RKHS are often known. Schaback added the doubling trick [Math. Comp. 68 (1999), pp. 201–216], which shows that functions having double the smoothness required by the RKHS (along with specific, albeit complicated boundary behavior) can be approximated with higher convergence rates than the optimal rates for the whole space. Other advances allowed interpolation of target functions which are less smooth, and different norms which measure interpolation error. The current state of the art of error analysis for RBF interpolation treats target functions having smoothness up to twice that of the native space, but error measured in norms which are weaker than that required for membership in the RKHS.</p> <p>Motivated by the fact that the kernels and the approximants they generate are smoother than required by the native space, this article extends the doubling trick to error which measures higher smoothness. This extension holds for a family of kernels satisfying easily checked hypotheses which we describe in this article, and includes many prominent RBFs. In the course of the proof, new convergence rates are obtained for the abstract operator considered by Devore and Ron in [Trans. Amer. Math. Soc. 362 (2010), pp. 6205–6229], and new Bernstein estimates are obtained relating high order smoothness norms to the native space norm.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"60 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931238","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 discrete ground states of rotating Bose–Einstein condensates","authors":"Patrick Henning, Mahima Yadav","doi":"10.1090/mcom/3962","DOIUrl":"https://doi.org/10.1090/mcom/3962","url":null,"abstract":"<p>The ground states of Bose–Einstein condensates in a rotating frame can be described as constrained minimizers of the Gross–Pitaevskii energy functional with an angular momentum term. In this paper we consider the corresponding discrete minimization problem in Lagrange finite element spaces of arbitrary polynomial order and we investigate the approximation properties of discrete ground states. In particular, we prove a priori error estimates of optimal order in the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L squared\"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding=\"application/x-tex\">L^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>- and <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H Superscript 1\"> <mml:semantics> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding=\"application/x-tex\">H^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-norm, as well as for the ground state energy and the corresponding chemical potential. A central issue in the analysis of the problem is the missing uniqueness of ground states, which is mainly caused by the invariance of the energy functional under complex phase shifts. Our error analysis is therefore based on an Euler–Lagrange functional that we restrict to certain tangent spaces in which we have local uniqueness of ground states. This gives rise to an error decomposition that is ultimately used to derive the desired a priori error estimates. We also present numerical experiments to illustrate various aspects of the problem structure.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"352 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140942064","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}