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Wasserstein Archetypal Analysis 瓦瑟斯坦原型分析
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2024-09-02 DOI: 10.1007/s00245-024-10175-w
Katy Craig, Braxton Osting, Dong Wang, Yiming Xu
{"title":"Wasserstein Archetypal Analysis","authors":"Katy Craig,&nbsp;Braxton Osting,&nbsp;Dong Wang,&nbsp;Yiming Xu","doi":"10.1007/s00245-024-10175-w","DOIUrl":"10.1007/s00245-024-10175-w","url":null,"abstract":"<div><p>Archetypal analysis is an unsupervised machine learning method that summarizes data using a convex polytope. In its original formulation, for fixed <i>k</i>, the method finds a convex polytope with <i>k</i> vertices, called archetype points, such that the polytope is contained in the convex hull of the data and the mean squared Euclidean distance between the data and the polytope is minimal. In the present work, we consider an alternative formulation of archetypal analysis based on the Wasserstein metric, which we call Wasserstein archetypal analysis (WAA). In one dimension, there exists a unique solution of WAA and, in two dimensions, we prove the existence of a solution, as long as the data distribution is absolutely continuous with respect to the Lebesgue measure. We discuss obstacles to extending our result to higher dimensions and general data distributions. We then introduce an appropriate regularization of the problem, via a Rényi entropy, which allows us to obtain the existence of solutions of the regularized problem for general data distributions, in arbitrary dimensions. We prove a consistency result for the regularized problem, ensuring that if the data are iid samples from a probability measure, then as the number of samples is increased, a subsequence of the archetype points converges to the archetype points for the limiting data distribution, almost surely. Finally, we develop and implement a gradient-based computational approach for the two-dimensional problem, based on the semi-discrete formulation of the Wasserstein metric. Detailed numerical experiments are provided to support our theoretical findings.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"90 2","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142409471","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}
引用次数: 0
On the One-Dimensional Singular Abreu Equations 关于一维奇异阿布鲁方程
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2024-09-02 DOI: 10.1007/s00245-024-10178-7
Young Ho Kim
{"title":"On the One-Dimensional Singular Abreu Equations","authors":"Young Ho Kim","doi":"10.1007/s00245-024-10178-7","DOIUrl":"10.1007/s00245-024-10178-7","url":null,"abstract":"<div><p>Singular fourth-order Abreu equations have been used to approximate minimizers of convex functionals subject to a convexity constraint in dimensions higher than or equal to two. For Abreu type equations, they often exhibit different solvability phenomena in dimension one and dimensions at least two. We prove the analogues of these results for the variational problem and singular Abreu equations in dimension one, and use the approximation scheme to obtain a characterization of limiting minimizers to the one-dimensional variational problem.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"90 2","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142409470","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}
引用次数: 0
On the Ill-posedness for the Navier–Stokes Equations in the Weakest Besov Spaces 论最弱贝索夫空间中的纳维-斯托克斯方程的假定性
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2024-08-31 DOI: 10.1007/s00245-024-10177-8
Yanghai Yu, Jinlu Li
{"title":"On the Ill-posedness for the Navier–Stokes Equations in the Weakest Besov Spaces","authors":"Yanghai Yu,&nbsp;Jinlu Li","doi":"10.1007/s00245-024-10177-8","DOIUrl":"10.1007/s00245-024-10177-8","url":null,"abstract":"<div><p>It was proved in Iwabuchi and Ogawa (J Elliptic Parabol Equ 7(2):571–587, 2021) that the Cauchy problem for the full compressible Navier–Stokes equations of the ideal gas is ill-posed in <span>(dot{B}_{p, q}^{2 / p}(mathbb {R}^2) times dot{B}_{p, q}^{2 / p-1}(mathbb {R}^2) times dot{B}_{p, q}^{2 / p-2}(mathbb {R}^2) )</span> with <span>(1le ple infty )</span> and <span>(1le q&lt;infty )</span>. In this paper, we aim to solve the end-point case left in [17] and prove that the Cauchy problem is ill-posed in <span>(dot{B}_{p, infty }^{d / p}(mathbb {R}^d) times dot{B}_{p, infty }^{d / p-1}(mathbb {R}^d) times dot{B}_{p, infty }^{d / p-2}(mathbb {R}^d))</span> with <span>(1le ple infty )</span> by constructing a sequence of initial data for showing discontinuity of the solution map at zero. As a by-product, we demonstrate that the Cauchy problem for the incompressible Navier–Stokes equations is also ill-posed in <span>(dot{B}_{p,infty }^{d/p-1}(mathbb {R}^d))</span>, which is an interesting open problem in itself.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"90 2","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142415142","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}
引用次数: 0
Hamilton–Jacobi–Bellman Approach for Optimal Control Problems of Sweeping Processes 扫频过程最优控制问题的汉密尔顿-雅各比-贝尔曼方法
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2024-08-21 DOI: 10.1007/s00245-024-10174-x
Cristopher Hermosilla, Michele Palladino, Emilio Vilches
{"title":"Hamilton–Jacobi–Bellman Approach for Optimal Control Problems of Sweeping Processes","authors":"Cristopher Hermosilla,&nbsp;Michele Palladino,&nbsp;Emilio Vilches","doi":"10.1007/s00245-024-10174-x","DOIUrl":"10.1007/s00245-024-10174-x","url":null,"abstract":"<div><p>This paper is concerned with a state constrained optimal control problem governed by a Moreau’s sweeping process with a controlled drift. The focus of this work is on the Bellman approach for an infinite horizon problem. In particular, we focus on the regularity of the value function and on the Hamilton–Jacobi–Bellman equation it satisfies. We discuss a uniqueness result and we make a comparison with standard state constrained optimal control problems to highlight a regularizing effect that the sweeping process induces on the value function.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"90 2","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142412708","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}
引用次数: 0
Boundary Stabilization of the Korteweg-de Vries-Burgers Equation with an Infinite Memory-Type Control and Applications: A Qualitative and Numerical Analysis 具有无限记忆型控制的 Korteweg-de Vries-Burgers 方程的边界稳定及其应用：定性与数值分析
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2024-08-14 DOI: 10.1007/s00245-024-10172-z
Boumediène Chentouf, Aissa Guesmia, Mauricio Sepúlveda Cortés, Rodrigo Véjar Asem
{"title":"Boundary Stabilization of the Korteweg-de Vries-Burgers Equation with an Infinite Memory-Type Control and Applications: A Qualitative and Numerical Analysis","authors":"Boumediène Chentouf,&nbsp;Aissa Guesmia,&nbsp;Mauricio Sepúlveda Cortés,&nbsp;Rodrigo Véjar Asem","doi":"10.1007/s00245-024-10172-z","DOIUrl":"10.1007/s00245-024-10172-z","url":null,"abstract":"<div><p>This article is intended to present a qualitative and numerical analysis of well-posedness and boundary stabilization problems of the well-known Korteweg-de Vries-Burgers equation. Assuming that the boundary control is of memory type, the history approach is adopted in order to deal with the memory term. Under sufficient conditions on the physical parameters of the system and the memory kernel of the control, the system is shown to be well-posed by combining the semigroups approach of linear operators and the fixed point theory. Then, energy decay estimates are provided by applying the multiplier method. An application to the Kuramoto-Sivashinsky equation will be also given. Lastly, we present a numerical analysis based on a finite difference method and provide numerical examples illustrating our theoretical results.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"90 2","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194747","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}
引用次数: 0
Optimal Control for Optical Solitons in Nematic Liquid Crystals 向列液晶中光学孤子的优化控制
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2024-08-10 DOI: 10.1007/s00245-024-10173-y
Constanza Sánchez de la Vega, Juan Pablo Borgna, Diego Rial
{"title":"Optimal Control for Optical Solitons in Nematic Liquid Crystals","authors":"Constanza Sánchez de la Vega,&nbsp;Juan Pablo Borgna,&nbsp;Diego Rial","doi":"10.1007/s00245-024-10173-y","DOIUrl":"10.1007/s00245-024-10173-y","url":null,"abstract":"<div><p>We study an optimal control problem for a coupled Schrödinger-elliptic evolution system that describes the propagation of a laser beam in nematic liquid crystals. We consider a bilinear control related to an electric field depending on the optical axis acting on the sample. This problem arises from the study of an optimal way to transform the input signal into a target signal by modifying a system parameter related to the bias electric field. We prove well-posedness, existence and first order necessary conditions for an optimal solution.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"90 2","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141919952","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}
引用次数: 0
On the Controllability of the “Complete” Boussinesq System 论 "完整 "布森斯克系统的可控性
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2024-08-07 DOI: 10.1007/s00245-024-10171-0
Enrique Fernández-Cara, Juan B. Límaco, Dany Nina-Huaman
{"title":"On the Controllability of the “Complete” Boussinesq System","authors":"Enrique Fernández-Cara,&nbsp;Juan B. Límaco,&nbsp;Dany Nina-Huaman","doi":"10.1007/s00245-024-10171-0","DOIUrl":"10.1007/s00245-024-10171-0","url":null,"abstract":"<div><p>This paper deals with the local null controllability of the complete Boussinesq system (where quadratic viscous terms are kept in the right hand side of the heat equation) with distributed controls supported in small sets.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"90 2","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141939879","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}
引用次数: 0
Sensitivity Analysis and Optimal Control for a Friction Problem in the Linear Elastic Model 线性弹性模型中摩擦问题的灵敏度分析和优化控制
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2024-08-01 DOI: 10.1007/s00245-024-10156-z
Loïc Bourdin, Fabien Caubet, Aymeric Jacob de Cordemoy
{"title":"Sensitivity Analysis and Optimal Control for a Friction Problem in the Linear Elastic Model","authors":"Loïc Bourdin,&nbsp;Fabien Caubet,&nbsp;Aymeric Jacob de Cordemoy","doi":"10.1007/s00245-024-10156-z","DOIUrl":"10.1007/s00245-024-10156-z","url":null,"abstract":"<div><p>This paper investigates, without any regularization procedure, the sensitivity analysis of a mechanical friction problem involving the (nonsmooth) Tresca friction law in the linear elastic model. To this aim a recent methodology based on advanced tools from convex and variational analyses is used. Precisely we express the solution to the so-called Tresca friction problem thanks to the proximal operator associated with the corresponding Tresca friction functional. Then, using an extended version of twice epi-differentiability, we prove the differentiability of the solution to the parameterized Tresca friction problem, characterizing its derivative as the solution to a boundary value problem involving tangential Signorini’s unilateral conditions. Finally our result is used to investigate and numerically solve an optimal control problem associated with the Tresca friction model.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"90 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142409204","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}
引用次数: 0
Lower Semicontinuity of Pullback Attractors for a Non-autonomous Coupled System of Strongly Damped Wave Equations 强阻尼波方程非自治耦合系统的回拉吸引子的下半连续性
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2024-07-29 DOI: 10.1007/s00245-024-10170-1
Everaldo M. Bonotto, Alexandre N. Carvalho, Marcelo J. D. Nascimento, Eric B. Santiago
{"title":"Lower Semicontinuity of Pullback Attractors for a Non-autonomous Coupled System of Strongly Damped Wave Equations","authors":"Everaldo M. Bonotto,&nbsp;Alexandre N. Carvalho,&nbsp;Marcelo J. D. Nascimento,&nbsp;Eric B. Santiago","doi":"10.1007/s00245-024-10170-1","DOIUrl":"10.1007/s00245-024-10170-1","url":null,"abstract":"<div><p>The aim of this paper is to study the robustness of the family of pullback attractors associated with a non-autonomous coupled system of strongly damped wave equations, which is a modified version of the well known Klein–Gordon–Zakharov system. Under appropriate hyperbolicity conditions, we establish the gradient-like structure of the limit pullback attractor associated with this evolution system, and we prove the continuity of the family of pullback attractors at zero.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"90 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142414959","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}
引用次数: 0
Differentiation with Respect to Domains of Boundary Integral Functionals Involving Support Functions 涉及支持函数的边界积分函数域的微分
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2024-07-29 DOI: 10.1007/s00245-024-10168-9
Abdesslam Boulkhemair, Abdelkrim Chakib, Azeddine Sadik
{"title":"Differentiation with Respect to Domains of Boundary Integral Functionals Involving Support Functions","authors":"Abdesslam Boulkhemair,&nbsp;Abdelkrim Chakib,&nbsp;Azeddine Sadik","doi":"10.1007/s00245-024-10168-9","DOIUrl":"10.1007/s00245-024-10168-9","url":null,"abstract":"<div><p>The aim of this paper is to establish a new formula for the computation of the shape derivative of boundary integral cost functionals using Minkowski deformation of star-shaped domains by convex ones. The formula is expressed by means of the support function of the convex domain. The proof uses some geometrical tools in addition to an analysis of star-shapedness involving gauge functions. Finally, in order to illustrate this result, the formula is applied for solving an optimal shape design problem of minimizing a surface cost functional constrained to elliptic boundary value problem, using the gradient method performed by the finite element approximation.\u0000</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"90 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142414866","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}
引用次数: 0
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