数学最新文献

{"title":"SIGEST","authors":"The Editors","doi":"10.1137/24n976006","DOIUrl":"https://doi.org/10.1137/24n976006","url":null,"abstract":"SIAM Review, Volume 66, Issue 4, Page 719-719, November 2024. <br/> The SIGEST article in this issue, “A Bridge between Invariant Theory and Maximum Likelihood Estimation,” by Carlos Améndola, Kathlén Kohn, Philipp Reichenbach, and Anna Seigal, uncovers the deep connections between geometric invariant theory and statistical methods, specifically maximum likelihood estimation (MLE) by connecting it to norm minimization over group orbits. The authors develop a dictionary relating stability notions in geometric invariant theory to the existence and uniqueness of MLEs, which applies to both Gaussian and log-linear models. In comparison to the original 2021 version of the paper that appeared in the SIAM Journal on Applied Algebra and Geometry, for the SIGEST version, the authors added new content on log-linear models, simplified technical proofs, removed detailed appendices, and incorporated new examples and figures for accessibility. In particular, the focus was primarily on Gaussian models, whereas this updated SIGEST version expands the coverage by incorporating results from the authors' companion paper on log-linear models. Furthermore, a new figure (Fig. 1) visually illustrates the two core concepts of invariant theory and MLE. Significant changes include the removal of technical details and appendices to streamline the content and make it more accessible to a broader audience. The introduction of examples, particularly for the Kempf--Ness Theorem, further aids understanding. This paper makes several key contributions of broad mathematical interest. MLE is a key statistical technique that is widely used. Having a new handle on its well-posedness analysis deepens the understanding of the mechanisms behind this technique as well as potentially paves the way to extending existing theory for MLE models. Also, on the computational side, algorithms from the optimization over orbits can be used for MLE, and vice versa, which could possibly lead to new and more efficient algorithms in both fields. Overall, the work beautifully highlights how techniques from one field can be applied to the other, with applications to generalization bounds, group actions, and optimization landscapes. In the last section of their SIGEST paper the authors discuss possible future research directions that capitalize on the dictionary they have uncovered.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":null,"pages":null},"PeriodicalIF":10.2,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142594681","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":"Survey and Review","authors":"Marlis Hochbruck","doi":"10.1137/24n975980","DOIUrl":"https://doi.org/10.1137/24n975980","url":null,"abstract":"SIAM Review, Volume 66, Issue 4, Page 617-617, November 2024. <br/> Neural oscillations are periodic activities of neurons in the central nervous system of eumetazoa. In an oscillatory neural network, neurons are modeled by coupled oscillators. Oscillatory networks are employed for describing the behavior of complex systems in biology or ecology with respect to the connectivity of the network components or the nonlinear dynamics of the individual units. Phase-locked periodic states and their instabilities are core features in the analysis of oscillatory networks. In “Oscillatory Networks: Insights from Piecewise-Linear Modeling,” Stephen Coombes, Mustafa Şayli, Rüdiger Thul, Rachel Nicks, Mason A. Porter, and Yi Ming Lai review techniques for studying coupled oscillatory networks. They first discuss phase reductions, phase-amplitude reductions, and the master stability function for smooth dynamical systems. Then they consider nonsmooth piecewise-linear (PWL) systems, for which periodic orbits are easily obtained. Saltation operators are used for modeling the propagation of perturbations through switching manifolds in the analysis of the dynamics and bifurcations at the network level. Applications to neural systems, cardiac systems, networks of electromechanical oscillators, and cooperation in cattle herds illustrate the power of these methods. PWL modeling has been applied for a long time in engineering. Recently, it has been introduced in other fields, such as social sciences, finance, and biology. For many modern applications in science, piecewise models are much more versatile than the classical smooth dynamical systems. In neuroscience, PWL functions enable explicit calculations which are infeasible in the original smooth system. This includes discontinuous dynamical systems, which are used to model impacting mechanical oscillators, integrate-and-fire models of spiking neurons, and cardiac oscillators. On the other hand, the price to pay is the retrieval of new conditions for the existence, uniqueness, and stability of solutions. The paper discusses the application of PWL models to a large variety of applications from engineering and biology. It will be of interest to many readers.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":null,"pages":null},"PeriodicalIF":10.2,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142594685","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":"Sigmoid Functions, Multiscale Resolution of Singularities, and $hp$-Mesh Refinement","authors":"Daan Huybrechs, Lloyd N. Trefethen","doi":"10.1137/23m1556629","DOIUrl":"https://doi.org/10.1137/23m1556629","url":null,"abstract":"SIAM Review, Volume 66, Issue 4, Page 683-693, November 2024. <br/> In this short, conceptual paper we observe that closely related mathematics applies in four contexts with disparate literatures: (1) sigmoidal and RBF approximation of smooth functions, (2) rational approximation of analytic functions with singularities, (3) $hpkern .7pt$-mesh refinement for solution of pdes, and (4) double exponential (DE) and generalized Gauss quadrature. The relationships start from the change of variables $s = log(x)$, and they suggest possibilities for new analyses and new methods in several areas. Concerning (2) and (3), we show that both problems feature the same effect of “linear tapering” near the singularity---of clustered poles in rational approximation and of polynomial orders in $hpkern .7pt$-mesh refinement. Concerning (4), we note that the tapering effect appears here too, and that the change of variables interpretation sheds new light on why the DE and generalized Gauss methods are effective at integrating arbitrary singularities.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":null,"pages":null},"PeriodicalIF":10.2,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142594683","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":"Research Spotlights","authors":"Stefan M. Wild","doi":"10.1137/24n975992","DOIUrl":"https://doi.org/10.1137/24n975992","url":null,"abstract":"SIAM Review, Volume 66, Issue 4, Page 681-681, November 2024. <br/> Logarithmic transformations are used broadly in data science, mathematics, and engineering, and yet they can still reveal surprising connections between seemingly unrelated disciplines. This issue's first research spotlight, “Sigmoid Functions, Multiscale Resolution of Singularities, and $hp$-Mesh Refinement,” illuminates how the change of variables $s = log(x)$ connects different areas of computational mathematics. Authors Daan Huybrechs and Lloyd “Nick” Trefethen show new relationships between smooth approximation, rational approximation theory, adaptive mesh refinement, and numerical quadrature. For example, the authors show that this change of variables can be naturally tied to a “linear tapering” effect near singularities, which is a common feature in both rational approximation and $hp$-mesh refinement. Through a number of effective examples, the authors illustrate the power of these relationships across areas that have seen relatively independent lines of development. In doing so, the authors suggest opportunities for developing and analyzing new methods by leveraging the new connections, including mesh refinement strategies, techniques for multivariate approximation, and hybrid approaches that combine the strengths of disparate methods. How well can information be recovered from water waves? This question is at the heart of this issue's second research spotlight, “Feynman's Inverse Problem.” Author Adrian Kirkeby is motivated by a thought experiment posed by the physicist and iconoclast Richard Feynman wherein an insect floating in a swimming pool wants to determine where and when others have jumped into the pool, causing the waves the insect observes. Kirkeby constructs and analyzes a linear 2D-3D system of partial differential equations (PDEs) for the forward model. Leveraging the nonlocality of this system of PDEs, Kirkeby shows conditions under which the insect can determine the source of the waves---in fact, uniquely---simply by observing the wave amplitude and water velocity in any small area of the surface. This model is then extended to capture settings where noisy observations and observations at a finite number of time and space points are collected, and establishes stability properties and error bounds for the reconstruction. The paper concludes with illustrative numerical experiments based on a nonharmonic Fourier inversion method. Kirkeby also highlights several avenues for future research, noting that inverse problems for water or other surface waves have received less attention than those involving acoustic or electromagnetic waves. As an added bonus, the referenced video of Feynman is not to be missed.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":null,"pages":null},"PeriodicalIF":10.2,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142594684","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":"Directional flow in perivascular networks: mixed finite elements for reduced-dimensional models on graphs.","authors":"Ingeborg G Gjerde, Miroslav Kuchta, Marie E Rognes, Barbara Wohlmuth","doi":"10.1007/s00285-024-02154-0","DOIUrl":"https://doi.org/10.1007/s00285-024-02154-0","url":null,"abstract":"<p><p>Flow of cerebrospinal fluid through perivascular pathways in and around the brain may play a crucial role in brain metabolite clearance. While the driving forces of such flows remain enigmatic, experiments have shown that pulsatility is central. In this work, we present a novel network model for simulating pulsatile fluid flow in perivascular networks, taking the form of a system of Stokes-Brinkman equations posed over a perivascular graph. We apply this model to study physiological questions concerning the mechanisms governing perivascular fluid flow in branching vascular networks. Notably, our findings reveal that even long wavelength arterial pulsations can induce directional flow in asymmetric, branching perivascular networks. In addition, we establish fundamental mathematical and numerical properties of these Stokes-Brinkman network models, with particular attention to increasing graph order and complexity. By introducing weighted norms, we show the well-posedness and stability of primal and dual variational formulations of these equations, and that of mixed finite element discretizations.</p>","PeriodicalId":50148,"journal":{"name":"Journal of Mathematical Biology","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142606867","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Equilibria of large random Lotka-Volterra systems with vanishing species: a mathematical approach.","authors":"Imane Akjouj, Walid Hachem, Mylène Maïda, Jamal Najim","doi":"10.1007/s00285-024-02155-z","DOIUrl":"https://doi.org/10.1007/s00285-024-02155-z","url":null,"abstract":"<p><p>Ecosystems with a large number of species are often modelled as Lotka-Volterra dynamical systems built around a large interaction matrix with random part. Under some known conditions, a global equilibrium exists and is unique. In this article, we rigorously study its statistical properties in the large dimensional regime. Such an equilibrium vector is known to be the solution of a so-called Linear Complementarity Problem. We describe its statistical properties by designing an Approximate Message Passing (AMP) algorithm, a technique that has recently aroused an intense research effort in the fields of statistical physics, machine learning, or communication theory. Interaction matrices based on the Gaussian Orthogonal Ensemble, or following a Wishart distribution are considered. Beyond these models, the AMP approach developed in this article has the potential to describe the statistical properties of equilibria associated to more involved interaction matrix models.</p>","PeriodicalId":50148,"journal":{"name":"Journal of Mathematical Biology","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142606872","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Bridge between Invariant Theory and Maximum Likelihood Estimation","authors":"Carlos Améndola, Kathlén Kohn, Philipp Reichenbach, Anna Seigal","doi":"10.1137/24m1661753","DOIUrl":"https://doi.org/10.1137/24m1661753","url":null,"abstract":"SIAM Review, Volume 66, Issue 4, Page 721-747, November 2024. <br/> We uncover connections between maximum likelihood estimation in statistics and norm minimization over a group orbit in invariant theory. We present a dictionary that relates notions of stability from geometric invariant theory to the existence and uniqueness of a maximum likelihood estimate. Our dictionary holds for both discrete and continuous statistical models: we discuss log-linear models and Gaussian models, including matrix normal models and directed Gaussian graphical models. Our approach reveals promising consequences of the interplay between invariant theory and statistics. For instance, algorithms from statistics can be used in invariant theory, and vice versa.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":null,"pages":null},"PeriodicalIF":10.2,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142594680","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 Operator Preconditioned Combined Field Integral Equation for Electromagnetic Scattering","authors":"Van Chien Le, Kristof Cools","doi":"10.1137/23m1581674","DOIUrl":"https://doi.org/10.1137/23m1581674","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2484-2505, December 2024. <br/> Abstract. This paper aims to address two issues of integral equations for the scattering of time-harmonic electromagnetic waves by a perfect electric conductor with Lipschitz continuous boundary: ill-conditioned boundary element Galerkin discretization matrices on fine meshes and instability at spurious resonant frequencies. The remedy to ill-conditioned matrices is operator preconditioning, and resonant instability is eliminated by means of a combined field integral equation. Exterior traces of single and double layer potentials are complemented by their interior counterparts for a purely imaginary wave number. We derive the corresponding variational formulation in the natural trace space for electromagnetic fields and establish its well-posedness for all wave numbers. A Galerkin discretization scheme is employed using conforming edge boundary elements on dual meshes, which produces well-conditioned discrete linear systems of the variational formulation. Some numerical results are also provided to support the numerical analysis.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142594676","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":"Developing Workforce with Mathematical Modeling Skills","authors":"Ariel Cintrón-Arias, Ryan Andrew Nivens, Anant Godbole, Calvin B. Purvis","doi":"10.1137/19m1277643","DOIUrl":"https://doi.org/10.1137/19m1277643","url":null,"abstract":"SIAM Review, Volume 66, Issue 4, Page 778-792, November 2024. <br/> Mathematicians have traditionally been a select group of academics who produce high-impact ideas enabling substantial results in several fields of science. Throughout the past 35 years, undergraduates enrolling in mathematics or statistics have represented a nearly constant proportion of approximately 1% of bachelor degrees awarded in the United States. Even within STEM majors, mathematics or statistics only constitute about 6% of undergraduate degrees awarded nationally. However, the need for STEM professionals continues to grow, and the list of required occupational skills rests heavily in foundational concepts of mathematical modeling curricula, where the interplay of data, computer simulation, and underlying theoretical frameworks takes center stage. It is not viable to expect a majority of these STEM undergraduates to pursue a double major that includes mathematics. Here we present our solution, some early results of its implementation, and a vision for possible nationwide adoption.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":null,"pages":null},"PeriodicalIF":10.2,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142594677","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":"Longtime Dynamics for a Class of Strongly Damped Wave Equations with Variable Exponent Nonlinearities","authors":"Yanan Li,&nbsp;Yamei Li,&nbsp;Zhijian Yang","doi":"10.1007/s00245-024-10193-8","DOIUrl":"10.1007/s00245-024-10193-8","url":null,"abstract":"<div><p>The paper investigates the global well-posedness and the longtime dynamics for a class of strongly damped wave equations with evolutional <i>p</i>(<i>x</i>, <i>t</i>)-Laplacian and <i>q</i>(<i>x</i>, <i>t</i>)-growth source term on a bounded domain <span>( Omega subset {mathbb {R}}^3: u_{tt}-nabla cdot (|nabla u|^{p(x, t)-2} nabla u)-lambda Delta u- Delta u_t+ |u|^{q(x, t)-2}u=g)</span>, together with the perturbed parameter <span>(lambda in [0,1])</span> and the Dirichlet boundary condition. We show that under rather relaxed conditions, (i) the model is global well-posed; (ii) for each <span>(lambda _0in (0,1])</span>, the related nonautonomous dynamical systems acting on the time-dependent phase spaces have a family of pullback <span>({mathscr {D}})</span>-exponential attractor <span>({mathcal {E}}_lambda ={E_lambda (t)}_{tin {mathbb {R}}}in {mathscr {D}})</span> which is Hölder continuous w.r.t. <span>(lambda )</span> at <span>(lambda _0)</span>; (iii) they have also a family of finite dimensional pullback <span>({mathscr {D}})</span>-attractors <span>({mathcal {A}}_lambda ={A_lambda (t)}_{tin {mathbb {R}}})</span> which are upper semicontinuous and residual continuous w.r.t. <span>(lambda in (0,1])</span>. In particular, when <span>(lambda in (0,1])</span> and without the <i>p</i>(<i>x</i>, <i>t</i>)-Laplacian, the above mentioned results can be greatly improved, in the concrete; (iv) the weak solutions of the corresponding model possess additionally partial regularity and the Hölder stability in stronger <span>(H^1times H^1)</span>-norm, the pullback <span>({mathscr {D}})</span>-attractor and pullback <span>({mathscr {D}})</span>-exponential attractor in weaker <span>({mathcal {Y}}_1)</span>-norm can be regularized to be those in stronger <span>(H^1times H^1)</span>-norm, which are also the standard ones in <span>({mathcal {H}}_t)</span>-norm. The method provided here allows overcoming the difficulties arising from variable exponent nonlinearities and extending the analysis and the results for these type of models with constant exponent nonlinearities.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142595257","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}