SIAM ReviewPub Date : 2023-05-01DOI: 10.1137/23n975673
Marlis Hochbruck
{"title":"Survey and Review","authors":"Marlis Hochbruck","doi":"10.1137/23n975673","DOIUrl":"https://doi.org/10.1137/23n975673","url":null,"abstract":"A point process is called self-exciting if the arrival of an event increases the probability of similar events for some period of time. Typical examples include earthquakes, which frequently cause aftershocks due to increased geological tension in their region; raised intrusion rates in the vicinity of a burglary; retweets in social media incited by some provocative posting; or trading frenzies following a huge stock order. A Hawkes process is a point process that models self-excitement among time events. In contrast to a Markov chain (in which the probability of each event depends only on the state attained in the previous event), chances of arrival of events are increased for some time period after the initial arrival in a Hawkes process. The first Survey and Review paper in this issue, “Hawkes Processes Modeling, Inference, and Control: An Overview,” by Rafael Lima, discusses recent advances in Hawkes process modeling and inference. The parametric, nonparametric, deep learning, and reinforcement learning approaches are covered. Current research challenges for the topic and the real-world limitations of each approach are also addressed. The paper should be of interest to experts in the field, but it also aims to be suitable for newcomers. The second Survey and Review paper, “Proximal Splitting Algorithms for Convex Optimization: A Tour of Recent Advances, with New Twists,” by Laurent Condat, Daichi Kitahara, Andrés Contreras, and Akira Hirabayashi, is dedicated to the solution of convex nonsmooth optimization problems in high-dimensional spaces. The objective function $f$ is assumed to be a sum of simple convex functions $f_j$ with the property that the minimization problem for each $f_j$ is simple, but for $f$ it is hard. For nonsmooth functions, gradient-based optimization algorithms are infeasible. In proximal algorithms, the gradient is replaced by the so-called proximity operator. While closed forms of proximity operators are known for many functions of practical interest, there is no general closed form for the proximity operator of a sum of functions. Therefore, splitting algorithms handle the proximity operators of the functions $f_j$ individually. The paper provides a constructive and self-contained introduction to the class of proximal splitting algorithms. New variants of the algorithms under consideration are developed. Existing convergence results are revisited, unified, and, in some cases, improved. Reading the paper will be rewarding for anyone interested in high-dimensional nonsmooth convex optimization.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136338576","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}
SIAM ReviewPub Date : 2023-05-01DOI: 10.1137/23n975697
None The Editors
{"title":"SIGEST","authors":"None The Editors","doi":"10.1137/23n975697","DOIUrl":"https://doi.org/10.1137/23n975697","url":null,"abstract":"The SIGEST article in this issue is “Nonlinear Perron--Frobenius Theorems for Nonnegative Tensors,” by Antoine Gautier, Francesco Tudisco, and Matthias Hein. Most computational and applied mathematicians will be aware of the results that Perron published in 1907 about the eigensystems of positive matrices, which were then extended by Frobenius in 1912 to the case of nonnegative matrices. This theory has impacted many areas of mathematics, including graph theory, Markov chains, and matrix computation, and it forms a fundamental component in the analysis of a range of models in areas such as demography, economics, wireless networking, and search engine optimization. Our SIGEST article, which first appeared in SIAM Journal on Matrix Analysis and Applications in 2019, extends Perron--Frobenius theory in two directions. First, the authors generalize from matrices to multidimensional arrays. This ties in with one of SIAM Review's most highly cited offerings: •Tensor decompositions and applications, T. G. Kolda and B. W. Bader, SIAM Review, 51 (3) (2009), pp. 455--500. It may also be viewed as extending the theory from graphs to hypergraphs---objects that are currently of much interest, as evidenced by several recent SIAM Review articles, including •Hypergraph cuts with general splitting functions, N. Veldt, A. R. Benson and J. Kleinberg, SIAM Review, 64 (3) (2022), pp. 650--685. By studying this higher-order setting, the authors open up new applications in network science, computer vision, and machine learning. The second major direction of the article is to develop and study nonlinear versions of the underlying spectral problems, and corresponding extensions of the traditional power method. This makes available new classes of iterations for which a comprehensive and satisfactory convergence theory is available. In preparing this SIGEST version, the authors have included new material. The introduction has been extended, and section 2 has been added to provide nontrivial examples of tensor eigenvalue problems in applications, including problems from computer vision and optimal transport. Moreover, subsection 4.1 includes a new nonlinear Perron--Frobenius theorem (Theorem 4.4) that builds on the previously known results in Theorems 4.2. and 4.3.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"123 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136338578","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}
SIAM ReviewPub Date : 2023-02-01DOI: 10.1137/23n975636
None The Editors
{"title":"SIGEST","authors":"None The Editors","doi":"10.1137/23n975636","DOIUrl":"https://doi.org/10.1137/23n975636","url":null,"abstract":"The SIGEST article in this issue, “Improbability of Collisions of Point-Vortices in Bounded Planar Domains,” by Martin Donati, is based on the 2022 SIAM Journal on Mathematics Analysis article “Two-Dimensional Point Vortex Dynamics in Bounded Domains: Global Existence for Almost Every Initial Data.” This work concerns point-vortex dynamics in a very general bounded domain in the plane. The main result is that the set of initial configurations which lead to finite-time collision, although nonempty, has Lebesgue measure zero. This is an elegant and highly nontrivial extension to general bounded domains with $C^{2, alpha}$ boundary of a result previously known only for three domains: the full plane, a disk, and the complement of a disk. It is notable that the result also extends to point-vortex dynamics in multiply connected domains. In preparing this SIGEST version, the author has added a new introductory section that explains the context of the new results. Moreover, some of the technical details from the original article have been replaced by more accessible high-level explanations. Recent references on the topics of vortex collisions and desingularization problems are also included. This SIGEST article will be of particular interest to researchers in fluid mechanics, partial differential equations, and applied analysis.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136252628","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}
SIAM ReviewPub Date : 2023-02-01DOI: 10.1137/22m1523340
M. Donati
{"title":"Improbability of Collisions of Point-Vortices in Bounded Planar Domains","authors":"M. Donati","doi":"10.1137/22m1523340","DOIUrl":"https://doi.org/10.1137/22m1523340","url":null,"abstract":"","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"24 1","pages":"227-257"},"PeriodicalIF":10.2,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86358421","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}
SIAM ReviewPub Date : 2022-11-03DOI: 10.1137/22n975573
The Editors
{"title":"SIGEST","authors":"The Editors","doi":"10.1137/22n975573","DOIUrl":"https://doi.org/10.1137/22n975573","url":null,"abstract":"SIAM Review, Volume 64, Issue 4, Page 989-989, November 2022. <br/> The SIGEST article in this issue is “A Proximal Markov Chain Monte Carlo Method for Bayesian Inference in Imaging Inverse Problems: When Langevin Meets Moreau,” by Alain Durmus, Éric Moulines, and Marcelo Pereyra. The authors provide new algorithms to sample from high-dimensional log-concave probability measures, where they combine Moreau--Yosida envelopes with the Euler--Maruyama discretization of Langevin diffusions. This allows for an efficient Markov chain Monte Carlo methodology that is applicable to inverse problems arising in imaging sciences. Asymptotic and nonasymptotic convergence results are provided, along with extensive computational experiments on realistic imaging problems involving deconvolution and tomographic reconstruction. The original article, which appeared in the SIAM Journal on Imaging Sciences in 2018, has attracted substantial interest. In preparing this highlighted SIGEST version, the authors have expanded the introduction to make it accessible to a wide audience. The final section also discusses follow-up work arising from the original publication.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"166 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2022-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138534647","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}
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