Acta NumericaPub Date : 2018-05-01DOI: 10.1017/S0962492917000113
C. Kelley
{"title":"Numerical methods for nonlinear equations","authors":"C. Kelley","doi":"10.1017/S0962492917000113","DOIUrl":"https://doi.org/10.1017/S0962492917000113","url":null,"abstract":"This article is about numerical methods for the solution of nonlinear equations. We consider both the fixed-point form $mathbf{x}=mathbf{G}(mathbf{x})$ and the equations form $mathbf{F}(mathbf{x})=0$ and explain why both versions are necessary to understand the solvers. We include the classical methods to make the presentation complete and discuss less familiar topics such as Anderson acceleration, semi-smooth Newton’s method, and pseudo-arclength and pseudo-transient continuation methods.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"27 1","pages":"207 - 287"},"PeriodicalIF":14.2,"publicationDate":"2018-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492917000113","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45893294","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}
Acta NumericaPub Date : 2018-01-30DOI: 10.1017/S0962492918000016
M. Benning, M. Burger
{"title":"Modern regularization methods for inverse problems","authors":"M. Benning, M. Burger","doi":"10.1017/S0962492918000016","DOIUrl":"https://doi.org/10.1017/S0962492918000016","url":null,"abstract":"Regularization methods are a key tool in the solution of inverse problems. They are used to introduce prior knowledge and allow a robust approximation of ill-posed (pseudo-) inverses. In the last two decades interest has shifted from linear to nonlinear regularization methods, even for linear inverse problems. The aim of this paper is to provide a reasonably comprehensive overview of this shift towards modern nonlinear regularization methods, including their analysis, applications and issues for future research. In particular we will discuss variational methods and techniques derived from them, since they have attracted much recent interest and link to other fields, such as image processing and compressed sensing. We further point to developments related to statistical inverse problems, multiscale decompositions and learning theory.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"27 1","pages":"1 - 111"},"PeriodicalIF":14.2,"publicationDate":"2018-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492918000016","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44566147","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}
Acta NumericaPub Date : 2017-11-14DOI: 10.1017/S0962492917000101
Nawaf Bou-Rabee, J. Sanz-Serna
{"title":"Geometric integrators and the Hamiltonian Monte Carlo method","authors":"Nawaf Bou-Rabee, J. Sanz-Serna","doi":"10.1017/S0962492917000101","DOIUrl":"https://doi.org/10.1017/S0962492917000101","url":null,"abstract":"This paper surveys in detail the relations between numerical integration and the Hamiltonian (or hybrid) Monte Carlo method (HMC). Since the computational cost of HMC mainly lies in the numerical integrations, these should be performed as efficiently as possible. However, HMC requires methods that have the geometric properties of being volume-preserving and reversible, and this limits the number of integrators that may be used. On the other hand, these geometric properties have important quantitative implications for the integration error, which in turn have an impact on the acceptance rate of the proposal. While at present the velocity Verlet algorithm is the method of choice for good reasons, we argue that Verlet can be improved upon. We also discuss in detail the behaviour of HMC as the dimensionality of the target distribution increases.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"27 1","pages":"113 - 206"},"PeriodicalIF":14.2,"publicationDate":"2017-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492917000101","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49018849","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}
Acta NumericaPub Date : 2017-05-05DOI: 10.1017/s096249291700006x
Onur Özyeşil, Vladislav Voroninski, Ronen Basri, Amit Singer
{"title":"A survey of structure from motion.","authors":"Onur Özyeşil, Vladislav Voroninski, Ronen Basri, Amit Singer","doi":"10.1017/s096249291700006x","DOIUrl":"https://doi.org/10.1017/s096249291700006x","url":null,"abstract":"The structure from motion (SfM) problem in computer vision is to recover the three-dimensional (3D) structure of a stationary scene from a set of projective measurements, represented as a collection of two-dimensional (2D) images, via estimation of motion of the cameras corresponding to these images. In essence, SfM involves the three main stages of (i) extracting features in images (<jats:italic>e.g.</jats:italic> points of interest, lines,<jats:italic>etc.</jats:italic>) and matching these features between images, (ii) camera motion estimation (<jats:italic>e.g.</jats:italic> using relative pairwise camera positions estimated from the extracted features), and (iii) recovery of the 3D structure using the estimated motion and features (<jats:italic>e.g.</jats:italic> by minimizing the so-called<jats:italic>reprojection error</jats:italic>). This survey mainly focuses on relatively recent developments in the literature pertaining to stages (ii) and (iii). More specifically, after touching upon the early factorization-based techniques for motion and structure estimation, we provide a detailed account of some of the recent camera<jats:italic>location</jats:italic>estimation methods in the literature, followed by discussion of notable techniques for 3D structure recovery. We also cover the basics of the<jats:italic>simultaneous localization and mapping</jats:italic>(SLAM) problem, which can be viewed as a specific case of the SfM problem. Further, our survey includes a review of the fundamentals of feature extraction and matching (<jats:italic>i.e.</jats:italic> stage (i) above), various recent methods for handling ambiguities in 3D scenes, SfM techniques involving relatively uncommon camera models and image features, and popular sources of data and SfM software.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"229 1","pages":""},"PeriodicalIF":14.2,"publicationDate":"2017-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138530597","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}
Acta NumericaPub Date : 2017-05-01DOI: 10.1017/S0962492917000058
R. Kannan, S. Vempala
{"title":"Randomized algorithms in numerical linear algebra","authors":"R. Kannan, S. Vempala","doi":"10.1017/S0962492917000058","DOIUrl":"https://doi.org/10.1017/S0962492917000058","url":null,"abstract":"This survey provides an introduction to the use of randomization in the design of fast algorithms for numerical linear algebra. These algorithms typically examine only a subset of the input to solve basic problems approximately, including matrix multiplication, regression and low-rank approximation. The survey describes the key ideas and gives complete proofs of the main results in the field. A central unifying idea is sampling the columns (or rows) of a matrix according to their squared lengths.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"26 1","pages":"95 - 135"},"PeriodicalIF":14.2,"publicationDate":"2017-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492917000058","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45036614","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}
Acta NumericaPub Date : 2017-05-01DOI: 10.1017/S0962492917000046
A. Quarteroni, A. Manzoni, C. Vergara
{"title":"The cardiovascular system: Mathematical modelling, numerical algorithms and clinical applications *","authors":"A. Quarteroni, A. Manzoni, C. Vergara","doi":"10.1017/S0962492917000046","DOIUrl":"https://doi.org/10.1017/S0962492917000046","url":null,"abstract":"Mathematical and numerical modelling of the cardiovascular system is a research topic that has attracted remarkable interest from the mathematical community because of its intrinsic mathematical difficulty and the increasing impact of cardiovascular diseases worldwide. In this review article we will address the two principal components of the cardiovascular system: arterial circulation and heart function. We will systematically describe all aspects of the problem, ranging from data imaging acquisition, stating the basic physical principles, analysing the associated mathematical models that comprise PDE and ODE systems, proposing sound and efficient numerical methods for their approximation, and simulating both benchmark problems and clinically inspired problems. Mathematical modelling itself imposes tremendous challenges, due to the amazing complexity of the cardiocirculatory system, the multiscale nature of the physiological processes involved, and the need to devise computational methods that are stable, reliable and efficient. Critical issues involve filtering the data, identifying the parameters of mathematical models, devising optimal treatments and accounting for uncertainties. For this reason, we will devote the last part of the paper to control and inverse problems, including parameter estimation, uncertainty quantification and the development of reduced-order models that are of paramount importance when solving problems with high complexity, which would otherwise be out of reach.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"26 1","pages":"365 - 590"},"PeriodicalIF":14.2,"publicationDate":"2017-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492917000046","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43213446","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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