Pub Date : 2018-05-01DOI: 10.1017/S0962492918000028
A. Kurganov
Shallow-water equations are widely used to model water flow in rivers, lakes, reservoirs, coastal areas, and other situations in which the water depth is much smaller than the horizontal length scale of motion. The classical shallow-water equations, the Saint-Venant system, were originally proposed about 150 years ago and still are used in a variety of applications. For many practical purposes, it is extremely important to have an accurate, efficient and robust numerical solver for the Saint-Venant system and related models. As their solutions are typically non-smooth and even discontinuous, finite-volume schemes are among the most popular tools. In this paper, we review such schemes and focus on one of the simplest (yet highly accurate and robust) methods: central-upwind schemes. These schemes belong to the family of Godunov-type Riemann-problem-solver-free central schemes, but incorporate some upwinding information about the local speeds of propagation, which helps to reduce an excessive amount of numerical diffusion typically present in classical (staggered) non-oscillatory central schemes. Besides the classical one- and two-dimensional Saint-Venant systems, we will consider the shallow-water equations with friction terms, models with moving bottom topography, the two-layer shallow-water system as well as general non-conservative hyperbolic systems.
{"title":"Finite-volume schemes for shallow-water equations","authors":"A. Kurganov","doi":"10.1017/S0962492918000028","DOIUrl":"https://doi.org/10.1017/S0962492918000028","url":null,"abstract":"Shallow-water equations are widely used to model water flow in rivers, lakes, reservoirs, coastal areas, and other situations in which the water depth is much smaller than the horizontal length scale of motion. The classical shallow-water equations, the Saint-Venant system, were originally proposed about 150 years ago and still are used in a variety of applications. For many practical purposes, it is extremely important to have an accurate, efficient and robust numerical solver for the Saint-Venant system and related models. As their solutions are typically non-smooth and even discontinuous, finite-volume schemes are among the most popular tools. In this paper, we review such schemes and focus on one of the simplest (yet highly accurate and robust) methods: central-upwind schemes. These schemes belong to the family of Godunov-type Riemann-problem-solver-free central schemes, but incorporate some upwinding information about the local speeds of propagation, which helps to reduce an excessive amount of numerical diffusion typically present in classical (staggered) non-oscillatory central schemes. Besides the classical one- and two-dimensional Saint-Venant systems, we will consider the shallow-water equations with friction terms, models with moving bottom topography, the two-layer shallow-water system as well as general non-conservative hyperbolic systems.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"27 1","pages":"289 - 351"},"PeriodicalIF":14.2,"publicationDate":"2018-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492918000028","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42810874","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}
Pub Date : 2018-05-01DOI: 10.1017/S096249291800003X
J. Oden
The use of computational models and simulations to predict events that take place in our physical universe, or to predict the behaviour of engineered systems, has significantly advanced the pace of scientific discovery and the creation of new technologies for the benefit of humankind over recent decades, at least up to a point. That ‘point’ in recent history occurred around the time that the scientific community began to realize that true predictive science must deal with many formidable obstacles, including the determination of the reliability of the models in the presence of many uncertainties. To develop meaningful predictions one needs relevant data, itself possessing uncertainty due to experimental noise; in addition, one must determine model parameters, and concomitantly, there is the overriding need to select and validate models given the data and the goals of the simulation. This article provides a broad overview of predictive computational science within the framework of what is often called the science of uncertainty quantification. The exposition is divided into three major parts. In Part 1, philosophical and statistical foundations of predictive science are developed within a Bayesian framework. There the case is made that the Bayesian framework provides, perhaps, a unique setting for handling all of the uncertainties encountered in scientific prediction. In Part 2, general frameworks and procedures for the calculation and validation of mathematical models of physical realities are given, all in a Bayesian setting. But beyond Bayes, an introduction to information theory, the maximum entropy principle, model sensitivity analysis and sampling methods such as MCMC are presented. In Part 3, the central problem of predictive computational science is addressed: the selection, adaptive control and validation of mathematical and computational models of complex systems. The Occam Plausibility Algorithm, OPAL, is introduced as a framework for model selection, calibration and validation. Applications to complex models of tumour growth are discussed.
{"title":"Adaptive multiscale predictive modelling","authors":"J. Oden","doi":"10.1017/S096249291800003X","DOIUrl":"https://doi.org/10.1017/S096249291800003X","url":null,"abstract":"The use of computational models and simulations to predict events that take place in our physical universe, or to predict the behaviour of engineered systems, has significantly advanced the pace of scientific discovery and the creation of new technologies for the benefit of humankind over recent decades, at least up to a point. That ‘point’ in recent history occurred around the time that the scientific community began to realize that true predictive science must deal with many formidable obstacles, including the determination of the reliability of the models in the presence of many uncertainties. To develop meaningful predictions one needs relevant data, itself possessing uncertainty due to experimental noise; in addition, one must determine model parameters, and concomitantly, there is the overriding need to select and validate models given the data and the goals of the simulation. This article provides a broad overview of predictive computational science within the framework of what is often called the science of uncertainty quantification. The exposition is divided into three major parts. In Part 1, philosophical and statistical foundations of predictive science are developed within a Bayesian framework. There the case is made that the Bayesian framework provides, perhaps, a unique setting for handling all of the uncertainties encountered in scientific prediction. In Part 2, general frameworks and procedures for the calculation and validation of mathematical models of physical realities are given, all in a Bayesian setting. But beyond Bayes, an introduction to information theory, the maximum entropy principle, model sensitivity analysis and sampling methods such as MCMC are presented. In Part 3, the central problem of predictive computational science is addressed: the selection, adaptive control and validation of mathematical and computational models of complex systems. The Occam Plausibility Algorithm, OPAL, is introduced as a framework for model selection, calibration and validation. Applications to complex models of tumour growth are discussed.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"27 1","pages":"353 - 450"},"PeriodicalIF":14.2,"publicationDate":"2018-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S096249291800003X","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47025376","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}
Pub Date : 2018-05-01DOI: 10.1017/S0962492917000113
C. Kelley
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.
{"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}
Pub Date : 2018-01-30DOI: 10.1017/S0962492918000016
M. Benning, M. Burger
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.
{"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}
Pub Date : 2017-11-14DOI: 10.1017/S0962492917000101
Nawaf Bou-Rabee, J. Sanz-Serna
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.
{"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}
Pub Date : 2017-05-05DOI: 10.1017/s096249291700006x
Onur Özyeşil, Vladislav Voroninski, Ronen Basri, Amit Singer
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 (e.g. points of interest, lines,etc.) and matching these features between images, (ii) camera motion estimation (e.g. using relative pairwise camera positions estimated from the extracted features), and (iii) recovery of the 3D structure using the estimated motion and features (e.g. by minimizing the so-calledreprojection error). 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 cameralocationestimation methods in the literature, followed by discussion of notable techniques for 3D structure recovery. We also cover the basics of thesimultaneous localization and mapping(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 (i.e. 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.
{"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}
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