Pub Date : 2012-04-19DOI: 10.1017/S0962492912000025
A. Abdulle, E. Weinan, B. Engquist, E. Vanden-Eijnden
The heterogeneous multiscale method (HMM), a general framework for designing multiscale algorithms, is reviewed. Emphasis is given to the error analysis that comes naturally with the framework. Examples of finite element and finite difference HMM are presented. Applications to dynamical systems and stochastic simulation algorithms with multiple time scales, spall fracture and heat conduction in microprocessors are discussed.
{"title":"The heterogeneous multiscale method*","authors":"A. Abdulle, E. Weinan, B. Engquist, E. Vanden-Eijnden","doi":"10.1017/S0962492912000025","DOIUrl":"https://doi.org/10.1017/S0962492912000025","url":null,"abstract":"The heterogeneous multiscale method (HMM), a general framework for designing multiscale algorithms, is reviewed. Emphasis is given to the error analysis that comes naturally with the framework. Examples of finite element and finite difference HMM are presented. Applications to dynamical systems and stochastic simulation algorithms with multiple time scales, spall fracture and heat conduction in microprocessors are discussed.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"21 1","pages":"1 - 87"},"PeriodicalIF":14.2,"publicationDate":"2012-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492912000025","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57445057","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 : 2012-04-19DOI: 10.1017/s0962492911999978
S. Chandler-Wilde, I. Graham, S. Langdon, O. Pironneau
{"title":"ANU volume 21 Cover and Front matter","authors":"S. Chandler-Wilde, I. Graham, S. Langdon, O. Pironneau","doi":"10.1017/s0962492911999978","DOIUrl":"https://doi.org/10.1017/s0962492911999978","url":null,"abstract":"","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"21 1","pages":"f1 - f6"},"PeriodicalIF":14.2,"publicationDate":"2012-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/s0962492911999978","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57444997","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 : 2012-04-19DOI: 10.1017/S0962492912000049
P. Deuflhard, A. Schiela, M. Weiser
This paper surveys the mathematics required for a typically challenging problem from computational medicine: cancer therapy planning in deep regional hyperthermia. In the course of many years of close cooperation with clinics, the medical problem has given rise to many subtle mathematical problems, some of which were unsolved when the project started. Efficiency of numerical algorithms, i.e., computational speed and monitored reliability, plays a decisive role in the medical treatment. Off-the-shelf software had turned out to be insufficient to meet the requirements of medicine. Instead, new mathematical theory as well as new numerical algorithms had to be developed. In order to make our algorithms useful in the clinical environment, new visualization software, i.e., a ‘virtual lab’, including three-dimensional geometry processing of individual virtual patients, had to be designed and implemented. Moreover, before the problems could be attacked by numerical algorithms, careful mathematical modelling had to be done. Finally, parameter identification and constrained optimization for the PDEs had to be newly analysed and realized over the individual patient's geometry. Our new techniques had an impact on the specificity of the treatment of individual patients and on the construction of an improved hyperthermia applicator.
{"title":"Mathematical cancer therapy planning in deep regional hyperthermia*","authors":"P. Deuflhard, A. Schiela, M. Weiser","doi":"10.1017/S0962492912000049","DOIUrl":"https://doi.org/10.1017/S0962492912000049","url":null,"abstract":"This paper surveys the mathematics required for a typically challenging problem from computational medicine: cancer therapy planning in deep regional hyperthermia. In the course of many years of close cooperation with clinics, the medical problem has given rise to many subtle mathematical problems, some of which were unsolved when the project started. Efficiency of numerical algorithms, i.e., computational speed and monitored reliability, plays a decisive role in the medical treatment. Off-the-shelf software had turned out to be insufficient to meet the requirements of medicine. Instead, new mathematical theory as well as new numerical algorithms had to be developed. In order to make our algorithms useful in the clinical environment, new visualization software, i.e., a ‘virtual lab’, including three-dimensional geometry processing of individual virtual patients, had to be designed and implemented. Moreover, before the problems could be attacked by numerical algorithms, careful mathematical modelling had to be done. Finally, parameter identification and constrained optimization for the PDEs had to be newly analysed and realized over the individual patient's geometry. Our new techniques had an impact on the specificity of the treatment of individual patients and on the construction of an improved hyperthermia applicator.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"21 1","pages":"307 - 378"},"PeriodicalIF":14.2,"publicationDate":"2012-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492912000049","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57445191","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 : 2012-04-19DOI: 10.1017/S0962492912000074
O. Pironneau
In this article we will focus on only a small part of financial mathematics, namely the use of partial differential equations for pricing futures. Even within this narrow range it is hard to be systematic and complete, or even to do better than existing books such as Wilmott, Howison and Dewynne (1995), Achdou and Pironneau (2005), or software manuals such as Lapeyre, Martini and Sulem (2010). So this article may be valuable only to the extent that it reflects ten years of teaching, conferences and interaction with the protagonists of financial mathematics. Also, because the theory of partial differential equations is not always well known, we have chosen a pragmatic approach and left out the details of the theory or the proofs of some results, and refer the reader to other books. The numerical algorithms, on the other hand, are given in detail.
{"title":"Pricing futures by deterministic methods*","authors":"O. Pironneau","doi":"10.1017/S0962492912000074","DOIUrl":"https://doi.org/10.1017/S0962492912000074","url":null,"abstract":"In this article we will focus on only a small part of financial mathematics, namely the use of partial differential equations for pricing futures. Even within this narrow range it is hard to be systematic and complete, or even to do better than existing books such as Wilmott, Howison and Dewynne (1995), Achdou and Pironneau (2005), or software manuals such as Lapeyre, Martini and Sulem (2010). So this article may be valuable only to the extent that it reflects ten years of teaching, conferences and interaction with the protagonists of financial mathematics. Also, because the theory of partial differential equations is not always well known, we have chosen a pragmatic approach and left out the details of the theory or the proofs of some results, and refer the reader to other books. The numerical algorithms, on the other hand, are given in detail.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"344 1","pages":"577 - 671"},"PeriodicalIF":14.2,"publicationDate":"2012-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492912000074","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57445418","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 : 2012-04-19DOI: 10.1017/S0962492912000037
S. Chandler-Wilde, I. Graham, S. Langdon, E. Spence
In this article we describe recent progress on the design, analysis and implementation of hybrid numerical-asymptotic boundary integral methods for boundary value problems for the Helmholtz equation that model time harmonic acoustic wave scattering in domains exterior to impenetrable obstacles. These hybrid methods combine conventional piecewise polynomial approximations with high-frequency asymptotics to build basis functions suitable for representing the oscillatory solutions. They have the potential to solve scattering problems accurately in a computation time that is (almost) independent of frequency and this has been realized for many model problems. The design and analysis of this class of methods requires new results on the analysis and numerical analysis of highly oscillatory boundary integral operators and on the high-frequency asymptotics of scattering problems. The implementation requires the development of appropriate quadrature rules for highly oscillatory integrals. This article contains a historical account of the development of this currently very active field, a detailed account of recent progress and, in addition, a number of original research results on the design, analysis and implementation of these methods.
{"title":"Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering*","authors":"S. Chandler-Wilde, I. Graham, S. Langdon, E. Spence","doi":"10.1017/S0962492912000037","DOIUrl":"https://doi.org/10.1017/S0962492912000037","url":null,"abstract":"In this article we describe recent progress on the design, analysis and implementation of hybrid numerical-asymptotic boundary integral methods for boundary value problems for the Helmholtz equation that model time harmonic acoustic wave scattering in domains exterior to impenetrable obstacles. These hybrid methods combine conventional piecewise polynomial approximations with high-frequency asymptotics to build basis functions suitable for representing the oscillatory solutions. They have the potential to solve scattering problems accurately in a computation time that is (almost) independent of frequency and this has been realized for many model problems. The design and analysis of this class of methods requires new results on the analysis and numerical analysis of highly oscillatory boundary integral operators and on the high-frequency asymptotics of scattering problems. The implementation requires the development of appropriate quadrature rules for highly oscillatory integrals. This article contains a historical account of the development of this currently very active field, a detailed account of recent progress and, in addition, a number of original research results on the design, analysis and implementation of these methods.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"21 1","pages":"89 - 305"},"PeriodicalIF":14.2,"publicationDate":"2012-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492912000037","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57445114","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 : 2012-04-19DOI: 10.1017/S0962492912000062
M. Lebrun, M. Colom, A. Buades, J. Morel
Digital images are matrices of equally spaced pixels, each containing a photon count. This photon count is a stochastic process due to the quantum nature of light. It follows that all images are noisy. Ever since digital images have existed, numerical methods have been proposed to improve the signal-to-noise ratio. Such ‘denoising’ methods require a noise model and an image model. It is relatively easy to obtain a noise model. As will be explained in the present paper, it is even possible to estimate it from a single noisy image.
{"title":"Secrets of image denoising cuisine*","authors":"M. Lebrun, M. Colom, A. Buades, J. Morel","doi":"10.1017/S0962492912000062","DOIUrl":"https://doi.org/10.1017/S0962492912000062","url":null,"abstract":"Digital images are matrices of equally spaced pixels, each containing a photon count. This photon count is a stochastic process due to the quantum nature of light. It follows that all images are noisy. Ever since digital images have existed, numerical methods have been proposed to improve the signal-to-noise ratio. Such ‘denoising’ methods require a noise model and an image model. It is relatively easy to obtain a noise model. As will be explained in the present paper, it is even possible to estimate it from a single noisy image.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"21 1","pages":"475 - 576"},"PeriodicalIF":14.2,"publicationDate":"2012-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492912000062","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57445363","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 : 2012-04-19DOI: 10.1017/S0962492912000050
J. Dongarra, A. Steen
This article describes the current state of the art of high-performance computing systems, and attempts to shed light on near-future developments that might prolong the steady growth in speed of such systems, which has been one of their most remarkable characteristics. We review the different ways devised to speed them up, both with regard to components and their architecture. In addition, we discuss the requirements for software that can take advantage of existing and future architectures.
{"title":"High-performance computing systems: Status and outlook*","authors":"J. Dongarra, A. Steen","doi":"10.1017/S0962492912000050","DOIUrl":"https://doi.org/10.1017/S0962492912000050","url":null,"abstract":"This article describes the current state of the art of high-performance computing systems, and attempts to shed light on near-future developments that might prolong the steady growth in speed of such systems, which has been one of their most remarkable characteristics. We review the different ways devised to speed them up, both with regard to components and their architecture. In addition, we discuss the requirements for software that can take advantage of existing and future architectures.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"21 1","pages":"379 - 474"},"PeriodicalIF":14.2,"publicationDate":"2012-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492912000050","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57445339","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 : 2011-04-28DOI: 10.1017/S0962492911000043
R. LeVeque, D. George, M. Berger
Numerical modelling of transoceanic tsunami propagation, together with the detailed modelling of inundation of small-scale coastal regions, poses a number of algorithmic challenges. The depth-averaged shallow water equations can be used to reduce this to a time-dependent problem in two space dimensions, but even so it is crucial to use adaptive mesh refinement in order to efficiently handle the vast differences in spatial scales. This must be done in a ‘wellbalanced’ manner that accurately captures very small perturbations to the steady state of the ocean at rest. Inundation can be modelled by allowing cells to dynamically change from dry to wet, but this must also be done carefully near refinement boundaries. We discuss these issues in the context of Riemann-solver-based finite volume methods for tsunami modelling. Several examples are presented using the GeoClaw software, and sample codes are available to accompany the paper. The techniques discussed also apply to a variety of other geophysical flows.
{"title":"Tsunami modelling with adaptively refined finite volume methods*","authors":"R. LeVeque, D. George, M. Berger","doi":"10.1017/S0962492911000043","DOIUrl":"https://doi.org/10.1017/S0962492911000043","url":null,"abstract":"Numerical modelling of transoceanic tsunami propagation, together with the detailed modelling of inundation of small-scale coastal regions, poses a number of algorithmic challenges. The depth-averaged shallow water equations can be used to reduce this to a time-dependent problem in two space dimensions, but even so it is crucial to use adaptive mesh refinement in order to efficiently handle the vast differences in spatial scales. This must be done in a ‘wellbalanced’ manner that accurately captures very small perturbations to the steady state of the ocean at rest. Inundation can be modelled by allowing cells to dynamically change from dry to wet, but this must also be done carefully near refinement boundaries. We discuss these issues in the context of Riemann-solver-based finite volume methods for tsunami modelling. Several examples are presented using the GeoClaw software, and sample codes are available to accompany the paper. The techniques discussed also apply to a variety of other geophysical flows.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"20 1","pages":"211 - 289"},"PeriodicalIF":14.2,"publicationDate":"2011-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492911000043","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57444880","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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