Pub Date : 2017-01-01DOI: 10.14658/PUPJ-DRNA-2017-SPECIAL_ISSUE-16
S. Marchi, A. Sommariva, M. Vianello, M. Caliari, L. Bos, A. Rossi, R. Cavoretto
The organizing committee is summarizing the main facts of the Fourth Dolomites Workshop on Constructive Approximation and Applications. The Fourth Dolomites Workshop on Constructive Approximation and Applications was held from September 8 till September 13, 2016 at the conference facilities of the University of Verona located in the heart of the Dolomite mountains at Alba di Canazei. Professor Annie Cuyt kindly arranged to have her 60th birthday this year and we took the opportunity to dedicate the conference to this happy event. Interested readers will find in this issue a biography of Annie’s scientific life provided to us by J.A.C. (Andre) Weideman. The complete details of the conference remain available on the webpage: http://events.math.unipd.it/dwcaa16/, but here is a summary of the main statistics: • 116 participants from 30 different countries and 5 continents. • Countries most represented (by affiliation!): Italy (34), Germany (18), Poland (7), Switzerland (6), Belgium (5). • 7 main Invited Lectures, 6 Sessions of Contributed Talks, 1 Poster Session. • Total Session Contributions: 70 talks plus an open problem session • Posters: 12 The Plenary Talks were given by • C. Conti (Florence) • A. Iske (Hamburg) • E. Larsson (Uppsala) • N. Levenberg (Bloomington) • G. Plonka (Goettingen) • J.A.C. Weideman (Stellenbosch) • G. Wright (Boise) and the Special Sessions were • Approximation theory in imaging science, Organizers: W. Erb (Luebeck) and A. Weinmann (Muenchen) • Meshless methods, Organizers: A. De Rossi (Turin) and E. Francomano (Palermo) • Multiresolution techniques: recent insights and applications, Organizers: M.A. Cotronei (Reggio Calabria), L. Romani (Milan) • Multivariate polynomial approximation and pluripotential theory, Organizers: L. Białas-Cież (Krakow) and M. Baran (Krakow) • Numerical integration, integral equations and transforms, Organizers: G. Milovanovic (Beograd) and D. Occorsio (Potenza) • Sparse approximation, Organizers: A. Cuyt (Antwerp) and W.-S. Lee (Antwerp) • Poster Session, Organizers: R. Cavoretto (Turin) and M. Vianello (Padua) We would like to thank the session organizers, the speakers and indeed all the participants for making the conference the success that it was. In any case we are of the opinion that Mathematics in a beautiful natural environment cannot be but inspiring! We also gratefully acknowledge support from the following organizations: • The University of Verona • The Department of Computer Science of the University of Verona • The Department of Mathematics “Tullio Levi-Civita”of the University of Padova
{"title":"The Fourth Dolomites Workshop on Constructive Approximation and Applications","authors":"S. Marchi, A. Sommariva, M. Vianello, M. Caliari, L. Bos, A. Rossi, R. Cavoretto","doi":"10.14658/PUPJ-DRNA-2017-SPECIAL_ISSUE-16","DOIUrl":"https://doi.org/10.14658/PUPJ-DRNA-2017-SPECIAL_ISSUE-16","url":null,"abstract":"The organizing committee is summarizing the main facts of the Fourth Dolomites Workshop on Constructive Approximation and Applications. The Fourth Dolomites Workshop on Constructive Approximation and Applications was held from September 8 till September 13, 2016 at the conference facilities of the University of Verona located in the heart of the Dolomite mountains at Alba di Canazei. Professor Annie Cuyt kindly arranged to have her 60th birthday this year and we took the opportunity to dedicate the conference to this happy event. Interested readers will find in this issue a biography of Annie’s scientific life provided to us by J.A.C. (Andre) Weideman. The complete details of the conference remain available on the webpage: http://events.math.unipd.it/dwcaa16/, but here is a summary of the main statistics: • 116 participants from 30 different countries and 5 continents. • Countries most represented (by affiliation!): Italy (34), Germany (18), Poland (7), Switzerland (6), Belgium (5). • 7 main Invited Lectures, 6 Sessions of Contributed Talks, 1 Poster Session. • Total Session Contributions: 70 talks plus an open problem session • Posters: 12 The Plenary Talks were given by • C. Conti (Florence) • A. Iske (Hamburg) • E. Larsson (Uppsala) • N. Levenberg (Bloomington) • G. Plonka (Goettingen) • J.A.C. Weideman (Stellenbosch) • G. Wright (Boise) and the Special Sessions were • Approximation theory in imaging science, Organizers: W. Erb (Luebeck) and A. Weinmann (Muenchen) • Meshless methods, Organizers: A. De Rossi (Turin) and E. Francomano (Palermo) • Multiresolution techniques: recent insights and applications, Organizers: M.A. Cotronei (Reggio Calabria), L. Romani (Milan) • Multivariate polynomial approximation and pluripotential theory, Organizers: L. Białas-Cież (Krakow) and M. Baran (Krakow) • Numerical integration, integral equations and transforms, Organizers: G. Milovanovic (Beograd) and D. Occorsio (Potenza) • Sparse approximation, Organizers: A. Cuyt (Antwerp) and W.-S. Lee (Antwerp) • Poster Session, Organizers: R. Cavoretto (Turin) and M. Vianello (Padua) We would like to thank the session organizers, the speakers and indeed all the participants for making the conference the success that it was. In any case we are of the opinion that Mathematics in a beautiful natural environment cannot be but inspiring! We also gratefully acknowledge support from the following organizations: • The University of Verona • The Department of Computer Science of the University of Verona • The Department of Mathematics “Tullio Levi-Civita”of the University of Padova","PeriodicalId":51943,"journal":{"name":"Dolomites Research Notes on Approximation","volume":"10 1","pages":"169-170"},"PeriodicalIF":1.3,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66792260","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-12-08DOI: 10.14658/pupj-drna-2017-Special_Issue-9
G. Santin, B. Haasdonk
Kernel-based methods provide flexible and accurate algorithms for the reconstruction of functions from meshless samples. A major question in the use of such methods is the influence of the samples locations on the behavior of the approximation, and feasible optimal strategies are not known for general problems. Nevertheless, efficient and greedy point-selection strategies are known. This paper gives a proof of the convergence rate of the data-independent textit{$P$-greedy} algorithm, based on the application of the convergence theory for greedy algorithms in reduced basis methods. The resulting rate of convergence is shown to be near-optimal in the case of kernels generating Sobolev spaces. As a consequence, this convergence rate proves that, for kernels of Sobolev spaces, the points selected by the algorithm are asymptotically uniformly distributed, as conjectured in the paper where the algorithm has been introduced.
{"title":"Convergence rate of the data-independent P-greedy algorithm in kernel-based approximation","authors":"G. Santin, B. Haasdonk","doi":"10.14658/pupj-drna-2017-Special_Issue-9","DOIUrl":"https://doi.org/10.14658/pupj-drna-2017-Special_Issue-9","url":null,"abstract":"Kernel-based methods provide flexible and accurate algorithms for the reconstruction of functions from meshless samples. A major question in the use of such methods is the influence of the samples locations on the behavior of the approximation, and feasible optimal strategies are not known for general problems. \u0000Nevertheless, efficient and greedy point-selection strategies are known. This paper gives a proof of the convergence rate of the data-independent textit{$P$-greedy} algorithm, based on the application of the convergence theory for greedy algorithms in reduced basis methods. The resulting rate of convergence is shown to be near-optimal in the case of kernels generating Sobolev spaces. \u0000As a consequence, this convergence rate proves that, for kernels of Sobolev spaces, the points selected by the algorithm are asymptotically uniformly distributed, as conjectured in the paper where the algorithm has been introduced.","PeriodicalId":51943,"journal":{"name":"Dolomites Research Notes on Approximation","volume":"10 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2016-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66792276","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-01-01DOI: 10.14658/PUPJ-DRNA-2016-SPECIAL_ISSUE-2
A. Rossi, E. Perracchione, E. Venturino
In this paper, the Partition of Unity (PU) method is performed by blending Radial Basis Functions (RBFs) as local approximants and using locally supported weights. In particular, we present a new multidimensional data structure which makes use of an integer-based scheme. This approach allows to perform an optimized space-partitioning structure. Moreover, because of its flexibility, it turns out to be extremely meaningful in the reconstruction of the attraction basins in dynamical systems.
{"title":"Fast strategy for PU interpolation: An application for the reconstruction of separatrix manifolds","authors":"A. Rossi, E. Perracchione, E. Venturino","doi":"10.14658/PUPJ-DRNA-2016-SPECIAL_ISSUE-2","DOIUrl":"https://doi.org/10.14658/PUPJ-DRNA-2016-SPECIAL_ISSUE-2","url":null,"abstract":"In this paper, the Partition of Unity (PU) method is performed by blending Radial Basis Functions (RBFs) as local approximants and using locally supported weights. In particular, we present a new multidimensional data structure which makes use of an integer-based scheme. This approach allows to perform an optimized space-partitioning structure. Moreover, because of its flexibility, it turns out to be extremely meaningful in the reconstruction of the attraction basins in dynamical systems.","PeriodicalId":51943,"journal":{"name":"Dolomites Research Notes on Approximation","volume":"9 1","pages":"3-12"},"PeriodicalIF":1.3,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66792212","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-01-01DOI: 10.14658/PUPJ-DRNA-2016-SPECIAL_ISSUE-4
S. Marchi, A. Iske, A. Sironi
Computerized tomography requires suitable numerical methods for the approximation of a bivariate function f from a finite set of discrete Radon data, each of whose data samples represents one line integral of f . In standard reconstruction methods, specific assumptions concerning the geometry of the Radon lines are usually made. In relevant applications of image reconstruction, however, such assumptions are often too restrictive. In this case, one would rather prefer to work with reconstruction methods allowing for arbitrary distributions of scattered Radon lines. This paper proposes a novel image reconstruction method for scattered Radon data, which combines kernel-based scattered data approximation with a well-adapted regularization of the Radon transform. This results in a very flexible numerical algorithm for image reconstruction, which works for arbitrary distributions of Radon lines. This is in contrast to the classical filtered back projection, which essentially relies on a regular distribution of the Radon lines, e.g. parallel beam geometry. The good performance of the kernel-based image reconstruction method is illustrated by numerical examples and comparisons.
{"title":"Kernel-based Image Reconstruction from Scattered Radon Data","authors":"S. Marchi, A. Iske, A. Sironi","doi":"10.14658/PUPJ-DRNA-2016-SPECIAL_ISSUE-4","DOIUrl":"https://doi.org/10.14658/PUPJ-DRNA-2016-SPECIAL_ISSUE-4","url":null,"abstract":"Computerized tomography requires suitable numerical methods for the approximation of a bivariate function f from a finite set of discrete Radon data, each of whose data samples represents one line integral of f . In standard reconstruction methods, specific assumptions concerning the geometry of the Radon lines are usually made. In relevant applications of image reconstruction, however, such assumptions are often too restrictive. In this case, one would rather prefer to work with reconstruction methods allowing for arbitrary distributions of scattered Radon lines. This paper proposes a novel image reconstruction method for scattered Radon data, which combines kernel-based scattered data approximation with a well-adapted regularization of the Radon transform. This results in a very flexible numerical algorithm for image reconstruction, which works for arbitrary distributions of Radon lines. This is in contrast to the classical filtered back projection, which essentially relies on a regular distribution of the Radon lines, e.g. parallel beam geometry. The good performance of the kernel-based image reconstruction method is illustrated by numerical examples and comparisons.","PeriodicalId":51943,"journal":{"name":"Dolomites Research Notes on Approximation","volume":"9 1","pages":"19-31"},"PeriodicalIF":1.3,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66792224","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-12-01DOI: 10.14658/PUPJ-DRNA-2015-SPECIAL_ISSUE-9
G. Suryanarayana, R. Cools, Dirk Nuyens
We study the properties of a special rank-1 point set in 2 dimensions — Fibonacci lattice points. We present the analysis of these point sets for cubature and approximation of bivariate periodic functions with decaying spectral coefficients. We are interested in truncating the frequency space into index sets based on different degrees of exactness. The numerical results support that the Lebesgue constant of these point sets grows like the conjectured optimal rate ln 2 (N), where N is the number of sample points.
{"title":"Integration and Approximation with Fibonacci lattice points","authors":"G. Suryanarayana, R. Cools, Dirk Nuyens","doi":"10.14658/PUPJ-DRNA-2015-SPECIAL_ISSUE-9","DOIUrl":"https://doi.org/10.14658/PUPJ-DRNA-2015-SPECIAL_ISSUE-9","url":null,"abstract":"We study the properties of a special rank-1 point set in 2 dimensions — Fibonacci lattice points. We present the analysis of these point sets for cubature and approximation of bivariate periodic functions with decaying spectral coefficients. We are interested in truncating the frequency space into index sets based on different degrees of exactness. The numerical results support that the Lebesgue constant of these point sets grows like the conjectured optimal rate ln 2 (N), where N is the number of sample points.","PeriodicalId":51943,"journal":{"name":"Dolomites Research Notes on Approximation","volume":"8 1","pages":"92-101"},"PeriodicalIF":1.3,"publicationDate":"2015-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66792173","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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