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Condition: the geometry of numerical algorithms - Lecture 1 Bürgisser, Peter | CIRM H

Post-edited

Research talks;Computer Science

The performance of numerical algorithms, both regarding stability and complexity, can be understood in a unified way in terms of condition numbers. This requires to identify the appropriate geometric settings and to characterize condition in geometric ways.
A probabilistic analysis of numerical algorithms can be reduced to a corresponding analysis of condition numbers, which leads to fascinating problems of geometric probability and integral geometry. The most well known example is Smale's 17th problem, which asks to find a solution of a given system of n complex homogeneous polynomial equations in $n$ + 1 unknowns. This problem can be solved in average (and even smoothed) polynomial time.
In the course we will explain the concepts necessary to state and solve Smale's 17th problem. We also show how these ideas lead to new numerical algorithms for computing eigenpairs of matrices that provably run in average polynomial time. Making these algorithms more efficient or adapting them to structured settings are challenging and rewarding research problems. We intend to address some of these issues at the end of the course.
The performance of numerical algorithms, both regarding stability and complexity, can be understood in a unified way in terms of condition numbers. This requires to identify the appropriate geometric settings and to characterize condition in geometric ways.
A probabilistic analysis of numerical algorithms can be reduced to a corresponding analysis of condition numbers, which leads to fascinating problems of geometric probability and integral ...

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Numerical analysis :proceedings of the 8th biennal conference held at dundee#June 26-29 Watson, G. A. | Springer-Verlag 1980

Congrès

- 184 p.
ISBN 978-3-540-09740-2

Lecture notes in mathematics , 0773

Localisation : Collection 1er étage

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Multigrid methods :proceedings of the conference#Nov. 23-27 Hackbusch, W. ; Trottenberg, U. | Springer-Verlag 1982

Congrès

- 652 p.
ISBN 978-3-540-11955-5

Lecture notes in mathematics , 0960

Localisation : Collection 1er étage

algorithme # analyse de Fourier # méthode multigrille

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Preconditioned conjugate gradient methodsproceedings of a conference held in Nijmegen, the Netherlands, June 19-21, 1989 Axelsson, Owe ; Kolotilina, L. Yu. | Springer-Verlag 1990

Congrès

ISBN 978-3-540-53515-7

Lecture notes in mathematics , 1457

Localisation : Collection 1er étage

algèbre linéaire numérique # analyse numérique # méthode de Rayleygh-Ritz-Galerkin # méthode finie # méthode itérative pour les systèmes linéaires # solution des équations aux différences # éléments finis

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Iterative methods in linear algebraproceedings of the IMACS international symposium on iterative methods in linear algebra, Brussels, Belgium, 2-4 April, 1991 Beauwens, R. ; de Groen, P. | North-Holland 1992

Congrès

ISBN 978-0-444-89248-5

Localisation : Colloque 1er étage (BRUS)

algèbre linéaire # méthode de collocation # méthode itérative pour les systèmes linéaires # méthode multigrille # méthode numérique en probabilité et statistique # méthode spectrale # semi conducteur # stabilité de Lyapunov

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Domain decomposition methods in scientific and engineering computing :proceedings of the seventh international conference on domain decomposition methods, which was hosted by the Pennsylvania State University#Oct. 27-30 Keyes, David E. ; Xu, Jinchao | American Mathematical Society 1994

Congrès

- 546 p.
ISBN 978-0-8218-5171-5

Contemporary mathematics , 0180

Localisation : Collection 1er étage

algorithme # aérodynamique # espace de Sobolev # informatique scientifique et d'ingénierie # méthode de décomposition de domaine # méthode multigrille # parallélisme # problème de Helmholtz # problème elliptique # système en rotation # équation aux dérivées partielles # équation de Bellman # équation de Navier-Stokes

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Sparse matrix techniquesadvanced course held at the technical university of DenmarkAug. 9 - 12 Barker, V. A. | Springer-Verlag 1977

Congrès

ISBN 978-3-540-08130-2

Lecture notes in mathematics , 0572

Localisation : Collection 1er étage

calcul de valeur propre ou de vecteur propre # méthode directe générale # méthode directe pour problème d'élément fini # méthode itérative # solution de système linéaire d'équation # technique de matrice éparpillée

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Recent advances in numerical methods for partial differential equations and applications :proceedings of the 2001 John H. Barrett memorial lectures trends in comptational mathematics held at the University of Tennessee#May 10-12 Feng, Xiaobing ; Schulze, Tim P. | American Mathematical Society 2002

Congrès

- 177 p.
ISBN 978-0-8218-2970-7

Contemporary mathematics , 0306

Localisation : Collection 1er étage

EDP # solution numérique des équations # méthode multi-échelle # condition limite # dynamique des fluides # modèle de flux à 2 phases # flux de stokes # changement de phase # solidification # multigrille # simulation

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Fast algorithms for structured matrices :theory and applications#AMS-IMS-SIAM joint summer research conference on fast algorithms in mathematics, computer science and engineering held at Mount Holyoke College#Aug. 5-9 Olshevsky, Vadim | American Mathematical Society 2003

Congrès

- 433 p.
ISBN 978-0-8218-3177-9

Contemporary mathematics , 0323

Localisation : Collection 1er étage

matrice # transformation de Fourier # algorithme # interpolation rationnelle # décodage d'algorithme # algèbre de Lie # optimisation # résolution d'équation matricielle

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Recent advances in matrix and operator theory#July 24-27 Ball, Joseph A. ; Eidelman, Yuli ; helton, J.William ; Olshevsky, Vadim | Birkhäuser 2007

Congrès

- 338 p.
ISBN 978-3-7643-8538-5

Operator theory: advances and applications , 0179

Localisation : Collection 1er étage

théorie des opérateurs # algèbre linéaire et multi-linéaire # théorie des matrices # groupes ordonnés # problèmes inverses # fonction spectrale # mesure spectrale # transformation non-linéaire # fonctions à variables complexes # théorie des perturbations

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Condition: the geometry of numerical algorithms - Lecture 2 Bürgisser, Peter | CIRM H

Multi angle

Research talks;Computer Science

The performance of numerical algorithms, both regarding stability and complexity, can be understood in a unified way in terms of condition numbers. This requires to identify the appropriate geometric settings and to characterize condition in geometric ways.
A probabilistic analysis of numerical algorithms can be reduced to a corresponding analysis of condition numbers, which leads to fascinating problems of geometric probability and integral geometry. The most well known example is Smale's 17th problem, which asks to find a solution of a given system of n complex homogeneous polynomial equations in $n$ + 1 unknowns. This problem can be solved in average (and even smoothed) polynomial time.
In the course we will explain the concepts necessary to state and solve Smale's 17th problem. We also show how these ideas lead to new numerical algorithms for computing eigenpairs of matrices that provably run in average polynomial time. Making these algorithms more efficient or adapting them to structured settings are challenging and rewarding research problems. We intend to address some of these issues at the end of the course.
The performance of numerical algorithms, both regarding stability and complexity, can be understood in a unified way in terms of condition numbers. This requires to identify the appropriate geometric settings and to characterize condition in geometric ways.
A probabilistic analysis of numerical algorithms can be reduced to a corresponding analysis of condition numbers, which leads to fascinating problems of geometric probability and integral ...

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Algebraic multigrid and subdivision Charina, Maria | CIRM H

Multi angle

Research talks

Multigrid is an iterative method for solving large linear systems of equations whose Toeplitz system matrix is positive definite. One of the crucial steps of any Multigrid method is based on multivariate subdivision. We derive sufficient conditions for convergence and optimality of Multigrid in terms of trigonometric polynomials associated with the corresponding subdivision schemes.
(This is a joint work with Marco Donatelli, Lucia Romani and Valentina Turati).
Multigrid is an iterative method for solving large linear systems of equations whose Toeplitz system matrix is positive definite. One of the crucial steps of any Multigrid method is based on multivariate subdivision. We derive sufficient conditions for convergence and optimality of Multigrid in terms of trigonometric polynomials associated with the corresponding subdivision schemes.
(This is a joint work with Marco Donatelli, Lucia Romani and ...

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Linear solvers for reservoir simulation Hénon, Pascal | CIRM H

Multi angle

Research talks

In this presentation, we will first present the main goals and principles of reservoir simulation. Then we will focus on linear systems that arise in such simulation. The main HPC challenge is to solve those systems efficiently on massively parallel computers. The specificity of those systems is that their convergence is mostly governed by the elliptic part of the equations and the linear solver needs to take advantage of it to be efficient. The reference method in reservoir simulation is CPR-AMG which usually relies on AMG to solve the quasi elliptic part of the system. We will present some works on improving AMG scalability for the reservoir linear systems (work done in collaboration with CERFACS). We will then introduce an on-going work with INRIA to take advantage of their enlarged Krylov method (EGMRES) in the CPR method. In this presentation, we will first present the main goals and principles of reservoir simulation. Then we will focus on linear systems that arise in such simulation. The main HPC challenge is to solve those systems efficiently on massively parallel computers. The specificity of those systems is that their convergence is mostly governed by the elliptic part of the equations and the linear solver needs to take advantage of it to be efficient. The ...

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Efficient iterative solvers: FETI methods with multiple search directions Roux, François-Xavier | CIRM H

Multi angle

Research talks

In domain decomposition methods, most of the computational cost lies in the successive solutions of the local problems in subdomains via forward-backward substitutions and in the orthogonalization of interface search directions. All these operations are performed, in the best case, via BLAS-1 or BLAS-2 routines which are inefficient on multicore systems with hierarchical memory. A way to improve the parallel efficiency of the method consists in working with several search directions, since multiple forward-backward substitutions and reorthogonalizations involve BLAS-3 routines. In the case of a problem with several right-hand-sides, using a block Krylov method is a straightforward way to work with multiple search directions. This will be illustrated with an application in electromagnetism using FETI-2LM method. For problems with a single right-hand-side, deriving several search directions that make sense from the optimal one constructed by the Krylov method is not so easy. The recently developed S-FETI method gives a very good approach that does not only improve parallel efficiency but can also reduce the global computational cost in the case of very heterogeneous problems. In domain decomposition methods, most of the computational cost lies in the successive solutions of the local problems in subdomains via forward-backward substitutions and in the orthogonalization of interface search directions. All these operations are performed, in the best case, via BLAS-1 or BLAS-2 routines which are inefficient on multicore systems with hierarchical memory. A way to improve the parallel efficiency of the method consists in ...

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Krylov subspace solvers and preconditioners Vuik, Kees | CIRM H

Multi angle

Research schools;Computer Science

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Time parallel time integration Gander, Martin | CIRM H

Multi angle

Research schools

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Applied iterative methods Hageman, Louis A. ; Young, David M. | Academic Press 1981

Ouvrage

- 386 p.
ISBN 978-0-12-313340-3

Computer science and applied mathematics

Localisation : Ouvrage RdC (HAGE)

65F10

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Basic numerical mathematics. Vol. 2 :numerical algebra Todd, John | Birkhäuser Verlag 1977

Ouvrage

- 216 p.
ISBN 978-3-7643-0811-7

I.S.N.M. , 0022

Localisation : Ouvrage RdC (TODD)

algèbre numérique # espace vectoriel # itération # norme de matrice # problème d'inverse

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Nine papers in analysis Brodskii, M. S. ; Ginzburg, Ju. P. | American Mathematical Society 1974

Ouvrage

- 203 p.

American mathematical society translations series 2 , 0103

Localisation : Collection 1er étage

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Multiple grid methods for equations of the second kind with applications in fluid mechanics Schippers, H. | Mathematisch Centrum 1985

Ouvrage

- 133 p.
ISBN 978-90-6196-260-1

Mathematical centre tracts , 0163

Localisation : Collection 1er étage

analyse numérique # équation de second degré # équation différentielle # équation intégrale # mécanique des fluides # problème aux limites # théorie distribution invariante

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