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Multi angle  Reconstruction methods for ill-posed inverse problems - Part 1
Siltanen, Samuli (Auteur de la Conférence) | CIRM (Editeur )


inverse problem - reconstruction - regularization - tomography - computation

65N21 ; 65N20 ; 35R25

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Z
ential equations (PDEs for short), a forward Fokker-Planck equation and a backward Hamilton-Jacobi-Bellman equation.
The latter system of PDEs has closed form solutions in very few cases only. Therefore, numerical simulation are crucial in order to address applications. The present mini-course will be devoted to numerical methods that can be used to approximate the systems of PDEs.
The numerical schemes that will be presented rely basically on monotone approximations of the Hamiltonian and on a suitable weak formulation of the Fokker-Planck equation.
These schemes have several important features:

- The discrete problem has the same structure as the continous one, so existence, energy estimates, and possibly uniqueness can be obtained with the same kind of arguments

- Monotonicity guarantees the stability of the scheme: it is robust in the deterministic limit

- convergence to classical or weak solutions can be proved

Finally, there are particular cases named variational MFGS in which the system of PDEs can be seen as the optimality conditions of some optimal control problem driven by a PDE. In such cases, augmented Lagrangian methods can be used for solving the discrete nonlinear system. The mini-course will be orgamized as follows

1. Introduction to the system of PDEs and its interpretation. Uniqueness of classical solutions.

2. Monotone finite difference schemes

3. Examples of applications

4. Variational MFG and related algorithms for solving the discrete system of nonlinear equations Recently, an important research activity on mean field games (MFGs for short) has been initiated by the pioneering works of Lasry and Lions: it aims at studying the asymptotic behavior of stochastic differential games (Nash equilibria) as the number $n$ of agents tends to infinity. The field is now rapidly growing in several directions, including stochastic optimal control, analysis of PDEs, calculus of variations, numerical analysis and ...

49K20 ; 49N70 ; 35K40 ; 35K55 ; 35Q84 ; 65K10 ; 65M06 ; 65M12 ; 91A23 ; 91A15

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Post-edited  Introduction to hierarchical tiling dynamical systems: Supertile construction methods
Frank, Natalie Priebe (Auteur de la Conférence) | CIRM (Editeur )

These lectures introduce the dynamical systems approach to tilings of Euclidean space, especially quasicrystalline tilings that have been constructed using a ‘supertile method’. Because tiling dynamics parallels one-dimensional symbolic dynamics, we discuss this case as well, highlighting the differences and similarities in the methods of study and the results that can be obtained.
In the first lecture we motivate the field with the discovery of quasicrystals, which led to D. Schectman’s winning the 2011 Nobel Prize in Chemistry. Then we set up the basics of tiling dynamics, describing tiling spaces, a tiling metric, and the shift or translation actions. Shift-invariant and ergodic measures are discussed, along with fundamental topological and dynamical properties.
The second lecture brings in the supertile construction methods, including symbolic substitutions, self-similar tilings, $S$-adic systems, and fusion rules. Numerous examples are given, most of which are not the “standard” examples, and we identify many commonalities and differences between these interrelated methods of construction. Then we compare and contrast dynamical results for supertile systems, highlighting those key insights that can be adapted to all cases.
In the third lecture we investigate one of the many current tiling research areas: spectral theory. Schectman made his Nobel-prize-winning discovery using diffraction analysis, and studying the mathematical version has been quite fruitful. Spectral theory of tiling dynamical systems is also of broad interest. We describe how these types of spectral analysis are carried out, give examples, and discuss what is known and unknown about the relationship between dynamical and diffraction analysis. Special attention is paid to the “point spectrum”, which is related to eigenfunctions and also to the bright spots that appear on diffraction images.
These lectures introduce the dynamical systems approach to tilings of Euclidean space, especially quasicrystalline tilings that have been constructed using a ‘supertile method’. Because tiling dynamics parallels one-dimensional symbolic dynamics, we discuss this case as well, highlighting the differences and similarities in the methods of study and the results that can be obtained.
In the first lecture we motivate the field with the discovery of ...

37B50 ; 37B10 ; 52C23

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Multi angle  Compter et optimiser avec les graphes unimodulaires - Cours 1
Bordenave, Charles (Auteur de la Conférence) | CIRM (Editeur )

L'objectif de ce mini-cours est de présenter de la façon la plus élémentaire possible la convergence faible locale des graphes introduite par Benjamini et Schramm en 2001 et développée par Aldous et Steele (2004), Aldous et Lyons (2007). Nous montrerons comment cette notion peut être utilisée dans des dénombrements asymptotiques et dans des problèmes d'optimisation combinatoire.

05C80 ; 60C05

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Multi angle  Compter et optimiser avec les graphes unimodulaires - Cours 2
Bordenave, Charles (Auteur de la Conférence) | CIRM (Editeur )

L’objectif de ce mini-cours est de présenter de la façon la plus élémentaire possible la convergence faible locale des graphes introduite par Benjamini et Schramm en 2001 et développée par Aldous et Steele (2004), Aldous et Lyons (2007). Nous montrerons comment cette notion peut être utilisée dans des dénombrements asymptotiques et dans des problèmes d’optimisation combinatoire.

05C80 ; 60C05

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Multi angle  Le diamant aztèque - Cours 1
Corteel, Sylvie (Auteur de la Conférence) | CIRM (Editeur )

Le but du mini-cours sera de faire un cours introductif à différentes méthodes énumeratives à travers l’exemple des pavages par dominos du diamant aztèque. On essaiera de voir les fonctions (super)-symétriques, les moments de polynômes bi-orthogonaux, les évaluations de determinants, les algorithmes de génération...

05A15 ; 33C45

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Multi angle  Le diamant aztèque - Cours 2
Corteel, Sylvie (Auteur de la Conférence) | CIRM (Editeur )

Le but du mini-cours sera de faire un cours introductif à différentes méthodes énumeratives à travers l’exemple des pavages par dominos du diamant aztèque. On essaiera de voir les fonctions (super)-symétriques, les moments de polynômes bi-orthogonaux, les évaluations de determinants, les algorithmes de génération...

05A15 ; 33C45