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## ADMM in imaging inverse problems: some history and recent advances Figueiredo, Mário | CIRM H

Post-edited

Research talks;Analysis and its Applications;Computer Science

The alternating direction method of multipliers (ADMM) is an optimization tool of choice for several imaging inverse problems, namely due its flexibility, modularity, and efficiency. In this talk, I will begin by reviewing our earlier work on using ADMM to deal with classical problems such as deconvolution, inpainting, compressive imaging, and how we have exploited its flexibility to deal with different noise models, including Gaussian, Poissonian, and multiplicative, and with several types of regularizers (TV, frame-based analysis, synthesis, or combinations thereof). I will then describe more recent work on using ADMM for other problems, namely blind deconvolution and image segmentation, as well as very recent work where ADMM is used with plug-in learned denoisers to achieve state-of-the-art results in class-specific image deconvolution. Finally, on the theoretical front, I will describe very recent work on tackling the infamous problem of how to adjust the penalty parameter of ADMM. The alternating direction method of multipliers (ADMM) is an optimization tool of choice for several imaging inverse problems, namely due its flexibility, modularity, and efficiency. In this talk, I will begin by reviewing our earlier work on using ADMM to deal with classical problems such as deconvolution, inpainting, compressive imaging, and how we have exploited its flexibility to deal with different noise models, including Gaussian, ...

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## Numerical methods for mean field games - Lecture 2: Monotone finite difference schemes Achdou, Yves | CIRM H

Post-edited

Research schools;Computer Science;Control Theory and Optimization;Partial Differential 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 computing, and the potential applications to economics and social sciences are numerous.
In the limit when $n \to +\infty$, a given agent feels the presence of the others through the statistical distribution of the states. Assuming that the perturbations of a single agent's strategy does not influence the statistical states distribution, the latter acts as a parameter in the control problem to be solved by each agent. When the dynamics of the agents are independent stochastic processes, MFGs naturally lead to a coupled system of two partial differential 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 ...

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## Symbolic-numeric computationjuly 19-21 Wang, Dongming ; Zhi, Lihong | Springer 2007

Congrès

- 394 p.
ISBN 978-3-7643-7983-4

Trends in mathematics

Localisation : Colloque 1er étage (XIAN)

analyse numérique # informatique # calcul formel # calcul algébrique # système polynômial # ensemble triangulaire # base de Gröbner # algorithme # optimisation

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## Constructive, experimental, and nonlinear analysis :papers from the International workshop held in honor of Professor Jonathan Borwein at the University of Limoges#Sept. 22-23 Thera, Michel | American Mathematical Society 2000

Congrès

- 289 p.
ISBN 978-0-8218-2167-1

C.M.S. conference proceedings , 0027

Localisation : Collection 1er étage

théorie des nombres # analyse numérique # calcul de variation # optimisation # analyse non-linéaire # analyse constructive # fonction de Lyapunov # algorithme # calcul formel # calcul numérique # théorie du contrôle

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## Control of boundaries and stabilizationproceedings of the IFIP WG 7.2 conferenceJune 20-23 Simon, J. | Springer-Verlag 1989

Congrès

ISBN 978-3-540-51239-4

Lecture notes in control and information sciences , 0125

Localisation : Colloque 1er étage (CLER)

analyse de forme et optimisation # contrôlabilité pour de grands temps # contrôle des frontières # contrôle optimal d'inégalités hémivariationnelles # dissipation d'énergie interne # domaine perforé # géophysique du pétrole # homogénéisation itérée # modèle de mouillage distribué # méthode de Lagrange # méthode de Newton # petit ruban dans les flots laminaires # plaque thermoélastique # poly cristal # problème d'optimisation de domaine # propriété spectrale et asymptotique # science de l'information # stabilisation # système en chaîne vibrant # système élastique linéaire # équation d'onde # équation de la chaleur analyse de forme et optimisation # contrôlabilité pour de grands temps # contrôle des frontières # contrôle optimal d'inégalités hémivariationnelles # dissipation d'énergie interne # domaine perforé # géophysique du pétrole # homogénéisation itérée # modèle de mouillage distribué # méthode de Lagrange # méthode de Newton # petit ruban dans les flots laminaires # plaque thermoélastique # poly cristal # problème d'optimisation de domaine # ...

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## Global minimization of nonconvex energy functions :molecular conformation and protein folding DIMACS workshop at Rutgers University#March 20-21 Pardalos, P. M. ; Shalloway, D. ; Xue, G. | American Mathematical Society 1996

Congrès

- 271 p.
ISBN 978-0-8218-0471-1

DIMACS series in discrete mathematics and theoretical computer science , 0023

Localisation : Collection 1er étage

analyse numérique # biochimie # contrôle # mathématique discrète # modèle pliant # méthode d'optimisation globale # pliure # programmation # programmation convexe # programmation linéaire # programmation mathématique # programmation non linéaire # recherche opérationnel # structure protéine # système stochastique # technique variationnelle

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## Constraint programming and large scale discrete optimization :DIMACS workshop on ...#held at DIMACS Center at Rutgers University#Sept. 14-17 Freuder, Engene C. ; Wallace, Richard J. | American Mathematical Society 2001

Congrès

- 175 p.
ISBN 978-0-8218-2710-9

DIMACS series in discrete mathematics and theoretical computer science , 0057

Localisation : Collection 1er étage

intelligence artificielle # analyse numérique # opitmisation # programmation par contrainte # résolution de problème # recherche opérationnelle # planification # théorie déterministe # théorie stochastique # optimisation combinatoriale

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## Mathematical programmingproceedings of the international congress on... held at rio de janeiro,brazil, 6-8 april,1981 Cottle, R. W. ; Kelmanson, M. L. ; Korte, B. | North-Holland 1984

Congrès

- 355 p.
ISBN 978-0-444-86821-3

Localisation : Colloque 1er étage (RIO)

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## Modern optimization modelling techniques.Papers based on the presentations at the advanced course "optimization theory, methods and applications"Barcelona # july 20-24, 2009 Cominetti, Roberto ; Facchinei, Francisco ; Lasserre, Jean Bernard ; Daniilidis, Aris ; Martinez-Legaz, Juan-Enrique | Birkhäuser 2012

Congrès

- viii; 269 p.
ISBN 978-3-0348-0290-1

Advanced courses in mathematics - CRM Barcelona

Localisation : Colloque 1er étage (BARC)

optimisation mathématique # économie # programmation mathématique # problème de traffic

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## Numerische methoden bei optimierungsaufgabenvortragsauszuge der tagung uber...im Mathematischen ForschungsinstitutNov. 14-20 Collatz, L. ; Wetterling, W. | Birkhäuser Verlag 1973

Congrès

ISBN 978-3-7643-0668-7

I.S.N.M. , 0017

Localisation : Colloque 1er étage (OBER)

approximation non linéaire # conduite optimale # controle non linéaire # discrétisation d'optimisation convexe # dualité # formule de quadrature de type Gauss # méthode d'intégration cartésienne # méthode de décomposition de Bender # méthode numérique # polynome défini optimal # problème d'optimisation # programmation linéaire stochastique # slalom # système d'inéquation linéaire # équation aux dérivées partielles

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## Numerische methoden bei optimierungsaufgaben. Band 2vortragsauszuge der tagung uber...am Mathematichen ForschungsinstitutNov. 18-24 Collatz, L. ; Wetterling, W. | Birkhäuser Verlag 1974

Congrès

ISBN 978-3-7643-0732-5

I.S.N.M. , 0023

Localisation : Colloque 1er étage (OBER)

champ de gravitation # controle de conduction de chaleur # cristallographie # erreur d'arrondi # fonction groupe-booléenne # minimalisation # méthode de métrique variable # méthode de pénalisation de courant # méthode des gradients conjugués # méthode nomographique # méthode numérique # optimisation dynamique # optimisation linéaire # planification de production # problème d'optimisation non linéaire

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## Optimal controlcalculus of variations, optimal control theory and numerical methodsVariationsrechnung und optimalsteuerungen was held at the mathematisches forschungsinstitut of the University of FreiburgMay 26 - June 1 Bulirsch, R. ; Miele, A. ; Stoer, J. | Birkhäuser Verlag 1993

Congrès

ISBN 978-3-7643-2887-0

I.S.N.M. , 0111

Localisation : Colloque 1er étage (FREI)

analyse et synthèse des systèmes non linéaires # calcul des variations # commande optimale # condition d'optimalité et algorithmes # controlabilité # méthode d'approximation successive # méthode de Rauleigh-Ritz et Galerkin # méthode numérique # observabilité # problème de minima # structure des systèmes # système mécaniques spatiaux

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## Algorithms for constrained minimization of smooth nonlinear functionstenth mathematical programming symposiumAug. Buckley, A. G. ; Goffin, J. L. | North-Holland Publishing Co. 1982

Congrès

ISBN 978-0-444-86203-7

Mathematical programming study , 0016

Localisation : Colloque 1er étage (MONT)

algorithme # algorithme convergent superlinéairement # amélioration de faisabilité # calcul de la direction recherche # contrainte non linéaire éparse # fonction non linéaire lisse # forcer la convergence # implémentation d'algorithme de Lagrange projeté # minimisation contrainte # méthode du gradient conjugué # méthode quasi-Newton réduite # optimisation contrainte non linéairement # programmation non linéaire # technique de chien de garde

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## Control and optimization with PDE constraints.Based on the international workshop on control and optimization of PDEsMariatrost # october 10-14, 2011 Bredies, Kristian ; Clason, Christian ; Kunisch, Karl ; von Winckel, Gregory | Birkhäuser 2013

Congrès

- x; 215 p.
ISBN 978-3-0348-0630-5

International series of numerical mathematics , 0164

Localisation : Colloque 1er étage (MARI)

EDP # équation aux dérivées partielles # optimisation mathématique # méthode numérique pour le calcul des variations

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## Computational solution of nonlinear systems of equations :proceedings of the 1988 AMS-SIAM summer seminar held at Colorado State University#July 18-29 Allgower, Eugene L. ; Georg, Kurt | Amercian Mathematical Society 1990

Congrès

- 762 p.
ISBN 978-0-8218-1131-3

Lectures in applied mathematics , 0026

Localisation : Collection 1er étage

EDO # EDP non linéaire # algorithme PL # algorithme d'homotopie simplitielle # automatisation de génération de code adjoint # collision # continuation polynomial # contrôle stochastique singulier # méthode de type Newton # optimisation sans contrainte # problème de modèle non linéaire # résonance # scission de séparatrice et chaos # solution informatique d'équation de système non linéaire # spectrocopie diélectrique # système non linéaire d'équation # théorème de Sard # transport contaminant # équation d'onde non linéaire EDO # EDP non linéaire # algorithme PL # algorithme d'homotopie simplitielle # automatisation de génération de code adjoint # collision # continuation polynomial # contrôle stochastique singulier # méthode de type Newton # optimisation sans contrainte # problème de modèle non linéaire # résonance # scission de séparatrice et chaos # solution informatique d'équation de système non linéaire # spectrocopie diélectrique # système non linéaire ...

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## Data-driven wildfire behavior modelling: focus on front level-set data assimilation Rochoux, Mélanie | CIRM H

Multi angle

Research talks;Mathematical Physics

A front data assimilation system named FIREFLY has been developed at CERFACS in collaboration with the University of Maryland to better estimate the environmental conditions (biomass properties, near-surface wind). We discuss the sequential application of the ensemble Kalman filter (EnKF) in FIREFLY for correcting in a spatially-distributed way, input parameters in order to better track the fire front position. In particular, using a polynomial chaos surrogate to mimic the wildfire spread model in the EnKF algorithm was found in collaboration with LIMSI to be a promising strategy to reduce the computational cost of FIREFLY.
We also discuss the way we represent the distance between simulated and observed fronts. In the CEMRACS project, a new discrepancy operator will be introduced to better represent the match (or mismatch) between simulated fronts and mid-infrared observations in collaboration with INRIA. This front level-set data assimilation derived from image processing and designed for electrophysiology will be extended to wildfire spread monitoring.
A front data assimilation system named FIREFLY has been developed at CERFACS in collaboration with the University of Maryland to better estimate the environmental conditions (biomass properties, near-surface wind). We discuss the sequential application of the ensemble Kalman filter (EnKF) in FIREFLY for correcting in a spatially-distributed way, input parameters in order to better track the fire front position. In particular, using a polynomial ...

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## A review of different geometries for the training of neural networks Malago, Luigi | CIRM H

Multi angle

Research talks;Computer Science;Geometry

Neural networks consist of a variegate class of computational models, used in machine learning for both supervised and unsupervised learning. Several topologies of networks have been proposed in the literature, since the preliminary work from the late 50s, including models based on undirected probabilistic graphical models, such as (Restricted) Boltzmann Machines, and on multi-layer feed-forward computational graphs. The training of a neural network is usually performed by the minimization of a cost function, such as the negative log-likelihood. During the talk we will review alternative geometries used to describe the space of the functions encoded by a neural network, parametrized by its connection weights, and the implications on the optimization of the cost function during training, from the perspective of Riemannian optimization. In the first part of the presentation, we will introduce a probabilistic interpretation for neural networks, which goes back to the work of Amari and coauthors from the 90s, and which is based on the use of the Fisher-Rao metric studied in Information Geometry. In this framework, the weights of a Boltzmann Machine, and similarly for feed-forward neural networks, are interpreted as the parameters of a (joint) statistical model for the observed, and possibly latent, variables. In the second part of the talk, we will review other approaches, motivated by invariant principles in neural networks and not explicitly based on probabilistic models, to the definition of alternative geometries for the space of the parameters of a neural network. The use of alternative non-Euclidean geometries has direct impact on the training algorithms, indeed by modeling the space of the functions associated to a neural network as a Riemannian manifold determines a dependence of the gradient on the choice of metric tensor. We conclude the presentation by reviewing some recently proposed training algorithm for neural networks, based on Riemannian optimization algorithms. Neural networks consist of a variegate class of computational models, used in machine learning for both supervised and unsupervised learning. Several topologies of networks have been proposed in the literature, since the preliminary work from the late 50s, including models based on undirected probabilistic graphical models, such as (Restricted) Boltzmann Machines, and on multi-layer feed-forward computational graphs. The training of a neural ...

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## Reconstruction by optimal transport: applications in cosmology and finance Loeper, Grégoire | CIRM H

Multi angle

Research talks;Control Theory and Optimization;Partial Differential Equations;Mathematics in Science and Technology

Following the seminal work by Benamou and Brenier on the time continuous formulation of the optimal transport problem, we show how optimal transport techniques can be used in various areas, ranging from "the reconstruction problem" cosmology to a problem of volatility calibration in finance.

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## Mean field type control with congestion Laurière, Mathieu | CIRM H

Multi angle

Research talks;Control Theory and Optimization;Partial Differential Equations

The theory of mean field type control (or control of MacKean-Vlasov) aims at describing the behaviour of a large number of agents using a common feedback control and interacting through some mean field term. The solution to this type of control problem can be seen as a collaborative optimum. We will present the system of partial differential equations (PDE) arising in this setting: a forward Fokker-Planck equation and a backward Hamilton-Jacobi-Bellman equation. They describe respectively the evolution of the distribution of the agents' states and the evolution of the value function. Since it comes from a control problem, this PDE system differs in general from the one arising in mean field games.
Recently, this kind of model has been applied to crowd dynamics. More precisely, in this talk we will be interested in modeling congestion effects: the agents move but try to avoid very crowded regions. One way to take into account such effects is to let the cost of displacement increase in the regions where the density of agents is large. The cost may depend on the density in a non-local or in a local way. We will present one class of models for each case and study the associated PDE systems. The first one has classical solutions whereas the second one has weak solutions. Numerical results based on the Newton algorithm and the Augmented Lagrangian method will be presented.
This is joint work with Yves Achdou.
The theory of mean field type control (or control of MacKean-Vlasov) aims at describing the behaviour of a large number of agents using a common feedback control and interacting through some mean field term. The solution to this type of control problem can be seen as a collaborative optimum. We will present the system of partial differential equations (PDE) arising in this setting: a forward Fokker-Planck equation and a backward Hamilto...

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## Signal processing for nonlinear diffractive imaging Kamilov, Ulugbek | CIRM H

Multi angle

Research schools

Can modern signal processing be used to overcome the diffraction limit? The classical diffraction limit states that the resolution of a linear imaging system is fundamentally limited by one half of the wavelength of light. This implies that conventional light microscopes cannot distinguish two objects placed within a distance closer than 0.5 × 400 = 200nm (blue) or 0.5 × 700 = 350nm (red). This significantly impedes biomedical discovery by restricting our ability to observe biological structure and processes smaller than 100nm. Recent progress in sparsity-driven signal processing has created a powerful paradigm for increasing both the resolution and overall quality of imaging by promoting model-based image acquisition and reconstruction. This has led to multiple influential results demonstrating super-resolution in practical imaging systems. To date, however, the vast majority of work in signal processing has neglected the fundamental nonlinearity of the object-light interaction and its potential to lead to resolution enhancement. As a result, modern theory heavily focuses on linear measurement models that are truly effective only when object-light interactions are weak. Without a solid signal processing foundation for understanding such nonlinear interactions, we undervalue their impact on information transfer in the image formation. This ultimately limits our capability to image a large class of objects, such as biological tissue, that generally are in large-volumes and interact strongly and nonlinearly with light.
The goal of this talk is to present the recent progress in model-based imaging under multiple scattering. We will discuss several key applications including optical diffraction tomography, Fourier Ptychography, and large-scale Holographic microscopy. We will show that all these application can benefit from models, such as the Rytov approximation and beam propagation method, that take light scattering into account. We will discuss the integration of such models into the state-of-the-art optimization algorithms such as FISTA and ADMM. Finally, we will describe the most recent work that uses learned-priors for improving the quality of image reconstruction under multiple scattering.
Can modern signal processing be used to overcome the diffraction limit? The classical diffraction limit states that the resolution of a linear imaging system is fundamentally limited by one half of the wavelength of light. This implies that conventional light microscopes cannot distinguish two objects placed within a distance closer than 0.5 × 400 = 200nm (blue) or 0.5 × 700 = 350nm (red). This significantly impedes biomedical discovery by ...

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