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Research schools;Numerical Analysis and Scientific Computing;Control Theory and Optimization;Probability and Statistics
This course presents an overview of modern Bayesian strategies for solving imaging inverse problems. We will start by introducing the Bayesian statistical decision theory framework underpinning Bayesian analysis, and then explore efficient numerical methods for performing Bayesian computation in large-scale settings. We will pay special attention to high-dimensional imaging models that are log-concave w.r.t. the unknown image, related to so-called “convex imaging problems”. This will provide an opportunity to establish connections with the convex optimisation and machine learning approaches to imaging, and to discuss some of their relative strengths and drawbacks. Examples of topics covered in the course include: efficient stochastic simulation and optimisation numerical methods that tightly combine proximal convex optimisation with Markov chain Monte Carlo techniques; strategies for estimating unknown model parameters and performing model selection, methods for calculating Bayesian confidence intervals for images and performing uncertainty quantification analyses; and new theory regarding the role of convexity in maximum-a-posteriori and minimum-mean-square-error estimation. The theory, methods, and algorithms are illustrated with a range of mathematical imaging experiments.
This course presents an overview of modern Bayesian strategies for solving imaging inverse problems. We will start by introducing the Bayesian statistical decision theory framework underpinning Bayesian analysis, and then explore efficient numerical methods for performing Bayesian computation in large-scale settings. We will pay special attention to high-dimensional imaging models that are log-concave w.r.t. the unknown image, related to ...
49N45 ; 65C40 ; 65C60 ; 65J22 ; 68U10 ; 62C10 ; 62F15 ; 94A08
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- 262 p.
Proceedings of the Steklov institute of mathematics, supplement
Localisation : Collection 1er étage
mécanique # système dynamique # contrôle optimal # problème inverse # problème mal-posé # programmation mthématique # Yurii Sergeevich Osipov
37-06 ; 49-06 ; 47A52 ; 49N45 ; 37N05
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- 224 p.
Proceedings of the Steklov institute of mathematics, supplement
Localisation : Collection 1er étage
controle # stabilité # problème inverse # dynamique
49-06 ; 93Dxx ; 49N45
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- vi, 443 p.
ISBN 978-3-7643-8754-9
Operator theory: advances and applications , 0186
Localisation : Collection 1er étage
analyse spectrale # graphe # Hamiltonien # approximation Ablowitz-Ladik # opérateur Schrödinger # opérateur Aharonov-Bohm # valeur propre # matrice Jacobi # estimation
05C25 ; 34A55 ; 34Bxx ; 34L40 ; 35J10 ; 35Kxx ; 35Pxx ; 37KXX ; 42C05 ; 47-00 ; 49N45 ; 81Qxx ; 81T99 ; 82B43
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Research talks;Control Theory and Optimization;Partial Differential Equations
We consider spectral optimization problems of the form
$\min\lbrace\lambda_1(\Omega;D):\Omega\subset D,|\Omega|=1\rbrace$
where $D$ is a given subset of the Euclidean space $\textbf{R}^d$. Here $\lambda_1(\Omega;D)$ is the first eigenvalue of the Laplace operator $-\Delta$ with Dirichlet conditions on $\partial\Omega\cap D$ and Neumann or Robin conditions on $\partial\Omega\cap\partial D$. The equivalent variational formulation
$\lambda_1(\Omega;D)=\min\lbrace\int_\Omega|\nabla u|^2dx+k\int_{\partial D}u^2d\mathcal{H}^{d-1}:$
$u\in H^1(D),u=0$ on $\partial\Omega\cap D,||u||_{L^2(\Omega)}=1\rbrace$
reminds the classical drop problems, where the first eigenvalue replaces the total variation functional. We prove an existence result for general shape cost functionals and we show some qualitative properties of the optimal domains. The case of Dirichlet condition on a $\textit{fixed}$ part and of Neumann condition on the $\textit{free}$ part of the boundary is also considered
We consider spectral optimization problems of the form
$\min\lbrace\lambda_1(\Omega;D):\Omega\subset D,|\Omega|=1\rbrace$
where $D$ is a given subset of the Euclidean space $\textbf{R}^d$. Here $\lambda_1(\Omega;D)$ is the first eigenvalue of the Laplace operator $-\Delta$ with Dirichlet conditions on $\partial\Omega\cap D$ and Neumann or Robin conditions on $\partial\Omega\cap\partial D$. The equivalent variational formulation
$\lam...
49Q10 ; 49J20 ; 49N45
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Research schools;Numerical Analysis and Scientific Computing;Control Theory and Optimization;Probability and Statistics
This course presents an overview of modern Bayesian strategies for solving imaging inverse problems. We will start by introducing the Bayesian statistical decision theory framework underpinning Bayesian analysis, and then explore efficient numerical methods for performing Bayesian computation in large-scale settings. We will pay special attention to high-dimensional imaging models that are log-concave w.r.t. the unknown image, related to so-called “convex imaging problems”. This will provide an opportunity to establish connections with the convex optimisation and machine learning approaches to imaging, and to discuss some of their relative strengths and drawbacks. Examples of topics covered in the course include: efficient stochastic simulation and optimisation numerical methods that tightly combine proximal convex optimisation with Markov chain Monte Carlo techniques; strategies for estimating unknown model parameters and performing model selection, methods for calculating Bayesian confidence intervals for images and performing uncertainty quantification analyses; and new theory regarding the role of convexity in maximum-a-posteriori and minimum-mean-square-error estimation. The theory, methods, and algorithms are illustrated with a range of mathematical imaging experiments.
This course presents an overview of modern Bayesian strategies for solving imaging inverse problems. We will start by introducing the Bayesian statistical decision theory framework underpinning Bayesian analysis, and then explore efficient numerical methods for performing Bayesian computation in large-scale settings. We will pay special attention to high-dimensional imaging models that are log-concave w.r.t. the unknown image, related to ...
49N45 ; 65C40 ; 65C60 ; 65J22 ; 68U10 ; 62C10 ; 62F15 ; 94A08
... Lire [+]
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Research schools;Numerical Analysis and Scientific Computing;Control Theory and Optimization;Probability and Statistics
This course presents an overview of modern Bayesian strategies for solving imaging inverse problems. We will start by introducing the Bayesian statistical decision theory framework underpinning Bayesian analysis, and then explore efficient numerical methods for performing Bayesian computation in large-scale settings. We will pay special attention to high-dimensional imaging models that are log-concave w.r.t. the unknown image, related to so-called “convex imaging problems”. This will provide an opportunity to establish connections with the convex optimisation and machine learning approaches to imaging, and to discuss some of their relative strengths and drawbacks. Examples of topics covered in the course include: efficient stochastic simulation and optimisation numerical methods that tightly combine proximal convex optimisation with Markov chain Monte Carlo techniques; strategies for estimating unknown model parameters and performing model selection, methods for calculating Bayesian confidence intervals for images and performing uncertainty quantification analyses; and new theory regarding the role of convexity in maximum-a-posteriori and minimum-mean-square-error estimation. The theory, methods, and algorithms are illustrated with a range of mathematical imaging experiments.
This course presents an overview of modern Bayesian strategies for solving imaging inverse problems. We will start by introducing the Bayesian statistical decision theory framework underpinning Bayesian analysis, and then explore efficient numerical methods for performing Bayesian computation in large-scale settings. We will pay special attention to high-dimensional imaging models that are log-concave w.r.t. the unknown image, related to ...
49N45 ; 65C40 ; 65C60 ; 65J22 ; 68U10 ; 62C10 ; 62F15 ; 94A08
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Research schools;Numerical Analysis and Scientific Computing;Control Theory and Optimization;Probability and Statistics
This course presents an overview of modern Bayesian strategies for solving imaging inverse problems. We will start by introducing the Bayesian statistical decision theory framework underpinning Bayesian analysis, and then explore efficient numerical methods for performing Bayesian computation in large-scale settings. We will pay special attention to high-dimensional imaging models that are log-concave w.r.t. the unknown image, related to so-called “convex imaging problems”. This will provide an opportunity to establish connections with the convex optimisation and machine learning approaches to imaging, and to discuss some of their relative strengths and drawbacks. Examples of topics covered in the course include: efficient stochastic simulation and optimisation numerical methods that tightly combine proximal convex optimisation with Markov chain Monte Carlo techniques; strategies for estimating unknown model parameters and performing model selection, methods for calculating Bayesian confidence intervals for images and performing uncertainty quantification analyses; and new theory regarding the role of convexity in maximum-a-posteriori and minimum-mean-square-error estimation. The theory, methods, and algorithms are illustrated with a range of mathematical imaging experiments.
This course presents an overview of modern Bayesian strategies for solving imaging inverse problems. We will start by introducing the Bayesian statistical decision theory framework underpinning Bayesian analysis, and then explore efficient numerical methods for performing Bayesian computation in large-scale settings. We will pay special attention to high-dimensional imaging models that are log-concave w.r.t. the unknown image, related to ...
49N45 ; 65C40 ; 65C60 ; 65J22 ; 68U10 ; 62C10 ; 62F15 ; 94A08
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ISBN 978-0-8218-2533-4
Memoirs of the american mathematical society , 0473
Localisation : Collection 1er étage
analyse globale # analyse sur les varietes # bicomplexe variationnel # calcul des variations # contrïle optimal # equation differentielle ordinaire sur les varietes # geometrie symplectique # jet pulverise # optimisation # principe variationnel # principe variationnel pour les equations differentielles ord # probleme inverse # probleme variationnel dans les espaces de dimension infinie # systeme differentiel exterieur # systeme dynamique # systeme hamiltonien et lagrangien # theorie generale des varietes differentiables
analyse globale # analyse sur les varietes # bicomplexe variationnel # calcul des variations # contrïle optimal # equation differentielle ordinaire sur les varietes # geometrie symplectique # jet pulverise # optimisation # principe variationnel # principe variationnel pour les equations differentielles ord # probleme inverse # probleme variationnel dans les espaces de dimension infinie # systeme differentiel exterieur # systeme dynamique # ...
49N45 ; 58E30 ; 58F05
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- 127 p.
ISBN 978-3-540-39942-1
Lecture notes in mathematics , 1894
Localisation : Collection 1er étage
théorie des opérateurs # opérateur non borné # théorie d'approximation d'opérateur # problème inverse # méthode de Tikhonov-Morogov
47-02 ; 47A58 ; 47A50 ; 47A52 ; 49N45 ; 65J20
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