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 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 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 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.[-]

The aim of this talk is to present a new variation formulation of the time-dependent many-body electronic Schrödinger equation with Coulombic singularities. More precisely, its solution can actually be expressed as the solution of a global space-time quadratic minimization problem that proves to be useful for several tasks:
1) first, it is amenable to Galerkin time-space discretization schemes, using an appropriate least-square formulation
2) it enables to yield a new variational principle for the construction dynamical low-rank approximations, that is different from the classical Dirac-Frenkel variational principle
3) it enables to obtain fully certified a posteriori error estimators between the exact solution and approximate solutions.
The present analysis can be applied to the electronic many-body time-dependent Schrödinger equation with an arbitrary number of electrons and interaction potentials with Coulomb singularities.[-]

we discuss classification problems in high dimension. We study classification problems using three classical notions: complexity of decision boundary, noise, and margin. We demonstrate that under suitable conditions on the decision boundary, classification problems can be very efficiently approximated, even in high dimensions. If a margin condition is assumed, then arbitrary fast approximation rates can be achieved, despite the problem being high-dimensional and discontinuous. We extend the approximation results ta learning results and show close ta optimal learning rates for empirical risk minimization in high dimensional classification. [-]

Modern machine learning architectures often embed their inputs into a lower-dimensional latent space before generating a final output. A vast set of empirical results---and some emerging theory---predicts that these lower-dimensional codes often are highly structured, capturing lower-dimensional variation in the data. Based on this observation, in this talk I will describe efforts in my group to develop lightweight algorithms that navigate, restructure, and reshape learned latent spaces. Along the way, I will consider a variety of practical problems in machine learning, including low-rank adaptation of large models, regularization to promote local latent structure, and efficient training/evaluation of generative models.[-]

The development of a suitable, efficient and accurate numerical method to solve wave problems is encountered in many academic and industrial applications. The Boundary Integral Equation (BIE) technique, whose discretization is known as the Boundary Element Method (BEM), is an appealing alternative to classical domain method because it allows to handle problems defined on the exterior of bounded domains as easily as those defined in the interior, without the introduction of an artificial boundary to truncate the computational domain. Very recently, an Isogeometric Analysis based Boundary Element Method (IgA-BEM) has been proposed in literature for the numerical solution of frequency-domain (Helmholtz) wave problems on 3D domains admitting a multi-patch representation of the boundary surface. While being powerful and applicable to many situations, this approach shares with standard BEMs a disadvantage which can easily become significant in the 3D setting. Indeed, when the required accuracy is increased, it can soon lead to large dense linear systems, whose numerical solution requires huge memory, resulting also in important computational cost. Recently the development of fast H-matrix based direct and iterative solvers for oscillatory kernels, as the Helmholtz one, has been studied. Here, we investigate the effectiveness of the H-matrix technique, along with a suitable GMRES iterative solver, when used in the context of multi-patch IgA-BEM.[-]