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Publications associées (7)

A multigrid method for PDE-constrained optimization with uncertain inputs

Fabio Nobile, Tommaso Vanzan

In this manuscript, we present a collective multigrid algorithm to solve efficiently the large saddle-point systems of equations that typically arise in PDE-constrained optimization under uncertainty, and develop a novel convergence analysis of collective ...

Springer Science and Business Media LLC2024

Preconditioners for robust optimal control problems under uncertainty

Fabio Nobile, Tommaso Vanzan

The discretization of robust quadratic optimal control problems under uncertainty using the finite element method and the stochastic collocation method leads to large saddle-point systems, which are fully coupled across the random realizations. Despite its ...

2023

Gradient flow dynamics of shallow ReLU networks for square loss and orthogonal inputs

Nicolas Henri Bernard Flammarion, Loucas Pillaud-Vivien, Etienne Patrice Boursier

The training of neural networks by gradient descent methods is a cornerstone of the deep learning revolution. Yet, despite some recent progress, a complete theory explaining its success is still missing. This article presents, for orthogonal input vectors, ...

NeurIPS2022

Polynomial Escape-Time From Saddle Points In Distributed Non-Convex Optimization

Ali H. Sayed, Stefan Vlaski

The diffusion strategy for distributed learning from streaming data employs local stochastic gradient updates along with exchange of iterates over neighborhoods. In this work we establish that agents cluster around a network centroid in the mean-fourth sen ...

IEEE2019

A Petrov-Galerkin reduced basis approximation of the Stokes equation in parameterized geometries

Assyr Abdulle, Ondrej Budác

We present a Petrov-Galerkin reduced basis (RB) approximation for the parameterized Stokes equation. Our method, which relies on a reduced solution space and a parameter-dependent test space, is shown to be stable (in the sense of Babuska) and algebraicall ...

Elsevier2015

Reduced basis method for parametrized elliptic optimal control problems

Alfio Quarteroni, Gianluigi Rozza, Andrea Manzoni, Federico Negri

We propose a suitable model reduction paradigm -- the certied reduced basis method (RB) -- for the rapid and reliable solution of parametrized optimal control problems governed by partial dierential equations (PDEs). In particular, we develop the methodolo ...

Society for Industrial and Applied Mathematics2013

Ground state properties of large atoms and quantum dots

We investigate the ground state properties of large atoms and quantum dots described by a d-dimensional N-body Hamiltonian of confinement ZV. In atoms, d = 3 and V is the Coulomb interaction; in dots, d = 2 and V is phenomenologically determined. We expres ...

EPFL2009