B-Spline-Based Exact Discretization of Continuous-Domain Inverse Problems With Generalized TV Regularization

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Parabolic stochastic PDEs on bounded domains with rough initial conditions: moment and correlation bounds

Le Chen, Cheuk Yin Lee, David Jean-Michel Candil

We consider nonlinear parabolic stochastic PDEs on a bounded Lipschitz domain driven by a Gaussian noise that is white in time and colored in space, with Dirichlet or Neumann boundary condition. We establish existence, uniqueness and moment bounds of the r ...

Springer2023

Retraction-based numerical methods for continuation, interpolation and time integration on manifolds

Axel Elie Joseph Séguin

The goal of this thesis is the development and the analysis of numerical methods for problems where the unknown is a curve on a smooth manifold. In particular, the thesis is structured around the three following problems: homotopy continuation, curve inter ...

EPFL2023

Hamiltonian Deep Neural Networks Guaranteeing Non-Vanishing Gradients by Design

Giancarlo Ferrari Trecate, Luca Furieri, Liang Xu, Clara Lucía Galimberti

Deep Neural Networks (DNNs) training can be difficult due to vanishing and exploding gradients during weight optimization through backpropagation. To address this problem, we propose a general class of Hamiltonian DNNs (H-DNNs) that stem from the discretiz ...

2023

An optimal preconditioned FFT-accelerated finite element solver for homogenization

Till Junge, Ali Falsafi

We generalize and provide a linear algebra-based perspective on a finite element (FE) ho-mogenization scheme, pioneered by Schneider et al. (2017)[1] and Leuschner and Fritzen (2018)[2]. The efficiency of the scheme is based on a preconditioned, well-scale ...

ELSEVIER SCIENCE INC2023

Low-Rank Tensor Methods for High-Dimensional Problems

Christoph Max Strössner

In this thesis, we propose and analyze novel numerical algorithms for solving three different high-dimensional problems involving tensors. The commonality of these problems is that the tensors can potentially be well approximated in low-rank formats. Ident ...

EPFL2023

A Statistical Framework to Investigate the Optimality of Signal-Reconstruction Methods

Michaël Unser, Pakshal Narendra Bohra, Pol del Aguila Pla

We present a statistical framework to benchmark the performance of reconstruction algorithms for linear inverse problems, in particular, neural-network-based methods that require large quantities of training data. We generate synthetic signals as realizati ...

2023

Outlier-free spline spaces for isogeometric discretizations of biharmonic and polyharmonic eigenvalue problems

Espen Sande

We present outlier-free isogeometric Galerkin discretizations of eigenvalue problems related to the biharmonic and the polyharmonic operator in the univariate setting. These are Galerkin discretizations in certain spline subspaces that provide accurate app ...

Elsevier Science Sa2023

Regularization of polynomial networks for image recognition

Volkan Cevher, Grigorios Chrysos, Bohan Wang

Deep Neural Networks (DNNs) have obtained impressive performance across tasks, however they still remain as black boxes, e.g., hard to theoretically analyze. At the same time, Polynomial Networks (PNs) have emerged as an alternative method with a promising ...

2023

Regularization of polynomial networks for image recognition

Volkan Cevher, Grigorios Chrysos, Bohan Wang

Ieee Computer Soc2023

Variational Methods For Continuous-Domain Inverse Problems: the Quest for the Sparsest Solution

Thomas Jean Debarre

The goal of this thesis is to study continuous-domain inverse problems for the reconstruction of sparse signals and to develop efficient algorithms to solve such problems computationally. The task is to recover a signal of interest as a continuous function ...

EPFL2022

Saddle-to-Saddle Dynamics in Diagonal Linear Networks

Nicolas Henri Bernard Flammarion, Scott William Pesme

In this paper we fully describe the trajectory of gradient flow over diagonal linear networks in the limit of vanishing initialisation. We show that the limiting flow successively jumps from a saddle of the training loss to another until reaching the minim ...

2023

On the reconstruction of the attenuation function of a return-stroke current from the Fourier Transform of finite-duration measurements

Farhad Rachidi-Haeri, Marcos Rubinstein

A correct representation of the lightning current is crucial when the electromagnetic field radiated to a point of interest has to be computed. Based on the engineering models of Transmission Line type, such representation involves the knowledge of the ret ...

2022