Publications associées à Bifurcation at isolated singular points of the Hadamard derivative

Explicit representations for Banach subspaces of Lizorkin distributions

The Lizorkin space is well suited to the study of operators like fractional Laplacians and the Radon transform. In this paper, we show that the space is unfortunately not complemented in the Schwartz space. In return, we show that it is dense in C0(Double- ...

2023

Stability of Image-Reconstruction Algorithms

Michaël Unser, Pol del Aguila Pla, Sebastian Jonas Neumayer

Robustness and stability of image-reconstruction algorithms have recently come under scrutiny. Their importance to medical imaging cannot be overstated. We review the known results for the topical variational regularization strategies ( ℓ2 and ℓ1 regulariz ...

2023

Propagation of singularities for the stochastic wave equation

Cheuk Yin Lee

We study the existence and propagation of singularities of the solution to a one-dimensional linear stochastic wave equation driven by an additive Gaussian noise that is white in time and colored in space. Our approach is based on a simultaneous law of the ...

ELSEVIER2022

Spectral analysis for transmission eigenvalue problems with and without the complementing conditions

Jean Louis-Alexandre Fornerod

The interior transmission eigenvalue problem is a system of partial differential equations equipped with Cauchy data on the boundary: the transmission conditions. This problem appears in the inverse scattering theory for inhomogeneous media when, for some ...

EPFL2022

An Efficient Sampling Algorithm for Non-smooth Composite Potentials

Nicolas Henri Bernard Flammarion

We consider the problem of sampling from a density of the form p(x) ? exp(-f (x) - g(x)), where f : Rd-+ R is a smooth function and g : R-d-+ R is a convex and Lipschitz function. We propose a new algorithm based on the Metropolis-Hastings framework. Under ...

Microtome Publishing2022

Algebra and geometry of link homology Lecture notes from the IHES 2021 Summer School

Oscar Salomon Kivinen

These notes cover the lectures of the first named author at 2021 IHES Summer School on "Enumerative Geometry, Physics and Representation Theory" with additional details and references. They cover the definition of Khovanov-Rozansky triply graded homology, ...

WILEY2022

Optimization Over Banach Spaces: A Unified View on Supervised Learning and Inverse Problems

Shayan Aziznejad

In this thesis, we reveal that supervised learning and inverse problems share similar mathematical foundations. Consequently, we are able to present a unified variational view of these tasks that we formulate as optimization problems posed over infinite-di ...

EPFL2022

Collapse-invariant properties of spaces equipped with signals or directions

Stefania Ebli

Collapsing cell complexes was first introduced in the 1930's as a way to deform a space into a topological-equivalent subspace with a sequence of elementary moves. Recently, discrete Morse theory techniques provided an efficient way to construct deformatio ...

EPFL2022

Gradient-based optimisation of the conditional-value-at-risk using the multi-level Monte Carlo method

Fabio Nobile, Sundar Subramaniam Ganesh

In this work, we tackle the problem of minimising the Conditional-Value-at-Risk (CVaR) of output quantities of complex differential models with random input data, using gradient-based approaches in combination with the Multi-Level Monte Carlo (MLMC) method ...

2023

Global bifurcation at isolated singular points of the Hadamard derivative

Charles Stuart

Consider F is an element of C(RxX,Y) such that F(lambda, 0) = 0 for all lambda is an element of R, where X and Y are Banach spaces. Bifurcation from the line Rx{0} of trivial solutions is investigated in cases where F(lambda, center dot ) need not be Frech ...

2021

Linear Lipschitz and C-1 extension operators through random projection

Federico Stra

We construct a regular random projection of a metric space onto a closed doubling subset and use it to linearly extend Lipschitz and C-1 functions. This way we prove more directly a result by Lee and Naor [5] and we generalize the C-l extension theorem by ...

ACADEMIC PRESS INC ELSEVIER SCIENCE2021

Convex optimization in sums of Banach spaces

Michaël Unser, Shayan Aziznejad

We characterize the solution of a broad class of convex optimization problems that address the reconstruction of a function from a finite number of linear measurements. The underlying hypothesis is that the solution is decomposable as a finite sum of compo ...

2021

Discretisation and continuity: The emergence of symbols in communication

Martin Alois Rohrmeier, Robert Lieck

Vocal signalling systems, as used by humans and various non-human animals, exhibit discrete and continuous properties that can naturally be used to express discrete and continuous information, such as distinct words to denote objects in the world and proso ...

2021

Local invertibility and sensitivity of atomic structure-feature mappings

Michele Ceriotti, Sergey Pozdnyakov

Background: The increasingly common applications of machine-learning schemes to atomic-scale simulations have triggered efforts to better understand the mathematical properties of the mapping between the Cartesian coordinates of the atoms and the variety o ...

2021

A critically degenerate elliptic Dirichlet problem, spectral theory and bifurcation

Charles Stuart

In an open, bounded subset Omega of R-N such that 0 is an element of Omega we consider the nonlinear eigenvalue problem -Sigma(N)(i,j,=1) partial derivative(i){A(ij)(x)partial derivative(j)u} + V(x)u + n(x,del u)+ g(x, u) = lambda u in Omega integral(Omega ...

2020

On the asymptotic behavior of the solutions to parabolic variational inequalities

Maria Colombo

We consider various versions of the obstacle and thin-obstacle problems, we interpret them as variational inequalities, with non-smooth constraint, and prove that they satisfy a new constrained Lojasiewicz inequality. The difficulty lies in the fact that, ...

2020

A Unifying Representer Theorem for Inverse Problems and Machine Learning

Michaël Unser

Regularization addresses the ill-posedness of the training problem in machine learning or the reconstruction of a signal from a limited number of measurements. The method is applicable whenever the problem is formulated as an optimization task. The standar ...

2021

Characterization of image spaces of Riemann-Liouville fractional integral operators on Sobolev spaces W-m,W-p (omega)

Jan Sickmann Hesthaven, Lijing Zhao

Fractional operators are widely used in mathematical models describing abnormal and nonlocal phenomena. Although there are extensive numerical methods for solving the corresponding model problems, theoretical analysis such as the regularity result, or the ...

SCIENCE PRESS2020

Group Approximation in Cayley Topology and Coarse Geometry, Part II: Fibred Coarse Embeddings

Masato Mimura

The objective of this series is to study metric geometric properties of disjoint unions of Cayley graphs of amenable groups by group properties of the Cayley accumulation points in the space of marked groups. In this Part II, we prove that a disjoint union ...

2019

Beyond Wiener's Lemma: Nuclear Convolution Algebras and the Inversion of Digital Filters

Michaël Unser, John Paul Ward, Julien René Pierre Fageot

A convolution algebra is a topological vector space X that is closed under the convolution operation. It is said to be inverse-closed if each element of X whose spectrum is bounded away from zero has a convolution inverse that is also part of the algebra. ...

SPRINGER BIRKHAUSER2019