We analyze constrained and unconstrained minimization problems on patches of tetrahedra sharing a common vertex with discontinuous piecewise polynomial data of degree p. We show that the discrete minimizers in the spaces of piecewise polynomials …

The proximal Galerkin finite element method is a high-order, low iteration complexity, nonlinear numerical method that preserves the geometric and algebraic structure of pointwise bound constraints in infinite-dimensional function spaces. This …

The theory of group classifications has undergone significant changes in the past 25 years. New methods have been introduced, some difficult problems have been solved and group classifications have become widely available through computer algebra …

We address the noncommutative version of the Edmonds’ problem, which asks to determine the inner rank of a matrix in noncommuting variables. We provide an algorithm for the calculation of this inner rank by relating the problem with the …

We present a discretization of the dynamic optimal transport problem for which we can obtain the convergence rate for the value of the transport cost to its continuous value when the temporal and spatial stepsize vanish. This convergence result …

We consider the problem of reconstructing a function $$f\in L^2({\mathbb R})$$ f ∈ L 2 ( R ) given phase-less samples of its Gabor transform, which is defined by $$\begin{aligned} {\mathcal {G}}f(x,y) :=2^{\frac{1}{4}} \int _{\mathbb R}f(t) …

The NEPv approach has been increasingly used lately for optimization on the Stiefel manifold arising from machine learning. General speaking, the approach first turns the first order optimality condition into a nonlinear eigenvalue problem with …

In this paper, we propose a general approach for explicit a posteriori error representation for convex minimization problems using basic convex duality relations. Exploiting discrete orthogonality relations in the space of element-wise constant …

Hamiltonian Monte Carlo (HMC) is a Markov chain Monte Carlo method that allows to sample high dimensional probability measures. It relies on the integration of the Hamiltonian dynamics to propose a move which is then accepted or rejected thanks to …

The Gromov–Wasserstein distance—a generalization of the usual Wasserstein distance—permits comparing probability measures defined on possibly different metric spaces. Recently, this notion of distance has found several applications in Data Science …

In this paper, we introduce the signed barcode, a new visual representation of the global structure of the rank invariant of a multi-parameter persistence module or, more generally, of a poset representation. Like its unsigned counterpart in …

We study two related problems concerning the number of homogeneous subsets of given size in graphs that go back to questions of Erdős. Most notably, we improve the upper bounds on the Ramsey multiplicity of $$K_4$$ K 4 and $$K_5$$ K 5 and settle …

Weighted digraphs are used to model a variety of natural systems and can exhibit interesting structure across a range of scales. In order to understand and compare these systems, we require stable, interpretable, multiscale descriptors. To this …

Gradient-based learning in multi-layer neural networks displays a number of striking features. In particular, the decrease rate of empirical risk is non-monotone even after averaging over large batches. Long plateaus in which one observes barely …

The exotic aromatic Butcher series were originally introduced for the calculation of order conditions for the high order numerical integration of ergodic stochastic differential equations in $$\mathbb {R} ^d$$ R d and on manifolds. We prove in …

We introduce a new class of numerical schemes which allow for low-regularity approximations to the expectation $$ \mathbb {E}(|u_{k}(t, v^{\eta })|^2)$$ E ( | u k ( t , v η ) | 2 ) , where $$u_k$$ u k denotes the k-th Fourier coefficient of the …

Rapid convergence of the shifted QR algorithm on symmetric matrices was shown more than 50 years ago. Since then, despite significant interest and its practical relevance, an understanding of the dynamics and convergence properties of the shifted …

We consider the Navier–Stokes–Fourier system governing the motion of a general compressible, heat conducting, Newtonian fluid driven by random initial/boundary data. Convergence of the stochastic collocation and Monte Carlo numerical methods is …

We describe fast algorithms for approximating the connection coefficients between a family of orthogonal polynomials and another family with a polynomially or rationally modified measure. The connection coefficients are computed via …