We introduce and analyze a numerical approximation of the porous medium equation with fractional potential pressure introduced by Caffarelli and Vázquez: $$ \partial _t u = \nabla \cdot (u^{m-1}\nabla (-\Delta )^{-\sigma }u) \qquad \text {for} \qquad m\ge 2 \quad \text {and} \quad \sigma \in (0,1). $$ Our scheme is for one space dimension and positive solutions u. It consists of...
This paper considers the problem of understanding the behavior of a general class of accelerated gradient methods on smooth nonconvex functions. Motivated by some recent works that have proposed effective algorithms, based on Polyak’s heavy ball method and the Nesterov accelerated gradient method, to achieve convergence to a local minimum of nonconvex functions, this work...
In this work we develop a discrete trace theory that covers non-conforming hybrid discretization methods and holds on polytopal meshes. A notion of discrete trace seminorm is defined, and trace and lifting results with respect to a discrete \(H^1\)-seminorm on the hybrid fully discrete space are proven. Finally, we conduct a numerical test in which we compute the proposed...
We introduce a simple initialization of the Maubach bisection routine for adaptive mesh refinement which applies to any conforming initial triangulation and terminates in linear time with respect to the number of initial vertices. We show that Maubach’s routine with this initialization always terminates and generates meshes that preserve shape regularity and satisfy the closure...
The bubble transform is a procedure to decompose differential forms, which are piecewise smooth with respect to a given triangulation of the domain, into a sum of local bubbles. In this paper, an improved version of a construction in the setting of the de Rham complex previously proposed by the authors is presented. The major improvement in the decomposition is that unlike the...
In 1991, Moore (Nonlinearity 4:199–230, 1991) raised a question about whether hydrodynamics is capable of performing computations. Similarly, in 2016, Tao (J Am Math Soc 29(3):601–674, 2016) asked whether a mechanical system, including a fluid flow, can simulate a universal Turing machine. In this expository article, we review the construction in Cardona et al. (Proc Natl Acad...
We propose a novel non-parametric learning paradigm for the identification of drift and diffusion coefficients of multi-dimensional non-linear stochastic differential equations, which relies upon discrete-time observations of the state. The key idea essentially consists of fitting a RKHS-based approximation of the corresponding Fokker–Planck equation to such observations...
Sharpness is an almost generic assumption in continuous optimization that bounds the distance from minima by objective function suboptimality. It facilitates the acceleration of first-order methods through restarts. However, sharpness involves problem-specific constants that are typically unknown, and restart schemes typically reduce convergence rates. Moreover, these schemes are...
We introduce an algorithm to decompose matrix representations of the symmetric group over the reals into irreducible representations, which as a by-product also computes the multiplicities of the irreducible representations. The algorithm applied to a d-dimensional representation of $$S_n$$ is shown to have a complexity of $${\mathcal {O}}(n^2 d^3)$$ operations for determining...
While decomposition of one-parameter persistence modules behaves nicely, as demonstrated by the algebraic stability theorem, decomposition of multiparameter modules is known to be unstable in a certain precise sense. Until now, it has not been clear that there is any way to get around this and build a meaningful stability theory for multiparameter module decomposition. We...
The criticality problem in nuclear engineering asks for the principal eigenpair of a Boltzmann operator describing neutron transport in a reactor core. Being able to reliably design, and control such reactors requires assessing these quantities within quantifiable accuracy tolerances. In this paper, we propose a paradigm that deviates from the common practice of approximately...
We develop Conley’s theory for multivalued maps on finite topological spaces. More precisely, for discrete-time dynamical systems generated by the iteration of a multivalued map which satisfies appropriate regularity conditions, we establish the notions of isolated invariant sets and index pairs, and use them to introduce a well-defined Conley index. In addition, we verify some...
We present a randomized, inverse-free algorithm for producing an approximate diagonalization of any $$n \times n$$ matrix pencil (A, B). The bulk of the algorithm rests on a randomized divide-and-conquer eigensolver for the generalized eigenvalue problem originally proposed by Ballard, Demmel and Dumitriu (Technical Report 2010). We demonstrate that this divide-and-conquer...
Given a domain $$\Omega \subset \mathbb {R}^n$$ , the de Rham complex of differential forms arises naturally in the study of problems in electromagnetism and fluid mechanics defined on $$\Omega $$ , and its discretization helps build stable numerical methods for such problems. For constructing such stable methods, one critical requirement is ensuring that the discrete subcomplex...
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 paper introduces the proximal Galerkin method and applies it to solve free boundary problems, enforce discrete maximum principles, and...
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 systems. This survey describes the state of the art of the group classification problem, its history, its recent advances and some open...
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 distribution of a basic object in free probability theory, namely operator-valued semicircular elements. We have to solve a matrix-valued...
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 does not require any regularity assumption on the measures, though experiments suggest that the rate is not sharp. Via an analysis of the...
We consider the problem of reconstructing a function $$f\in L^2({\mathbb R})$$ given phase-less samples of its Gabor transform, which is defined by $$\begin{aligned} {\mathcal {G}}f(x,y) :=2^{\frac{1}{4}} \int _{\mathbb R}f(t) e^{-\pi (t-x)^2} e^{-2\pi i y t}\,\text{ d }t,\quad (x,y)\in {\mathbb R}^2. \end{aligned}$$ More precisely, given sampling positions $$\Omega \subseteq...
In this paper, we propose a general approach for explicit a posteriori error representation for convex minimization problems using basic convex duality relations. Exploiting discrete orthogonality relations in the space of element-wise constant vector fields as well as a discrete integration-by-parts formula between the Crouzeix–Raviart and the Raviart–Thomas element, all convex...
The Gromov–Wasserstein distance—a generalization of the usual Wasserstein distance—permits comparing probability measures defined on possibly different metric spaces. Recently, this notion of distance has found several applications in Data Science and in Machine Learning. With the goal of aiding both the interpretability of dissimilarity measures computed through the Gromov...
In this paper, we introduce the signed barcode, a new visual representation of the global structure of the rank invariant of a multi-parameter persistence module or, more generally, of a poset representation. Like its unsigned counterpart in one-parameter persistence, the signed barcode decomposes the rank invariant as a $${\mathbb {Z}}$$ -linear combination of rank invariants of...
We study two related problems concerning the number of homogeneous subsets of given size in graphs that go back to questions of Erdős. Most notably, we improve the upper bounds on the Ramsey multiplicity of $$K_4$$ and $$K_5$$ and settle the minimum number of independent sets of size 4 in graphs with clique number at most 4. Motivated by the elusiveness of the symmetric Ramsey...
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 end, we propose grounded persistent path homology (GrPPH)—a new, functorial, topological descriptor that describes the structure of an...