The Galois/monodromy group of a family of geometric problems or equations is a subtle invariant that encodes the structure of the solutions. We give numerical methods to compute the Galois group and study it when it is not the full symmetric group. One algorithm computes generators, while the other studies its structure as a permutation group. We illustrate these algorithms with examples using a Macaulay2 package we are developing that relies upon Bertini to perform monodromy computations.

We introduce and analyze a framework for function interpolation using compressed sensing. This framework - which is based on weighted $\ell^1$ minimization - does not require a priori bounds on the expansion tail in either its implementation or its theoretical guarantees, and in the absence of noise leads to genuinely interpolatory approximations. We also establish a new recovery guarantee for compressed sensing with weighted $\ell^1$ minimization based on this framework. This guarantee has several key benefits. First, unlike existing results, it is sharp (up to constants and log factors) for large classes of functions regardless of the choice of weights. Second, by examining the measurement condition in the recovery guarantee, we are able to suggest a good overall strategy for selecting the weights. In particular, when applied to the important case of multivariate approximation with orthogonal polynomials, this weighting strategy leads to provably optimal estimates on the number of measurements required, whenever the support set of the significant coefficients is a so-called lower set. Finally, this guarantee can also be used to theoretically confirm the benefits of alternative weighting strategies where the weights are chosen based on prior support information. This provides a theoretical basis for a number of recent numerical studies showing the effectiveness of such approaches.

We design a two-scale finite element method (FEM) for linear elliptic PDEs in non-divergence form $$A(x) : D^2 u(x) = f(x)$$ in a bounded but not necessarily convex domain $$\Omega $$ and study it in the max norm. The fine scale is given by the meshsize h, whereas the coarse scale $$\epsilon $$ is dictated by an integro-differential approximation of the PDE. We show that the FEM satisfies the discrete maximum principle for any uniformly positive definite matrix A provided that the mesh is face weakly acute. We establish a discrete Alexandroff–Bakelman–Pucci (ABP) estimate which is suitable for finite element analysis. Its proof relies on a discrete Alexandroff estimate which expresses the min of a convex piecewise linear function in terms of the measure of its sub-differential, and thus of jumps of its gradient. The discrete ABP estimate leads, under suitable regularity assumptions on A and u, to pointwise error estimates of the form $$\begin{aligned} \Vert \,u - u^{\epsilon }_h\,\Vert _{L^{\infty }(\Omega )} \le \, C(A,u) \, h^{2\alpha /(2 + \alpha )} \big | \ln h \big | \qquad 0< \alpha \le 2, \end{aligned}$$ provided $$\epsilon \approx h^{2/(2+\alpha )}$$ . Such a convergence rate is at best of order $$ h \big | \ln h \big |$$ , which turns out to be quasi-optimal.

We present an algorithmic framework for producing Delaunay triangulations of manifolds. The input to the algorithm is a set of sample points together with coordinate patches indexed by those points. The transition functions between nearby coordinate patches are required to be bi-Lipschitz with a constant close to 1. The primary novelty of the framework is that it can accommodate abstract manifolds that are not presented as submanifolds of Euclidean space. The output is a manifold simplicial complex that is the Delaunay complex of a perturbed set of points on the manifold. The guarantee of a manifold output complex demands no smoothness requirement on the transition functions, beyond the bi-Lipschitz constraint. In the smooth setting, when the transition functions are defined by common coordinate charts, such as the exponential map on a Riemannian manifold, the output manifold is homeomorphic to the original manifold, when the sampling is sufficiently dense.

Metropolis algorithms for approximate sampling of probability measures on infinite dimensional Hilbert spaces are considered and a generalization of the preconditioned Crank-Nicolson (pCN) proposal is introduced. The new proposal is able to incorporate information of the measure of interest. A numerical simulation of a Bayesian inverse problem indicates that a Metropolis algorithm with such a proposal performs independent of the state space dimension and the variance of the observational noise. Moreover, a qualitative convergence result is provided by a comparison argument for spectral gaps. In particular, it is shown that the generalization inherits geometric ergodicity from the Metropolis algorithm with pCN proposal.

The largest eigenvalue of a matrix is always larger or equal than its largest diagonal entry. We show that for a class of random Laplacian matrices with independent off-diagonal entries, this bound is essentially tight: the largest eigenvalue is, up to lower order terms, often the size of the largest diagonal. entry. Besides being a simple tool to obtain precise estimates on the largest eigenvalue of a class of random Laplacian matrices, our main result settles a number of open problems related to the tightness of certain convex relaxation-based algorithms. It easily implies the optimality of the semidefinite relaxation approaches to problems such as \({{\mathbb {Z}}}_2\) Synchronization and stochastic block model recovery. Interestingly, this result readily implies the connectivity threshold for Erdős–Renyi graphs and suggests that these three phenomena are manifestations of the same underlying principle. The main tool is a recent estimate on the spectral norm of matrices with independent entries by van Handel and the author.

In this paper we study multidimensional persistence modules [5,13] via what we call tame functors and noise systems. A noise system leads to a pseudo-metric topology on the category of tame functors. We show how this pseudo-metric can be used to identify persistent features of compact multidimensional persistence modules. To count such features we introduce the feature counting invariant and prove that assigning this invariant to compact tame functors is a 1-Lipschitz operation. For 1-dimensional persistence, we explain how, by choosing an appropriate noise system, the feature counting invariant identifies the same persistent features as the classical barcode construction.

Hierarchical tensors can be regarded as a generalisation, preserving many crucial features, of the singular value decomposition to higher-order tensors. For a given tensor product space, a recursiv...

Complexes of discrete distributional differential forms are introduced into finite element exterior calculus. Thus, we generalize a notion of Braess and Schoberl, originally studied for a posteriori error estimation. We construct isomorphisms between the simplicial homology groups of the triangulation, the discrete harmonic forms of the finite element complex, and the harmonic forms of the distributional finite element complexes. As an application, we prove that the complexes of finite element exterior calculus have cohomology groups isomorphic to the de Rham cohomology, including the case of partial boundary conditions. Poincare–Friedrichs-type inequalities will be studied in a subsequent contribution.

Historically, much of the theory and practice in nonlinear optimization has revolved around the quadratic models. Though quadratic functions are nonlinear polynomials, they are well structured and easy to deal with. Limitations of the quadratics, however, become increasingly binding as higher degree nonlinearity is imperative in modern applications of optimization. In the recent years, one observes a surge of research activities in polynomial optimization, and modeling with quartic or higher order polynomial functions has been more commonly accepted. On the theoretical side, there are also major recent progresses on polynomial functions and optimization. For instance, Ahmadi et al. [2] proved that checking the convexity of a quartic polynomial function is strongly NP-hard in general, which settles a long-standing open question. In this paper we proceed to studying six fundamentally important convex cones of quartic functions in the space of symmetric quartic tensors, including the cone of nonnegative quartic polynomials, the sum of squared polynomials, the convex quartic polynomials, and the sum of fourth powered polynomials. It turns out that these convex cones coagulate into a chain in decreasing order. The complexity status of these cones is sorted out as well. Finally, potential applications of the new results to solve highly nonlinear and/or combinatorial optimization problems are discussed.