Related papers: Optimal Re-Embeddings of Border Basis Schemes
In this paper, we study ideals spanned by polynomials or overconvergent series in a Tate algebra. With state-of-the-art algorithms for computing Tate Gr{\"o}bner bases, even if the input is polynomials, the size of the output grows with the…
We provide lower error bounds for randomized algorithms that approximate integrals of functions depending on an unrestricted or even infinite number of variables. More precisely, we consider the infinite-dimensional integration problem on…
The Bandwidth Problem seeks for a simultaneous permutation of the rows and columns of the adjacency matrix of a graph such that all nonzero entries are as close as possible to the main diagonal. This work focuses on investigating novel…
In this paper we study {\em terminal embeddings}, in which one is given a finite metric $(X,d_X)$ (or a graph $G=(V,E)$) and a subset $K \subseteq X$ of its points are designated as {\em terminals}. The objective is to embed the metric into…
In this work, we present an efficiently computational approach for designing material micro-structures by means of topology optimization. The central idea relies on using the isogeometric analysis integrated with the parameterized level set…
We consider the problem of computing a grevlex Gr\"obner basis for the set $F_r(M)$ of minors of size $r$ of an $n\times n$ matrix $M$ of generic linear forms over a field of characteristic zero or large enough. Such sets are not regular…
A novel boundary element method (BEM) removes the classical dependence on explicit fundamental solutions and extends quasi-optimal BEM discretisations to strongly elliptic operators with variable coefficients. The approach constructs a…
We prove that for any real-valued matrix $X \in \R^{m \times n}$, and positive integers $r \ge k$, there is a subset of $r$ columns of $X$ such that projecting $X$ onto their span gives a $\sqrt{\frac{r+1}{r-k+1}}$-approximation to best…
We consider the problem of computing critical points of the restriction of a polynomial map to an algebraic variety. This is of first importance since the global minimum of such a map is reached at a critical point. Thus, these points…
In scattered data approximation, the span of a finite number of translates of a chosen radial basis function is used as approximation space and the basis of translates is used for representing the approximate. However, this natural choice…
In this paper, we describe a new method to compute the minimum of a real polynomial function and the ideal defining the points which minimize this polynomial function, assuming that the minimizer ideal is zero-dimensional. Our method is a…
The computation of amoebas has been a challenging open problem for the last dozen years. The most natural approach, namely to compute an amoeba via its boundary, has not been practical so far since only a superset of the boundary, the…
The reconstruction of an unknown quantity from noisy measurements is a mathematical problem relevant in most applied sciences, for example, in medical imaging, radar inverse scattering, or astronomy. This underlying mathematical problem is…
Low-degree polynomials have emerged as a powerful paradigm for providing evidence of statistical-computational gaps across a variety of high-dimensional statistical models [Wein25]. For detection problems -- where the goal is to test a…
We develop a new optimisation technique that combines multiresolution subdivision surfaces for boundary description with immersed finite elements for the discretisation of the primal and adjoint problems of optimisation. Similar to wavelets…
The present paper investigates properties of quasi-stable ideals and of Borel-fixed ideals in a polynomial ring $k[x_0,\dots,x_n]$, in order to design two algorithms: the first one takes as input $n$ and an admissible Hilbert polynomial…
We consider the problem of numerically approximating the solutions to an elliptic partial differential equation (PDE) for which the boundary conditions are lacking. To alleviate this missing information, we assume to be given measurement…
High-dimensional planted problems, such as finding a hidden dense subgraph within a random graph, often exhibit a gap between statistical and computational feasibility. While recovering the hidden structure may be statistically possible, it…
We provide a polynomial time algorithm for computing the universal Gr\"obner basis of any polynomial ideal having a finite set of common zeros in fixed number of variables. One ingredient of our algorithm is an effective construction of the…
We present an indirect higher order boundary element method utilising NURBS mappings for exact geometry representation and an interpolation-based fast multipole method for compression and reduction of computational complexity, to counteract…