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Jon Eivind Vatne

Publications and source records attributed to Jon Eivind Vatne.

14 recordsLinked to original sources
arXiv2026

Approximation properties of double complexes

We consider the simplicial de Rham complex and the Čech-de Rham complex, two bigraded Hilbert complexes whose Hodge-Laplace problems govern spatially coupled problems in mixed dimension and homogeneous dimension, respectively. The former complex can be realized as a subcomplex of the latter. In this paper, we quantify how close these complexes are to each other by constructing bounded cochain complexes between them, and thus we quantify how close a mixed-dimensional formulation of a problem is to an equidimensionally coupled formulation of the same problem. From this construction, we derive a priori- and a posteriori error estimates between the associated Hodge-Laplace problems on the two complexes. These estimates represent the error which is introduced by treating a spatially coupled problem as mixed-dimensional, rather than an equidimensional problem with thin overlaps.

math.NA↗
arXiv2024

An abstract approximation tool for mixed-dimensional and equidimensional modeling

Many coupled problems in engineering and science can be described by elliptic partial differential equations on adjacent domains, where the coupling can be considered either as a thin equidimensional overlap between the model domains, or as a lower-dimensional interface. Thereby we distinguish equidimensional and mixed-dimensional models of the same system, and the relationship between these modeling approaches is of natural interest. In this paper, we construct an overlapping open cover for a class of simplicial geometries and construct a bounded cochain map from the simplicial de Rham complex to the Čech-de Rham complex associated with the overlapping cover. Thus, we establish an isomorphism between simplicial de Rham complexes (i.e. functions and forms on mixed-dimensional partitions and their differentials) and subcomplexes of Čech-de Rham complexes (i.e. functions and forms on equidimensional partitions and their differentials), which serves as an abstract approximation tool for comparing mixed-dimensional problems to the equidimensional version of the same problem.

math.AT↗
arXiv2024

On a vectorized basic linear algebra package for prototyping codes in MATLAB

When writing high-performance code for numerical computation in a scripting language like MATLAB, it is crucial to have the operations in a large for-loop vectorized. If not, the code becomes too slow to use, even for a moderately large problem. However, in the process of vectorizing, the code often loses its original structure and becomes less readable. This is particularly true in the case of a finite element implementation, even though finite element methods are inherently structured. A basic remedy to this is the separation of the vectorization part from the mathematics part of the code, which is easily achieved through building the code on top of the basic linear algebra subprograms that are already vectorized codes, an idea that has been used in a series of papers over the last fifteen years, developing codes that are fast and still structured and readable. We discuss the vectorized basic linear algebra package and introduce a formalism using multi-linear algebra to explain and define formally the functions in the package, as well as MATLAB pagetime functions. We provide examples from computations of varying complexity, including the computation of normal vectors, volumes, and finite element methods. Benchmarking shows that we also get fast computations. Using the library, we can write codes that closely follow our mathematical thinking, making writing, following, reusing, and extending the code easier.

cs.MS↗
arXiv2023

The Hodge-Laplacian on the Čech-de Rham complex governs coupled problems

By endowing the Čech-de Rham complex with a Hilbert space structure, we obtain a Hilbert complex with sufficient properties to allow for well-posed Hodge-Laplace problems. We observe that these Hodge-Laplace equations govern a class of coupled problems arising from physical systems including elastically attached strings, multiple-porosity flow systems and 3D-1D coupled flow models.

math.AP↗
arXiv2018

The minimum angle condition for $d$-simplices

In this note we present a natural generalization of the minimum angle condition, commonly used in the finite element analysis for planar triangulations, to the case of simplicial meshes in any space dimension. The equivalence of this condition with some other mesh regularity conditions is proved.

math.NA↗
arXiv2018

On generalizations of the Synge-Křížek maximum angle condition for $d$-simplices

In this note we present a generalization of the maximum angle condition, proposed by J. L. Synge in 1957 and M. Křížek in 1992 for triangular and tetrahedral elements, respectively, for the case of higher-dimensional simplicial finite elements. Its relations to the other angle-type conditions commonly used in finite element methods are analysed.

math.NA↗
arXiv2016

The sequence of middle divisors is unbounded

The sequence of middle divisors is shown to be unbounded. For a given number $n$, $a_{n,0}$ is the number of divisors of $n$ in between $\sqrt{n/2}$ and $\sqrt{2n}$. We explicitly construct a sequence of numbers $n(i)$ and a list of divisors in the interesting range, so that the length of the list goes to infinity as $i$ increases.

math.NT↗
arXiv2010

Artin-Schelter regular algebras of dimension five

We show that there are exactly three types of Hilbert series of Artin-Schelter regular algebras of dimension five with two generators. One of these cases (the most extreme) may not be realized by an enveloping algebra of a graded Lie algebra. This is a new phenomenon compared to lower dimensions, where all resolution types may be realized by such enveloping algebras.

math.RA↗
arXiv2006

PBW-deformations of N-Koszul algebras

For a quotient algebra $U$ of the tensor algebra we give explicit conditions on its relations for $U$ being a PBW-deformation of an $N$-Koszul algebra $A$. We show there is a one-one correspondence between such deformations and a class of $A_\infty$-structures on the Yoneda algebra $Ext_A^*(k,k)$ of $A$. We compute the PBW-deformations of the algebra whose relations are the anti-symmetrizers of degree $N$ and also of cubic Artin-Schelter algebras.

math.RA↗
arXiv2005

The Operad Quad is Koszul

The purpose of this paper is to prove the koszulity of the operad ${\mathcal Quad}$, governing quadri-algebras. That ${\mathcal Quad}$ is Koszul was conjectured by Aguiar and Loday. The operad ${\mathcal Dend}$, governing dendriform algebras, is known to be Koszul, by work of Loday, and ${\mathcal Quad}$ is its second black square power. We find a new complex, based on the associahedron, which captures the structure of ${\mathcal Dend}$. This complex behaves well with respect to the black squaring process, and allows us to conclude. Also, this proves koszulity of higher powers of ${\mathcal Dend}$.

math.QA↗
arXiv2004

(Bi-)Cohen-Macaulay simplicial complexes and their associated coherent sheaves

Via the BGG correspondence a simplicial complex Delta on [n] is transformed into a complex of coherent sheaves on P^n-1. We show that this complex reduces to a coherent sheaf F exactly when the Alexander dual Delta^* is Cohen-Macaulay. We then determine when both Delta and Delta^* are Cohen-Macaulay. This corresponds to F being a locally Cohen-Macaulay sheaf. Lastly we conjecture for which range of invariants of such Delta it must be a cone.

math.AG↗
arXiv2002

Monomial Multiple Structures

In this paper we study monomial multiple structures on a linear subspace of codimension two in projective space. We show that these structures determine smooth points in their respective Hilbert schemes, with (smooth) neighbourhoods of two such points intersecting if their Hilbert functions are equal. We generalize a construction for multiple structures on points in the plane to this setting, giving a kind of product of monomial multiple structures.

math.AG↗
arXiv2002

Multiple Structures

We give a systematic approach to constructing non-reduced, locally Cohen-Macaulay schemes with reduced support a smooth projective variety. The hierarchy of such structures includes a lot of information about the underlying variety, its embeddings in projective space and the behaviour of its vector bundles. For instance, Hartshorne's conjecture on complete intersections in codimension two is reformulated in terms of existence of certain schemes of degrees two and three. There are many examples, and classifications of multiple structures with special properties (like low degree).

math.AG↗