Related papers: Simplification of complexes for persistent homolog…

An abstract construction of coarse spaces for non-Hermitian problems and non-Hermitian domain decomposition preconditioners based on extended generalized eigenproblems was proposed in [Nataf and Parolin, arXiv:2404.02758] and analyzed on…

Patch-based relaxation refers to a family of methods for solving linear systems which partitions the matrix into smaller pieces often corresponding to groups of adjacent degrees of freedom residing within patches of the computational…

0-dimensional persistent homology is known, from a computational point of view, as the easy case. Indeed, given a list of $n$ edges in non-decreasing order of filtration value, one only needs a union-find data structure to keep track of the…

The biggest cost of computing with large matrices in any modern computer is related to memory latency and bandwidth. The average latency of modern RAM reads is 150 times greater than a clock step of the processor. Throughput is a little…

We introduce a set of combinatorial techniques for studying the simplicial bounded cohomology of semi-simplicial sets, simplicial complexes and posets. We apply these methods to prove several new bounded acyclicity results for…

In this paper, we review a method for computing and parameterizing the set of homotopy classes of chain maps between two chain complexes. This is then applied to finding topologically meaningful maps between simplicial complexes, which in…

In this paper, we show that contraction operations preserve the homology of $n$D generalized maps, under some conditions. Removal and contraction operations are used to propose an efficient algorithm that compute homology generators of $n$D…

In this article, we design fast algorithms for the computation of approximant bases in shifted Popov normal form. We first recall the algorithm known as PM-Basis, which will be our second fundamental engine after polynomial matrix…

We discuss and review recent developments in the area of applied algebraic topology, such as persistent homology and barcodes. In particular, we discuss how these are related to understanding more about manifold learning from random point…

Simplicial complexes are higher-order combinatorial structures which have been used to represent real-world complex systems. In this paper, we concentrate on the local patterns in simplicial complexes called simplets, a generalization of…

Complex valued systems with an indefinite matrix term arise in important applications such as for certain time-harmonic partial differential equations such as the Maxwell's equation and for the Helmholtz equation. Complex systems with…

Protein structure prediction remains a challenge in the field of computational biology. Traditional protein structure prediction approaches include template-based modelling (say, homology modelling, and threading), and ab initio. A…

Higher-order tensor methods were recently proposed for minimizing smooth convex and nonconvex functions. Higher-order algorithms accelerate the convergence of the classical first-order methods thanks to the higher-order derivatives used in…

Large language models produce powerful text embeddings, but their causal attention mechanism restricts the flow of information from later to earlier tokens, degrading representation quality. While recent methods attempt to solve this by…

We examine interpolatory model reduction methods that are well-suited for treating large scale port-Hamiltonian differential-algebraic systems in a way that is able to preserve and indeed, take advantage of the underlying structural…

This thesis addresses the theory of topological spaces and the foundations of persistence theory. We will discuss chain complexes and the associated simplicial homology groups, as well as their relationship with singular homology theory.…

The aim of this short note is to develop a (co)homology theory for topological spaces together with the specialisation preorder. A known way to construct such a (co)homology is to define a partial order on the topological space starting…

Humans often think of complex tasks as combinations of simpler subtasks in order to learn those complex tasks more efficiently. For example, a backflip could be considered a combination of four subskills: jumping, tucking knees, rolling…

In this paper we utilize the Proper Orthogonal Decomposition (POD) method for model order reduction in application to Smoluchowski aggregation equations with source and sink terms. In particular, we show in practice that there exists a…

Optimizing deep learning models is generally performed in two steps: (i) high-level graph optimizations such as kernel fusion and (ii) low level kernel optimizations such as those found in vendor libraries. This approach often leaves…