Related papers: Interval Decomposition of Persistence Modules over…
We present a domain decomposition formulation based on hybridization which is inspired by hybridized discontinuous Galerkin (HDG) methods, that enhance mixed domain decomposition methods by incorporating stabilization terms. Unlike…
We focus on two central themes in this dissertation. The first one is on decomposing polytopes and polynomials in ways that allow us to perform nonlinear optimization. We start off by explaining important results on decomposing a polytope…
The Koszul homology of modules of the polynomial ring $R$ is a central object in commutative algebra.It is strongly related with the minimal free resolution of these modules, and thus with regularity, Hilbert functions, etc. Here we…
Given a polynomial ring $C$ over a field and proper ideals $I$ and $J$ whose generating sets involve disjoint variables, we determine how to embed the associated primes of each power of $I+J$ into a collection of primes described in terms…
We establish the essential normality of a large new class of homogeneous submodules of the finite rank d-shift Hilbert module. The main idea is a notion of essential decomposability that determines when an arbitrary submodule can be…
We characterize structures such as monotonicity, convexity, and modality in smooth regression curves using persistent homology. Persistent homology is a key tool in topological data analysis that detects higher-dimensional topological…
Local conditions for the direct summands of a persistence module to belong to a certain class of indecomposables have been proposed in the 2-parameter setting, notably for the class of indecomposables called block modules, which plays a…
Motivated by the need to relate the biparameter persistence module induced by a pair of scalar functions with the monoparameter persistence modules induced by each function separately, we introduce a construction that defines a kind of…
We present a parallelizable algorithm for computing the persistent homology of a filtered chain complex. Our approach differs from the commonly used reduction algorithm by first computing persistence pairs within local chunks, then…
In standard persistent homology, a persistent cycle born and dying with a persistence interval (bar) associates the bar with a concrete topological representative, which provides means to effectively navigate back from the barcode to the…
Persistent homology is a method for computing the topological features present in a given data. Recently, there has been much interest in the integration of persistent homology as a computational step in neural networks or deep learning. In…
Segmenting curvilinear structures in medical images is essential for analyzing morphological patterns in clinical applications. Integrating topological properties, such as connectivity, improves segmentation accuracy and consistency.…
Barcodes form a complete set of invariants for interval decomposable persistence modules and are an important summary in topological data analysis. The set of barcodes is equipped with a canonical one-parameter family of metrics, the…
The theory of persistence modules on the commutative ladders $CL_n(\tau)$ provides an extension of persistent homology. However, an efficient algorithm to compute the generalized persistence diagrams is still lacking. In this work, we view…
Let Q be an affine semigroup generating Z^d, and fix a finitely generated Z^d-graded module M over the semigroup algebra k[Q] for a field k. We provide an algorithm to compute a minimal Z^d-graded injective resolution of M up to any desired…
The implementation of discontinuous functions occurs in many of today's state-of-the-art partial differential equation solvers. However, in finite element methods, this poses an inherent difficulty: efficient quadrature rules available when…
Cohomological ideas have recently been injected into persistent homology and have for example been used for accelerating the calculation of persistence diagrams by the software Ripser. The cup product operation which is available at…
Physics-informed neural networks (PINNs) have shown promise in solving partial differential equations (PDEs) relevant to multiscale modeling, but they often fail when applied to materials with discontinuous coefficients, such as media with…
We propose a notion of minimal free resolutions for differential modules, and we prove existence and uniqueness results for such resolutions. We also take the first steps toward studying the structure of minimal free resolutions of…
Persistent homology is a tool of topological data analysis that has been used in a variety of settings to characterize different dimensional holes in data. However, persistent homology computations can be memory intensive with a…