arXiv subjects

Anil N. Hirani

Publications and source records attributed to Anil N. Hirani.

At least 19 recordsLinked to original sources

arXiv2026

Discrete Vector Bundles with Connection

We develop a combinatorial theory of vector bundles with connection on locally ordered simplicial complexes. This is a first step towards a discrete exterior calculus for bundle-valued forms. The basic building block is the discrete exterior covariant derivative, a forward-difference operator defined on bundle-valued cochains. Many standard objects in differential geometry (e.g., curvature, connection 1-forms, gauge transformations) can be understood via the discrete covariant derivative operator, with their defining formulas identical to the smooth setting. These discrete objects satisfy all of the expected algebraic identities, such as naturality with respect to simplicial maps, and a Bianchi identity for discrete curvature. We also show that flat discrete connections determine a cochain complex that computes twisted de Rham cohomology in a local coefficient system determined by the discrete vector bundle, with twisted Poincare duality (of densities) being one application. Finally, a coarsening operation applied to bundle-valued cochains provides a direct and concrete comparison with the recent framework for discrete bundles of Christiansen and Hu.

math.DG↗

arXiv2026

Discrete Poincare and Bogovskii operators on cochains and Whitney forms

Smooth Poincare operators are a tool used to show the vanishing of smooth de Rham cohomology on contractible manifolds and have found use in the analysis of finite element methods based on the Finite Element Exterior Calculus (FEEC). We construct analogous discrete Poincare operators acting on cochains and Whitney forms. We provide explicit, constructive realizations of these operators under various assumptions on the underlying domain or simplicial complex. In particular, we provide simple constructions for the discrete Poincare operators on simplicial complexes which are collapsible and those with underlying domain being star-shaped with respect to a point. We then provide more abstract constructions on simplicial complexes which are discrete contractible and domains which are Lipschitz contractible. We also modify the discrete Poincare operator on star-shaped domains to construct a discrete Bogovskii operator which satisfies the requisite homotopy identity while preserving homogeneous boundary conditions. Applications arise in the construction of discrete scalar and vector potentials and in the discrete wedge product of Discrete Exterior Calculus (DEC).

math.NA↗

arXiv2026

Convergence and Stability of Discrete Exterior Calculus for the Hodge Laplace Problem in Two Dimensions

We prove convergence and stability of the discrete exterior calculus (DEC) solutions for the Hodge-Laplace problems in two dimensions for families of meshes that are non-degenerate Delaunay and shape regular. We do this by relating the DEC solutions to the lowest order finite element exterior calculus (FEEC) solutions. A Poincaré inequality and a discrete inf-sup condition for DEC are part of this proof. We also prove that under appropriate geometric conditions on the mesh the DEC and FEEC norms are equivalent. Only one side of the norm equivalence is needed for proving stability and convergence and this allows us to relax the conditions on the meshes.

math.NA↗

arXiv2023

Averaging Property of Wedge Product and Naturality in Discrete Exterior Calculus

In exterior calculus on smooth manifolds, the exterior derivative and wedge product are natural with respect to smooth maps between manifolds, that is, these operations commute with pullback. In discrete exterior calculus (DEC), simplicial cochains play the role of discrete forms, the coboundary operator serves as the discrete exterior derivative, and the antisymmetrized cup product provides a discrete wedge product. We show that these discrete operations in DEC are natural with respect to abstract simplicial maps. A second contribution is a new averaging interpretation of the discrete wedge product in DEC. We also show that this wedge product is the same as Wilson's cochain product defined using Whitney and de Rham maps.

math.NA↗

arXiv2022

Discrete exterior calculus discretization of two-phase incompressible Navier-Stokes equations with a conservative phase field method

We present a discrete exterior calculus (DEC) based discretization scheme for incompressible two-phase flows. Our physically-compatible exterior calculus discretization of single phase flow is extended to simulate immiscible two-phase flows with discontinuous changes in fluid properties such as density and viscosity across the interface. The two-phase incompressible Navier-Stokes equations and conservative phase field equation for interface capturing are first transformed into the exterior calculus framework. The discrete counter part of these smooth equations is obtained by substituting with discrete differential forms and discrete exterior operators. We prove the boundedness of the method for the first order Euler forward and predictor-corrector time integration schemes in the DEC framework. With a proper choice of two free parameters, the scheme remains phase field bounded without the requirement of any ad hoc mass redistribution. We verify the scheme against several standard test cases (for interface capturing) comprising not only the flat domains but also the curved domains, leveraging the advantage that DEC operators are independent of the coordinate system. The results show excellent properties of boundedness, mass conservation and convergence. Moreover, we demonstrate the ability of the scheme towards handling large density and viscosity ratios as well as surface tension in the simulation of various two phase flow physical phenomena on flat or curved surfaces.

physics.flu-dyn↗

arXiv2022

Conservative Integrators for Piecewise Smooth Systems with Transversal Dynamics

We introduce conservative integrators for long term integration of piecewise smooth systems with transversal dynamics and piecewise smooth conserved quantities. In essence, for a piecewise dynamical system with piecewise defined conserved quantities such that its trajectories cross transversally to its interface, we combine Mannshardt's transition scheme and the Discrete Multiplier Method to obtain conservative integrators capable of preserving conserved quantities up to machine precision and accuracy order. We prove that the order of accuracy of the conservative integrators is preserved after crossing the interface in the case of codimension one number of conserved quantities. Numerical examples illustrate the preservation of accuracy order and conserved quantities across the interface.

math.NA↗

arXiv2021

GraphSAT -- a decision problem connecting satisfiability and graph theory

Satisfiability of boolean formulae (SAT) has been a topic of research in logic and computer science for a long time. In this paper we are interested in understanding the structure of satisfiable and unsatisfiable sentences. In previous work we initiated a new approach to SAT by formulating a mapping from propositional logic sentences to graphs, allowing us to find structural obstructions to 2SAT (clauses with exactly 2 literals) in terms of graphs. Here we generalize these ideas to multi-hypergraphs in which the edges can have more than 2 vertices and can have multiplicity. This is needed for understanding the structure of SAT for sentences made of clauses with 3 or more literals (3SAT), which is a building block of NP-completeness theory. We introduce a decision problem that we call GraphSAT, as a first step towards a structural view of SAT. Each propositional logic sentence can be mapped to a multi-hypergraph by associating each variable with a vertex (ignoring the negations) and each clause with a hyperedge. Such a graph then becomes a representative of a collection of possible sentences and we can then formulate the notion of satisfiability of such a graph. With this coarse representation of classes of sentences one can then investigate structural obstructions to SAT. To make the problem tractable, we prove a local graph rewriting theorem which allows us to simplify the neighborhood of a vertex without knowing the rest of the graph. We use this to deduce several reduction rules, allowing us to modify a graph without changing its satisfiability status which can then be used in a program to simplify graphs. We study a subclass of 3SAT by examining sentences living on triangulations of surfaces and show that for any compact surface there exists a triangulation that can support unsatisfiable sentences, giving specific examples of such triangulations for various surfaces.

math.CO↗

arXiv2019

The complete set of minimal simple graphs that support unsatisfiable 2-CNFs

A propositional logic sentence in conjunctive normal form that has clauses of length two (a 2-CNF) can be associated with a multigraph in which the vertices correspond to the variables and edges to clauses. We first show that every such sentence that has been reduced, that is, which is unchanged under application of certain tautologies, is equisatisfiable to a 2-CNF whose associated multigraph is, in fact, a simple graph. Our main result is a complete characterization of graphs that can support unsatisfiable 2-CNF sentences. We show that a simple graph can support an unsatisfiable reduced 2-CNF sentence if and only if it contains any one of four specific small graphs as a topological minor. Equivalently, all reduced 2-CNF sentences supported on a given simple graph are satisfiable if and only if all subdivisions of those four graphs are forbidden as subgraphs of of the original graph. We conclude with a discussion of why the Robertson-Seymour graph minor theorem does not apply in our approach.

math.CO↗

arXiv2018

Numerical Convergence of Discrete Exterior Calculus on Arbitrary Surface Meshes

Discrete exterior calculus (DEC) is a structure-preserving numerical framework for partial differential equations solution, particularly suitable for simplicial meshes. A longstanding and widespread assumption has been that DEC requires special (Delaunay) triangulations, which complicated the mesh generation process especially on curved surfaces. This paper presents numerical evidences demonstrating that this restriction is unnecessary. Convergence experiments are carried out for various physical problems using both Delaunay and non-Delaunay triangulations. Signed diagonal definition for the key DEC operator (Hodge star) is adopted. The errors converge as expected for all considered meshes and experiments. This relieves the DEC paradigm from unnecessary triangulation limitation.

math.NA↗

arXiv2017

Delaunay Hodge Star

We define signed dual volumes at all dimensions for circumcentric dual meshes. We show that for pairwise Delaunay triangulations with mild boundary assumptions these signed dual volumes are positive. This allows the use of such Delaunay meshes for Discrete Exterior Calculus (DEC) because the discrete Hodge star operator can now be correctly defined for such meshes. This operator is crucial for DEC and is a diagonal matrix with the ratio of primal and dual volumes along the diagonal. A correct definition requires that all entries be positive. DEC is a framework for numerically solving differential equations on meshes and for geometry processing tasks and has had considerable impact in computer graphics and scientific computing. Our result allows the use of DEC with a much larger class of meshes than was previously considered possible.

cs.CG↗

arXiv2016

Discrete exterior calculus discretization of incompressible Navier-Stokes equations over surface simplicial meshes

A conservative discretization of incompressible Navier-Stokes equations is developed based on discrete exterior calculus (DEC). A distinguishing feature of our method is the use of an algebraic discretization of the interior product operator and a combinatorial discretization of the wedge product. The governing equations are first rewritten using the exterior calculus notation, replacing vector calculus differential operators by the exterior derivative, Hodge star and wedge product operators. The discretization is then carried out by substituting with the corresponding discrete operators based on the DEC framework. Numerical experiments for flows over surfaces reveal a second order accuracy for the developed scheme when using structured-triangular meshes, and first order accuracy for otherwise unstructured meshes. By construction, the method is conservative in that both mass and vorticity are conserved up to machine precision. The relative error in kinetic energy for inviscid flow test cases converges in a second order fashion with both the mesh size and the time step.

physics.flu-dyn↗

arXiv2013

Edge Contractions and Simplicial Homology

We study the effect of edge contractions on simplicial homology because these contractions have turned to be useful in various applications involving topology. It was observed previously that contracting edges that satisfy the so called link condition preserves homeomorphism in low dimensional complexes, and homotopy in general. But, checking the link condition involves computation in all dimensions, and hence can be costly, especially in high dimensional complexes. We define a weaker and more local condition called the p-link condition for each dimension p, and study its effect on edge contractions. We prove the following: (i) For homology groups, edges satisfying the p- and (p-1)-link conditions can be contracted without disturbing the p-dimensional homology group. (ii) For relative homology groups, the (p-1)-, and the (p-2)-link conditions suffice to guarantee that the contraction does not introduce any new class in any of the resulting relative homology groups, though some of the existing classes can be destroyed. Unfortunately, the surjection in relative homolgy groups does not guarantee that no new relative torsion is created. (iii) For torsions, edges satisfying the p-link condition alone can be contracted without creating any new relative torsion and the p-link condition cannot be avoided. The results on relative homology and relative torsion are motivated by recent results on computing optimal homologous chains, which state that such problems can be solved by linear programming if the complex has no relative torsion. Edge contractions that do not introduce new relative torsions, can safely be availed in these contexts.

cs.CG↗

arXiv2012

A posteriori error estimates for finite element exterior calculus: The de Rham complex

Finite element exterior calculus (FEEC) has been developed over the past decade as a framework for constructing and analyzing stable and accurate numerical methods for partial differential equations by employing differential complexes. The recent work of Arnold, Falk and Winther \cite{ArFaWi2010} includes a well-developed theory of finite element methods for Hodge Laplace problems, including a priori error estimates. In this work we focus on developing a posteriori error estimates in which the computational error is bounded by some computable functional of the discrete solution and problem data. More precisely, we prove a posteriori error estimates of residual type for Arnold-Falk-Winther mixed finite element methods for Hodge-de Rham Laplace problems. While a number of previous works consider a posteriori error estimation for Maxwell's equations and mixed formulations of the scalar Laplacian, the approach we take is distinguished by unified treatment of the various Hodge Laplace problems arising in the de Rham complex, consistent use of the language and analytical framework of differential forms, and the development of a posteriori error estimates for harmonic forms and the effects of their approximation on the resulting numerical method for the Hodge Laplacian.

math.NA↗

arXiv2012

PyDEC: Software and Algorithms for Discretization of Exterior Calculus

This paper describes the algorithms, features and implementation of PyDEC, a Python library for computations related to the discretization of exterior calculus. PyDEC facilitates inquiry into both physical problems on manifolds as well as purely topological problems on abstract complexes. We describe efficient algorithms for constructing the operators and objects that arise in discrete exterior calculus, lowest order finite element exterior calculus and in related topological problems. Our algorithms are formulated in terms of high-level matrix operations which extend to arbitrary dimension. As a result, our implementations map well to the facilities of numerical libraries such as NumPy and SciPy. The availability of such libraries makes Python suitable for prototyping numerical methods. We demonstrate how PyDEC is used to solve physical and topological problems through several concise examples.

math.NA↗

arXiv2011

Cohomologous Harmonic Cochains

We describe algorithms for finding harmonic cochains, an essential ingredient for solving elliptic partial differential equations in exterior calculus. Harmonic cochains are also useful in computational topology and computer graphics. We focus on finding harmonic cochains cohomologous to a given cocycle. Amongst other things, this allows localization near topological features of interest. We derive a weighted least squares method by proving a discrete Hodge-deRham theorem on the isomorphism between the space of harmonic cochains and cohomology. The solution obtained either satisfies the Whitney form finite element exterior calculus equations or the discrete exterior calculus equations for harmonic cochains, depending on the discrete Hodge star used.

cs.CG↗

arXiv2011

Least Squares Ranking on Graphs

Given a set of alternatives to be ranked, and some pairwise comparison data, ranking is a least squares computation on a graph. The vertices are the alternatives, and the edge values comprise the comparison data. The basic idea is very simple and old: come up with values on vertices such that their differences match the given edge data. Since an exact match will usually be impossible, one settles for matching in a least squares sense. This formulation was first described by Leake in 1976 for rankingfootball teams and appears as an example in Professor Gilbert Strang's classic linear algebra textbook. If one is willing to look into the residual a little further, then the problem really comes alive, as shown effectively by the remarkable recent paper of Jiang et al. With or without this twist, the humble least squares problem on graphs has far-reaching connections with many current areas ofresearch. These connections are to theoretical computer science (spectral graph theory, and multilevel methods for graph Laplacian systems); numerical analysis (algebraic multigrid, and finite element exterior calculus); other mathematics (Hodge decomposition, and random clique complexes); and applications (arbitrage, and ranking of sports teams). Not all of these connections are explored in this paper, but many are. The underlying ideas are easy to explain, requiring only the four fundamental subspaces from elementary linear algebra. One of our aims is to explain these basic ideas and connections, to get researchers in many fields interested in this topic. Another aim is to use our numerical experiments for guidance on selecting methods and exposing the need for further development.

math.NA↗

arXiv2011

Numerical method for Darcy flow derived using Discrete Exterior Calculus

We derive a numerical method for Darcy flow, hence also for Poisson's equation in mixed (first order) form, based on discrete exterior calculus (DEC). Exterior calculus is a generalization of vector calculus to smooth manifolds and DEC is one of its discretizations on simplicial complexes such as triangle and tetrahedral meshes. DEC is a coordinate invariant discretization, in that it does not depend on the embedding of the simplices or the whole mesh. We start by rewriting the governing equations of Darcy flow using the language of exterior calculus. This yields a formulation in terms of flux differential form and pressure. The numerical method is then derived by using the framework provided by DEC for discretizing differential forms and operators that act on forms. We also develop a discretization for spatially dependent Hodge star that varies with the permeability of the medium. This also allows us to address discontinuous permeability. The matrix representation for our discrete non-homogeneous Hodge star is diagonal, with positive diagonal entries. The resulting linear system of equations for flux and pressure are saddle type, with a diagonal matrix as the top left block. The performance of the proposed numerical method is illustrated on many standard test problems. These include patch tests in two and three dimensions, comparison with analytically known solution in two dimensions, layered medium with alternating permeability values, and a test with a change in permeability along the flow direction. We also show numerical evidence of convergence of the flux and the pressure. A convergence experiment is also included for Darcy flow on a surface. A short introduction to the relevant parts of smooth and discrete exterior calculus is included in this paper. We also include a discussion of the boundary condition in terms of exterior calculus.

math.NA↗

arXiv2011

Numerical Experiments for Darcy Flow on a Surface Using Mixed Exterior Calculus Methods

There are very few results on mixed finite element methods on surfaces. A theory for the study of such methods was given recently by Holst and Stern, using a variational crimes framework in the context of finite element exterior calculus. However, we are not aware of any numerical experiments where mixed finite elements derived from discretizations of exterior calculus are used for a surface domain. This short note shows results of our preliminary experiments using mixed methods for Darcy flow (hence scalar Poisson's equation in mixed form) on surfaces. We demonstrate two numerical methods. One is derived from the primal-dual Discrete Exterior Calculus and the other from lowest order finite element exterior calculus. The programming was done in the language Python, using the PyDEC package which makes the code very short and easy to read. The qualitative convergence studies seem to be promising.

math.NA↗