Hodge Decomposition Research Articles

Decomposition of $L^{2}$-Vector Fields on Lipschitz Surfaces: Characterization via Null-Spaces of the Scalar Potential

For $\partial\Omega$ the boundary of a bounded and connected strongly Lipschitz domain in $\mathbb{R}^{d}$ with $d\geq3$, we prove that any field $f\in L^{2}(\partial\Omega ; \mathbb{R}^{d})$ decomposes, in a unique way, as the sum of three invisible vector fields---fields whose magnetic potential vanishes in one or both components of $\mathbb{R}^d\setminus\partial\Omega$. Moreover, this decomposition is orthogonal if and only if $\partial\Omega$ is a sphere. We also show that any $f$ in $L^{2}(\partial\Omega ; \mathbb{R}^{d})$ is uniquely the sum of two invisible fields and a Hardy function, in which case the sum is orthogonal regardless of $\partial\Omega$; we express the corresponding orthogonal projections in terms of layer potentials. When $\partial\Omega$ is a sphere, both decompositions coincide and match what has been called the Hardy--Hodge decomposition in the literature.

A Hodge theoretic extension of Shapley axioms

Lloyd S. Shapley \cite{Shapley1953a, Shapley1953} introduced a set of axioms in 1953, now called the {\em Shapley axioms}, and showed that the axioms characterize a natural allocation among the players who are in grand coalition of a {\em cooperative game}. Recently, \citet{StTe2019} showed that a cooperative game can be decomposed into a sum of {\em component games}, one for each player, whose value at the grand coalition coincides with the {\em Shapley value}. The component games are defined by the solutions to the naturally defined system of least squares linear equations via the framework of the {\em Hodge decomposition} on the hypercube graph. In this paper we propose a new set of axioms which characterizes the component games given by Stern and Tettenhorst, thereby suggesting that the component values for every coalition state may also serve for a valid measure of fair allocation among the players in each coalition. Our axioms may be seen as a completion of Shapley's in view of this characterization of the Hodge-theoretic component games. In addition, we provide a path integral representation of the component games which may be seen as an extension of the {\em Shapley formula}.

Cartoon and Texture Image Decomposition Driven by Weighted Curvature

In this paper, we propose a curvature-guided model with divergence-free constraints to facilitate image decomposition. Since basic TV regularization has difficulty in processing edge geometry information of cartoon image, in order to preserve the edge features, we introduce a level set curvature term to smooth the uniform area, and use the edge indicator function as the weighted regularization to preserve the edge. In addition, to control the smoothness, a gradient function is proposed to balance the edge indicator function and the curvature term. On the other hand, we find that the existing partial decomposition models measure the oscillation function with the $H^{-1}$ functional, but this functional ignores the divergence-free vector field during the Hodge decomposition process, then the texture will lose some vector direction information, and then affects the decomposition results. By analyzing the theory of divergence-free vector field, a new decomposition model with the constraint of divergence-free vector field is proposed in this paper. Numerical experiments show that the proposed model can well preserves the edges of cartoon and protects the texture of repetitive patterns.

Trading networks and Hodge theory ∗

The problem of analyzing interconnectedness is one of today’s premier challenges in understanding systemic risk. Connections can both stabilize networks and provide pathways for contagion. The central problem in such networks is establishing global behavior from local interactions. Jiang-Lim-Yao-Ye (Jiang et al 2011 Mathematical Programming 127 1 203–244) recently introduced the use of the Hodge decomposition (see Lim 2020 SIAM Review 62 685–715 for a review), a fundamental tool from algebraic geometry, to construct global rankings from local interactions (see Barbarossa et al 2018 (2018 IEEE Data Science Workshop (DSW), IEEE) pp 51–5; Haruna and Fujiki 2016 Frontiers in Neural Circuits 10 77; Jia et al 2019 (Proc. of the XXV ACM SIGKDD International Conf. on Knowledge Discovery & Data Mining, pp 761–71 for other applications). We apply this to a study of financial networks, starting from the Eisenberg-Noe (Eisenberg and Noe 2001 Management Science 47 236–249) setup of liabilities and endowments, and construct a network of defaults. We then use Jiang-Lim-Yao-Ye to construct a global ranking from the defaults, which yields one way of quantifying ‘systemic importance’.

Hodge decomposition of string topology

AbstractLetXbe a simply connected closed oriented manifold of rationally elliptic homotopy type. We prove that the string topology bracket on the$S^1$-equivariant homology$ {\overline {\text {H}}}_\ast ^{S^1}({\mathcal {L}} X,{\mathbb {Q}}) $of the free loop space ofXpreserves the Hodge decomposition of$ {\overline {\text {H}}}_\ast ^{S^1}({\mathcal {L}} X,{\mathbb {Q}}) $, making it a bigraded Lie algebra. We deduce this result from a general theorem on derived Poisson structures on the universal enveloping algebras of homologically nilpotent finite-dimensional DG Lie algebras. Our theorem settles a conjecture of [7].

Small Gauge Transformations and Universal Geometry in Heterotic Theories

The first part of this paper describes in detail the action of small gauge transformations in heterotic supergravity. We show a convenient gauge fixing is 'holomorphic gauge' together with a condition on the holomorphic top form. This gauge fixing, combined with supersymmetry and the Bianchi identity, allows us to determine a set of non-linear PDEs for the terms in the Hodge decomposition. Although solving these in general is highly non-trivial, we give a prescription for their solution perturbatively in α and apply this to the moduli space metric. The second part of this paper relates small gauge transformations to a choice of connection on the moduli space. We show holomorphic gauge is related to a choice of holomorphic structure and Lee form on a 'universal bundle'. Connections on the moduli space have field strengths that appear in the second order deformation theory and we point out it is generically the case that higher order deformations do not commute.

Geometric decomposition, potential-based representation and integrability of non-linear systems

Abstract This paper considers the problem of representing a sufficiently smooth non-linear dynamical [system] as a structured potential-driven system. The proposed method is based on a decomposition of a differential one-form associated to a given vector field into its exact and anti-exact components, and into its co -exact and anti-coexact components. The decomposition method, based on the Hodge decomposition theorem, is rendered constructive by introducing a dual operator to the standard homotopy operator. The dual operator inverts locally the co-differential operator, and is used in the present paper to identify the symplectic structure of the dynamics. Applications of the proposed approach to gradient systems, Hamiltonian systems and generalized Hamiltonian systems are given to illustrate the proposed approach. Finally, integrability conditions for generalized Hamiltonian systems are established using the proposed decomposition.

Linear stability of Schwarzschild spacetime: Decay of metric coefficients

In this paper, we study the theory of linearized gravity and prove the linear stability of Schwarzschild black holes as solutions of the vacuum Einstein equations. In particular, we prove that solutions to the linearized vacuum Einstein equations centered at a Schwarzschild metric, with suitably regular initial data, remain uniformly bounded and decay to a linearized Kerr metric on the exterior region. We employ Hodge decomposition to split the solution into closed and co-closed portions, respectively identified with even-parity and odd-parity solutions in the physics literature. For the co-closed portion, we extend previous results by the first two authors, deriving Regge–Wheeler type equations for two gauge-invariant master quantities without the earlier paper’s need of axisymmetry. For the closed portion, we build upon earlier work of Zerilli and Moncrief, wherein the authors derive an equation for a gauge-invariant master quantity in a spherical harmonic decomposition. We work with gauge-invariant quantities at the level of perturbed connection coefficients, with the initial value problem formulated on Cauchy data sets. With the choice of an appropriate gauge in each of the two portions, decay estimates on these decoupled quantities are used to establish decay of the metric coefficients of the solution, completing the proof of linear stability. Our result differs from that of Dafermos–Holzegel–Rodnianski, both in our choice of gauge and in our identification and utilization of lower-level gauge-invariant master quantities.

Ψec: A local spectral exterior calculus

We introduce $\Psi \mathrm{ec}$, a discretization of Cartan's exterior calculus of differential forms using wavelets. Our construction consists of differential $r$-form wavelets with flexible directional localization that provide tight frames for the spaces $\Omega^r(\mathbb{R}^n)$ of forms in $\mathbb{R}^2$ and $\mathbb{R}^3$. By construction, the wavelets satisfy the de Rahm co-chain complex, the Hodge decomposition, and that the $k$-dimensional integral of an $r$-form is an $(r-k)$-form. They also verify Stokes' theorem for differential forms, with the most efficient finite dimensional approximation attained using directionally localized, curvelet- or ridgelet-like forms. The construction of $\Psi \mathrm{ec}$ builds on the geometric simplicity of the exterior calculus in the Fourier domain. We establish this structure by extending existing results on the Fourier transform of differential forms to a frequency description of the exterior calculus, including, for example, a Plancherel theorem for forms and a description of the symbols of all important operators.

On a curious variant of the $S_n$-module Lie$_n$

We introduce a variant of the much-studied $Lie$ representation of the symmetric group $S_n$, which we denote by $Lie_n^{(2)}.$ Our variant gives rise to a decomposition of the regular representation as a sum of {exterior} powers of modules $Lie_n^{(2)}.$ This is in contrast to the theorems of Poincar\'e-Birkhoff-Witt and Thrall which decompose the regular representation into a sum of symmetrised $Lie$ modules. We show that nearly every known property of $Lie_n$ has a counterpart for the module $Lie_n^{(2)},$ suggesting connections to the cohomology of configuration spaces via the character formulas of Sundaram and Welker, to the Eulerian idempotents of Gerstenhaber and Schack, and to the Hodge decomposition of the complex of injective words arising from Hochschild homology, due to Hanlon and Hersh.

Application of Helmholtz–Hodge decomposition to the study of certain vector fields

Smooth vector fields on can be decomposed into the sum of a gradient vector field and divergence-free (solenoidal) vector field under suitable hypotheses. This is called the Helmholtz–Hodge decomposition (HHD), which has been applied to analyze the topological features of vector fields. In this study, we apply the HHD to study certain types of vector fields. In particular, we investigate the existence of strictly orthogonal HHDs, which assure an effective analysis. The first object of the study is linear vector fields. We demonstrate that a strictly orthogonal HHD for a vector field of the form F(x) = Ax can be obtained by solving an algebraic Riccati equation. Subsequently, a method to explicitly construct a Lyapunov function is established. In particular, if A is normal, there exists an easy solution to this equation. Next, we study planar vector fields. In this case, the HHD yields a complex potential, which is a generalization of the notion in hydrodynamics with the same name. We demonstrate the convenience of the complex potential formalism by analyzing vector fields given by homogeneous quadratic polynomials.

Money Flow Network Among Firms’ Accounts in a Regional Bank of Japan

In this study, we investigate the flow of money among bank accounts possessed by firms in a region by employing an exhaustive list of all the bank transfers in a regional bank in Japan, to clarify how the network of money flow is related to the economic activities of the firms. The network statistics and structures are examined and shown to be similar to those of a nationwide production network. Specifically, the bowtie analysis indicates what we refer to as a "walnut" structure with core and upstream/downstream components. To quantify the location of an individual account in the network, we used the Hodge decomposition method and found that the Hodge potential of the account has a significant correlation to its position in the bowtie structure as well as to its net flow of incoming and outgoing money and links, namely the net demand/supply of individual accounts. In addition, we used non-negative matrix factorization to identify important factors underlying the entire flow of money; it can be interpreted that these factors are associated with regional economic activities.One factor has a feature whereby the remittance source is localized to the largest city in the region, while the destination is scattered. The other factors correspond to the economic activities specific to different local places.This study serves as a basis for further investigation on the relationship between money flow and economic activities of firms.

Linear Stability of Higher Dimensional Schwarzschild Spacetimes: Decay of Master Quantities

In this paper, we study solutions to the linearized vacuum Einstein equations centered at higher-dimensional Schwarzschild metrics. We employ Hodge decomposition to split solutions into scalar, co-vector, and two-tensor pieces; the first two portions respectively correspond to the closed and co-closed, or polar and axial, solutions in the case of four spacetime dimensions, while the two-tensor portion is a new feature in the higher-dimensional setting. Rephrasing earlier work of Kodama-Ishibashi-Seto in the language of our Hodge decomposition, we produce decoupled gauge-invariant master quantities satisfying Regge-Wheeler type wave equations in each of the three portions. The scalar and co-vector quantities respectively generalize the Moncrief-Zerilli and Regge-Wheeler quantities found in the setting of four spacetime dimensions; beyond these quantities, we discover a higher-dimensional analog of the Cunningham-Moncrief-Price quantity in the co-vector portion. In addition, our work provides the first verification that the scalar master quantity satisfies its putative Regge-Wheeler equation. In the analysis of the master quantities, we strengthen the mode stability result of Kodama-Ishibashi to a uniform boundedness estimate in all dimensions; further, we prove decay estimates in the case of six or fewer spacetime dimensions. In the case of more than six spacetime dimensions, we discover an obstruction to Morawetz type estimates arising from negative potential terms growing quadratically in spacetime dimension. Finally, we provide a rigorous argument that linearized solutions of low angular frequency are decomposable as a sum of pure gauge solution and linearized Myers-Perry solution, the latter solutions generalizing the linearized Kerr solutions in four spacetime dimensions.

Illuminance Compensation and Texture Enhancement via the Hodge Decomposition

Image brightness in color representation has caught active research interests, while its influence to texture features is relatively rarely studied. In this paper, we address the issue of illuminance, or brightness, interference to texture descriptors, especially to the Gabor filter based approaches. Firstly, we reveal the fact that the Gabor filter response linearly drifts with the illuminance. Secondly, the Hodge decomposition is introduced, and a linear regression model is presented to interpret the linear drift phenomenon. An interesting connection with the intrinsic image decomposition is established. Finally, a partial differential equation based method is proposed to compensate the drift issue. Consequently, the texture of the compensated image is enhanced. Extensive experiments are conducted to verify the proposed statements. The proposed linear regression model is verified empirically, and the compensation performances are demonstrated. Classification results on various databases indicate that the compensation can effectively improve the capacity of many existing texture description methods. Especially, the results show that the boosting effect on some deep learning based methods is also valid.

Relationship between Macroeconomic Indicators and Economic Cycles in U.S.

We analyze monthly time series of 57 US macroeconomic indicators (18 leading, 30 coincident, and 9 lagging) and 5 other trade/money indexes. Using novel methods, we confirm statistically significant co-movements among these time series and identify noteworthy economic events. The methods we use are Complex Hilbert Principal Component Analysis (CHPCA) and Rotational Random Shuffling (RRS). We obtain significant complex correlations among the US economic indicators with leads/lags. We then use the Hodge decomposition to obtain the hierarchical order of each time series. The Hodge potential allows us to better understand the lead/lag relationships. Using both CHPCA and Hodge decomposition approaches, we obtain a new lead/lag order of the macroeconomic indicators and perform clustering analysis for positively serially correlated positive and negative changes of the analyzed indicators. We identify collective negative co-movements around the Dot.com bubble in 2001 as well as the Global Financial Crisis (GFC) in October 2008. We also identify important events such as the Hurricane Katrina in August 2005 and the Oil Price Crisis in July 2008. Additionally, we demonstrate that some coincident and lagging indicators actually show leading indicator characteristics. This suggests that there is a room for existing indicators to be improved.

Influence of Thermal Expansion on Potential and Rotational Components of Turbulent Velocity Field Within and Upstream of Premixed Flame Brush

Direct Numerical Simulation (DNS) data obtained earlier from two statistically stationary, 1D, planar, weakly turbulent premixed flames are analyzed in order to examine the influence of combustion-induced thermal expansion on the flow structure within the mean flame brushes and upstream of them. The two flames are associated with the flamelet combustion regime and are characterized by significantly different density ratios, i.e. $$\sigma = 7.53$$ and 2.5. The Helmholtz–Hodge decomposition is applied to the DNS data in order to extract rotational and potential velocity fields. Comparison of the two fields shows that combustion-induced thermal expansion can significantly change the local structure of the incoming constant-density turbulent flow of unburned reactants by significantly increasing the relative magnitude of the potential velocity fluctuations when compared to the rotational velocity fluctuations in the flow. Such effects are documented not only within the mean flame brush, but also well upstream of it. The effect magnitude is increased by the density ratio $$\sigma$$ , with the effects being well (weakly) pronounced at $$\sigma = 7.53$$ (2.5, respectively). Moreover, the potential and rotational velocity fields can cause opposite variations of the local area of an iso-scalar surface $$c\left( {{\mathbf{x}},t} \right) = {\text{const}}$$ within flamelets by generating the local strain rates of opposite signs.

Exploring the worldline formulation of the Potts model

We revisit the issue of worldline formulations for the q-state Potts model and discuss a worldline representation in arbitrary dimensions which also allows for magnetic terms. For vanishing magnetic field we implement a Hodge decomposition for resolving the constraints with dual variables, which in two dimensions implies self-duality as a simple corollary. We present exploratory 2-d Monte Carlo simulations in terms of the worldlines, based on worm algorithms. We study both, vanishing and non-zero magnetic field, and explore q between q=2 and q=30, i.e., Potts models with continuous, as well as strong first order transitions.

Meshfree methods on manifolds for hydrodynamic flows on curved surfaces: A Generalized Moving Least-Squares (GMLS) approach

We utilize generalized moving least squares (GMLS) to develop meshfree techniques for discretizing hydrodynamic flow problems on manifolds. We use exterior calculus to formulate incompressible hydrodynamic equations in the Stokesian regime and handle the divergence-free constraints via a generalized vector potential. This provides less coordinate-centric descriptions and enables the development of efficient numerical methods and splitting schemes for the fourth-order governing equations in terms of a system of second-order elliptic operators. Using a Hodge decomposition, we develop methods for manifolds having spherical topology. We show the methods exhibit high-order convergence rates for solving hydrodynamic flows on curved surfaces. The methods also provide general high-order approximations for the metric, curvature, and other geometric quantities of the manifold and associated exterior calculus operators. The approaches also can be utilized to develop high-order solvers for other scalar-valued and vector-valued problems on manifolds.

Discrete Vector Calculus and Helmholtz Hodge Decomposition for Classical Finite Difference Summation by Parts Operators

In this article, discrete variants of several results from vector calculus are studied for classical finite difference summation by parts operators in two and three space dimensions. It is shown that existence theorems for scalar/vector potentials of irrotational/solenoidal vector fields cannot hold discretely because of grid oscillations, which are characterised explicitly. This results in a non-vanishing remainder associated to grid oscillations in the discrete Helmholtz Hodge decomposition. Nevertheless, iterative numerical methods based on an interpretation of the Helmholtz Hodge decomposition via orthogonal projections are proposed and applied successfully. In numerical experiments, the discrete remainder vanishes and the potentials converge with the same order of accuracy as usual in other first order partial differential equations. Motivated by the successful application of the Helmholtz Hodge decomposition in theoretical plasma physics, applications to the discrete analysis of magnetohydrodynamic (MHD) wave modes are presented and discussed.

Untangling the complexity of market competition in consumer goods - A complex Hilbert PCA analysis

Today's consumer goods markets are rapidly evolving with significant growth in the number of information media as well as the number of competitive products. In this environment, obtaining a quantitative grasp of heterogeneous interactions of firms and customers, which have attracted interest of management scientists and economists, requires the analysis of extremely high-dimensional data. Existing approaches in quantitative research could not handle such data without any reliable prior knowledge nor strong assumptions. Alternatively, we propose a novel method called complex Hilbert principal component analysis (CHPCA) and construct a synchronization network using Hodge decomposition. CHPCA enables us to extract significant comovements with a time lead/delay in the data, and Hodge decomposition is useful for identifying the time-structure of correlations. We apply this method to the Japanese beer market data and reveal comovement of variables related to the consumer choice process across multiple products. Furthermore, we find remarkable customer heterogeneity by calculating the coordinates of each customer in the space derived from the results of CHPCA. Lastly, we discuss the policy and managerial implications, limitations, and further development of the proposed method.