Finite Element Exterior Calculus with Applications to the Numerical Solution of the Green–Naghdi Equations
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Combinatorial Laplacian and Rank Aggregation
Combinatorial Laplacian and Rank Aggregation Combinatorial Laplacian and Rank Aggregation Yuan Yao Stanford University ICIAM, Z¨urich,July 16–20, 2007 Joint work with Lek-Heng Lim Combinatorial Laplacian and Rank Aggregation Outline 1 Two Motivating Examples 2 Reflections on Ranking Ordinal vs. Cardinal Global, Local, vs. Pairwise 3 Discrete Exterior Calculus and Combinatorial Laplacian Discrete Exterior Calculus Combinatorial Laplacian Operator 4 Hodge Theory Cyclicity of Pairwise Rankings Consistency of Pairwise Rankings 5 Conclusions and Future Work Combinatorial Laplacian and Rank Aggregation Two Motivating Examples Example I: Customer-Product Rating Example (Customer-Product Rating) m×n m-by-n customer-product rating matrix X ∈ R X typically contains lots of missing values (say ≥ 90%). The first-order statistics, mean score for each product, might suffer from most customers just rate a very small portion of the products different products might have different raters, whence mean scores involve noise due to arbitrary individual rating scales Combinatorial Laplacian and Rank Aggregation Two Motivating Examples From 1st Order to 2nd Order: Pairwise Rankings The arithmetic mean of score difference between product i and j over all customers who have rated both of them, P k (Xkj − Xki ) gij = , #{k : Xki , Xkj exist} is translation invariant. If all the scores are positive, the geometric mean of score ratio over all customers who have rated both i and j, !1/#{k:Xki ,Xkj exist} Y Xkj gij = , Xki k is scale invariant. Combinatorial Laplacian and Rank Aggregation Two Motivating Examples More invariant Define the pairwise ranking gij as the probability that product j is preferred to i in excess of a purely random choice, 1 g = Pr{k : X > X } − . -
Laplacians in Geometric Analysis
LAPLACIANS IN GEOMETRIC ANALYSIS Syafiq Johar syafi[email protected] Contents 1 Trace Laplacian 1 1.1 Connections on Vector Bundles . .1 1.2 Local and Explicit Expressions . .2 1.3 Second Covariant Derivative . .3 1.4 Curvatures on Vector Bundles . .4 1.5 Trace Laplacian . .5 2 Harmonic Functions 6 2.1 Gradient and Divergence Operators . .7 2.2 Laplace-Beltrami Operator . .7 2.3 Harmonic Functions . .8 2.4 Harmonic Maps . .8 3 Hodge Laplacian 9 3.1 Exterior Derivatives . .9 3.2 Hodge Duals . 10 3.3 Hodge Laplacian . 12 4 Hodge Decomposition 13 4.1 De Rham Cohomology . 13 4.2 Hodge Decomposition Theorem . 14 5 Weitzenb¨ock and B¨ochner Formulas 15 5.1 Weitzenb¨ock Formula . 15 5.1.1 0-forms . 15 5.1.2 k-forms . 15 5.2 B¨ochner Formula . 17 1 Trace Laplacian In this section, we are going to present a notion of Laplacian that is regularly used in differential geometry, namely the trace Laplacian (also called the rough Laplacian or connection Laplacian). We recall the definition of connection on vector bundles which allows us to take the directional derivative of vector bundles. 1.1 Connections on Vector Bundles Definition 1.1 (Connection). Let M be a differentiable manifold and E a vector bundle over M. A connection or covariant derivative at a point p 2 M is a map D : Γ(E) ! Γ(T ∗M ⊗ E) 1 with the properties for any V; W 2 TpM; σ; τ 2 Γ(E) and f 2 C (M), we have that DV σ 2 Ep with the following properties: 1. -
Physics-Compatible Discretization Techniques on Single and Dual Grids, with Application to the Poisson Equation of Volume Forms
Physics-compatible discretization techniques on single and dual grids, with application to the Poisson equation of volume forms Artur Palhaa, Pedro Pinto Rebelob, Rene´ Hiemstrab, Jasper Kreeftc, Marc Gerritsmab,∗ aDelft University of Technology, Faculty of Aerospace Engineering, Wind Energy Group P.O. Box 5058, 2600 GB Delft, The Netherlands bDelft University of Technology, Faculty of Aerospace Engineering, Aerodynamics Group P.O. Box 5058, 2600 GB Delft, The Netherlands cShell Global Solutions, The Netherlands Abstract This paper introduces the basic concepts for physics-compatible discretization techniques. The paper gives a clear distinction between vectors and forms. Based on the difference between forms and pseudo-forms and the ?-operator which switches between the two, a dual grid description and a single grid description are presented. The dual grid method resembles a staggered finite volume method, whereas the single grid approach shows a strong resemblance with a finite element method. Both approaches are compared for the Poisson equation for volume forms. Keywords: Mimetic discretization, differential forms, single grid, dual grid, geometric flexibility. 1. INTRODUCTION Mimetic methods aim to preserve essential physical/mathematical structures in a discrete setting. Many of such structures are topological, i.e. independent of metric, and involve integral relations. Since integration will play an important role and integration of differential forms is a metric-free operation, we will work with differential forms. Formally, differentials forms are linear functionals on multi-vectors, but Flanders, [17, p.1], refers to them as ‘things which occur under integral signs’. Such would not be the case if we were to use vectors, because integration of vector quantities is a metric operation. -
Statistical Ranking and Combinatorial Hodge Theory
STATISTICAL RANKING AND COMBINATORIAL HODGE THEORY XIAOYE JIANG, LEK-HENG LIM, YUAN YAO, AND YINYU YE Abstract. We propose a number of techniques for obtaining a global ranking from data that may be incomplete and imbalanced | characteristics that are almost universal to modern datasets coming from e-commerce and internet applications. We are primarily interested in cardinal data based on scores or ratings though our methods also give specific insights on ordinal data. From raw ranking data, we construct pairwise rankings, represented as edge flows on an appropriate graph. Our statistical ranking method exploits the graph Helmholtzian, which is the graph theoretic analogue of the Helmholtz operator or vector Laplacian, in much the same way the graph Laplacian is an ana- logue of the Laplace operator or scalar Laplacian. We shall study the graph Helmholtzian using combinatorial Hodge theory, which provides a way to un- ravel ranking information from edge flows. In particular, we show that every edge flow representing pairwise ranking can be resolved into two orthogonal components, a gradient flow that represents the l2-optimal global ranking and a divergence-free flow (cyclic) that measures the validity of the global ranking obtained | if this is large, then it indicates that the data does not have a good global ranking. This divergence-free flow can be further decomposed or- thogonally into a curl flow (locally cyclic) and a harmonic flow (locally acyclic but globally cyclic); these provides information on whether inconsistency in the ranking data arises locally or globally. When applied to statistical ranking problems, Hodge decomposition sheds light on whether a given dataset may be globally ranked in a meaningful way or if the data is inherently inconsistent and thus could not have any reasonable global ranking; in the latter case it provides information on the nature of the inconsistencies. -
Space-Time Finite-Element Exterior Calculus and Variational Discretizations of Gauge Field Theories
21st International Symposium on Mathematical Theory of Networks and Systems July 7-11, 2014. Groningen, The Netherlands Space-Time Finite-Element Exterior Calculus and Variational Discretizations of Gauge Field Theories Joe Salamon1, John Moody2, and Melvin Leok3 Abstract— Many gauge field theories can be described using a the space-time covariant (or multi-Dirac) perspective can be multisymplectic Lagrangian formulation, where the Lagrangian found in [1]. density involves space-time differential forms. While there has An example of a gauge symmetry arises in Maxwell’s been much work on finite-element exterior calculus for spatial and tensor product space-time domains, there has been less equations of electromagnetism, which can be expressed in done from the perspective of space-time simplicial complexes. terms of the scalar potential φ, the vector potential A, the One critical aspect is that the Hodge star is now taken with electric field E, and the magnetic field B. respect to a pseudo-Riemannian metric, and this is most natu- @A @ rally expressed in space-time adapted coordinates, as opposed E = −∇φ − ; r2φ + (r · A) = 0; to the barycentric coordinates that the Whitney forms (and @t @t their higher-degree generalizations) are typically expressed in @φ terms of. B = r × A; A + r r · A + = 0; @t We introduce a novel characterization of Whitney forms and their Hodge dual with respect to a pseudo-Riemannian metric where is the d’Alembert (or wave) operator. The following that is independent of the choice of coordinates, and then apply gauge transformation leaves the equations invariant, it to a variational discretization of the covariant formulation of Maxwell’s equations. -
The Language of Differential Forms
Appendix A The Language of Differential Forms This appendix—with the only exception of Sect.A.4.2—does not contain any new physical notions with respect to the previous chapters, but has the purpose of deriving and rewriting some of the previous results using a different language: the language of the so-called differential (or exterior) forms. Thanks to this language we can rewrite all equations in a more compact form, where all tensor indices referred to the diffeomorphisms of the curved space–time are “hidden” inside the variables, with great formal simplifications and benefits (especially in the context of the variational computations). The matter of this appendix is not intended to provide a complete nor a rigorous introduction to this formalism: it should be regarded only as a first, intuitive and oper- ational approach to the calculus of differential forms (also called exterior calculus, or “Cartan calculus”). The main purpose is to quickly put the reader in the position of understanding, and also independently performing, various computations typical of a geometric model of gravity. The readers interested in a more rigorous discussion of differential forms are referred, for instance, to the book [22] of the bibliography. Let us finally notice that in this appendix we will follow the conventions introduced in Chap. 12, Sect. 12.1: latin letters a, b, c,...will denote Lorentz indices in the flat tangent space, Greek letters μ, ν, α,... tensor indices in the curved manifold. For the matter fields we will always use natural units = c = 1. Also, unless otherwise stated, in the first three Sects. -
Arxiv:1905.00851V1 [Cs.CV] 2 May 2019 Sibly Nonconvex) Continuous Function on X × Y × R That Is Polyconvex (See Def.2) in the Jacobian Matrix Ξ
Lifting Vectorial Variational Problems: A Natural Formulation based on Geometric Measure Theory and Discrete Exterior Calculus Thomas Möllenhoff and Daniel Cremers Technical University of Munich {thomas.moellenhoff,cremers}@tum.de Abstract works [30, 31, 29, 55, 39,9, 10, 14] we consider the way less explored continuous (infinite-dimensional) setting. Numerous tasks in imaging and vision can be formu- Our motivation partly stems from the fact that formula- lated as variational problems over vector-valued maps. We tions in function space are very general and admit a variety approach the relaxation and convexification of such vecto- of discretizations. Finite difference discretizations of con- rial variational problems via a lifting to the space of cur- tinuous relaxations often lead to models that are reminis- rents. To that end, we recall that functionals with poly- cent of MRFs [70], while piecewise-linear approximations convex Lagrangians can be reparametrized as convex one- are related to discrete-continuous MRFs [71], see [17, 40]. homogeneous functionals on the graph of the function. More recently, for the Kantorovich relaxation in optimal This leads to an equivalent shape optimization problem transport, approximations with deep neural networks were over oriented surfaces in the product space of domain and considered and achieved promising performance, for exam- codomain. A convex formulation is then obtained by relax- ple in generative modeling [2, 54]. ing the search space from oriented surfaces to more gen- We further argue that fractional (non-integer) solutions eral currents. We propose a discretization of the resulting to a careful discretization of the continuous model can infinite-dimensional optimization problem using Whitney implicitly approximate an “integer” continuous solution. -
On a Class of Einstein Space-Time Manifolds
Publ. Math. Debrecen 67/3-4 (2005), 471–480 On a class of Einstein space-time manifolds By ADELA MIHAI (Bucharest) and RADU ROSCA (Paris) Abstract. We deal with a general space-time (M,g) with usual differen- tiability conditions and hyperbolic metric g of index 1, which carries 3 skew- symmetric Killing vector fields X, Y , Z having as generative the unit time-like vector field e of the hyperbolic metric g. It is shown that such a space-time (M,g) is an Einstein manifold of curvature −1, which is foliated by space-like hypersur- faces Ms normal to e and the immersion x : Ms → M is pseudo-umbilical. In addition, it is proved that the vector fields X, Y , Z and e are exterior concur- rent vector fields and X, Y , Z define a commutative Killing triple, M admits a Lorentzian transformation which is in an orthocronous Lorentz group and the distinguished spatial 3-form of M is a relatively integral invariant of the vector fields X, Y and Z. 0. Introduction Let (M,g) be a general space-time with usual differentiability condi- tions and hyperbolic metric g of index 1. We assume in this paper that (M,g) carries 3 skew-symmetric Killing vector fields (abbr. SSK) X, Y , Z having as generative the unit time-like vector field e of the hyperbolic metric g (see[R1],[MRV].Therefore,if∇ is the Levi–Civita connection and ∧ means the wedge product of vector Mathematics Subject Classification: 53C15, 53D15, 53C25. Key words and phrases: space-time, skew-symmetric Killing vector field, exterior con- current vector field, orthocronous Lorentz group. -
VISUALIZING EXTERIOR CALCULUS Contents Introduction 1 1. Exterior
VISUALIZING EXTERIOR CALCULUS GREGORY BIXLER Abstract. Exterior calculus, broadly, is the structure of differential forms. These are usually presented algebraically, or as computational tools to simplify Stokes' Theorem. But geometric objects like forms should be visualized! Here, I present visualizations of forms that attempt to capture their geometry. Contents Introduction 1 1. Exterior algebra 2 1.1. Vectors and dual vectors 2 1.2. The wedge of two vectors 3 1.3. Defining the exterior algebra 4 1.4. The wedge of two dual vectors 5 1.5. R3 has no mixed wedges 6 1.6. Interior multiplication 7 2. Integration 8 2.1. Differential forms 9 2.2. Visualizing differential forms 9 2.3. Integrating differential forms over manifolds 9 3. Differentiation 11 3.1. Exterior Derivative 11 3.2. Visualizing the exterior derivative 13 3.3. Closed and exact forms 14 3.4. Future work: Lie derivative 15 Acknowledgments 16 References 16 Introduction What's the natural way to integrate over a smooth manifold? Well, a k-dimensional manifold is (locally) parametrized by k coordinate functions. At each point we get k tangent vectors, which together form an infinitesimal k-dimensional volume ele- ment. And how should we measure this element? It seems reasonable to ask for a function f taking k vectors, so that • f is linear in each vector • f is zero for degenerate elements (i.e. when vectors are linearly dependent) • f is smooth These are the properties of a differential form. 1 2 GREGORY BIXLER Introducing structure to a manifold usually follows a certain pattern: • Create it in Rn. -
On Riemannian Manifolds Endowed with a Locally Conformal Cosymplectic Structure
ON RIEMANNIAN MANIFOLDS ENDOWED WITH A LOCALLY CONFORMAL COSYMPLECTIC STRUCTURE ION MIHAI, RADU ROSCA, AND VALENTIN GHIS¸OIU Received 18 September 2004 and in revised form 7 September 2005 We deal with a locally conformal cosymplectic manifold M(φ,Ω,ξ,η,g)admittingacon- formal contact quasi-torse-forming vector field T. The presymplectic 2-form Ω is a lo- cally conformal cosymplectic 2-form. It is shown that T is a 3-exterior concurrent vector field. Infinitesimal transformations of the Lie algebra of ∧M are investigated. The Gauss map of the hypersurface Mξ normal to ξ is conformal and Mξ × Mξ is a Chen submanifold of M × M. 1. Introduction Locally conformal cosymplectic manifolds have been investigated by Olszak and Rosca [7] (see also [6]). In the present paper, we consider a (2m + 1)-dimensional Riemannian manifold M(φ, Ω,ξ,η,g) endowed with a locally conformal cosymplectic structure. We assume that M admits a principal vector field (or a conformal contact quasi-torse-forming), that is, ∇T = sdp + T ∧ ξ = sdp + η ⊗ T − T ⊗ ξ, (1.1) with ds = sη. First, we prove certain geometrical properties of the vector fields T and φT. The exis- tence of T and φT is determined by an exterior differential system in involution (in the sense of Cartan [3]). The principal vector field T is 3-exterior concurrent (see also [8]), it defines a Lie relative contact transformation of the co-Reeb form η, and the Lie differential of T with respect to T is conformal to T.ThevectorfieldφT is an infinitesimal transfor- mation of generators T and ξ.Thevectorfieldsξ, T,andφT commute and the distri- bution DT ={T,φT,ξ} is involutive. -
Nonlocal Exterior Calculus on Riemannian Manifolds
The Pennsylvania State University The Graduate School Department of Mathematics NONLOCAL EXTERIOR CALCULUS ON RIEMANNIAN MANIFOLDS A Dissertation in Mathematics by Thinh Duc Le c 2013 Thinh Duc Le Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2013 ii The dissertation of Thinh Duc Le was reviewed and approved* by the following: Qiang Du Verne M. Willaman Professor of Mathematics Dissertation Adviser Chair of Committee Long-Qing Chen Distinguished Professor of Material Sciences and Engineering Ping Xu Distinguished Professor of Mathematics Mathieu Stienon Associate Professor of Mathematics Svetlana Katok Director of Graduate Studies, Department of Mathematics *Signatures are on file in the Graduate School. iii Abstract Exterior calculus and differential forms are basic mathematical concepts that have been around for centuries. Variations of these concepts have also been made over the years such as the discrete exterior calculus and the finite element exterior calculus. In this work, motivated by the recent studies of nonlocal vector calculus we develop a nonlocal exterior calculus framework on Riemannian manifolds which mimics many properties of the standard (local/smooth) exterior calculus. However the key difference is that nonlocal \interactions" (functions, operators, fields,...) are not required to be smooth. Also any point/particle can interact directly with any other point/particle in the studied domain (at least in principle). Just as in the standard context, we introduce all necessary elements of exterior calculus such as forms, vector fields, exterior derivatives, etc. We point out the relationships between these elements with the known ones in (local) exterior calcu- lus, discrete exterior calculus, etc. -
An Introduction to Space–Time Exterior Calculus
mathematics Article An Introduction to Space–Time Exterior Calculus Ivano Colombaro 1,* , Josep Font-Segura 1 and Alfonso Martinez 1 1 Department of Information and Communication Technologies, Universitat Pompeu Fabra, 08018 Barcelona, Spain; [email protected] (J.F.-S.); [email protected] (A.M.) * Correspondence: [email protected]; Tel.: +34-93-542-1496 Received: 21 May 2019; Accepted: 18 June 2019; Published: 21 June 2019 Abstract: The basic concepts of exterior calculus for space–time multivectors are presented: Interior and exterior products, interior and exterior derivatives, oriented integrals over hypersurfaces, circulation and flux of multivector fields. Two Stokes theorems relating the exterior and interior derivatives with circulation and flux, respectively, are derived. As an application, it is shown how the exterior-calculus space–time formulation of the electromagnetic Maxwell equations and Lorentz force recovers the standard vector-calculus formulations, in both differential and integral forms. Keywords: exterior calculus, exterior algebra, electromagnetism, Maxwell equations, differential forms, tensor calculus 1. Introduction Vector calculus has, since its introduction by J. W. Gibbs [1] and Heaviside, been the tool of choice to represent many physical phenomena. In mechanics, hydrodynamics and electromagnetism, quantities such as forces, velocities and currents are modeled as vector fields in space, while flux, circulation, divergence or curl describe operations on the vector fields themselves. With relativity theory, it was observed that space and time are not independent but just coordinates in space–time [2] (pp. 111–120). Tensors like the Faraday tensor in electromagnetism were quickly adopted as a natural representation of fields in space–time [3] (pp.