WUCHEN LI AND STANLEY OSHER
|
|
- Beverly Hill
- 5 years ago
- Views:
Transcription
1 CONSTRAINED DYNAMICAL OPTIMAL TRANSPORT AND ITS LAGRANGIAN FORMULATION WUCHEN LI AND STANLEY OSHER Abstract. We propose ynamical optimal transport (OT) problems constraine in a parameterize probability subset. In applie problems such as eep learning, the probability istribution is often generate by a parameterize mapping function. In this case, we erive a formulation for the constraine ynamical OT. 1. Introuction Dynamical optimal transport problems play vital roles in flui ynamics [12] an mean fiel games [8]. They provie a type of statistical istance an interesting ifferential structures in the set of probability ensities [14]. The full probability set is often intractable, when the imension of sample space is large. For this reason, parameterize probability subsets have been wiely consiere, especially in machine learning problems an information geometry [1, 2, 4, 7, 11, 13]. We are intereste in stuying ynamical OT problems over a parameterize probability subset. In this note, we follow a series of work foun in [6, 9,?, 10,?], an introuce general constraine ynamical OT problems in a parametrize probability subset. As in eep learning [3], the probability subset is often constructe by a parameterize mapping. In these cases, we emonstrate that the constraine ynamical OT problems exhibit simple variational structures. We arrange this note as follows: In section 2, we briefly review the ynamical OT in both Eulerian an Lagrangian coorinates 1. Using Eulerian coorinates, we propose the constraine ynamical OT over a parameterize probability subset. In section 3, we next erive an equivalent Lagrangian formulation for the constraine problem. Key wors an phrases. Dynamical optimal transport; Mean fiel games; Information geometry; Machine learning. The research is supporte by AFOSR MURI proposal number 18RT In flui ynamics, the Eulerian coorinates represent the evolution of probability ensity function of particles, while the Lagrangian coorinates escribe the motion of particles. In learning problems, the Eulerian coorinates naturally connect with the minimization problem in term of probability ensities, while Lagrangian coorinate refers to the variational problem formulate in samples, whose analog are particles. In learning, we moel problems in Eulerian, an compute them in Lagrangian. In other wors, we often write the objective function in term of ensities an compute them via samples. 1
2 2 LI, OSHER 2. Constraine ynamical OT In this section, we briefly review the ynamical OT in a full probability set via both Eulerian an Lagrangian formalisms. Using the Eulerian coorinates, we propose the ynamical OT in a parameterize probability subset. For the simplicity of exposition, { all our erivations assume smoothness. Consier ensities ρ 0, ρ 1 P + () = ρ(x) C (): ρ(x) > 0, ρ(x)x = 1, where is a n- imensional sample space. Here can be R n, or a convex compact region in R n with zero flux conitions or perioic bounary conitions. Dynamical OT stuies a variational problem in ensity space, known as the Benamou- Brenier formula [5]. Given a Lagrangian function L: T [0, + ) uner suitable conitions, consier C(ρ 0, ρ 1 ) := inf v t 1 0 EL(X t (ω), v(t, X t (ω)))t, where E is the expectation operator over realizations ω in event space an the infimum is taken over all vector fiels v t = v(t, ), such that (1a) Ẋ t (ω) = v(t, X t (ω)), X 0 ρ 0, X 1 ρ 1. (1b) Here X i ρ i represents that X i (ω) satisfies the probability ensity ρ i (x), for i = 0, 1. Equivalently, enote the ensity of particles X t (ω) at space x an time t by ρ(t, x). Then problem (1) refers to a variational problem in ensity space: 1 C(ρ 0, ρ 1 ) := inf v t 0 L(x, v(t, x))ρ(t, x)xt, where the infimum is taken over all Borel vector fiels v t = v(t, ), such that the ensity function ρ(t, x) satisfies the continuity equation: ρ(t, x) t (2a) + (ρ(t, x)v(t, x)) = 0, ρ(0, x) = ρ 0 (x), ρ(1, x) = ρ 1 (x). (2b) Here is the ivergence operator in. In the language of flui ynamics, problem (1) refers to the Lagrangian formalism, while problem (2) is the associate Eulerian formalism; see etails in [14]. The Lagrangian formalism focuses on the motion of each iniviual particles, while the Eulerian formalism escribes the global behavior of all particles. Here (1) an (2) are equivalent since they represent the same variational problem using ifferent coorinate systems. In aition, one often consiers L(x, v) = v p, p 1, where is the Eucliean norm. In this case, the optimal value of the variational problem efines a istance function in the set of probability space. Denote W p (ρ 0, ρ 1 ) := C(ρ 0, ρ 1 ) 1 p, where W p is calle the L p -Wasserstein istance. We next stuy the variational problem (2) constraine on a parameterize probability ensity set. In other wors, consier a parameter space Θ R with { P Θ = ρ(θ, x) C (): θ Θ, ρ(θ, x)x = 1, ρ(θ, x) > 0, x.
3 LAGRANGIAN FORMULATION 3 Here we assume that ρ: Θ P + () is an injective mapping 2. We introuce the constraine ynamical OT as follows: 1 c(θ 0, θ 1 ) := inf L(x, v(t, x))ρ(θ t, x)xt, (3a) v t 0 where θ t = θ(t) Θ, t [0, 1], is a path in parameter space, an the infimum is taken over the Borel vector fiels v t = v(t, ), such that the constraine continuity equation hols: t ρ(θ t, x) + (ρ(θ t, x)v(t, x)) = 0, θ 0, θ 1 are fixe. (3b) We notice that the infimum of problem (3) is taken over ensity paths lying in the parameterize probability set, i.e. ρ(θ t, ) ρ(θ). Here the changing ratio of ensity is reuce into a finite imensional irection, i.e. t ρ(θ t, x) = ( θt ρ(θ t, x), θt t ), where (, ) is an inner prouct in R. A natural question arises. Variational problem (2), together with its constraine problem (3), are written in Eulerian coorinates. They evolve unavoiably with the entire probability ensity functions. For practical reasons, can we fin the Lagrangian coorinates for constraine problem (3)? In other wors, what are analogs of (1) in ρ(θ)? We next emonstrate an answer to this question. We show that there is an expression for the motion of particles, whose ensity path moves accoring to the constraine continuity equation (3b). 3. Lagrangian formulations In this section, we show the main result of this note, that is the constraine ynamical OT (3) has a simple Lagrangian formulation in Proposition 1. Consier a parameterize mapping or implicit generative moel as follows. Given a input space R n 1, n 1 n, let g θ :, x = g(θ, z), for z. Here g θ is a mapping function epening on parameters θ Θ. Given realizations ω in event space, we assume that the ranom variable z(ω) satisfies a ensity function µ(z) P + (), an enote x(ω) = g(θ, z(ω)) satisfying the ensity function ρ(θ, x). This means that the map g θ pushes forwar µ(z) to ρ(θ, x), enote by ρ(θ, x) = g θ µ(z): f(g(θ, z))µ(z)z = f(x)ρ(θ, x)x, for any f Cc (). (4) In this case, the parameterize probability set is given as follows: { ρ(θ) = ρ(θ, x) C (): θ Θ, ρ(θ, x) = g θ µ(z). We next present the constraine ynamical OT in Lagrangian coorinates. We notice a fact that, the vector fiel in optimal ensity path of problem (2) or (3) satisfies v(t, x) = D p H(x, Φ(t, x)), 2 We abuse the notation of ρ. Notice that ρ(θ, x) is a probability istribution parameterize by θ Θ, while ρ(x) is a probability istribution function in the full probability set.
4 4 LI, OSHER where H(x, p) = sup pv L(x, v) v T x is the Hamiltonian function associate with L. Proposition 1 (Constraine ynamical OT in Lagrangian formulation). The constraine ynamical OT has the following formulation: { 1 c(θ 0, θ 1 ) = inf E z µ L(g(θ t, z), t g(θ t, z))t: 0 t g(θ t, z) = D p H(g(θ t, z), x Φ(t, g(θ t, z))), θ(0) = θ 0, θ(1) = θ 1, (5) where the infimum is taken over all feasible potential functions Φ: [0, 1] R an parameter paths θ : [0, 1] R. Proof. Denote t g(θ t, z) = v(t, g(θ, z)), with v(t, g(θ t, z)) = D p H(g(θ t, z), Φ(t, g(θ t, z))). We show that the probability ensity transition equation of g(θ t, z) satisfies the constraine continuity equation t ρ(θ t, x) + (ρ(θ t, x)v(t, x)) = 0, (6) an E z µ L(g(θ t, z), t g(θ t, z)) = L(x, v(t, x))ρ(θ t, x)x. (7) On the one han, consier f C c (), then t E z µf(g(θ t, z)) = f(g(θ t, z))µ(z)z t = f(x)ρ(θ t, x)x t = f(x) t ρ(θ t, x)x, (8) where the secon equality hols from the push forwar relation (4).
5 On the other han, consier LAGRANGIAN FORMULATION 5 t E z µf(g(θ t, z)) = lim E f(g(θ t+ t, z) f(g(θ t, z)) z µ t 0 t f(g(θ t+ t, z)) f(g(θ t, z)) = lim µ(z)z t 0 t = f(g(θ t, z)) t g(θ t, z)µ(z)z = f(g(θ t, z))v(t, g(θ t, z))µ(z)z = f(x)v(t, x)ρ(θ t, x)x = f(x) (v(t, x)ρ(θ t, x))x, where, are graient an ivergence operators w.r.t. x. The secon to last equality hols from the push forwar relation (4), an the last equality hols using the integration by parts w.r.t. x. Since (8) = (9) for any f C c (), we have proven (6). In aition, by the efinition of the push forwar operator (4), we have E z µ L(g(θ t, z), t g(θ t, z)) = L(g(θ t, z), v(t, g(θ t, z)))µ(z)z = L(x, v(t, x))ρ(θ t, x)x. Thus we prove (7). (9) It is interesting to compare variational problems (1) with (5). We can view g(θ t, z) as parameterize particles, whose ensity function is constraine in the parameterize probability set ρ(θ). Their motions result at the evolution of probability transition ensities in ρ(θ), satisfying the constraine continuity equation (3b). For this reason, we call (5) the Lagrangian formalism of constraine ynamical OT. It is also worth noting that each movement of g(θ t, z) results a motion in ensity path ρ(θ t, x). The change of ensity path will ientify a potential function Φ(t, x) epening on θ t. In aiton, the cost functional in ynamical OT can involve general potential energies, such as linear potential energy: V(ρ) = V (x)ρ(x)x, an interaction energy: W(ρ) = w(x, y)ρ(x)ρ(y)xy.
6 6 LI, OSHER Here V (x) is a linear potential, an w(x, y) = w(y, x) is a symmetric interaction potential. If ρ(θ, x) ρ(θ), then V(ρ(θ, )) = V (x)ρ(θ, x)x = V (g(θ, z))µ(z)z an W(ρ(θ, )) = = =E z µ V (g(θ, z)), w(x, y)ρ(θ, x)ρ(θ, y)xy w(g(θ, z 1 ), g(θ, z 2 ))µ(z 1 )µ(z 2 )z 1 z 2 =E (z1,z 2 ) µ µw(g(θ, z 1 ), g(θ, z 2 )), where each secon equality in the above two formulas hol because of the constraine mapping relation (4) an µ µ represents an inepenent joint ensity function supporte on with marginals µ(z 1 ), µ(z 2 ). Similarly in proposition 1, we have { 1 [ ] c(θ 0, θ 1 ) = inf L(x, v(t, x))ρ(θ t, x)x V(ρ(θ t, )) W(ρ(θ t, )) t: v t, θ(0)=θ 0, θ(1)=θ 1 0 t ρ(θ t, x) + (ρ(θ t, x)v(t, x)) = 0 { 1 [ ] = inf L(x, D p H(x, Φ(t, x)))ρ(θ t, x)x V(ρ(θ t, )) W(ρ(θ t, )) t: Φ t, θ t θ(0)=θ 0, θ(1)=θ 1 0 t ρ(θ t, x) + (ρ(θ t, x)v(t, x)) = 0 { 1 = inf [E z µ L(g(θ t, z), Φ t, θ t, θ(0)=θ 0, θ(1)=θ 1 0 t g(θ t, z)) E z µ V (g(θ t, z)) E (z1,z 2 ) µ µw(g(θ t, z 1 ), g(θ t, z 2 ))]t: t g(θ t, z) = D p H(g(θ t, z), x Φ(t, g(θ t, z))) We next emonstrate an example of constraine ynamical OT problems. Example 1 (Constraine L 2 -Wasserstein istance). Let L(x, v) = v 2 an enote W2 (θ 0, θ 1 ) = c(θ 0, θ 1 ) 1 2, then { W2 (θ 0, θ 1 ) 2 1 = inf E z µ t g(θ t, z)) 2 t: t g(θ t, z) = Φ(t, g(θ t, z)), θ(0) = θ 0, θ(1) = θ 1. 0 Observe that (5) forms a geometric action energy function in parameter space Θ, in which the metric tensor can be extracte explicitly. In other wors, enote G(θ) R by θ T G(θ) θ = θ T E µ ( θ g(θ, z) θ g(θ, z) T ) θ,. with the constraint ( θ, θ g(θ, z)) = x Φ(g(θ, z)).
7 Here θ g(θ, z) R n, Φ is a potential function satisfying LAGRANGIAN FORMULATION 7 (ρ(θ, x) Φ(x)) = ( θ ρ(θ, x), θ), an G(θ) = E z µ ( θ g(θ, z) θ g(θ, z) T ) R is a semi-positive efinite matrix. References [1] S.-i. A. Natural Graient Works Efficiently in Learning. Neural Computation, 10(2): , [2] S. Amari. Information Geometry an Its Applications. Number volume 194 in Applie mathematical sciences. Springer, Japan, [3] M. Arjovsky, S. Chintala, an L. Bottou. Wasserstein GAN. arxiv: [cs, stat], [4] N. Ay, J. Jost, H. V. Lê, an L. J. Schwachhöfer. Information Geometry. Ergebnisse er Mathematik un ihrer Grenzgebiete of moern surveys in mathematics$l3. Folge, volume 64. Springer, Cham, [5] J.-D. Benamou an Y. Brenier. A computational flui mechanics solution to the Monge-Kantorovich mass transfer problem. Numerische Mathematik, 84(3): , [6] Y. Chen an W. Li. Natural graient in Wasserstein statistical manifol. arxiv: [cs, math], [7] A. Haler an T. T. Georgiou. Graient Flows in Uncertainty Propagation an Filtering of Linear Gaussian Systems. arxiv: [cs, math], [8] J.-M. Lasry an P.-L. Lions. Mean fiel games. Japanese Journal of Mathematics, 2(1): , [9] W. Li. Geometry of probability simplex via optimal transport. arxiv: [math], [10] W. Li an G. Montufar. Natural graient via optimal transport I. arxiv: [cs, math], [11] K. Moin. Geometry of matrix ecompositions seen through optimal transport an information geometry. Journal of Geometric Mechanics, 9(3): , [12] E. Nelson. Derivation of the Schröinger Equation from Newtonian Mechanics. Physical Review, 150(4): , [13] G. Pistone. Lagrangian Function on the Finite State Space Statistical Bunle. Entropy, 20(2):139, [14] C. Villani. Optimal Transport: Ol an New. Number 338 in Grunlehren er mathematischen Wissenschaften. Springer, Berlin, aress: wcli@math.ucla.eu aress: sjo@math.ucla.eu Department of Mathematics, University of California, Los Angeles.
6 General properties of an autonomous system of two first order ODE
6 General properties of an autonomous system of two first orer ODE Here we embark on stuying the autonomous system of two first orer ifferential equations of the form ẋ 1 = f 1 (, x 2 ), ẋ 2 = f 2 (, x
More informationSchrödinger s equation.
Physics 342 Lecture 5 Schröinger s Equation Lecture 5 Physics 342 Quantum Mechanics I Wenesay, February 3r, 2010 Toay we iscuss Schröinger s equation an show that it supports the basic interpretation of
More informationThe total derivative. Chapter Lagrangian and Eulerian approaches
Chapter 5 The total erivative 51 Lagrangian an Eulerian approaches The representation of a flui through scalar or vector fiels means that each physical quantity uner consieration is escribe as a function
More informationSwitching Time Optimization in Discretized Hybrid Dynamical Systems
Switching Time Optimization in Discretize Hybri Dynamical Systems Kathrin Flaßkamp, To Murphey, an Sina Ober-Blöbaum Abstract Switching time optimization (STO) arises in systems that have a finite set
More informationChapter 6: Energy-Momentum Tensors
49 Chapter 6: Energy-Momentum Tensors This chapter outlines the general theory of energy an momentum conservation in terms of energy-momentum tensors, then applies these ieas to the case of Bohm's moel.
More informationA new proof of the sharpness of the phase transition for Bernoulli percolation on Z d
A new proof of the sharpness of the phase transition for Bernoulli percolation on Z Hugo Duminil-Copin an Vincent Tassion October 8, 205 Abstract We provie a new proof of the sharpness of the phase transition
More informationSecond order differentiation formula on RCD(K, N) spaces
Secon orer ifferentiation formula on RCD(K, N) spaces Nicola Gigli Luca Tamanini February 8, 018 Abstract We prove the secon orer ifferentiation formula along geoesics in finite-imensional RCD(K, N) spaces.
More informationIntroduction to the Vlasov-Poisson system
Introuction to the Vlasov-Poisson system Simone Calogero 1 The Vlasov equation Consier a particle with mass m > 0. Let x(t) R 3 enote the position of the particle at time t R an v(t) = ẋ(t) = x(t)/t its
More informationCalculus of Variations
16.323 Lecture 5 Calculus of Variations Calculus of Variations Most books cover this material well, but Kirk Chapter 4 oes a particularly nice job. x(t) x* x*+ αδx (1) x*- αδx (1) αδx (1) αδx (1) t f t
More informationTopic 2.3: The Geometry of Derivatives of Vector Functions
BSU Math 275 Notes Topic 2.3: The Geometry of Derivatives of Vector Functions Textbook Sections: 13.2 From the Toolbox (what you nee from previous classes): Be able to compute erivatives scalar-value functions
More informationLecture 2 Lagrangian formulation of classical mechanics Mechanics
Lecture Lagrangian formulation of classical mechanics 70.00 Mechanics Principle of stationary action MATH-GA To specify a motion uniquely in classical mechanics, it suffices to give, at some time t 0,
More informationTensors, Fields Pt. 1 and the Lie Bracket Pt. 1
Tensors, Fiels Pt. 1 an the Lie Bracket Pt. 1 PHYS 500 - Southern Illinois University September 8, 2016 PHYS 500 - Southern Illinois University Tensors, Fiels Pt. 1 an the Lie Bracket Pt. 1 September 8,
More informationPhysics 5153 Classical Mechanics. The Virial Theorem and The Poisson Bracket-1
Physics 5153 Classical Mechanics The Virial Theorem an The Poisson Bracket 1 Introuction In this lecture we will consier two applications of the Hamiltonian. The first, the Virial Theorem, applies to systems
More informationMath Notes on differentials, the Chain Rule, gradients, directional derivative, and normal vectors
Math 18.02 Notes on ifferentials, the Chain Rule, graients, irectional erivative, an normal vectors Tangent plane an linear approximation We efine the partial erivatives of f( xy, ) as follows: f f( x+
More informationProblem set 2: Solutions Math 207B, Winter 2016
Problem set : Solutions Math 07B, Winter 016 1. A particle of mass m with position x(t) at time t has potential energy V ( x) an kinetic energy T = 1 m x t. The action of the particle over times t t 1
More informationThe continuity equation
Chapter 6 The continuity equation 61 The equation of continuity It is evient that in a certain region of space the matter entering it must be equal to the matter leaving it Let us consier an infinitesimal
More informationEuler equations for multiple integrals
Euler equations for multiple integrals January 22, 2013 Contents 1 Reminer of multivariable calculus 2 1.1 Vector ifferentiation......................... 2 1.2 Matrix ifferentiation........................
More informationGeneralization of the persistent random walk to dimensions greater than 1
PHYSICAL REVIEW E VOLUME 58, NUMBER 6 DECEMBER 1998 Generalization of the persistent ranom walk to imensions greater than 1 Marián Boguñá, Josep M. Porrà, an Jaume Masoliver Departament e Física Fonamental,
More informationINDEPENDENT COMPONENT ANALYSIS VIA
INDEPENDENT COMPONENT ANALYSIS VIA NONPARAMETRIC MAXIMUM LIKELIHOOD ESTIMATION Truth Rotate S 2 2 1 0 1 2 3 4 X 2 2 1 0 1 2 3 4 4 2 0 2 4 6 4 2 0 2 4 6 S 1 X 1 Reconstructe S^ 2 2 1 0 1 2 3 4 Marginal
More informationWasserstein GAN. Juho Lee. Jan 23, 2017
Wasserstein GAN Juho Lee Jan 23, 2017 Wasserstein GAN (WGAN) Arxiv submission Martin Arjovsky, Soumith Chintala, and Léon Bottou A new GAN model minimizing the Earth-Mover s distance (Wasserstein-1 distance)
More informationDarboux s theorem and symplectic geometry
Darboux s theorem an symplectic geometry Liang, Feng May 9, 2014 Abstract Symplectic geometry is a very important branch of ifferential geometry, it is a special case of poisson geometry, an coul also
More informationAgmon Kolmogorov Inequalities on l 2 (Z d )
Journal of Mathematics Research; Vol. 6, No. ; 04 ISSN 96-9795 E-ISSN 96-9809 Publishe by Canaian Center of Science an Eucation Agmon Kolmogorov Inequalities on l (Z ) Arman Sahovic Mathematics Department,
More informationSOME RESULTS ON THE GEOMETRY OF MINKOWSKI PLANE. Bing Ye Wu
ARCHIVUM MATHEMATICUM (BRNO Tomus 46 (21, 177 184 SOME RESULTS ON THE GEOMETRY OF MINKOWSKI PLANE Bing Ye Wu Abstract. In this paper we stuy the geometry of Minkowski plane an obtain some results. We focus
More informationHyperbolic Moment Equations Using Quadrature-Based Projection Methods
Hyperbolic Moment Equations Using Quarature-Base Projection Methos J. Koellermeier an M. Torrilhon Department of Mathematics, RWTH Aachen University, Aachen, Germany Abstract. Kinetic equations like the
More informationThe derivative of a function f(x) is another function, defined in terms of a limiting expression: f(x + δx) f(x)
Y. D. Chong (2016) MH2801: Complex Methos for the Sciences 1. Derivatives The erivative of a function f(x) is another function, efine in terms of a limiting expression: f (x) f (x) lim x δx 0 f(x + δx)
More informationFLUCTUATIONS IN THE NUMBER OF POINTS ON SMOOTH PLANE CURVES OVER FINITE FIELDS. 1. Introduction
FLUCTUATIONS IN THE NUMBER OF POINTS ON SMOOTH PLANE CURVES OVER FINITE FIELDS ALINA BUCUR, CHANTAL DAVID, BROOKE FEIGON, MATILDE LALÍN 1 Introuction In this note, we stuy the fluctuations in the number
More informationarxiv: v1 [math.mg] 10 Apr 2018
ON THE VOLUME BOUND IN THE DVORETZKY ROGERS LEMMA FERENC FODOR, MÁRTON NASZÓDI, AND TAMÁS ZARNÓCZ arxiv:1804.03444v1 [math.mg] 10 Apr 2018 Abstract. The classical Dvoretzky Rogers lemma provies a eterministic
More informationarxiv: v2 [math.dg] 16 Dec 2014
A ONOTONICITY FORULA AND TYPE-II SINGULARITIES FOR THE EAN CURVATURE FLOW arxiv:1312.4775v2 [math.dg] 16 Dec 2014 YONGBING ZHANG Abstract. In this paper, we introuce a monotonicity formula for the mean
More informationarxiv: v1 [math-ph] 5 May 2014
DIFFERENTIAL-ALGEBRAIC SOLUTIONS OF THE HEAT EQUATION VICTOR M. BUCHSTABER, ELENA YU. NETAY arxiv:1405.0926v1 [math-ph] 5 May 2014 Abstract. In this work we introuce the notion of ifferential-algebraic
More informationPartial Differential Equations
Chapter Partial Differential Equations. Introuction Have solve orinary ifferential equations, i.e. ones where there is one inepenent an one epenent variable. Only orinary ifferentiation is therefore involve.
More information1 Heisenberg Representation
1 Heisenberg Representation What we have been ealing with so far is calle the Schröinger representation. In this representation, operators are constants an all the time epenence is carrie by the states.
More informationLagrangian and Hamiltonian Dynamics
Lagrangian an Hamiltonian Dynamics Volker Perlick (Lancaster University) Lecture 1 The Passage from Newtonian to Lagrangian Dynamics (Cockcroft Institute, 22 February 2010) Subjects covere Lecture 2: Discussion
More informationarxiv: v1 [math.dg] 30 May 2012
VARIATION OF THE ODUUS OF A FOIATION. CISKA arxiv:1205.6786v1 [math.dg] 30 ay 2012 Abstract. The p moulus mo p (F) of a foliation F on a Riemannian manifol is a generalization of extremal length of plane
More informationANALYSIS OF A GENERAL FAMILY OF REGULARIZED NAVIER-STOKES AND MHD MODELS
ANALYSIS OF A GENERAL FAMILY OF REGULARIZED NAVIER-STOKES AND MHD MODELS MICHAEL HOLST, EVELYN LUNASIN, AND GANTUMUR TSOGTGEREL ABSTRACT. We consier a general family of regularize Navier-Stokes an Magnetohyroynamics
More informationCHAPTER 1 : DIFFERENTIABLE MANIFOLDS. 1.1 The definition of a differentiable manifold
CHAPTER 1 : DIFFERENTIABLE MANIFOLDS 1.1 The efinition of a ifferentiable manifol Let M be a topological space. This means that we have a family Ω of open sets efine on M. These satisfy (1), M Ω (2) the
More informationNOTES ON EULER-BOOLE SUMMATION (1) f (l 1) (n) f (l 1) (m) + ( 1)k 1 k! B k (y) f (k) (y) dy,
NOTES ON EULER-BOOLE SUMMATION JONATHAN M BORWEIN, NEIL J CALKIN, AND DANTE MANNA Abstract We stuy a connection between Euler-MacLaurin Summation an Boole Summation suggeste in an AMM note from 196, which
More informationOn the Aloha throughput-fairness tradeoff
On the Aloha throughput-fairness traeoff 1 Nan Xie, Member, IEEE, an Steven Weber, Senior Member, IEEE Abstract arxiv:1605.01557v1 [cs.it] 5 May 2016 A well-known inner boun of the stability region of
More informationSection 2.7 Derivatives of powers of functions
Section 2.7 Derivatives of powers of functions (3/19/08) Overview: In this section we iscuss the Chain Rule formula for the erivatives of composite functions that are forme by taking powers of other functions.
More informationLagrangian and Hamiltonian Mechanics
Lagrangian an Hamiltonian Mechanics.G. Simpson, Ph.. epartment of Physical Sciences an Engineering Prince George s Community College ecember 5, 007 Introuction In this course we have been stuying classical
More informationApproximate Reduction of Dynamical Systems
Proceeings of the 4th IEEE Conference on Decision & Control Manchester Gran Hyatt Hotel San Diego, CA, USA, December 3-, 6 FrIP.7 Approximate Reuction of Dynamical Systems Paulo Tabuaa, Aaron D. Ames,
More informationApproximate reduction of dynamic systems
Systems & Control Letters 57 2008 538 545 www.elsevier.com/locate/sysconle Approximate reuction of ynamic systems Paulo Tabuaa a,, Aaron D. Ames b, Agung Julius c, George J. Pappas c a Department of Electrical
More informationAn extension of Alexandrov s theorem on second derivatives of convex functions
Avances in Mathematics 228 (211 2258 2267 www.elsevier.com/locate/aim An extension of Alexanrov s theorem on secon erivatives of convex functions Joseph H.G. Fu 1 Department of Mathematics, University
More informationTRAJECTORY TRACKING FOR FULLY ACTUATED MECHANICAL SYSTEMS
TRAJECTORY TRACKING FOR FULLY ACTUATED MECHANICAL SYSTEMS Francesco Bullo Richar M. Murray Control an Dynamical Systems California Institute of Technology Pasaena, CA 91125 Fax : + 1-818-796-8914 email
More informationSome Examples. Uniform motion. Poisson processes on the real line
Some Examples Our immeiate goal is to see some examples of Lévy processes, an/or infinitely-ivisible laws on. Uniform motion Choose an fix a nonranom an efine X := for all (1) Then, {X } is a [nonranom]
More informationCentrum voor Wiskunde en Informatica
Centrum voor Wiskune en Informatica Moelling, Analysis an Simulation Moelling, Analysis an Simulation Conservation properties of smoothe particle hyroynamics applie to the shallow water equations J.E.
More informationA note on asymptotic formulae for one-dimensional network flow problems Carlos F. Daganzo and Karen R. Smilowitz
A note on asymptotic formulae for one-imensional network flow problems Carlos F. Daganzo an Karen R. Smilowitz (to appear in Annals of Operations Research) Abstract This note evelops asymptotic formulae
More informationAPPROXIMATE SOLUTION FOR TRANSIENT HEAT TRANSFER IN STATIC TURBULENT HE II. B. Baudouy. CEA/Saclay, DSM/DAPNIA/STCM Gif-sur-Yvette Cedex, France
APPROXIMAE SOLUION FOR RANSIEN HEA RANSFER IN SAIC URBULEN HE II B. Bauouy CEA/Saclay, DSM/DAPNIA/SCM 91191 Gif-sur-Yvette Ceex, France ABSRAC Analytical solution in one imension of the heat iffusion equation
More informationON ISENTROPIC APPROXIMATIONS FOR COMPRESSIBLE EULER EQUATIONS
ON ISENTROPIC APPROXIMATIONS FOR COMPRESSILE EULER EQUATIONS JUNXIONG JIA AND RONGHUA PAN Abstract. In this paper, we first generalize the classical results on Cauchy problem for positive symmetric quasilinear
More informationAsymptotic estimates on the time derivative of entropy on a Riemannian manifold
Asymptotic estimates on the time erivative of entropy on a Riemannian manifol Arian P. C. Lim a, Dejun Luo b a Nanyang Technological University, athematics an athematics Eucation, Singapore b Key Lab of
More information7.1 Support Vector Machine
67577 Intro. to Machine Learning Fall semester, 006/7 Lecture 7: Support Vector Machines an Kernel Functions II Lecturer: Amnon Shashua Scribe: Amnon Shashua 7. Support Vector Machine We return now to
More informationKey words. Optimal mass transport, Wasserstein distance, gradient flow, Schrödinger bridge.
ENTROPIC INTERPOLATION AND GRADIENT FLOWS ON WASSERSTEIN PRODUCT SPACES YONGXIN CHEN, TRYPHON GEORGIOU AND MICHELE PAVON Abstract. We show that the entropic interpolation between two given marginals provie
More informationThe Three-dimensional Schödinger Equation
The Three-imensional Schöinger Equation R. L. Herman November 7, 016 Schröinger Equation in Spherical Coorinates We seek to solve the Schröinger equation with spherical symmetry using the metho of separation
More informationChapter 4. Electrostatics of Macroscopic Media
Chapter 4. Electrostatics of Macroscopic Meia 4.1 Multipole Expansion Approximate potentials at large istances 3 x' x' (x') x x' x x Fig 4.1 We consier the potential in the far-fiel region (see Fig. 4.1
More informationTable of Common Derivatives By David Abraham
Prouct an Quotient Rules: Table of Common Derivatives By Davi Abraham [ f ( g( ] = [ f ( ] g( + f ( [ g( ] f ( = g( [ f ( ] g( g( f ( [ g( ] Trigonometric Functions: sin( = cos( cos( = sin( tan( = sec
More informationNoether s theorem applied to classical electrodynamics
Noether s theorem applie to classical electroynamics Thomas B. Mieling Faculty of Physics, University of ienna Boltzmanngasse 5, 090 ienna, Austria (Date: November 8, 207) The consequences of gauge invariance
More informationLecture Introduction. 2 Examples of Measure Concentration. 3 The Johnson-Lindenstrauss Lemma. CS-621 Theory Gems November 28, 2012
CS-6 Theory Gems November 8, 0 Lecture Lecturer: Alesaner Mąry Scribes: Alhussein Fawzi, Dorina Thanou Introuction Toay, we will briefly iscuss an important technique in probability theory measure concentration
More informationGradient flow of the Chapman-Rubinstein-Schatzman model for signed vortices
Graient flow of the Chapman-Rubinstein-Schatzman moel for signe vortices Luigi Ambrosio, Eoaro Mainini an Sylvia Serfaty Deicate to the memory of Michelle Schatzman (1949-2010) Abstract We continue the
More informationChapter 2 Lagrangian Modeling
Chapter 2 Lagrangian Moeling The basic laws of physics are use to moel every system whether it is electrical, mechanical, hyraulic, or any other energy omain. In mechanics, Newton s laws of motion provie
More informationCONSERVATION PROPERTIES OF SMOOTHED PARTICLE HYDRODYNAMICS APPLIED TO THE SHALLOW WATER EQUATIONS
BIT 0006-3835/00/4004-0001 $15.00 200?, Vol.??, No.??, pp.?????? c Swets & Zeitlinger CONSERVATION PROPERTIES OF SMOOTHE PARTICLE HYROYNAMICS APPLIE TO THE SHALLOW WATER EQUATIONS JASON FRANK 1 an SEBASTIAN
More informationMonotonicity of facet numbers of random convex hulls
Monotonicity of facet numbers of ranom convex hulls Gilles Bonnet, Julian Grote, Daniel Temesvari, Christoph Thäle, Nicola Turchi an Florian Wespi arxiv:173.31v1 [math.mg] 7 Mar 17 Abstract Let X 1,...,
More informationLower Bounds for the Smoothed Number of Pareto optimal Solutions
Lower Bouns for the Smoothe Number of Pareto optimal Solutions Tobias Brunsch an Heiko Röglin Department of Computer Science, University of Bonn, Germany brunsch@cs.uni-bonn.e, heiko@roeglin.org Abstract.
More informationθ x = f ( x,t) could be written as
9. Higher orer PDEs as systems of first-orer PDEs. Hyperbolic systems. For PDEs, as for ODEs, we may reuce the orer by efining new epenent variables. For example, in the case of the wave equation, (1)
More information12.5. Differentiation of vectors. Introduction. Prerequisites. Learning Outcomes
Differentiation of vectors 12.5 Introuction The area known as vector calculus is use to moel mathematically a vast range of engineering phenomena incluing electrostatics, electromagnetic fiels, air flow
More informationWitten s Proof of Morse Inequalities
Witten s Proof of Morse Inequalities by Igor Prokhorenkov Let M be a smooth, compact, oriente manifol with imension n. A Morse function is a smooth function f : M R such that all of its critical points
More informationTractability results for weighted Banach spaces of smooth functions
Tractability results for weighte Banach spaces of smooth functions Markus Weimar Mathematisches Institut, Universität Jena Ernst-Abbe-Platz 2, 07740 Jena, Germany email: markus.weimar@uni-jena.e March
More informationConvergence rates of moment-sum-of-squares hierarchies for optimal control problems
Convergence rates of moment-sum-of-squares hierarchies for optimal control problems Milan Kora 1, Diier Henrion 2,3,4, Colin N. Jones 1 Draft of September 8, 2016 Abstract We stuy the convergence rate
More informationL p Theory for the Multidimensional Aggregation Equation
L p Theory for the Multiimensional Aggregation Equation ANDREA L. BERTOZZI University of California - Los Angeles THOMAS LAURENT University of California - Los Angeles AND JESÚS ROSADO Universitat Autònoma
More informationLecture 2 - First order linear PDEs and PDEs from physics
18.15 - Introuction to PEs, Fall 004 Prof. Gigliola Staffilani Lecture - First orer linear PEs an PEs from physics I mentione in the first class some basic PEs of first an secon orer. Toay we illustrate
More informationMean field games via probability manifold II
Mean field games via probability manifold II Wuchen Li IPAM summer school, 2018. Introduction Consider a nonlinear Schrödinger equation hi t Ψ = h2 1 2 Ψ + ΨV (x) + Ψ R d W (x, y) Ψ(y) 2 dy. The unknown
More informationExistence of equilibria in articulated bearings in presence of cavity
J. Math. Anal. Appl. 335 2007) 841 859 www.elsevier.com/locate/jmaa Existence of equilibria in articulate bearings in presence of cavity G. Buscaglia a, I. Ciuperca b, I. Hafii c,m.jai c, a Centro Atómico
More informationNöether s Theorem Under the Legendre Transform by Jonathan Herman
Nöether s Theorem Uner the Legenre Transform by Jonathan Herman A research paper presente to the University of Waterloo in fulfilment of the research paper requirement for the egree of Master of Mathematics
More informationNumerical Integrator. Graphics
1 Introuction CS229 Dynamics Hanout The question of the week is how owe write a ynamic simulator for particles, rigi boies, or an articulate character such as a human figure?" In their SIGGRPH course notes,
More informationLeast-Squares Regression on Sparse Spaces
Least-Squares Regression on Sparse Spaces Yuri Grinberg, Mahi Milani Far, Joelle Pineau School of Computer Science McGill University Montreal, Canaa {ygrinb,mmilan1,jpineau}@cs.mcgill.ca 1 Introuction
More informationREVERSIBILITY FOR DIFFUSIONS VIA QUASI-INVARIANCE. 1. Introduction We look at the problem of reversibility for operators of the form
REVERSIBILITY FOR DIFFUSIONS VIA QUASI-INVARIANCE OMAR RIVASPLATA, JAN RYCHTÁŘ, AND BYRON SCHMULAND Abstract. Why is the rift coefficient b associate with a reversible iffusion on R given by a graient?
More informationMany problems in physics, engineering, and chemistry fall in a general class of equations of the form. d dx. d dx
Math 53 Notes on turm-liouville equations Many problems in physics, engineering, an chemistry fall in a general class of equations of the form w(x)p(x) u ] + (q(x) λ) u = w(x) on an interval a, b], plus
More informationTime-of-Arrival Estimation in Non-Line-Of-Sight Environments
2 Conference on Information Sciences an Systems, The Johns Hopkins University, March 2, 2 Time-of-Arrival Estimation in Non-Line-Of-Sight Environments Sinan Gezici, Hisashi Kobayashi an H. Vincent Poor
More informationCapacity Analysis of MIMO Systems with Unknown Channel State Information
Capacity Analysis of MIMO Systems with Unknown Channel State Information Jun Zheng an Bhaskar D. Rao Dept. of Electrical an Computer Engineering University of California at San Diego e-mail: juzheng@ucs.eu,
More informationAssignment 1. g i (x 1,..., x n ) dx i = 0. i=1
Assignment 1 Golstein 1.4 The equations of motion for the rolling isk are special cases of general linear ifferential equations of constraint of the form g i (x 1,..., x n x i = 0. i=1 A constraint conition
More informationRobust Forward Algorithms via PAC-Bayes and Laplace Distributions. ω Q. Pr (y(ω x) < 0) = Pr A k
A Proof of Lemma 2 B Proof of Lemma 3 Proof: Since the support of LL istributions is R, two such istributions are equivalent absolutely continuous with respect to each other an the ivergence is well-efine
More informationHomework 2 Solutions EM, Mixture Models, PCA, Dualitys
Homewor Solutions EM, Mixture Moels, PCA, Dualitys CMU 0-75: Machine Learning Fall 05 http://www.cs.cmu.eu/~bapoczos/classes/ml075_05fall/ OUT: Oct 5, 05 DUE: Oct 9, 05, 0:0 AM An EM algorithm for a Mixture
More informationOn Kelvin-Voigt model and its generalizations
Nečas Center for Mathematical Moeling On Kelvin-Voigt moel an its generalizations M. Bulíček, J. Málek an K. R. Rajagopal Preprint no. 1-11 Research Team 1 Mathematical Institute of the Charles University
More informationLectures - Week 10 Introduction to Ordinary Differential Equations (ODES) First Order Linear ODEs
Lectures - Week 10 Introuction to Orinary Differential Equations (ODES) First Orer Linear ODEs When stuying ODEs we are consiering functions of one inepenent variable, e.g., f(x), where x is the inepenent
More informationLower bounds on Locality Sensitive Hashing
Lower bouns on Locality Sensitive Hashing Rajeev Motwani Assaf Naor Rina Panigrahy Abstract Given a metric space (X, X ), c 1, r > 0, an p, q [0, 1], a istribution over mappings H : X N is calle a (r,
More informationLie symmetry and Mei conservation law of continuum system
Chin. Phys. B Vol. 20, No. 2 20 020 Lie symmetry an Mei conservation law of continuum system Shi Shen-Yang an Fu Jing-Li Department of Physics, Zhejiang Sci-Tech University, Hangzhou 3008, China Receive
More informationRelative Entropy and Score Function: New Information Estimation Relationships through Arbitrary Additive Perturbation
Relative Entropy an Score Function: New Information Estimation Relationships through Arbitrary Aitive Perturbation Dongning Guo Department of Electrical Engineering & Computer Science Northwestern University
More informationGLOBAL SOLUTIONS FOR 2D COUPLED BURGERS-COMPLEX-GINZBURG-LANDAU EQUATIONS
Electronic Journal of Differential Equations, Vol. 015 015), No. 99, pp. 1 14. ISSN: 107-6691. URL: http://eje.math.txstate.eu or http://eje.math.unt.eu ftp eje.math.txstate.eu GLOBAL SOLUTIONS FOR D COUPLED
More informationOn the Inclined Curves in Galilean 4-Space
Applie Mathematical Sciences Vol. 7 2013 no. 44 2193-2199 HIKARI Lt www.m-hikari.com On the Incline Curves in Galilean 4-Space Dae Won Yoon Department of Mathematics Eucation an RINS Gyeongsang National
More informationLecture XII. where Φ is called the potential function. Let us introduce spherical coordinates defined through the relations
Lecture XII Abstract We introuce the Laplace equation in spherical coorinates an apply the metho of separation of variables to solve it. This will generate three linear orinary secon orer ifferential equations:
More informationLECTURE 1: BASIC CONCEPTS, PROBLEMS, AND EXAMPLES
LECTURE 1: BASIC CONCEPTS, PROBLEMS, AND EXAMPLES WEIMIN CHEN, UMASS, SPRING 07 In this lecture we give a general introuction to the basic concepts an some of the funamental problems in symplectic geometry/topology,
More informationStable and compact finite difference schemes
Center for Turbulence Research Annual Research Briefs 2006 2 Stable an compact finite ifference schemes By K. Mattsson, M. Svär AND M. Shoeybi. Motivation an objectives Compact secon erivatives have long
More informationAn Eulerian level set method for partial differential equations on evolving surfaces
Computing an Visualization in Science manuscript No. (will be inserte by the eitor) G. Dziuk C. M. Elliott An Eulerian level set metho for partial ifferential equations on evolving surfaces Receive: ate
More informationarxiv: v4 [math.pr] 27 Jul 2016
The Asymptotic Distribution of the Determinant of a Ranom Correlation Matrix arxiv:309768v4 mathpr] 7 Jul 06 AM Hanea a, & GF Nane b a Centre of xcellence for Biosecurity Risk Analysis, University of Melbourne,
More informationA PAC-Bayesian Approach to Spectrally-Normalized Margin Bounds for Neural Networks
A PAC-Bayesian Approach to Spectrally-Normalize Margin Bouns for Neural Networks Behnam Neyshabur, Srinah Bhojanapalli, Davi McAllester, Nathan Srebro Toyota Technological Institute at Chicago {bneyshabur,
More informationMonte Carlo Methods with Reduced Error
Monte Carlo Methos with Reuce Error As has been shown, the probable error in Monte Carlo algorithms when no information about the smoothness of the function is use is Dξ r N = c N. It is important for
More informationarxiv:nlin/ v1 [nlin.cd] 21 Mar 2002
Entropy prouction of iffusion in spatially perioic eterministic systems arxiv:nlin/0203046v [nlin.cd] 2 Mar 2002 J. R. Dorfman, P. Gaspar, 2 an T. Gilbert 3 Department of Physics an Institute for Physical
More informationLecture 2: Correlated Topic Model
Probabilistic Moels for Unsupervise Learning Spring 203 Lecture 2: Correlate Topic Moel Inference for Correlate Topic Moel Yuan Yuan First of all, let us make some claims about the parameters an variables
More informationLecture 1b. Differential operators and orthogonal coordinates. Partial derivatives. Divergence and divergence theorem. Gradient. A y. + A y y dy. 1b.
b. Partial erivatives Lecture b Differential operators an orthogonal coorinates Recall from our calculus courses that the erivative of a function can be efine as f ()=lim 0 or using the central ifference
More informationSeparation of Variables
Physics 342 Lecture 1 Separation of Variables Lecture 1 Physics 342 Quantum Mechanics I Monay, January 25th, 2010 There are three basic mathematical tools we nee, an then we can begin working on the physical
More information1 M3-4-5A16 Assessed Problems # 1: Do 4 out of 5 problems
D. D. Holm M3-4-5A16 Assesse Problems # 1 Due 1 Nov 2012 1 1 M3-4-5A16 Assesse Problems # 1: Do 4 out of 5 problems Exercise 1.1 (Poisson brackets for the Hopf map) Figure 1: The Hopf map. In coorinates
More information