# Quasi-static evolution and congested transport

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Quasi-static evolution and congested transport Inwon Kim Joint with Damon Alexander, Katy Craig and Yao Yao UCLA, UW Madison

2 Hard congestion in crowd motion The following crowd motion model is proposed by Maury, Roundneff-Chupin and Santambrogio (2010): ρ(x, t): pedestrian population density, which cannot exceed a certain maximal value (which we assume to be 1). Φ(x): The desired velocity field for an individual located at x. It may not be achieved due to the constraint ρ(, t) 1. In the saturated zone {ρ(, t) = 1}, the actual velocity field v(, t) must satisfy v(, t) 0, in order to not increase the density. (If so, we say v is feasible).

3 Hard congestion in crowd motion In [MRS], they consider the PDE system (P) { ρt + (ρv) = 0 v(, t) = P Cρ Φ, where P Cρ is the projection towards the space of feasible velocity fields in L 2 sense. They link this system with the following gradient flow: For ρ P 2 (R d ), let ˆ ρ(x)φ(x)dx ρ 1 E [ρ] = R d + otherwise.

4 Gradient flow approach for hard congestion Let ρ (, t) be the gradient flow of E with respect to Wasserstein distance W 2 with initial data ρ 0. Existence and uniqueness of ρ can be shown following the book by Ambrosio, Gigli and Savaré (2005). [MRSV] showed that ρ is a weak solution to (P), however uniqueness of the weak solution to (P) is unknown since the velocity field v = P Cρ Φ is only in L 2. Let Φ 0. Then formally speaking, the velocity field pulls the particles together. In this setting we will show a unique characterization of the velocity field v for ρ.

5 Modification of the velocity We expect the modified velocity to be of the form v = p + Φ. The continuity equation for ρ then is expected to be of the form ρ t (ρ( p + Φ)) = 0, where p is the pressure generated by the constraint, supported on the congested set {ρ = 1}.

6 A remark on the assumption Φ 0 When Φ 0, we expect the pressure to be nonzero in all of {ρ = 1}. Moreover the modified velocity should be incompressible in the congested region since the original velocity field only tries to compress the density. Thus p should solve ( p + Φ) = 0 or p = Φ in {ρ = 1}.

7 Evolution of the congested region Suppose ρ solves the modified discontinuity equation with p discontinuously changing to zero across Ω(t). Then denoting ρ = ρ I χ Ω(t) + ρ O we have 0 = t [ Ω(t) (ρi ) t dx + R n Ω(t) ρo dx] = Ω(t) (ρi ) t + R n Ω(t) (ρo ) t dx + Ω(t) V (ρi ρ O )ds = Ω(t) (ρi ( p + Φ) + Ω(t) (ρ O Φ) + Ω(t) V C (ρi ρ O )d = Ω(t) ρi ν p + (ρ I ρ O )( ν Φ + V )ds, where V = V x,t denotes the (outward) normal velocity of Ω(t).

8 Evolution of ρ when Φ 0 Above calculation suggests the following evolution for ρ(, t) = χ Ωt + ρ O, where the congested set Ω(t) := {p(, t) > 0} is determined by the following free boundary problem for p 0: p(, t) = Φ in {p(, t) > 0}; V = 1 (1 ρ O ) ( νp) ν Φ on {p(, t) > 0}.

9 Note that the velocity law V = 1 1 ρ O Dp νφ indicates that there is a generic discontinuity of ρ across Ω(t). The well-posedness of the free boundary problem can be shown by viscosity solutions theory. We are interested in connecting this problem with the gradient flow solution ρ. In addition to the assumption Φ 0, we will also assume that the initial data is patch.

10 Quasi-static evolution: patch case Suppose ρ 0 = χ Ω0, and consider the following free boundary problem that p solves with the initial data {p(, 0) > 0} = Ω 0 : (FB) { p(, t) = Φ in {p(, t) > 0} =: Ωt ; V = Dp ν Φ on Ω t. Our goal is to prove that the gradient flow solution ρ with initial data ρ 0 satisfies ρ (, t) = χ Ωt.

11 Approximation by Porous Medium Equation Let ρ m solve the following porous medium equation with drift: ρ t = ρ m + (ρ Φ), with initial data ρ(, 0) = χ Ω0. It is well known (Otto, 2001) that ρ m is the gradient flow for E m [ρ] = 1 ˆ ˆ ρ m dx + ρφdx. m We will show that ρ m converging to ρ as m, with a rate. We will also show that ρ m converges to χ Ωt, yielding the desired statement, ρ = χ Ωt.

12 Convergence as m Theorem (Alexander-K-Yao., 2013) Let Ω 0 be a compact set in R d with locally Lipschitz boundary, then (a) Assuming Φ 0. Then there is a unique family of compact sets Ω t evolving with (FB). As m, ρ m χ Ωt locally uniformly away from Ω(t). (b) Assume D 2 Φ and inf Φ is finite. Then ρ m (, t) converges to ρ (, t) in W 2 distance uniformly in t [0, T ], with convergence rate sup W 2 (ρ m (t), ρ (t)) 1 t [0,T ] m 1/24.

13 Convergence as m Corollary Since ρ m (, t) converges to both χ Ωt and ρ (, t) as m, χ Ωt and ρ (, t) must be equal almost everywhere. While the gradient flow approach cannot directly deal with general nonconvex bounded domains (with e.g. Neumann boundary conditions), the viscosity solution approach still applies and we have ρ m χ Ωt. On the other hand, without the condition Φ 0, the gradient flow approach still works and we still have ρ m (, t) ρ (, t) but the characterization of the modified velocity remains open.

14 ρ m χ Ωt : Heuristics First let us discuss the convergence of ρ m to χ Ωt. The following heuristics suggest that ρ m (, t) should converge to χ Ωt as m. The main trick is to consider the limit of the corresponding pressure variable p m = m m 1 ρm 1 m rather than ρ m, which remains continuous as m. The equation for p m is (ρ m ) t (ρ m ( p m + Φ) = 0 One then considers the corresponding PDE for p m, and show the convergence of p m to p by barrier (viscosity solutions) argument.

15 Convergence of ρ m, p m as m : Relevant work When Φ = 0 but with source term, in the patch case: Gil and Quirós (1998), K(2003). Weak solutions theory is developed for general initial data in Perthame-Quiros-Vazquez (2013), where they study the m limit of ρ t (ρ m ) = ρg(p).

16 ρ m χ Ωt We first define viscosity solution for (FB), and prove the comparison principle. Then we construct the lower limit of {p m } m as m : u 2 (x, t) := lim inf n m n (x,t) (y,s) <1/n p m (y, s), and use comparison arguments to show that u 2 is a supersolution of the Hele-Shaw problem (P) with the initial pressure satisfying {p 0 > 0} = Ω 0.

17 ρ m χ Ωt Roughly speaking, the strategy is then to show that the corresponding upper limit u 1 of {p m } m is a subsolution of (P) with the initial data p 0. Then due to the comparison principle we can conclude that u 1 (u 2 ). Since u 1 u 2 by definition, it follows that u 1 = (u 2 ) and (u 1 ) = u 2. Let Ω t := {u 1 (, t) > 0}. Then (Ω t ) 0 = (Ω t ) 0 and ρ m uniformly converges to χ Ωt away from Ω t with initial data u 0 = χ Ω0.

18 ρ m ρ : based on the JKO scheme Next we proceed to show ρ m ρ. The goal is to show W 2 (ρ m, ρ ) Cm 1/24, where W 2 is the 2-Wasserstein distance. To this end, let us first compare discrete-time solutions over one time step. Let µ m and µ be the respective minimizer of the following JKO scheme for one time step: µ m = argmin E m [ρ] + 1 ρ P 2 (R d ) 2 t W 2 2 (ρ, ρ 0 ) µ = argmin E [ρ] + 1 ρ P 2 (R d ) 2 t W 2 2 (ρ, ρ 0 ) We want to estimate W 2 (µ m, µ ): the main difficulty is that µ m may not be in L.

19 ρ m ρ Towards a contradiction, suppose W 2 (µ m, µ ) is large; in this case we want to find a better competitor µ, such that E m [ µ] t W 2 2 ( µ, ρ 0 ) + E [ µ] t W 2 2 ( µ, ρ 0 ) <E m [µ m ] t W 2 2 (µ m, ρ 0 ) + E [µ ] t W 2 2 (µ, ρ 0 ) This means µ would at least beat one of µ m and µ! How do we find such µ? First guess: Choose µ as the midpoint (along the generalized geodesics) of µ m and µ. This choice saves the distance. But E [ µ] may be infinite.

20 ρ m ρ A suitable competitor µ can be found as follows: 1 Even though the maximum density of µ m may exceed 1, (µm 1) + dx m 1/2 for m > 2. 2 Thus we can find some η m 1, such that W 2 (η m, µ m ) m 1/4, and E m [η m ] E m [µ m ]. 3 One can then choose µ as the midpoint (along the generalized geodesics) of η m and µ. (Note that µ 1.) 4 µ would be better than either µ m or µ if W 2 (µ m, µ ) m 1/8.

21 ρ m ρ :Controlling the distance for multiple time steps m m ρ 3 m m ρ 2 m ρ 0 m ρ1 m ρ 1 d m η 2 1 δ d 2 d 3 δ ρ 2 m η 3 δ ρ 3 m

22 Confinement and long time behavior For 1 < m, if Φ(x) as x, using the comparison principle, we know that if the initial data is compactly supported, the discrete solution to JKO scheme will be uniformly confined for all time steps. If Φ is strictly convex, ρ converges to the global minimizer (which is χ {Φ(x) C} for some C) exponentially fast in W 2 distance. χ Ωt t equilibrium profile χ {Φ(x) C}

23 Many open questions remain. The ultimate goal would be to try to generalize the continuity equation with L constraint, possibly characterized as the singular limit of the porous medium equation with drift ρ t (ρ m ) + ( vρ) = 0. as m.

24 Part II: Aggregation with Height constraint Next we discuss a different but relevant problem, which can be formally viewed as the singular limit of Patlak-Keller-Segel (PKS) equation: ρ t (ρ m ) (ρ (ρ N )) = 0. This is joint work with Katy Craig (UCLA) and Yao Yao (UW Madison).

25 Interaction energy Let ρ (, t) be the gradient flow of { Ẽ [ρ] = R ρ(ρ N )dx for ρ otherwise where N is the Newtonian potential. Uniqueness of the minimizer follows from certain convexity properties of Ẽ given by Carrillo, Lisini and Mainini (2012). The stability of the JKO scheme is much weaker due to the weak convexity properties of the energy Ẽ.

26 Open questions: Does the discrete-time solutions converge to a gradient flow solution in a stable way? Can one characterize the corresponding gradient flow solution? As t, does ρ eventually converge to the characteristic function of a ball? yes if n=2 Does the solutions of (PKS) converge to ρ as m? Open

27 Convexity property of E To show the convergence of discrete-time scheme with E, we recall the notion of ω-convexity: Definition (Carrillo-Lisini-Mainini) E is called ω- convex if for ρ 1, ρ 2 P 2,ac (R d ) we have E(ρ t ) (1 t)e(ρ 0 ) + te(ρ 1 ) +C[(1 t)ω(t 2 W 2 2 (ρ 0, ρ 1 )) + tω((1 t) 2 W 2 2 (ρ 0, ρ 1 ))], where ω(x) = x ln x for small x.

28 Contraction Inequality for ω-convex energy We then have the following contraction inequality. Theorem ( K. Craig) Ẽ is ω-convex. Suppose E is ω-convex, then the corresponding solutions of the JKO scheme satisfies f τ (W 2 (µ τ, ν τ )) W 2 2 (µ 0, ν 0 ) + Cτ 2 ln τ, where f τ (x) = x cτω(x). Based on above inequality, one can follow the argument of Crandall-Liggett to obtain a recursive inequality to estimate W 2 (µ τ, ν h ), and thus to show that JKO scheme converges to a unique limit ρ in Wasserstein distance.

29 Characterization of the gradient flow solution Next we consider characterizing ρ with a free boundary problem. The corresponding approximating energy is ˆ 1 Ẽ m (ρ) := m ρm + ρn ρ, But then we realize that the corresponding gradient flow of above energy is hard to analyze due to the lack of convexity properties of Ẽ m. The main difficulty lies in the lack of L -bound for the discrete-time gradient flow solutions. In fact it is open whether the discrete-time solutions converges to the continuum solutions of (PKS) in spite of their formal connection.

30 Characterization of the gradient flow solution Hence we will use instead the following energy ˆ 1 E m (ρ) := m ρm + ρn ρ, which is ω-convex, and show that the corresponding gradient flow solution ρ m converges.

31 aggregation with density constraint: preliminary results As before, we only consider the patch case, i.e. when ρ 0 = χ Ω0. Here the corresponding free boundary problem is: (FB) p = Φ = 1 in {p(, t) > 0} =: Ω(t); V = Dp ν Φ on Ω(t), where Φ = χ Ω(t) N. Theorem (Craig-K-Yao) ρ m converges to ρ = χ Ω(t) where Ω(t) solves (FB).

32 Long time behavior of ρ Using the free boundary formulation for ρ, we can show the following: Theorem (Craig-K-Yao) When n = 2, Ω t converges to a ball as t. We prove this by showing that the second moment ρ (x, t) x 2 dx decreases in time unless ρ = χ B.

33 Computing the second moment The evolution of M 2 [ρ(t)] is given by ˆ ˆ d dt M 2[ρ(t)] = ρ (ρ N ) xdx ρ p xdx R 2 R 2 = 1 ˆ ˆ ˆ ( x y) x ρ(x)ρ(y) 2π R 2 R 2 x y 2 dydx = 1 ˆ ˆ ˆ ρ(x)ρ(y)dydx + 2 p(x)dx 4π R 2 R 2 Ω(t) = 1 ˆ 4π Ω(t) p(x)dx. Ω(t) The quantity above is negative unless Ω(t) is a ball due to [Talenti, 1976]. Ω(t) p xdx

34 References D. Alexander, I. Kim and Y. Yao, Quasi-static evolution and crowded transport, Nonlinearity (2014), Vol. 27 no. 4, J. A.Carrillo, S. Lisini, and E. Mainini. Uniqueness for Keller-Segel-type chemotaxis models. Discrete Contin. Dyn. Syst., 34(4): , B. Maury, A. Roudneff-Chupin, F. Santambrogio and J. Venel, Handling Congestion in Crowd Motion Modeling. Networks and Heterogeneous Media (2011), Vol 6, no. 3, pp B. Maury, A. Roudneff-Chupin and F. Santambrogio, A macroscopic Crowd Motion Model of the gradient-flow type, M3AS (2010), Vol 20 no. 10, pp Thank you for your attention!

### 1 if 1 x 0 1 if 0 x 1

Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

### Notes V General Equilibrium: Positive Theory. 1 Walrasian Equilibrium and Excess Demand

Notes V General Equilibrium: Positive Theory In this lecture we go on considering a general equilibrium model of a private ownership economy. In contrast to the Notes IV, we focus on positive issues such

### 6. Metric spaces. In this section we review the basic facts about metric spaces. d : X X [0, )

6. Metric spaces In this section we review the basic facts about metric spaces. Definitions. A metric on a non-empty set X is a map with the following properties: d : X X [0, ) (i) If x, y X are points

### FIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper.

FIRST YEAR CALCULUS WWLCHENW L c WWWL W L Chen, 1982, 2008. 2006. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It It is is

### An optimal transportation problem with import/export taxes on the boundary

An optimal transportation problem with import/export taxes on the boundary Julián Toledo Workshop International sur les Mathématiques et l Environnement Essaouira, November 2012..................... Joint

### Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1

Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1 J. Zhang Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing

### EXIT TIME PROBLEMS AND ESCAPE FROM A POTENTIAL WELL

EXIT TIME PROBLEMS AND ESCAPE FROM A POTENTIAL WELL Exit Time problems and Escape from a Potential Well Escape From a Potential Well There are many systems in physics, chemistry and biology that exist

### Example 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum. asin. k, a, and b. We study stability of the origin x

Lecture 4. LaSalle s Invariance Principle We begin with a motivating eample. Eample 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum Dynamics of a pendulum with friction can be written

### Research Article Stability Analysis for Higher-Order Adjacent Derivative in Parametrized Vector Optimization

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 510838, 15 pages doi:10.1155/2010/510838 Research Article Stability Analysis for Higher-Order Adjacent Derivative

### Math 317 HW #5 Solutions

Math 317 HW #5 Solutions 1. Exercise 2.4.2. (a) Prove that the sequence defined by x 1 = 3 and converges. x n+1 = 1 4 x n Proof. I intend to use the Monotone Convergence Theorem, so my goal is to show

### The Heat Equation. Lectures INF2320 p. 1/88

The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)

### FIELDS-MITACS Conference. on the Mathematics of Medical Imaging. Photoacoustic and Thermoacoustic Tomography with a variable sound speed

FIELDS-MITACS Conference on the Mathematics of Medical Imaging Photoacoustic and Thermoacoustic Tomography with a variable sound speed Gunther Uhlmann UC Irvine & University of Washington Toronto, Canada,

### 10. Proximal point method

L. Vandenberghe EE236C Spring 2013-14) 10. Proximal point method proximal point method augmented Lagrangian method Moreau-Yosida smoothing 10-1 Proximal point method a conceptual algorithm for minimizing

### GenOpt (R) Generic Optimization Program User Manual Version 3.0.0β1

(R) User Manual Environmental Energy Technologies Division Berkeley, CA 94720 http://simulationresearch.lbl.gov Michael Wetter MWetter@lbl.gov February 20, 2009 Notice: This work was supported by the U.S.

### Lecture 12 Basic Lyapunov theory

EE363 Winter 2008-09 Lecture 12 Basic Lyapunov theory stability positive definite functions global Lyapunov stability theorems Lasalle s theorem converse Lyapunov theorems finding Lyapunov functions 12

### Sequences and Convergence in Metric Spaces

Sequences and Convergence in Metric Spaces Definition: A sequence in a set X (a sequence of elements of X) is a function s : N X. We usually denote s(n) by s n, called the n-th term of s, and write {s

### Sensitivity analysis of utility based prices and risk-tolerance wealth processes

Sensitivity analysis of utility based prices and risk-tolerance wealth processes Dmitry Kramkov, Carnegie Mellon University Based on a paper with Mihai Sirbu from Columbia University Math Finance Seminar,

### Reference: Introduction to Partial Differential Equations by G. Folland, 1995, Chap. 3.

5 Potential Theory Reference: Introduction to Partial Differential Equations by G. Folland, 995, Chap. 3. 5. Problems of Interest. In what follows, we consider Ω an open, bounded subset of R n with C 2

### Valuation and Optimal Decision for Perpetual American Employee Stock Options under a Constrained Viscosity Solution Framework

Valuation and Optimal Decision for Perpetual American Employee Stock Options under a Constrained Viscosity Solution Framework Quan Yuan Joint work with Shuntai Hu, Baojun Bian Email: candy5191@163.com

### Course 221: Analysis Academic year , First Semester

Course 221: Analysis Academic year 2007-08, First Semester David R. Wilkins Copyright c David R. Wilkins 1989 2007 Contents 1 Basic Theorems of Real Analysis 1 1.1 The Least Upper Bound Principle................

### Extremal equilibria for reaction diffusion equations in bounded domains and applications.

Extremal equilibria for reaction diffusion equations in bounded domains and applications. Aníbal Rodríguez-Bernal Alejandro Vidal-López Departamento de Matemática Aplicada Universidad Complutense de Madrid,

### Notes on weak convergence (MAT Spring 2006)

Notes on weak convergence (MAT4380 - Spring 2006) Kenneth H. Karlsen (CMA) February 2, 2006 1 Weak convergence In what follows, let denote an open, bounded, smooth subset of R N with N 2. We assume 1 p

### A Simple Model of Price Dispersion *

Federal Reserve Bank of Dallas Globalization and Monetary Policy Institute Working Paper No. 112 http://www.dallasfed.org/assets/documents/institute/wpapers/2012/0112.pdf A Simple Model of Price Dispersion

### 2.3 Convex Constrained Optimization Problems

42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions

### OPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN. E.V. Grigorieva. E.N. Khailov

DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS Supplement Volume 2005 pp. 345 354 OPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN E.V. Grigorieva Department of Mathematics

### Stochastic Inventory Control

Chapter 3 Stochastic Inventory Control 1 In this chapter, we consider in much greater details certain dynamic inventory control problems of the type already encountered in section 1.3. In addition to the

### Random graphs with a given degree sequence

Sourav Chatterjee (NYU) Persi Diaconis (Stanford) Allan Sly (Microsoft) Let G be an undirected simple graph on n vertices. Let d 1,..., d n be the degrees of the vertices of G arranged in descending order.

### Motion of a Leaky Tank Car

1 Problem Motion of a Leaky Tank Car Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princeton, NJ 8544 (December 4, 1989; updated October 1, 214) Describe the motion of a tank car initially

### Estimating the Degree of Activity of jumps in High Frequency Financial Data. joint with Yacine Aït-Sahalia

Estimating the Degree of Activity of jumps in High Frequency Financial Data joint with Yacine Aït-Sahalia Aim and setting An underlying process X = (X t ) t 0, observed at equally spaced discrete times

### Some stability results of parameter identification in a jump diffusion model

Some stability results of parameter identification in a jump diffusion model D. Düvelmeyer Technische Universität Chemnitz, Fakultät für Mathematik, 09107 Chemnitz, Germany Abstract In this paper we discuss

### Pacific Journal of Mathematics

Pacific Journal of Mathematics GLOBAL EXISTENCE AND DECREASING PROPERTY OF BOUNDARY VALUES OF SOLUTIONS TO PARABOLIC EQUATIONS WITH NONLOCAL BOUNDARY CONDITIONS Sangwon Seo Volume 193 No. 1 March 2000

### t := maxγ ν subject to ν {0,1,2,...} and f(x c +γ ν d) f(x c )+cγ ν f (x c ;d).

1. Line Search Methods Let f : R n R be given and suppose that x c is our current best estimate of a solution to P min x R nf(x). A standard method for improving the estimate x c is to choose a direction

### Error Bound for Classes of Polynomial Systems and its Applications: A Variational Analysis Approach

Outline Error Bound for Classes of Polynomial Systems and its Applications: A Variational Analysis Approach The University of New South Wales SPOM 2013 Joint work with V. Jeyakumar, B.S. Mordukhovich and

### Properties of BMO functions whose reciprocals are also BMO

Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a non-negative BMO-function w, whose reciprocal is also in BMO, belongs to p> A p,and

### Solvability of Fractional Dirichlet Problems with Supercritical Gradient Terms.

Solvability of Fractional Dirichlet Problems with Supercritical Gradient Terms. Erwin Topp P. Universidad de Santiago de Chile Conference HJ2016, Rennes, France May 31th, 2016 joint work with Gonzalo Dávila

### Shape Optimization Problems over Classes of Convex Domains

Shape Optimization Problems over Classes of Convex Domains Giuseppe BUTTAZZO Dipartimento di Matematica Via Buonarroti, 2 56127 PISA ITALY e-mail: buttazzo@sab.sns.it Paolo GUASONI Scuola Normale Superiore

### Online Learning, Stability, and Stochastic Gradient Descent

Online Learning, Stability, and Stochastic Gradient Descent arxiv:1105.4701v3 [cs.lg] 8 Sep 2011 September 9, 2011 Tomaso Poggio, Stephen Voinea, Lorenzo Rosasco CBCL, McGovern Institute, CSAIL, Brain

### 10.2 Series and Convergence

10.2 Series and Convergence Write sums using sigma notation Find the partial sums of series and determine convergence or divergence of infinite series Find the N th partial sums of geometric series and

### THE BANACH CONTRACTION PRINCIPLE. Contents

THE BANACH CONTRACTION PRINCIPLE ALEX PONIECKI Abstract. This paper will study contractions of metric spaces. To do this, we will mainly use tools from topology. We will give some examples of contractions,

### Chap2: The Real Number System (See Royden pp40)

Chap2: The Real Number System (See Royden pp40) 1 Open and Closed Sets of Real Numbers The simplest sets of real numbers are the intervals. We define the open interval (a, b) to be the set (a, b) = {x

### ARBITRAGE-FREE OPTION PRICING MODELS. Denis Bell. University of North Florida

ARBITRAGE-FREE OPTION PRICING MODELS Denis Bell University of North Florida Modelling Stock Prices Example American Express In mathematical finance, it is customary to model a stock price by an (Ito) stochatic

### The Goldberg Rao Algorithm for the Maximum Flow Problem

The Goldberg Rao Algorithm for the Maximum Flow Problem COS 528 class notes October 18, 2006 Scribe: Dávid Papp Main idea: use of the blocking flow paradigm to achieve essentially O(min{m 2/3, n 1/2 }

### Universidad de Montevideo Macroeconomia II. The Ramsey-Cass-Koopmans Model

Universidad de Montevideo Macroeconomia II Danilo R. Trupkin Class Notes (very preliminar) The Ramsey-Cass-Koopmans Model 1 Introduction One shortcoming of the Solow model is that the saving rate is exogenous

### CONSTANT-SIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE p-laplacian. Pasquale Candito and Giuseppina D Aguí

Opuscula Math. 34 no. 4 2014 683 690 http://dx.doi.org/10.7494/opmath.2014.34.4.683 Opuscula Mathematica CONSTANT-SIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE p-laplacian Pasquale

### Convex analysis and profit/cost/support functions

CALIFORNIA INSTITUTE OF TECHNOLOGY Division of the Humanities and Social Sciences Convex analysis and profit/cost/support functions KC Border October 2004 Revised January 2009 Let A be a subset of R m

Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650

### Section 3 Sequences and Limits

Section 3 Sequences and Limits Definition A sequence of real numbers is an infinite ordered list a, a 2, a 3, a 4,... where, for each n N, a n is a real number. We call a n the n-th term of the sequence.

### Fuzzy Differential Systems and the New Concept of Stability

Nonlinear Dynamics and Systems Theory, 1(2) (2001) 111 119 Fuzzy Differential Systems and the New Concept of Stability V. Lakshmikantham 1 and S. Leela 2 1 Department of Mathematical Sciences, Florida

### k=1 k2, and therefore f(m + 1) = f(m) + (m + 1) 2 =

Math 104: Introduction to Analysis SOLUTIONS Alexander Givental HOMEWORK 1 1.1. Prove that 1 2 +2 2 + +n 2 = 1 n(n+1)(2n+1) for all n N. 6 Put f(n) = n(n + 1)(2n + 1)/6. Then f(1) = 1, i.e the theorem

### First Welfare Theorem

First Welfare Theorem Econ 2100 Fall 2015 Lecture 17, November 2 Outline 1 First Welfare Theorem 2 Preliminaries to Second Welfare Theorem Last Class Definitions A feasible allocation (x, y) is Pareto

### CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.

CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,

### ON FIBER DIAMETERS OF CONTINUOUS MAPS

ON FIBER DIAMETERS OF CONTINUOUS MAPS PETER S. LANDWEBER, EMANUEL A. LAZAR, AND NEEL PATEL Abstract. We present a surprisingly short proof that for any continuous map f : R n R m, if n > m, then there

### The Black-Scholes-Merton Approach to Pricing Options

he Black-Scholes-Merton Approach to Pricing Options Paul J Atzberger Comments should be sent to: atzberg@mathucsbedu Introduction In this article we shall discuss the Black-Scholes-Merton approach to determining

### arxiv:1112.0829v1 [math.pr] 5 Dec 2011

How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman Thomas P. Hayes arxiv:1112.0829v1 [math.pr] 5 Dec 2011 Abstract Consider a gambling game in which we are allowed to repeatedly

### x if x 0, x if x < 0.

Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the

### 15 Limit sets. Lyapunov functions

15 Limit sets. Lyapunov functions At this point, considering the solutions to ẋ = f(x), x U R 2, (1) we were most interested in the behavior of solutions when t (sometimes, this is called asymptotic behavior

### Lecture 13 Linear quadratic Lyapunov theory

EE363 Winter 28-9 Lecture 13 Linear quadratic Lyapunov theory the Lyapunov equation Lyapunov stability conditions the Lyapunov operator and integral evaluating quadratic integrals analysis of ARE discrete-time

### Chapter 15 Introduction to Linear Programming

Chapter 15 Introduction to Linear Programming An Introduction to Optimization Spring, 2014 Wei-Ta Chu 1 Brief History of Linear Programming The goal of linear programming is to determine the values of

### 1 The basic equations of fluid dynamics

1 The basic equations of fluid dynamics The main task in fluid dynamics is to find the velocity field describing the flow in a given domain. To do this, one uses the basic equations of fluid flow, which

### 14.451 Lecture Notes 10

14.451 Lecture Notes 1 Guido Lorenzoni Fall 29 1 Continuous time: nite horizon Time goes from to T. Instantaneous payo : f (t; x (t) ; y (t)) ; (the time dependence includes discounting), where x (t) 2

### Finite covers of a hyperbolic 3-manifold and virtual fibers.

Claire Renard Institut de Mathématiques de Toulouse November 2nd 2011 Some conjectures. Let M be a hyperbolic 3-manifold, connected, closed and oriented. Theorem (Kahn, Markovic) The fundamental group

### CHAPTER 7 APPLICATIONS TO MARKETING. Chapter 7 p. 1/54

CHAPTER 7 APPLICATIONS TO MARKETING Chapter 7 p. 1/54 APPLICATIONS TO MARKETING State Equation: Rate of sales expressed in terms of advertising, which is a control variable Objective: Profit maximization

### Notes on metric spaces

Notes on metric spaces 1 Introduction The purpose of these notes is to quickly review some of the basic concepts from Real Analysis, Metric Spaces and some related results that will be used in this course.

### Metric Spaces. Chapter 1

Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence

### Differential Relations for Fluid Flow. Acceleration field of a fluid. The differential equation of mass conservation

Differential Relations for Fluid Flow In this approach, we apply our four basic conservation laws to an infinitesimally small control volume. The differential approach provides point by point details of

### MATH 425, PRACTICE FINAL EXAM SOLUTIONS.

MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator

### ON CERTAIN DOUBLY INFINITE SYSTEMS OF CURVES ON A SURFACE

i93 c J SYSTEMS OF CURVES 695 ON CERTAIN DOUBLY INFINITE SYSTEMS OF CURVES ON A SURFACE BY C H. ROWE. Introduction. A system of co 2 curves having been given on a surface, let us consider a variable curvilinear

### Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

### Math 563 Measure Theory Project 1 (Funky Functions Group) Luis Zerón, Sergey Dyachenko

Math 563 Measure Theory Project (Funky Functions Group) Luis Zerón, Sergey Dyachenko 34 Let C and C be any two Cantor sets (constructed in Exercise 3) Show that there exists a function F: [,] [,] with

### 2DI36 Statistics. 2DI36 Part II (Chapter 7 of MR)

2DI36 Statistics 2DI36 Part II (Chapter 7 of MR) What Have we Done so Far? Last time we introduced the concept of a dataset and seen how we can represent it in various ways But, how did this dataset came

### LECTURE 2: Stress Conditions at a Fluid-fluid Interface

LETURE 2: tress onditions at a Fluid-fluid Interface We proceed by deriving the normal and tangential stress boundary conditions appropriate at a fluid-fluid interface characterized by an interfacial tension

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22 RAVI VAKIL CONTENTS 1. Discrete valuation rings: Dimension 1 Noetherian regular local rings 1 Last day, we discussed the Zariski tangent space, and saw that it

### Differentiating under an integral sign

CALIFORNIA INSTITUTE OF TECHNOLOGY Ma 2b KC Border Introduction to Probability and Statistics February 213 Differentiating under an integral sign In the derivation of Maximum Likelihood Estimators, or

### 14. Dirichlet polygons: the construction

14. Dirichlet polygons: the construction 14.1 Recap Let Γ be a Fuchsian group. Recall that a Fuchsian group is a discrete subgroup of the group Möb(H) of all Möbius transformations. In the last lecture,

### Taylor Polynomials and Taylor Series Math 126

Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will

### Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions

Natalia Lazzati Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions Note 5 is based on Madden (1986, Ch. 1, 2, 4 and 7) and Simon and Blume (1994, Ch. 13 and 21). Concave functions

### IMPLEMENTING ARROW-DEBREU EQUILIBRIA BY TRADING INFINITELY-LIVED SECURITIES

IMPLEMENTING ARROW-DEBREU EQUILIBRIA BY TRADING INFINITELY-LIVED SECURITIES Kevin X.D. Huang and Jan Werner DECEMBER 2002 RWP 02-08 Research Division Federal Reserve Bank of Kansas City Kevin X.D. Huang

### Using Generalized Forecasts for Online Currency Conversion

Using Generalized Forecasts for Online Currency Conversion Kazuo Iwama and Kouki Yonezawa School of Informatics Kyoto University Kyoto 606-8501, Japan {iwama,yonezawa}@kuis.kyoto-u.ac.jp Abstract. El-Yaniv

### Critical points of once continuously differentiable functions are important because they are the only points that can be local maxima or minima.

Lecture 0: Convexity and Optimization We say that if f is a once continuously differentiable function on an interval I, and x is a point in the interior of I that x is a critical point of f if f (x) =

### 1. (First passage/hitting times/gambler s ruin problem:) Suppose that X has a discrete state space and let i be a fixed state. Let

Copyright c 2009 by Karl Sigman 1 Stopping Times 1.1 Stopping Times: Definition Given a stochastic process X = {X n : n 0}, a random time τ is a discrete random variable on the same probability space as

### Low upper bound of ideals, coding into rich Π 0 1 classes

Low upper bound of ideals, coding into rich Π 0 1 classes Antonín Kučera the main part is a joint project with T. Slaman Charles University, Prague September 2007, Chicago The main result There is a low

### A Network Flow Approach in Cloud Computing

1 A Network Flow Approach in Cloud Computing Soheil Feizi, Amy Zhang, Muriel Médard RLE at MIT Abstract In this paper, by using network flow principles, we propose algorithms to address various challenges

### The integrating factor method (Sect. 2.1).

The integrating factor method (Sect. 2.1). Overview of differential equations. Linear Ordinary Differential Equations. The integrating factor method. Constant coefficients. The Initial Value Problem. Variable

### Duality of linear conic problems

Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least

### Notes on Elastic and Inelastic Collisions

Notes on Elastic and Inelastic Collisions In any collision of 2 bodies, their net momentus conserved. That is, the net momentum vector of the bodies just after the collision is the same as it was just

### {f 1 (U), U F} is an open cover of A. Since A is compact there is a finite subcover of A, {f 1 (U 1 ),...,f 1 (U n )}, {U 1,...

44 CHAPTER 4. CONTINUOUS FUNCTIONS In Calculus we often use arithmetic operations to generate new continuous functions from old ones. In a general metric space we don t have arithmetic, but much of it

### ECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE

ECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE YUAN TIAN This synopsis is designed merely for keep a record of the materials covered in lectures. Please refer to your own lecture notes for all proofs.

### Elasticity Theory Basics

G22.3033-002: Topics in Computer Graphics: Lecture #7 Geometric Modeling New York University Elasticity Theory Basics Lecture #7: 20 October 2003 Lecturer: Denis Zorin Scribe: Adrian Secord, Yotam Gingold

### Hydrodynamic Limits of Randomized Load Balancing Networks

Hydrodynamic Limits of Randomized Load Balancing Networks Kavita Ramanan and Mohammadreza Aghajani Brown University Stochastic Networks and Stochastic Geometry a conference in honour of François Baccelli

### Section 3 Sequences and Limits, Continued.

Section 3 Sequences and Limits, Continued. Lemma 3.6 Let {a n } n N be a convergent sequence for which a n 0 for all n N and it α 0. Then there exists N N such that for all n N. α a n 3 α In particular

### Systems with Persistent Memory: the Observation Inequality Problems and Solutions

Chapter 6 Systems with Persistent Memory: the Observation Inequality Problems and Solutions Facts that are recalled in the problems wt) = ut) + 1 c A 1 s ] R c t s)) hws) + Ks r)wr)dr ds. 6.1) w = w +

### Online Convex Optimization

E0 370 Statistical Learning heory Lecture 19 Oct 22, 2013 Online Convex Optimization Lecturer: Shivani Agarwal Scribe: Aadirupa 1 Introduction In this lecture we shall look at a fairly general setting

### Math 317 HW #7 Solutions

Math 17 HW #7 Solutions 1. Exercise..5. Decide which of the following sets are compact. For those that are not compact, show how Definition..1 breaks down. In other words, give an example of a sequence

### General Theory of Differential Equations Sections 2.8, 3.1-3.2, 4.1

A B I L E N E C H R I S T I A N U N I V E R S I T Y Department of Mathematics General Theory of Differential Equations Sections 2.8, 3.1-3.2, 4.1 Dr. John Ehrke Department of Mathematics Fall 2012 Questions

### Unraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets

Unraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets Nathaniel Hendren January, 2014 Abstract Both Akerlof (1970) and Rothschild and Stiglitz (1976) show that

### Ri and. i=1. S i N. and. R R i

The subset R of R n is a closed rectangle if there are n non-empty closed intervals {[a 1, b 1 ], [a 2, b 2 ],..., [a n, b n ]} so that R = [a 1, b 1 ] [a 2, b 2 ] [a n, b n ]. The subset R of R n is an

### Variational approach to restore point-like and curve-like singularities in imaging

Variational approach to restore point-like and curve-like singularities in imaging Daniele Graziani joint work with Gilles Aubert and Laure Blanc-Féraud Roma 12/06/2012 Daniele Graziani (Roma) 12/06/2012

### Chapter 7. Sealed-bid Auctions

Chapter 7 Sealed-bid Auctions An auction is a procedure used for selling and buying items by offering them up for bid. Auctions are often used to sell objects that have a variable price (for example oil)