Stochastic partial differential equation models for highway traffic

UNIVERSITY OF CALIFORNIA Santa Barbara Stochastic Partial Differential Equation Models for Highway Traffic A Dissertation submitted in partial satisfaction of the requirement for the degree of Doctor of Philosophy in Mathematics by Gunnar Gunnarsson Committee in charge: Professor Guillaume Bonnet, Committee Chairman Professor Michael Crandall, Committee Chairman Professor Bj¨orn Birnir September 2006 UMI Number: 3233741 UMI Microform 3233741 Copyright 2006 by ProQuest Information and Learning Company All rights reserved This microform edition is protected against unauthorized copying under Title 17, United States Code ProQuest Information and Learning Company 300 North Zeeb Road P.O Box 1346 Ann Arbor, MI 48106-1346 The dissertation of Gunnar Gunnarsson is approved Bj¨orn Birnir Michael Crandall, Committee Chairman Guillaume Bonnet, Committee Chairman September 2006 Stochastic Partial Differential Equation Models for Highway Traffic Copyright c 2006 by Gunnar Gunnarsson iii Dedication To my family iv Acknowledgements There are many people who I would like to thank for making my stay at the University of California, Santa Barbara possible, and as great as it was First I like to thank my family My parents for supporting my ambition to get higher education And my wife, Solla, for supporting me the whole time and taking care of our wonderful daughter after she was born Academically, I would like to especially thank professor Guillaume Bonnet for his endless support and help while I often found myself lost in the world of stochastics I am endebted to professor Michael Crandall for offering his help, at a crucial time, when I switched fields of research and to professor Bj¨orn Birnir for getting me in touch with Guillaume and for his guidance and assistance in many other matters Lastly, I thank the wonderful staff and faculty at the department of mathematics as well as my fellow graduate students My stay here, would certainly not have been this nice if it were not for the good times I spent in Medina Teel’s office, or with Bill Lyons in our shared office or on the balcony Also, I might have completely lost touch with the world of soccer, were it not for professor Darren Long Finally, I thank professor Daryl Cooper for selflessly taking over my teaching load in time of need Thank you all v Vita of Gunnar Gunnarsson Education PhD Mathematics, Sept 2006, University of California, Santa Barbara MA Mathematics, March 2002, University of California, Santa Barbara BSc Mathematics, May 2000, University of Iceland, Reykjavik Work Experience Lecturer and Teaching Assistant, 2000 - 2006, UC, Santa Barbara, USA Researcher, 2002, Decode Genetics, Reykjavik, Iceland Programmer, 1998 - 2000, K¨ogun hf., Reykjavik, Iceland Programmer, 1999, RISC, Research Inst of Symbolic Computing, Linz, Austria Teaching Assistant, 2001 - 2003, University of Iceland, Reykjavik, Iceland Awards and Fellowships Graduate Division Dissertation Fellowship, Fall 2005 UCSB Affiliates Dissertation Fellowship, Fall 2005 Memorial Fund of Helga Jonsdottir, Fall 2004 Teaching Award, Spring 2004 Thor Thors Fellowship, Spring 2001 Raymond L Wilder Award, Spring 2001 Fulbright Fellowship, June 2000 vi Abstract Stochastic Partial Differential Equation Models for Highway Traffic by Gunnar Gunnarsson We introduce a new model for multi-lane highway traffic, based on stochastic partial differential equations We prove that the model is well-posed; has one and only one solution We prove the existence constructively and thus derive a numerical scheme to compute the solution We examine measured traffic data and introduce a new method and algorithm to estimate the fundamental diagram, an integral part of almost every macroscopic highway traffic model vii Contents Background 1.1 Introduction 1.2 Notation 1.3 Models 1.3.1 Macroscopic Models 1.3.2 Microscopic Models 13 SPDE Model 18 2.1 Introduction 18 2.2 The SPDE Model 23 2.3 Discrete System 26 2.4 Convergence to the SPDE System 35 2.4.1 Mild Formulation 36 2.4.2 Tightness 39 2.4.3 Martingale Problem Representation 45 2.5 Uniqueness 49 2.6 Conclusions and Remarks 54 Calibration and Simulation 56 viii 3.1 3.2 Data Analysis 56 3.1.1 Viewing the Data 57 3.1.2 Fundamental Diagram 60 3.1.3 Lane Shifting 64 Numerical Simulations 65 Bibliography 67 A Relative Compactness Criteria 71 A.1 Arzela-Ascoli 71 A.2 Relative Compactness in L2γ 72 B The approximate heat kernels GN (t, x, y) ix 73 The middle picture is of the range near where congestion starts occuring We see the presence of two slopes, one positive and one negative The positive one represents unjammed, synchronized traffic (unstable state) while the negative one represents congested traffic (stable state) So we see that at the same density, the traffic is sometimes jammed and sometimes not One of the extensions we mentioned in Section 2.6 is to make Q stochastic That extension aims to capture this phenomenon, i.e in the critical region (around in our case) Q fluctuates between the two states, generating the so called inverse lambda shape, see for example [9] Preliminary numerical experiments with this extension have been very succesful in predicting stop-and-go waves Our algorithm works as follows: We isolate lines from a range of densities, see Figure 3.5, and then we find the “best slope” that fits the linear trend in the x 10 ρ ∈ [.1, 12] x 10 10 10 9 8 7 6 t t 4 3 2 1 ρ ∈ [.29, 3] x 10 ρ ∈ [.4, 45] t 0 500 1000 x 1500 2000 500 1000 x 1500 2000 500 1000 x 1500 2000 Figure 3.5: Density ranges data Our algorithm for that purpose is novel as far as we know It works in the following manner: • Fix N large 61 • For each, θ ∈ {0, π/N, 2π/N, , π} we calculate the distance of each point to a fixed line that cuts the x-axis at an angle of θ • For each θ make a histogram of the distances • Calculate a score for each histogram In our case we use the l2 norm of the histogram • The histogram with the highest score comes from the θ in which direction the data is distributed To clarify, see Figure 3.6 The choice of scoring function is by no means unique Best line is solid Distances from best line Density in [0.45,0.5] 30 0.8 25 0.6 0.4 20 0.2 15 −0.2 10 −0.4 −0.6 −0.8 −1 −0.5 0.5 −1 −1 Score −0.5 0.5 Distances from the dotted line 0.18 10 0.16 score 0.14 0.12 0.1 0.08 0.06 0.04 0.5 1.5 2.5 3.5 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 angle Figure 3.6: Finding the linear trend One can for example imagine using a linear combination of properly chosen lp 62 norms Finally, if we now plot 1/slope we get an estimate of the derivative of the fundamental diagram, see Figure 3.7, which suggests that we consider using triangular fundamental diagrams, see Figure 3.8 Estimate of derivative of fundamental diagram 0.05 0.04 0.03 0.02 0.01 −0.01 −0.02 −0.03 −0.04 −0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Figure 3.7: Inverse of estimated slopes In Section 3.2 below, we will use such a fundamental diagram Before moving to lane shifting, we mention that our algorithm is useful in finding trends in any data that has hidden parallel lines whose slope needs to be found An example of such data could be locations of landmine explosions In that case, if one assumes that the mines have been placed in parallel lines, the algorithm will determine the most likely direction of the mine laying, making it easier to disarm or remove the mines 63 −3 Estimated fundamental diagram x 10 Flux q∗ 0.1 0.2 0.3 0.4 0.5 Density 0.6 0.7 0.8 0.9 Figure 3.8: Estimate of fundamental diagram 3.1.3 Lane Shifting Lane shifting is a very random process We will show below that a large part of all lane shifts are made for no apparent reason We acknowledge two types of valid reasons for laneshifts, avoiding high density and moving towards/away from an exit or entry In Figure 3.9 we have plotted the densities surrounding vehicles that switch lanes, i.e the density on the lane that the vehicle is leaving (From Density) vs the density on the lane the vehicle is entering (To Density) Note that we have discounted all shifts from vehicles that either started or ended outside the highway As we can see, only approximately 60% of the 360 lane shifts go from higher to lower density, strongly suggesting that there should be a random element in any model that models lane shifting directly 64 820: Percentage of high to low 59% 0.9 0.8 To Density 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.2 0.4 0.6 From Density 0.8 Figure 3.9: Densities around lane switches (the diagonal is only for show) 3.2 Numerical Simulations Our extension of (1.3.5) consists of adding a random element to the lane shifting mechanism What we hope to accomplish with this, is that the random lane shifting will produce stop-and-go waves We look at the simplest case possible We consider a two lane ring road, M = 2, with the initial density equal and constant on both lanes In the deterministic setting, nothing happens since all densities are equal This is not the case in the stochastic setting as we see in Figure 3.10 The noise is mainly small, but sometimes it gets pushed up enough to produce a wave travelling backwards More extensive numerical experiments are expected in later publications 65 Lane simulated 10 0.5 Time in seconds 0.45 0.4 0.35 0.3 0 Length along road in ft Figure 3.10: Density on one of the lanes 66 10 Bibliography [1] Robert A Adams Sobolev spaces Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975 Pure and Applied Mathematics, Vol 65 [2] A Aw, A Klar, T Materne, and M Rascle Derivation of continuum traffic flow models from microscopic follow-the-leader models SIAM Journal on Applied Mathematics, 63(1):259–278, 2002 [3] A Aw and M Rascle Resurrection of ”second order” models of traffic flow SIAM Journal on Applied Mathematics, 60(3):916–938, 2000 [4] Sandra Cerrai Stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term Probab Theory Relat Fields, 2002 [5] Carlos F Daganzo Requiem for 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complete metric space, with norm · S Then a subset A ⊂ C([0, T ]; S) is relatively compact if and only if the following two conditions hold A.1-a the images of the coordinate projections are relatively compact, i.e πt (A) = {f (t) : f ∈ A} is relatively compact for all t ∈ [0, T ], A.1-b the functions are equicontinuous, i.e lim sup sup ω T (f, δ) = δ→0 f ∈A |t−s| the following three conditions hold A.2-a There exists an M > such that f L2γ < M for all f ∈ A A.2-b There exists a δ > such that for all f ∈ A, |h| < δ we have f (· + h) − f (·) L2γ we have R |GN (t + δ, x, y) − GN (t, x, y)| dy ≤ Kδ + O(δ ) where K is a constant independent of t and x and O(δ 2) is uniform in x and independent of t and corresponding bounds on the norms of space increments B-a’ GN (t, x + h, ·) − GN (t, x, ·) p ≤ Chρ , for ρ < 2p − B-b’ G′N (t, x + h, ·) − G′N (t, x, ·) p ≤ Chρ , for ρ < 2p −1 All the above inequalities are standard in the literature except (3) We include a proof of it for completeness Proof of (3) We will use the notation from Section 2.4.1, more precisely the 74 notation around (2.4.1) R |GN (t + δ, x, y) − GN (t, x, y)| dy = y∈XN |P(Yt+δ = y|Y0 = x) − P(Yt = y|Y0 = x)| = y z y z = P(Yt = z|Y0 = x)P(Yt+δ = y|Yt = z) − pN (t, x, y) pN (t, x, z)pN (δ, z, y) − pN (t, x, y) |pN (t, x, y + )δ + pN (t, x, y)(1 − 2δ) = y + pN (t, x, y − )δ − pN (t, x, y) + O(δ 2)| =δ y pN (t, x, y+) − 2pN (t, x, y) + pN (t, x, y − ) + O(δ ) ≤ Kδ + O(δ ) where we have used the time homogeneity of the process Y to replace P(Yt+δ = y|Yt = z) with pN (δ, z, y) and the space homogeneity to get the uniformity in x of O(δ ) 75 ... Abstract Stochastic Partial Differential Equation Models for Highway Traffic by Gunnar Gunnarsson We introduce a new model for multi-lane highway traffic, based on stochastic partial differential equations... Committee Chairman Guillaume Bonnet, Committee Chairman September 2006 Stochastic Partial Differential Equation Models for Highway Traffic Copyright c 2006 by Gunnar Gunnarsson iii Dedication To my... fluid 1.3 Models In this section we take a closer look at some of the aforementioned models 1.3.1 Macroscopic Models Conservation of Mass A basis for almost every kind of PDE model for fluid