Efficient parameter estimation in a macroscopic traffic

Transportmetrica A: Transport Science, 2015 Vol. 11, No. 8, 702–715, http://dx.doi.org/10.1080/23249935.2015.1063022

Efficient parameter estimation in a macroscopic traffic flow model by discrete mollification Carlos D. Acostaa , Raimund Bürgerb∗ and Carlos E. Mejíac

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a Departamento

de Matemáticas y Estadística, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Colombia — Sede Manizales, Manizales, Colombia; b CI 2 MA and Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción, Casilla 160-C, Concepción, Chile; c Facultad de Ciencias, Escuela de Matemáticas, Universidad Nacional de Colombia — Sede Medellín, Medellín, Colombia (Received 21 June 2014; accepted 13 June 2015) Our concern is the numerical identification of traffic flow parameters in a macroscopic onedimensional model whose governing equation is strongly degenerate parabolic. The unknown parameters determine the flux and the diffusion terms. The parameters are estimated by repeatedly solving the corresponding direct problem under variation of the parameter values, starting from an initial guess, with the aim of minimizing the distance between a time-dependent observation and the corresponding numerical solution. The direct problem is solved by a modification of a well-known monotone finite difference scheme obtained by discretizing the nonlinear diffusive term by a formula that involves a discrete mollification operator. The mollified scheme occupies a larger stencil but converges under a less restrictive CourantFriedrichs-Lewy (CFL) condition, which allows one to employ a larger time step. The ability of the proposed procedure for the identification of traffic flow parameters is illustrated by a numerical experiment. Keywords: traffic flow; inverse problem; degenerate parabolic equation; parameter estimation; discrete mollification

1. Introduction Traffic modelling is an interdisciplinary area of research whose goal is to describe as accurately as possible the interactions between vehicles, drivers, infrastructures and regulations. According to the level of detail, there are three different approaches (Leclercq and Moutari 2007): microscopic, focused on the vehicles, macroscopic, based on the traffic stream and hybrid, combination of both. Our goal is to offer an efficient alternative, based on Acosta, Bürger, and Mejía (2014), for the solution of inverse problems (IPs) arising in a diffusively corrected Lighthill-WhithamRichards (DCLWR) traffic flow model, which is of macroscopic nature (Nelson 2000, 2002; Bürger and Karlsen 2003). The Lighthill-Whitham-Richards (LWR) model (Lighthill and Whitham 1955; Richards 1956) is, mathematically, a scalar hyperbolic conservation law. Several modifications and applications of this model have been proposed, for instance, a multiclass LWR model in which drivers differ according to their preferential velocity and a system of hyperbolic conservation laws is studied (Wong and Wong 2002; Benzoni-Gavage and Colombo 2003; Gupta and Katiyar 2007; Ngoduy 2010, 2011; Bürger, Chalons, and Villada 2013a), a diffusively corrected multiclass LWR *Corresponding author. Email: [email protected] c 2015 Hong Kong Society for Transportation Studies Limited "

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model (Bürger, Mulet, and Villada 2013b,c) where additional properties like anticipation lengths and reaction times are taken into account, a multi-commodity LWR model (Jin 2013) whose main distinction is between weaving and non-weaving vehicles, a model of traffic in an urban network (Berrone et al. 2012) with its negative effects on the population via different kinds of pollution, the LWR model on inhomogeneous highways (Sun et al. 2011) and the introduction of highway entries and exits (Bagnerini, Colombo, and Corli 2006) through the consideration of source terms in the system of hyperbolic conservation laws. This paper is concerned with the efficient numerical solution of an IP by repeatedly solving the direct problem by an explicit finite difference scheme enhanced by discrete mollification. We propose a fast and reliable technique for the numerical identification of traffic flow parameters that has been successful for the estimation of parameters in a sedimentation-consolidation model that share a similar set of equations (see Acosta, Bürger, and Mejía 2014 for details). The main components of our contribution are: (1) The governing equation is a nonlinear initial-boundary value problem (IBVP) for a strongly degenerate parabolic equation in one space dimension. (2) The unknown parameters appear in the flux and the diffusion terms. (3) The IP consists on the identification of the unknown parameters. The required overposed data is a future time observation of the density of cars (cars per mile). The iterative procedure consists on comparing the overposed data with the repeated solutions of the direct problem and adjust the parameters accordingly. (4) The direct problem is solved by an explicit finite difference scheme (Evje and Karlsen 2000; Bürger and Karlsen, 2004) enhanced by discrete mollification (Murio 1993, 2002; Acosta, Bürger, and Mejía 2012, 2014) in order to accelerate computations but preserving stability. The rest of the paper is organized as follows: Section 2 is dedicated to the mathematical setting, that is, the IBVP in which the traffic model is based. In Section 3, we outline the traffic flow model (Section 3.1), describe discrete mollification (Section 3.2) and introduce the basic and mollified numerical schemes for the approximation of Equation (1a) (Section 3.3). Section 4 presents the numerical solution of the parameter identification problem. It consists in the formulation of the continuous IP (Section 4.1), its discrete version (Section 4.2) and the numerical algorithm (Section 4.3), including the Nelder-Mead simplex method to successively minimize the cost functional. An illustrative numerical example and some conclusions are presented in Section 5. 2. Mathematical setting We are interested in the numerical identification of unknown parameters appearing in the flux and diffusion terms of the following IBVP for a strongly degenerate parabolic equation in one space dimension: ∂ 2 A(u) ∂u ∂f (u) , (x, t) ∈ "T := (0, L) × (0, T], T > 0, + = ∂t ∂x ∂x2 u(x, 0) = u0 (x), x ∈ [0, L], ∂A(u) |x=0 = ψ0 (t), ∂x ∂A(u) f (u) − |x=L = ψL (t), ∂x f (u) −

(1a) (1b)

t ∈ (0, T],

(1c)

t ∈ (0, T],

(1d)

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where A is an integrated diffusion coefficient, that is,


A(u) =

!

0

u

a(s) ds,

a(u) ≥ 0.

(2)

The diffusion function a is assumed to be integrable and is allowed to vanish on u-intervals of positive length, on which Equation (1a) turns into a first-order hyperbolic conservation law, so that (1a) is strongly degenerate parabolic. On the other hand, we assume that f is piecewise smooth and Lipschitz continuous. Under suitable choices of the functions f, a, ψ0 and ψL , the IBVP (1) describes a variety of real processes of interest in engineering, for instance, the sedimentation–consolidation process of a solid-liquid suspension (Bürger and Karlsen 2001). In this paper, we focus on Equation (1) as a model of the evolution of the local car density on a finite road segment for a diffusively corrected kinematic traffic model (Nelson 2000, 2002; Bürger and Karlsen 2003; Bürger, Mulet, and Villada 2013b,c). Due to its strongly degenerate parabolic nature and the nonlinearity of the convective flux, solutions of Equation (1a) are, in general, discontinuous even if u0 is smooth, and need to be defined as weak solutions along with an entropy condition to select the physically relevant solution, the entropy solution. For the definition, existence and uniqueness of entropy solutions of Equation (1) we refer to Bürger and Karlsen (2001). For the original LWR model ∂u ∂f (u) + = 0, ∂t ∂x

(3)

to which the present governing Equation (1a) is reduced either partially (namely for those u-values for which a(u) = 0, usually for 0 ≤ u ≤ uc , where uc is a critical density whose significance is discussed below) or totally (if we assume that a ≡ 0 and hence A ≡ 0), the entropy condition is also known as driver’s ride impulse or speedup impulse (Ansorge 1990; Velan and Florian 2002; Bürger et al. 2008b), which compels that in certain situations that allow several weak solutions, that solution is selected that allows drivers to speed up. For the parameter estimation, we compute repeated numerical solutions of the direct problem Equation (1) under successive variation of parameters appearing in f and a. The goal is to minimize a certain cost function defined below. A numerical scheme suitable for the solution of the direct problem is the explicit, conservative monotone finite difference scheme first introduced by Evje and Karlsen (2000) for initial value problems of Equation (1a), and then adapted to IBVP (Bürger and Karlsen 2001). To describe the essential advantage in using mollified versions of this scheme, assume that $t and $x are the corresponding time and space steps of a Cartesian grid introduced on "T , and define λ := $t/$x and µ := $t/$x2 = λ/$x. Then the scheme by Evje and Karlsen (2000), henceforth called basic scheme, converges to the unique entropy solution of Equation (1) provided that $t and $x are always chosen such that the following Courant-Friedrichs-Lewy (CFL) condition is satisfied: λ)f * )∞ + 2µ)a)∞ ≤ 1.

(4)

In the basic scheme, the term ∂ 2 A(u)/∂x2 is discretized conservatively by standard second finite differences of A(u). The discrete mollification method provides an alternative conservative, centered discretization of ∂ 2 A(u)/∂x2 on a stencil of in total 2η + 1 points, where η ∈ N is a parameter. Acosta, Bürger, and Mejía (2012) show that this device, which defines what we call mollified scheme, preserves monotonicity and convergence (to the entropy solution) of the basic

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λ)f * )∞ + 2µεη )a)∞ ≤ 1,

(5)

where εη < 1. (For the particular mollification weights considered herein, we get ε3 = 0.7130, ε5 = 0.3969 and ε8 = 0.1960.) Clearly, condition (5) is more favorable than Equation (4) since it shows that for a given value of $x, mollified schemes may proceed by larger time steps. As was shown by Acosta, Bürger, and Mejía (2012), these schemes are even competitive in efficiency in terms of error reduction per CPU time despite the additional effort for the evaluation of wider stencils, and the slightly larger numerical diffusion. For this reason, mollified schemes are an attractive option for computations that are usually time consuming, such as parameter identification problems since the same IBVP (but with varied parameters) has to be solved repeatedly. More efficient high-resolution schemes such as weighted essentially non-oscillatory or discontinuous Galerkin methods are available to discretize the original LWR model Equation (3) or its multiclass variants (not considered herein), see for example, Shu (1998) and Le Veque (2002) for general introductions to these schemes and Zhang et al. (2003) and Zhang and Liu (2005) (and references cited in these works) for the application to traffic flow. However, in view of the discretization of the diffusive term, whose basic or mollified versions (discussed in Section 3.3) are of second-order accuracy, high-order accuracy is lost if these methods are employed for Equation (1). Moreover, convergence to an entropy solution of a strongly degenerate parabolic equation such as Equation (1a) can currently be ensured for monotone, and therefore first-order schemes. This convergence property, sometimes called ‘entropicity’, is essential in light of the physical interpretation of entropy solutions in the context of traffic modelling, and to embed the numerical treatment into the well-posedness theory for entropy solutions of Equation (1a). Entropicity is also ensured for semi-implicit schemes for Equation (1a) with an implicit treatment of the diffusion term (cf. Evje and Karlsen 1999; Bürger, Coronel, and Sepúlveda 2006), for which the CFL condition analogous to Equation (4) imposes a limitation of the type λ)f * )∞ ≤ const. This condition is of course, far more advantageous than Equations (4) or (5) when $x → 0. However, semi-implicit schemes require in each time step the approximate solution of nonlinear systems of equations, which needs to be done by iterative techniques such as Newton-Raphson (NR) method. The effort of implementation is considerable, and convergence of the NR method is not ensured (cf. the discussion by Bürger, Mulet, and Villada 2013c). We therefore believe that explicit but mollified schemes are a competitive compromise between entropicity, efficiency, robustness and ease of implementation for the class of problems at hand.

3. The direct problem 3.1. Traffic model For our model, u = u(x, t) denotes the local density of cars (measured e.g. in cars per mile), and the function f is given by one of the many semi-empirical approaches that relate traffic velocity V = V (u) to the local density u via f (u) = uV (u). We employ the Dick-Greenberg expression (Dick 1996; Greenberg 1959)

V (u) = vmax ·

   1

&

' −1 , for 0 ≤ u ≤ u∗ := umax exp C

(u )   C ln max for u∗ < u ≤ umax , u

(6)

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where C > 0 is a parameter, umax is a maximal density and vmax is the preferential (maximal) velocity a vehicle would attain on a free highway, which yields   u ( u ) for 0 ≤ u ≤ u∗ , max for u∗ < u ≤ umax , (7) f (u) = Cu ln  u  0 otherwise.


Assume now that τ is a reaction time and La is an anticipation length, where the latter may depend on V (u). As in Bürger and Karlsen (2003), we use the following formula (Nelson 2002): * + (V (u))2 , Lmin , (8) La (u) = max 2α

where Lmin is a minimal anticipation distance and α denotes a deceleration, so that the first argument in (8) denotes the distance required to decelerate from speed V (u) to full stop at deceleration α. One may then argue that the velocity of a vehicle at position x at time t does not depend on the density seen at the same point (x, t), as in the standard LWR model, but rather on the density at position x + La − V τ at time t − τ . An appropriate expansion of u evaluated at this displaced argument around (x, t) (Nelson 2002; Bürger and Karlsen 2003) leads to the conclusion that to within an O(τ 2 + L2a ) error in consistency, u = u(x, t) is given by Equation (1a) (instead of the first-order conservation law (3) of the LWR model) with A given by Equation (2), where a(u) = −uV * (u)(La (u) + τ uV * (u)).

(9)

The function a given by Equation (9) in combination with the particular function (6) satisfies a(u) = 0 for u < uc := u∗ and u = umax , and therefore Equation (1a) is indeed strongly degenerate parabolic. Other functions V still give rise to a strongly degenerate parabolic equation if we assume that reaction times and anticipation lengths are effective only whenever the local traffic density u exceeds a critical value uc (Rouvre and Gagneux 1999; Bürger and Karlsen 2003). If the expressions (6) and (8) are employed, then the function A obtained via Equation (2) can be expressed in closed form (Bürger and Karlsen 2003) namely A(u) = 0 for 0 ≤ u ≤ uc and A(u) = vmax (R(u) − R(uc )) for uc < u ≤ umax , where the function R is defined as follows. Fix a bound of integration u˜ 0 ∈ (0, uc ), and let L0 = v2max C 2 /(2α) and u∗ = umax exp(−(Lmin /L0 )1/2 ). If u∗ > uc , then * K(u) for uc ≤ u ≤ u∗ , R(u) = ∗ 2 ∗ K(u ) + (CLmin − C τ vmax )(u − u ) for u > u∗ , where we define the following function (where ζ := ln umax ):

2 ˜ 0 ). K(u) = CL0 {[ζ 2 + 2ζ + 2]s − 2(ζ + 1)s ln s + s(ln s)2 }|s=u s=˜u0 − C τ vmax (u − u

3.2.

The discrete mollification operator

The discrete mollification method (Murio 1993, 2002) is based on replacing a discrete set of data y = {yj }j∈Z , which may consist of evaluations or cell averages of a real function y = y(x) at equidistant grid points xj = x0 + j$x, $x > 0, j ∈ Z, by its mollified version Jη y, where Jη is the discrete mollification operator defined by [Jη y]j :=

η ,

i=−η

wi yj−i ,

j ∈ Z.

Here, the support parameter η ∈ N indicates the width of the mollification stencil, and the weights wi satisfy wi = w−i and 0 ≤ wi ≤ wi−1 for i = 1, . . . , η along with w−η + · · · + wη−1 + wη = 1.

707



Table 1.

Sample discrete mollification weights wi = w−i , i = 0, . . . , η.

η

i=0

i=1

0 1 2 3 4 5 6 7 8 η 7 8

1 0.84272 0.60387 0.45556 0.36266 0.30028 0.25585 0.22270 0.19708 i=7 1.0697e-4 4.9782e-4

0.07864 0.19262 0.23772 0.24003 0.22625 0.20831 0.19058 0.17444 i=8

i=2

i=3

i=4

i=5

i=6

5.4438e-3 3.3291e-2 6.9440e-2 9.6723e-2 0.11241 0.11942 0.12097

1.2099e-3 8.7275e-3 2.3430e-2 4.0192e-2 5.4793e-2 6.5725e-2

4.7268e-4 3.2095e-3 9.5154e-3 1.8403e-2 2.7973e-2

2.4798e-4 1.4905e-3 4.5234e-3 9.3255e-3

1.5434e-4 8.1342e-4 2.4348e-3

7.9691e-5

The weights wi are obtained by numerical integration of a suitable truncated Gaussian kernel. As an illustration, Table 1 reports the values of wi for several values of η. See Acosta and Mejía (2008), Mejía, Acosta„ and Saleme (2011), Acosta and Bürger (2012) and Acosta, Bürger, and Mejía (2012) for more details. 3.3. Discretization of the direct problem The domain "T is discretized by a standard Cartesian grid by setting xj := j$x, j = 0, . . . , N , where N $x = L, and tn := n$t, n = 0, . . . , M, where M$t = T. We assume that $x and $t satisfy the respective CFL conditions (4) and (5) of the methods to be introduced. We denote by unj an approximate value of the cell average of u = u(x, t) over the cell [xj , xj+1 ] at time t = tn , and correspondingly set u0j =

1 $x

!

xj

xj+1

u0 (x) dx,

j = 0, . . . , N − 1.

For the numerical solution of Equation (1) we consider two convergent finite difference methods, namely the basic scheme (Evje and Karlsen 2000; Bürger and Karlsen 2001) and alternatively, its mollified version introduced by Acosta, Bürger, and Mejía (2012). The first one has the following form: = unj − λ$+ F EO (unj−1 , unj ) + µ(A(unj+1 ) − 2A(unj ) + A(unj−1 )). un+1 j

(10)

The scheme (10) is monotone and convergent under the CFL condition (4). Here F EO stands for the well-known numerical flux by Engquist and Osher (1981). The mollified scheme is also monotone and convergent and takes the form = unj − λ$+ F EO (unj−1 , unj ) + 2µCη ([Jη A(un )]j − A(unj )), un+1 j

(11)

where Cη := [2(η2 wη + (η − 1)2 wη−1 + · · · + w1 ) + w0 ]−1 . This is an explicit method, however, it is equipped with the convenient CFL condition (5), where εη := Cη (1 − w0 ). The basic scheme is taken as a reference scheme, for comparison purposes.

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4. Parameter identification 4.1. Inverse Problem For the formulation of the IP we emulate the unconstrained least-squares minimization problems that arise when solving inverse identification problems based on distributed parameter systems. Given observation data uobs = uobs (t) at position xr and the functions u0 , ψ1 and ψ2 , find the flux f and the diffusion function a such that the entropy solution u(xr , t) of problem Equation (1) is as close as possible to uobs in some suitable norm. If J = J(u(xr , ·)) is a cost function that measures the distance between u(xr , ·) and uobs , then the IP can be formulated as follows: minimize J(u(xr , ·)) with respect to the functions f and a,


where u is the entropy solution to (1).

(IP)

A common choice for the cost function J is 1 J(u(xr , ·)) = 2

!

T 0

|u(xr , t) − uobs (t)|2 dt.

(12)

The functions f and a are given in semi-empirical parametric forms via Equations (7)–(9), so that (IP) is reduced to finding a vector p of a finite number of parameters such that the solution u of Equation (1) calculated for the corresponding functions f and a minimizes J(u(xr , ·)). Thus, (IP) is replaced by the parameter identification problem minimize J(u(xr , ·)) with respect to the parameter vector p, where u is the entropy solution of (1) found with f and a

(PI)

associated to the current values of p. In what follows, we will write J(p) instead of J(u(xr , ·)). We briefly comment on some theoretical results related to (IP) and (IP). First of all, we recall that the existence and uniqueness of weak solutions to the direct problem Equation (1) rely on the appropriate entropy solution concept (see Section 2 and Bürger and Karlsen 2001). The existence of solutions for the IP is a consequence of the continuous dependence of entropy solutions to Equation (1) with respect to the functions f and A (Cockburn and Gripenberg 1999; Evje, Karlsen, and Risebro 2001; Coronel, James, and Sepúlveda 2003). Uniqueness of solutions to the IP cannot be ensured due to the hyperbolic behaviour of Equation (1a) for φ ≤ φc , see James and Sepúlveda (1994, 1999) for a discussion of this point. This means that the problem (IP) is, in general, ill-posed. Nevertheless, it is very plausible to make the effort to solve the parameter identification problem (IP) with suitable methods. We mention that Equation (1) is very similar to a model of sedimentation of suspensions, to which a variety of techniques to solve (IP) have been applied successfully. These include a discrete adjoint method (Coronel, James, and Sepúlveda 2003), a numerical descent method (Bürger, Coronel, and Sepúlveda 2008a), and the sensitivity methodology (Bürger, Coronel, and Sepúlveda 2009), besides the method applied in the present work (Acosta, Bürger, and Mejía 2014). 4.2. Discretization of the parameter identification problem We define the piecewise constant function u$ on "T by u$ (x, t) = unj for x ∈ [xj , xj+1 ) and t ∈ [tn , tn+1 ) for j = 0, . . . , N − 1 and n = 0, . . . , M − 1, and replace uobs by a piecewise constant

Transportmetrica A: Transport Science function uobs,$ formed by cell averages of uobs as follows: ! tn+1 1 obs,$ obs (t) = ur := uobs (τ ) dτ u $t tn

for t ∈ [tn , tn+1 ),

709

(13)

where n = 0, . . . , M − 1. We define the following discrete analogue of Equation (12): J $ (p) :=

1 2

!

0

T

|u$ (xr , τ ) − uobs,$ (τ )|2 dτ =

M−1 $t , n 2 |u − uobs r | . 2 n=0 r

(14)

The discrete version of (IP) can now be formulated as follows: minimize J $ (p) with respect to the parameter vector p,


where u$ is the numerical solution of (1) found with f and a

(PI$ )

associated to the current values of p. Note that each evaluation of J $ (p) (for one value of p) requires the numerical solution of the direct problem Equation (1). 4.3. Nelder-Mead simplex method The optimization process of (PI$ ) is carried out by a restarted version of the Nelder-Mead simplex method (fminsearch in MATLAB). This is a derivative-free optimization method that is widely used by researchers in different fields, is very well documented but with convergence limitations (Lagarias et al. 1998). Due to the lack of convergence in some cases, many modifications have been proposed. For instance, Kelley (1999), Luersen and Le Riche (2004) and Zhao et al. (2009) consider different ways of updating the current simplex and different restarting procedures for obtaining a descendent and deterministic method. Our restarted strategy ends when no substantial variation of the values of the parameters is achieved. It takes the following form, where we assume that pj = (p1j , . . . , pKj ), that is, we assume that K different parameters are sought: (1) Input p0 , , (2) for j = 1 to M (a) pj = fminsearch(J $ , pj−1 ) (b) if max |(pkj − pkj−1 )/pkj−1 | ! , then break, end (3) end

1≤k≤K

Finally, we note that only an initial value p0 for the parameter vector p is needed as input, from which the fminsearch routine builds the initial simplex following a criterion developed by L. Pfeffer, see Price, Cooper, and Byatt (2002). 5. Numerical example The reference solution is generated by the corresponding numerical scheme (10) or (11) on a very fine grid. Since there is a temporal observation, we choose a suitable CFL for working on a whole set of parameter values. So, the solutions can be computed with the same discretization parameters ($x, $t). We solve a parameter recognition problem, which means that the observation uobs is generated ‘synthetically’ not from real-world experimental data, but by a numerical solution with known parameters, possibly perturbed by the addition of some ‘noise’.


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5.1. Traffic model, effect of a time-dependent observation We consider a traffic ‘platoon’ with density of 50 cars/mi, entering an initially empty road segment of length L = 3.5 mi at x = 0. At arriving to the point x = 1 a traffic light changes to red. From that instant, we assume the traffic obeys Equation (1) with a flux function f and the diffusion function a of the respective forms Equations (7) and (9). The parameter values used are the same as those used by Bürger and Karlsen (2003) and references cited therein, namely C = e/7, τ = 2 s, Lmin = 0.05 mi, umax = 200 cars/mi and α = 0.1g, where g = 9.81 m/s2 is the acceleration of gravity. The targets for our identification procedure are C and τ . The restarting parameter and the tolerance parameter for the optimization are M = 10 and , = 10−4 , respectively. The traffic light changes from red to green and vice versa every 30 s. When the light shows ‘red’, we will consider the road divided into the segments [0, 1] and [1, 3.5]. The traffic light works as an impermeable boundary condition between these two regions. At the point x = 0 we assume that the platoon continues entering with the same density. Beyond the point x = 3.5 we suppose the road continues empty. When the light changes to green we work with the whole domain [0, 3.5]. Finally, for computing Jη A(un ) with red light, on [0, 1] we extend the density beyond 1.0 by the constant umax , so the drivers cannot advance. On the domain [1, 3.5] we extend the density by zero to the left of x = 1 because the drivers will feel nobody is coming from there. All these assumptions rely on the framework of Equation (1) with ψ0 = 0 and ψL = 0. Figure 1 shows the reference solution of the direct problem. For the experiments the temporal observation is done at x = 1.25 from 0 to 2 s, starting with the red light, see Figure 2. The selected spatial resolution was $x = 1/128 and by A, B, C and D we

Figure 1.

Reference solution of the direct problem.

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Temporal profile Critical density

35

30

u[cars/mi]

25

20


15

10

5

0 0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

t[h]

Figure 2.

Temporal profile uobs for the definition of the cost functional.

Table 2. Results for the basic scheme (10) and the mollified scheme (11) with η = 3, 5 and 8. Basic scheme (10)

Mollified scheme (11) with η = 3 Mollified scheme (11) with η = 5 Mollified scheme (11) with η = 8

IG

j

pj

EJ

e∞

cpu [s]

A B C D A B C D A B C D A B C D

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

(3.8833e-1, 5.0000e-2) (3.8833e-1, 5.5558e-4) (5.5555e-4, 3.8833e-1, 5.0001e-2) (5.5554e-4, 3.8833e-1, 5.0000e-2) (3.8833e-1, 4.9999e-2) (3.8833e-1, 5.5558e-4) (5.5550e-4, 3.8833e-1, 4.9997e-2) (5.5552e-4, 3.8833e-1, 4.9999e-2) (3.8833e-1, 4.9999e-2) (3.8833e-1, 5.5557e-4) (5.5557e-4, 3.8833e-1, 5.0001e-2) (5.5565e-4, 3.8833e-1, 5.0004e-2) (3.8833e-1, 4.9999e-2) (3.8833e-1, 5.5553e-4) (5.5554e-4, 3.8833e-1, 5.0001e-2) (5.5554e-4, 3.8832e-1, 4.9998e-2)

55 54 84 81 52 52 83 83 50 55 84 78 50 50 72 73

2.3502e-5 4.7648e-5 2.3494e-5 3.8126e-5 3.4670e-5 4.7648e-5 2.4995e-5 3.1224e-5 1.6282e-3 5.2485e-6 1.5059e-5 2.8235e-5 1.9759e-4 1.4369e-5 4.1872e-5 1.8799e-5

403 417 766 790 365 368 744 690 204 218 417 368 101 127 217 221

denote the choices of the initial guesses (IG) (0.7C, 1.3Lmin ), (1.3C, 1.3τ ), (0.7τ , 0.7C, 1.3Lmin ) and (1.7τ , 0.6C, 0.7Lmin ), respectively. The results are summarized in Table 2. Here, j denotes the number of calls of the fminsearch algorithm, pj is the found vector of parameter values, EJ

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(a)

(b)

h=8

Basic

2

2

1.5

JD

JD

1.5


1

1

0.5

0.5

0

0 0.3

(c)

0.4 C

0.5

0.3

0.4 C

x 10–3

Basic

x 10–3

8

8

7

7

6

6

5

5

4

4

3

3

2

2

1

1

0

0

0.5

1.5

2 t [s]

2.5

1.5

h=8

2 t [s]

2.5

(d)

Figure 3. One-parameter cost functional for (a) C and (b) τ ; two-parameter cost functional for the parameter set (C, τ ) (c) for the basic scheme and (d) for the mollified scheme.

is the required number of computed solutions of the direct problem, e∞ is the maximum relative error in the result for each parameter and cpu denotes the total CPU time of each run. 5.2. Conclusions This example illustrates the applicability of the proposed identification procedure to a strongly degenerate problem. From the results displayed in Table 2 it becomes evident that mollified versions produce better CPU time results while keeping competent accuracy (error level), and that the CPU times consistently decrease as η is increased. As suggested by the convex shape of the cost function (Figure 3), computationally, the procedure behaves in a well-posed fashion. However, a more extensive sensitivity analysis and an identifiability study should be carried out to confirm this. It would be of interest to extend the present methodology to related important topics like multiclass LWR models (Wong and Wong 2002; Benzoni-Gavage and Colombo 2003)


713

or multiclass DCLWR models (Bürger, Mulet, and Villada 2013b,c) in which different kinds of drivers are taken into consideration.

Disclosure statement No potential conflict of interest was reported by the authors.


Funding CDA and CEM acknowledge support by Universidad Nacional de Colombia through the project Mathematics and Computation, Hermes code 20305. RB acknowledges support by Conicyt Anillo project ACT 1118 (ANANUM), Fondecyt project 1130154, BASAL project CMM at Universidad de Chile, and Centro de Investigación en Ingeniería Matemática (CI2 MA), Universidad de Concepción and Red Doctoral REDOC.CTA, project UCO 1202 at U. de Concepción.

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