So first of all, I would like to thank the organizing committee and the scientific committee

for inviting me here.

So I'm definitely a PDE person and when thinking about what to propose as a talk here, I thought

that the most suitable subject was the work of one of my former PhD students, Alexandra

Wurz, which is by the way now in the Fraunhofer in Kaiserslautern, and which I co-advised

with my colleague, Michele Bignois, which is an expert in statistical optimization.

So the work focuses on the calibration of microscopic traffic flow models.

So these models basically are meant to describe the spatiotemporal evolution of some aggregate

quantities like the mean traffic density, rho, which is roughly speaking the number

of vehicles per unit space.

Then we can consider the mean velocity V, which is the average distance covered by the

vehicles per unit time, and the traffic flow, which is the product of the two and gives

the number of vehicles per unit time.

This quantity can be measured on the road and the more traditional data sources are

magnetic loop detectors, which counts the vehicle passing at fixed positions on the

road.

And you can notice sometimes the traces on the pavement.

Nowadays there are many more sources of data which are in principle available, like video

recordings, flotist car data, and other sources, but for this work we consider magnetic loop

detectors, so traditional counting at fixed positions.

So to give you an idea of how this data looks like, this is an example taken from the highway

network close to Marseille that was provided by DIRMED, which is the public administrator

of this sector.

And these are the raw data, so not treated data, which basically mark the time at which

a vehicle crosses a given loop detector.

So here you have the ID of the loop.

Here you have its position given by the distance from some milestone.

Here is the day, the date, and the hour, the lane.

And then, so for this type of loop detectors, which are double, we also can retry the speed

at which the vehicle passes through.

And its length, we want to distinguish which type of vehicle is it.

There are also other sensor types that are single loop detectors that cannot measure

the speed, but instead can measure the occupancy, which is basically the ratio of time the loop

is occupied by a vehicle, and this is somehow proportional to the density of the vehicle.

So from this, we have a direct measure of the flow, a measure either of the density

or of the speed, and we can retry the third component, either the density and the speed,

by considering this formula here.

So moving to the modeling, the basic assumption of microscopy model is the mass conservation.

So we start from a conservation equation of this form, so it's a partial differential

equation, which says that the partial derivative of the density plus the partial derivative

of the flux is equal to zero.

So to close the model, one of the first suggestions was given by Lighting-Gridham and Richard,

and was to associate to each density value a given speed value, so to write the speed

as a function of the density.

So the basic assumption is that the speed must be decreasing with density or increasing

with density, and corresponding fundamental diagram, which is the product of the density

and the speed, is usually a concave down function.

So this is the most simple traffic flow model, microscopy traffic flow model you can deal

with, and it's referred to as a first order traffic flow model.

Nevertheless, when you look at data and how they look like if you plot a speed density