We aim at developing a theory allowing the inverse design in the absence of **backward uniqueness** to be addressed both for linear and nonlinear problems. We shall take advantage of the fact that initial data recovered by backward **weak but not entropic solutions** can lead to the desired target by the forward entropic flow. This leads to the interesting and non-standard question of building numerical schemes to approximate non-entropic weak solutions. This program will also be developed in the context of **Hamilton-Jacobi** equations where, among the wide class of weak solutions, the physical ones are characterized by the viscosity criterion.

#### Available tutorials:

#### Inverse problem for Hamilton-Jacobi equations

In this tutorial we study the inverse design problem for time-evolution Hamilton-Jacobi equations. More precisely, for a given observation of the viscosity solution at time $T>0$, we construct all the possible initial data that could have led the solution to the observed state. We note that these initial data are not in general unique.

#### POD and DMD Reduced Order Models for a 2D Burgers Equation

A short python implementation of POD and DMD for a 2D Burgers equation using FEniCS and Scipy

#### An eulerian-lagrangian scheme for the problem of the inverse design of hyperbolic transport equations

The inverse design of hyperbolic transport equations can be addressed by using gradient-adjoint methodologies. Recently, Morales-Hernandez and Zuazua [1] investigated the convenience of using low order numerical schemes for the adjoint resolution in the gradient-adjoint method. They focused on hyperbolic transport scalar equations with an heterogeneous time-independent vector field.

#### Inverse design for the one-dimensional Burgers equation

Our aim is to study an optimal control problem which consists in minimizing the difference between the predictions of the Burgers equation and the observations of the system at a final time in $L^2(\mathbb{R})$ norm.

#### Sparse sources identification through adjoint localization algorithm

The aim of this work is to recover the initial sparse sources that lead to a given final measurements using the diffusion equation. It is assumed that the initial condition can be written down as a linear combination of unitary deltas and their weights. In that context, an algorithm that combines the adjoint methodology with least squares is presented. In particular, the adjoint methodology is used to find the localization of the sparse sources and least squares to find the corresponding intensities.

#### Reconstruction of traffic state using autonomous vehicles

Autonomous Vehicles (briefly AVs) have already been tested on urban and highway networks and, for the future they may be used to monitor and manage the vehicular traffic. The impact of autonomous vehicles on vehicular traffic is modeled by a PDE+ODE models with moving point-flux constraint. The PDE is a nonlinear hyperbolic conservation laws and the ODE represents the trajectory of autonomous vehicles. We show that the data collected by autonomous vehicles' sensors can be used to reconstruct the traffic state.

#### 2D inverse design of linear transport equations on unstructured grids

Various discrete adjoint methodologies are discussed for the inverse design of linear transport equations in 2 space dimensions

#### Conservation laws in the presence of shocks

Tracking control of 1D scalar conservation laws in the presence of shocks

#### Numerical aspects of LTHC of Burgers equation

Numerical approximation of the inverse design problem for the Burgers equation