Inverse Design for Hamilton-Jacobi Equations

Inverse Design For Hamilton-Jacobi Equations

In many evolution models, the reconstruction of the initial state given an observation of the system at time $T>0$ represents a major challenge in mathematical modelling.

Especially if it involves irreversible processes, where sometimes, different initial conditions can lead the system to the same state at time $T>0$ . In these cases, the goal would be to describe all the possible initial states that agree with the given observation.

Here we consider the following initial-value problem for a time-dependent Hamilton-Jacobi equation:

$\left\{ \begin{array}{ll} \partial_t u + H(\nabla_x u) = 0, & \text{in} \ [0,T]\times \mathbb{R}^n, \\ u(0,x) = u_0(x), & \text{in} \ \mathbb{R}^n, \end{array}\right.$

where $u_0\in C^{0,1}(\mathbb{R}^n)$ and $H: \mathbb{R}^n\rightarrow\mathbb{R}$ is a superlinear convex Hamiltonian. From now on, we will denote by $S_T^+ u_0$ the solution to (1) at time $t=T$ .

In this case, due to the presence of the nonlinear term $H(\nabla_x u)$ in the equation, one cannot in general expect the existence of a regular $C^1$ solution, even if the initial condition $u_0$ is assumed to be very smooth. As we shall see, this loss of regularity results in a loss of information of the initial condition and the impossibility of uniquely determining $u_0$ from the observed solution at time $T$ .

Our goal is, for a given observation $u_T\in C^{0,1}(\mathbb{R}^n)$ , determine all the initial conditions $u_0$ for which the solution $u$ to (1) satisfies $u(T,\cdot)= u_T$ , or equivalently, $S_T^+u_0 =u_T$ .

We split the problem in three steps:

Step 1: We first determine if the observation $u_T$ is admissible, i.e. if there exists at least one $u_0$ satisfying $S_T^+ u_0=u_T$ .

The natural candidate is the one obtained by reversing the time in the equation (1), considering $u_T$ as terminal condition.

We denote by $S_T^-u_T$ the backward in time solution at time $t=0$ . In fact, it is proved in [1] that $u_T$ is admissible if and only if the initial condition $\tilde{u}_0 = S_T^- u_T$ satisfies $S_T^+ \tilde{u}_T =u_T$ .

Step 2: Secondly, if the observation $u_T$ is not admissible (it might be a consequence of noise effects or errors in the measurements) we compute the following projection of $u_T$ on the set of admissible observations:

$u_T^\ast : = S_T^+(S_T^- u_T).$

Observe that, for any $u_T$ , the projection $u_T^\ast$ is, by definition, admissible. In [1], it is proven that, under suitable hypotheses on $H$ , the function $u_T^\ast$ is the unique viscosity solution to the nonlinear obstacle problem.

\min \left\{ v - u_T, \ -\lambda_n \left[ D^2 v - \dfrac{[H_{pp}(D v)]^{-1}}{T}\right] \right\} = 0.

Here, for a symmetric $n\times n$ matrix, $\lambda_n[X]$ represents its greatest eigenvalue. The projection $u_T^\ast$ is, in fact, the smallest admissible observation bounded from below by $u_T$ .

Step 3: Finally, given an admissible observation $u_T$ , or its projection if it were not admissible, we construct all the initial conditions $u_0$ satisfying $S_T^+ u_0=u_T$ .

This construction is done in [1], and needs of only two ingredients:

(1) The backward solution $\tilde{u}_0 = S_T^- u_T$ , obtained by reversing the time in (1);

(2) and the following subset of $\mathbb{R}^n$ :

X_T(u_T) : = \left\{ z- T\, H_p (\nabla u_T(z)); \ \forall z\in\mathbb{R}^n \ \text{such that} \ u_T(\cdot) \ \text{is differentiable at} \ z\right\},

obtained from the differentiability points of $u_T$ .

Once these two elements are obtained, one can characterize the initial conditions $u_0$ satisfying $S_T^+ u_0 =u_T$ as the functions that are bounded from below by $\tilde{u}_0$ in all $\mathbb{R}^n$ and are equal to $\tilde{u}_0$ in $X_T(u_T)$ .

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If, for an admissible observation $u_T$ , we denote

I_T(u_T):= \left\{ u_0 \in C^{0,1}(\mathbb{R}^n)\, ; \ S_T^+ u_0 = u_T\right\},

we can characterize this set of initial conditions as follows:

I_T(u_T) = \left\{ \tilde{u}_0 + \varphi \, ; \, \varphi\in C^{0,1}(\mathbb{R}^n)\ \text{such that} \ \varphi\geq 0\ \text{and}\ \text{supp} (\varphi) \subset \mathbb{R}^n\setminus X_T(u_T)\right\}.

Inverse Design for Hamilton-Jacobi Equation

We point out that, given an admissible observation $u_T$ , the initial state $u_0$ can be uniquely determined if and only if $X_T(u_T)=\mathbb{R}^n$ .

This is the case when the solution is $C^1$ in $[0,T]$ , however, as soon as the solution fails to be $C^1$ in $[0,T]$ , the set $X_T(u_T)$ is a proper subset of $\mathbb{R}^n$ and then the initial condition can no longer be uniquely determined in the region $\mathbb{R}^n\setminus X_T(u_T)$ (white region in the right-hand-side plot).