Work Packages Inverse design of time-irreversible models

Inverse design for the one-dimensional Burgers equation

Authors: - 12 December 2019

The problem

We consider the following one-dimensional Burgers equation

where $u$ is the state, $u_0$ is the initial state and the flux function $f$ is defined by $f(u)=\frac{u^2}{2}$. Kruzkov’s theory provides existence and uniqueness of a solution of \eqref{eq} with initial datum $u_0 \in L^{\infty}(\mathbb R)$. This solution is called a weak-entropy solution, denoted by $(t,x) \to S_t^+(u_0)(x)$. For a given target function $u^T$, we introduce the backward entropy solution $(t,x) \to S^-_{t}(u^T)(x)$ as follows: for every $t\in [0,T]$, for a.e $x\in \mathbb R$,

We study the problem of inverse design for \eqref{eq}. This problem consists in identifying the set of initial data evolving to a given target at a final time.

Due to the time-irreversibility of the Burgers equation, some target functions are unattainable from weak-entropy solutions of this equation, making the inverse problem under consideration ill-posed. To get around this issue, we introduce the following optimal control problem

where $u^T$ is a given target function and the class of admissible initial data $\mathcal{U}^0_{\text{ad}}$ in \eqref{opt2} is defined by

Above, stands for functions of bounded variation and $C>0$ is a constant large enough. The study of \eqref{opt2} is motivated by the minimization of the sonic boom effects generated by supersonic aircrafts [2].

To solve the optimal control problem \eqref{opt2}, some difficulties arise from a theoretical and numerical point of view.

  • Since the entropy solution $u$ of \eqref{eq} may contain shocks even if the initial datum is a smooth function, this generates important added difficulties that have been the object of intensive study in the past, see [3,4] and the references therein. In particular, the authors make sense of the derivative of $J_0$ in \eqref{opt2} in a weak way by requiring strong conditions on the set of initial data. This leads to require that entropy solutions of \eqref{eq} have a finite number of non-interacting jumps.
  • When $J_0$ is weakly differentiable, gradient descent methods have been implemented in [1,5,6] to solve numerically the optimal problem \eqref{opt2}. In the cases where it was applied successfully, only one possible initial datum emerges, namely the backward entropy solution $S_T^-(u^T)$. This is mainly due to the numerical viscosity that numerical schemes introduce to gain stability. To find some multiple minimizers, the authors in [8] use a filtering step in the backward adjoint solution.

Dans [9], we fully characterize the set of minimizers of the optimal control problem \eqref{opt2}.

Theorem 1. Let $u^T\in BV(\mathbb R)$. The optimal control problem \eqref{opt2} admits multiple optimal solutions. Moreover, for a.e $T>0$, the initial datum $u_0\in BV(\mathbb R)$ is an optimal solution of \eqref{opt2} if and only if $u_0 \in BV(\mathbb R)$ verifies $S_T^+(u_0)=S_T^+ (S_T^-(u^T))$.

A characterisation of the set is given in [7]. An illustration of Theorem 1. is given in Figure 1.

Figure 1: The backward-forward solution $S_T^+(S_T^-(u^T))$ is the projection of $u^T$ onto the set of attainable target functions. The shaded area in red at time $t=0$ represents the set of minimizers of \eqref{opt2}

The proof of Theorem 1 is structured as follows. From [7, Theorem 3.1, Corollary 3.2] or [8, Corollary 1], there exists $u_0\in BV(\mathbb R)$ such that if and only if satisfies the one-sided Lipschitz condition, i.e Thus, the optimal problem \eqref{opt2} can be rewritten as follows.

where the admissible set $\mathcal{U}^T_{\text{ad}}$ is defined by

Above, $K_1$ an open bounded interval large enough. Note that the optimal problem \eqref{opt5} is not related to the PDE model \eqref{eq}. We prove that $q=S_T^+ (S_T^-(u^T))$ is a critical point of \eqref{opt5} using the first-order optimality conditions applied to \eqref{opt5} and the full characterization of the set given in [9, Theorem A.2].

Numerical simulations

In [9,Section 3], we implement a wave-front tracking algorithm to construct numerically the set of minimizers of \eqref{opt2}. We consider for instance, a target function defined by

From Theorem 1, the backward solution is an optimal solution of \eqref{opt2} and is an optimal solution of \eqref{opt2} if and only if . In Figure 2, the target function , the backward solution and the backward-forward solution are plotted.

$x\to S^{-}_T(u^T)(x)$
$(t,x)\to S^{+}_t(S^{-}_T(u^T))(x)$

$u^T$ and $\color{red}{x\to S^{+}_T(S^{-}_T(u^T))(x)}$

$ $

Figure 2. Plotting of the target function $u^T$ defined in \eqref{uT}, the optimal solution $S_T^-(u^T)$ and the backward-forward solution $\color{red}{S^{+}_T(S^{-}_T(u^T))}$.

Note that has four different shocks located at $x=1.1$, $x=3.1$, $x=5.3$ and $x=7.2$. If we use a conservative numerical method as Godunov scheme, the approximate solution of doesn’t have shocks because of numerical viscosity that numerical schemes introduced, see Video 1.

Video 1. Approximate solution of $S_T^+(u_0)$ with $u_0$ an $N$-wave constructed with $ \color{red}{\text{a wave-front tracking algorithm}}$ and $\color{blue}{\text{a Godunov scheme}}$

This implies that only one minimizer of \eqref{opt2} can be constructed using a Godunov scheme, which is the backward entropy solution $S_T^-(u^T)$. When a wave-front tracking algorithm is implemented, the approximate solution of $S_T^+(S_T^-(u^T))$ has shocks since we track the possible discontinuities from $u^T$ to $S_T^+(S_T^-(u^T))$. This implies that all initial data $u_0$ that coincide with the approximate solution of $S_T^+ (S_T^-(u^T))$ can be recovered, see [9,Section 3].

In Video 2, we show that the weak-entropy solution of \eqref{eq} with initial data $S_T^-(u^T)$ coincides with $S_T^+ (S_T^-(u^T))$ at time $T$.

Video 2. Approximate solution of $(t,x) \to S_t^+(S_T^-(u^T))(x)$ using a wave-front tracking algorithm.

In Video 3, three other approximate optimal solutions $u_0$ of \eqref{opt2} are constructed. In particular, we show that $S_T^+ (u_0)=S_T^+ (S_T^-(u^T))$.

Video 3. Three approximate optimal solutions of \eqref{opt2} constructed using a wave-front tracking algorithm

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[9] Thibault Liard and Enrique Zuazua. Inverse design for the one-dimensional Burgers equation. Submitted (2019).