Iterative Methods for Navier--Stokes Inverse Problems

Abstract

Even when the partial differential equation underlying a physical process can be evolved forward in time, the retrospective (backward in time) inverse problem often has its own challenges and applications. Direct Adjoint Looping (DAL) is the defacto approach for solving retrospective inverse problems, but it has not been applied to deterministic retrospective Navier--Stokes inverse problems in 2D or 3D. In this paper, we demonstrate that DAL is ill-suited for solving retrospective 2D Navier--Stokes inverse problems. Alongside DAL, we study two other iterative methods: Simple Backward Integration (SBI) and the Quasi-Reversible Method (QRM). Our iterative SBI approach is novel while iterative QRM has previously been used. Using these three iterative methods, we solve two retrospective inverse problems: 1D Korteweg--de Vries--Burgers (decaying nonlinear wave) and 2D Navier--Stokes (unstratified Kelvin--Helmholtz vortex). In both cases, SBI and QRM reproduce the target final states more accurately and in fewer iterations than DAL. We attribute this performance gap to additional terms present in SBI and QRM's respective backward integrations which are absent in DAL.

Figures (14)

• Figure 1: Conceptual illustration of iterative methods. Purple: the target solution $\overline{u}(x, t)$ is known only at the final time $t_f$. Blue: we evolve a trial state $u(x, 0)$ to $u(x, t_f)$ where we then initialize one of three backward integration systems using a single compatibility condition. Each backward integration gives a solution at $t=0$ which we use to update our trial initial condition. We seek the trial solution's deviation $u'=u-\overline{u}$ at the initial time $t=0$. Yellow: Direct Adjoint Looping (DAL) is a linear backward integration. For advective systems such as KdVB and Navier--Stokes, DAL advects the final deviation $u'(x,t_f)$ by the trial solution $u$. Orange: Simple Backward Integration (SBI) is a hybrid method, which supplements the linear DAL system with two additional advective terms. Red: the Quasi-Reversible Method (QRM) aims to compute the deviation by introducing a small hyperdiffusion term with coefficient $\varepsilon$. The QRM backward integration is ill-posed when $\varepsilon=0$ and numerically unstable as $\varepsilon\to 0$.
• Figure 2: Target KdVB solution $\overline{u}(x,t)$ (left) plotted as a function of space $x$ and time $t$. We aim to recover the initial condition of a nonlinear wave which propagates right at an initial speed of one. The problem's spatial domain is 1D periodic. We include diffusion ($a=0.04$) such that its speed decreases as the wave propagates. Beginning at $t=3\pi$, SBI (center) evolves $\overline{u}(x,t_f)$ backward to $t=0$ by reversing the sign of the diffusive term. The wave diffuses during the forward and backward integrations, such that its amplitude and speed of propagation decrease in both directions. We also perform the QRM backward integration (right) on $t:3\pi\to 0$. Unlike SBI, QRM repopulates small-scale modes which are lost during forward integration. As the wave's peak regains its former amplitude, its speed of propagation increases mimicking $\overline{u}(x,t)$.
• Figure 3: Evolution of the KdVB trial initial conditions $u_n(x,0)$ using an initial guess $u_0(x,0)=0$ (shown in green). We compare two DAL runs (top row) with SBI (bottom left) and QRM (bottom right). (Top left) gradient descent approaches the target initial state gradually. (Top right) DAL minimization of $\mathcal{J}^{u}_f$ using L-BFGS approaches the target state more rapidly, but this method introduces undesired medium-scale features. SBI captures the target state faster and more directly than DAL. QRM resembles the target state immediately $(n=1)$. However, this method introduces low-amplitude small-scale errors which do not subside at $n=200$.
• Figure 4: Deviation of SBI and QRM trial initial conditions, plotted as a function of $x$ at iteration $n=200$. Both curves have prominent peaks which overlap near $x=\pi$. The SBI deviation has a slightly larger magnitude while the QRM deviation oscillates at a particular wavenumber.
• Figure 5: Initial errors ($\mathcal{J}^{u}_0$, top) and final errors ($\mathcal{J}^{u}_f$, bottom) for the 1D KdVB inverse problem plotted as a function of iteration $n$. We compare DAL, SBI, and QRM. DAL performs slightly better when paired with the second-order sparse minimization routine L-BFGS (LB $\mathcal{J}^{u}_f$), whereas gradient descent (GD $\mathcal{J}^{u}_f$) performs the worst. Our iterative applications of SBI and QRM minimize $\mathcal{J}^{u}_0$ and $\mathcal{J}^{u}_f$ more effectively than DAL. QRM minimizes both $\mathcal{J}^{u}_0$ and $\mathcal{J}^{u}_f$ more rapidly than SBI.