Hamiltonian Interface Dynamics for Reduced-Order Optimization of Incompressible Mixing

Technical Summary: Hamiltonian Interface Dynamics for Reduced-Order Optimization of Incompressible Mixing

Problem Statement
The paper addresses the optimal control of mixing in two-dimensional incompressible flows. The governing model is the transport equation $\partial_t \theta + v \cdot \nabla \theta = 0$ for a passive scalar $\theta$ advected by a divergence-free velocity field $v$ in a bounded planar domain $\Omega$. The objective is to design a stirring field $v$ (derived from a stream function $\psi$) that maximizes mixing efficiency.

Traditionally, mixing is quantified using negative Sobolev norms, specifically the homogeneous $\dot{H}^{-1}(\Omega)$ norm. While effective, optimizing directly with respect to this norm requires solving the full transport partial differential equation (PDE) and its adjoint at every iteration, which is computationally expensive. The authors propose a geometric alternative: instead of optimizing the scalar field directly, they optimize the length of a material interface advected by the flow. This approach leverages the fact that in 2D incompressible flows, the velocity field admits a stream-function representation ($v = \nabla^\perp \psi$), rendering the characteristic dynamics Hamiltonian.

Methodology
The authors develop a reduced-order framework that replaces the Eulerian PDE optimization with a Lagrangian Hamiltonian control problem.

1. Hamiltonian Interface Dynamics:
   The method tracks a material interface $\gamma_0$ parameterized by $s$. The position $X(t, s)$ and tangent vector $a(t, s) = \partial_s X(t, s)$ evolve according to a coupled system of ordinary differential equations (ODEs) driven by the stream function $\psi$:
   $$\partial_t X = J \nabla_x \psi(t, X), \quad \partial_t a = J \nabla^2_x \psi(t, X) a$$
   where $J$ is the symplectic matrix. The objective functional is the final length of this interface, $L(\psi; T) = \int_0^1 |a(T, s)| ds$.

2. Optimal Control Formulation:
   The stream function is restricted to a finite-dimensional subspace spanned by smooth basis functions $\{h_k\}_{k=1}^N$. The control variables are the time-dependent coefficients $u(t) \in \mathbb{R}^N$. The optimization problem minimizes a regularized cost functional:
   $$J_{\lambda, \epsilon}(\psi) = -\int_0^1 \ell_\epsilon(a(T, s)) ds + \frac{\lambda}{2} \int_0^T \|\psi(t, \cdot)\|_{L^2(\Omega)}^2 dt$$
   where $\ell_\epsilon$ is a smooth approximation of the Euclidean norm to ensure differentiability. Existence of a minimizer is proven for this constrained problem.

3. Adjoint-Based Gradient Computation:
   The authors derive the continuous adjoint system $(p, q)$ for the interface position and tangent vector. This allows for the computation of the reduced gradient with respect to the control coefficients $u$ without solving the full PDE. The gradient is expressed explicitly in terms of the adjoint variables transported along the marker trajectories.

4. Numerical Discretization:
   The forward and adjoint ODEs are discretized using the implicit midpoint rule, a symplectic and self-adjoint integrator. This choice ensures that the discrete adjoint is algebraically consistent with the derivative of the fully discrete objective function. The optimization is performed using a Polak–Ribièr conjugate-gradient method with an Armijo line search and projection onto an $L^2$-ball constraint.

Key Contributions

• Geometric Reduced Model: The paper establishes that 2D incompressible mixing can be modeled via a closed system of ODEs governing the interface position and tangent, replacing the full Eulerian transport PDE.
• Adjoint-Based Optimization on Reduced Model: A rigorous framework is presented for optimizing the interface length within a finite-dimensional stream-function ansatz, including proofs of existence and derivation of the continuous adjoint system.
• Symplectic Discrete Adjoint: The authors implement a "discretize-then-optimize" approach using the implicit midpoint rule. This guarantees that the discrete gradient is algebraically exact with respect to the discrete cost function, up to the tolerance of the nonlinear solver.
• Validation and Comparison: The reduced controls are evaluated on the full transport PDE. The method is systematically compared against a state-of-the-art Eulerian Sobolev-norm optimizer.

Results
Numerical experiments were conducted on two benchmarks: a stationary cellular flow and a Doswell frontogenesis model.

• Stationary Flows: Both benchmarks exhibited polynomial (algebraic) growth of interface length and polynomial decay of the $\dot{H}^{-1}$ norm, consistent with theoretical expectations for autonomous 2D flows.
• Optimized Time-Dependent Flows: The adjoint-based optimizer successfully designed time-dependent controls that induced near-exponential stretching of the interface and exponential decay of the $\dot{H}^{-1}$ norm.
• Computational Efficiency: When compared to the Eulerian Sobolev-norm optimizer (which solves the full 2D PDE and adjoint PDE at each step), the interface-length optimizer was significantly faster. In cellular flow benchmarks, the interface method reduced computation time by a factor of 13–21 while achieving comparable or superior $\dot{H}^{-1}$ decay rates. In Doswell benchmarks, the speedup was approximately 2.7–2.9 times.
• Limitations of the Surrogate: An experiment increasing the control basis from $N=2$ to $N=4$ modes revealed a limitation. While the $N=4$ optimizer achieved a much longer interface length (approx. 16 times longer), the resulting $\dot{H}^{-1}$ decay rate improved only moderately. This indicates that while interface length is an effective proxy for mixing, it is not fully faithful in geometrically complex regimes where local stretching may not translate to global mixing efficiency.

Significance and Claims
The authors claim this is the first framework to combine a Hamiltonian marker/tangent reduced model, adjoint-based optimization of interface length, a symplectic discrete adjoint consistent with the discrete cost, and a systematic matched-condition validation against an Eulerian optimizer.

The paper positions the method not as a universal surrogate for mix-norm optimization but as a computationally efficient companion. It aligns with Glowinski's "two-grid philosophy," where optimization on a reduced model capturing essential geometric mechanisms can yield controls that perform well on the original PDE at a fraction of the cost. The work highlights that interface length is a powerful geometric proxy for mixing but cautions that its predictive power must be assessed a posteriori on the full transport equation, particularly in regimes with complex geometric structures.