Sparse Optimal Control of Viscous Cahn-Hilliard Systems

A high-performance computational framework for solving sparse optimal control problems governed by the Viscous Cahn-Hilliard (vCH) equation with a logarithmic potential.

This project implements a Proximal Gradient Descent (PGD) algorithm to steer phase separation processes. It features robust forward solvers (Newton-Raphson), adjoint-based gradient computation, and an interactive configuration system.

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🎥 Demonstration

1D Control Evolution	2D Target Steering
Evolution of phi under Optimal Control	Controlled Phase Separation

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Note: The graphics displayed above are for demonstration purposes and may differ slightly from current outputs due to code updates. All default parameters are defined in config.py; you can modify them directly in that file or adjust them interactively by running the main driver scripts.

🌟 Key Features

• Multi-Dimensional: Fully supported solvers for both 1D and 2D domains.
• Physics-Fidelity:
  • Models Viscous Cahn-Hilliard dynamics (inertial effects).
  • Uses the Logarithmic Flory-Huggins potential ensuring physical bounds ($-1 < \varphi < 1$).
• Robust Numerics:
  • Forward Solver: Semi-implicit Crank-Nicolson scheme with convex-concave splitting and monolithic Newton-Raphson iteration.
  • Adjoint Solver: Efficient backpropagation-through-time for exact gradient computation.
• Sparse Control: Implements Proximal Gradient Descent (ISTA) to handle non-smooth $L^1$ regularization, promoting energy-efficient, sparse actuation.
• Interactive CLI: User-friendly prompts to configure grid size, weights, targets, and time steps without touching code.
• Verification: Includes automated checks for KKT conditions (first-order) and Coercivity (second-order sufficient conditions).

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🧠 Mathematical Model

1. The State System

We control the phase-field variable $\varphi$ and chemical potential $\mu$ via an external control $u$, filtered through an auxiliary variable $w$. As detailed in the project report:

$$ \begin{cases} \partial_t \varphi - \Delta \mu = 0 \\ \tau \partial_t \varphi - \kappa \Delta \varphi + f'(\varphi) = \mu + w \\ \gamma \partial_t w + w = u \end{cases} $$

• $\tau$: Viscosity/relaxation parameter.
• $\kappa$: Gradient energy coefficient (interface width).
• $f'(\varphi)$: Derivative of the logarithmic double-well potential.

2. The Optimization Problem

We minimize a cost functional $J(\varphi, u)$ composed of four terms:

$$ \min_{u} J(\varphi, u) = \underbrace{\frac{b_1}{2} ||\varphi - \varphi_Q||_{L^2}^2}_{\text{Trajectory Tracking}} + \underbrace{\frac{b_2}{2} ||\varphi(T) - \varphi_\Omega||_{L^2}^2}_{\text{Terminal Target}} + \underbrace{\frac{b_3}{2} ||u||_{L^2}^2}_{\text{Control Energy}} + \underbrace{\kappa_{spar} ||u||_{L^1}}_{\text{Sparsity}} $$

The non-differentiable $L^1$ term is handled via Soft-Thresholding, ensuring the control $u(x,t)$ is zero where actuation is not strictly necessary.

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🚀 Installation

Prerequisites (Min)

• Python 3.11+
• RAM: ~4GB+ recommended for 2D simulations.

Setup

1. Clone the repository:

   git clone [https://github.com/YOUR_USERNAME/Viscous_Cahn_Hilliard.git](https://github.com/YOUR_USERNAME/Viscous_Cahn_Hilliard.git)
   cd Viscous_Cahn_Hilliard

2. Create a virtual environment (Recommended):

   conda create -n ch-opt python=3.11 -y
   conda activate ch-opt

3. Install dependencies:

   pip install -r requirements.txt

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💻 Usage

The simulators are designed to be run from the project root directory.

Interactive Mode

Run the driver script. [cite_start]You will see a "Forward Preview" first, followed by prompts to configure the optimization (weights, steps, targets).

To run the 1D Simulator:

python 1d/Vch_control_1D/GD_1D.py

To run the 2D Simulator:

python 1d/Vch_control_2D/GD2_configured.py

🧪 Testing & Verification

This codebase is rigorously tested to ensure solver stability and numerical accuracy. We use pytest to validate the Newton solvers, discrete gradients, and optimization routines.

Running Tests

1. Install pytest (if not already installed):

   pip install pytest

2. Run the Test Suite: Navigate to the folder 1D or 2D and simply run the test file present in each 'test cases':(e.g., test_1d_*.py or test_2d_*.py).

📚 References

1. P. Colli, J. Sprekels, and F. Tröltzsch, "Optimality Conditions for Sparse Optimal Control of Viscous Cahn-Hilliard Systems with Logarithmic Potential," Applied Mathematics & Optimization, 2024.