[Paper Review] Convergence rates for an optimally controlled Ginzburg-Landau equation

Mattias Sandberg|ArXiv.org|Sep 10, 2008

Numerical methods in inverse problems15 references3 citations

TL;DR

This paper establishes quadratic convergence of the spatially discretized value function to the true value function for an optimal control problem governed by the Ginzburg-Landau equation, using viscosity solution theory for infinite-dimensional Hamilton-Jacobi equations. Time discretization via the Symplectic Euler method achieves first-order convergence with a constant independent of spatial resolution, under a reasonable stability condition.

ABSTRACT

An optimal control problem related to the probability of transition between stable states for a thermally driven Ginzburg-Landau equation is considered. The value function for the optimal control problem with a spatial discretization is shown to converge quadratically to the value function for the original problem. This is done by using that the value functions solve similar Hamilton-Jacobi equations, the equation for the original problem being defined on an infinite dimensional Hilbert space. Time discretization is performed using the Symplectic Euler method. Imposing a reasonable condition this method is shown to be convergent of order one in time, with a constant independent of the spatial discretization.

Motivation & Objective

• To analyze the convergence of spatial and temporal discretizations for an optimal control problem associated with the stochastic Ginzburg-Landau equation.
• To establish that the value function of the spatially discrete problem converges quadratically to the value function of the original infinite-dimensional problem.
• To demonstrate first-order convergence in time using the Symplectic Euler method, with a constant independent of spatial discretization.
• To provide a rigorous error analysis grounded in viscosity solution theory for infinite-dimensional Hamilton-Jacobi equations.
• To support numerical computation of transition probabilities in thermally driven phase transitions via optimal control.

Proposed method

• Formulates the optimal control problem by minimizing a cost functional involving a penalty on the final state, with the state governed by a controlled Ginzburg-Landau PDE.
• Uses mild solutions defined via the semigroup generated by the Laplacian with Dirichlet boundary conditions.
• Applies viscosity solution theory for infinite-dimensional Hamilton-Jacobi equations to analyze convergence of the value function.
• Implements spatial semidiscretization and proves quadratic convergence of the discrete value function to the continuous one.
• Uses the Symplectic Euler method for time discretization and establishes first-order convergence under a stability condition.
• Employs contraction mapping arguments and energy estimates to ensure existence, uniqueness, and boundedness of solutions in appropriate function spaces.

Experimental results

Research questions

• RQ1 How fast does the value function of a spatially semidiscretized optimal control problem converge to the value function of the original infinite-dimensional problem?
• RQ2 What is the convergence rate of the Symplectic Euler method for time discretization in the context of this optimal control problem?
• RQ3 Can the value function of the controlled Ginzburg-Landau equation be rigorously analyzed using viscosity solution theory in infinite dimensions?
• RQ4 Does the time discretization error remain bounded independently of the spatial discretization parameter?
• RQ5 Under what conditions does the solution remain within the original potential well, ensuring the validity of the control formulation?

Key findings

• The spatial discretization of the value function converges quadratically to the true value function, as measured in the supremum norm.
• The Symplectic Euler method achieves first-order convergence in time for the optimal control problem, with the error constant independent of the spatial discretization parameter.
• The value function of the original problem satisfies an infinite-dimensional Hamilton-Jacobi equation in a viscosity sense, enabling the convergence analysis.
• Existence and uniqueness of mild solutions to the controlled PDE are established in $ C(t_0, T; H_0^1) $ for initial data in $ H_0^1 $, with bounded controls.
• Energy estimates and Sobolev embedding ensure that solutions remain bounded in the supremum norm, allowing the use of the original double-well potential without modification.
• The error bound for the spatial discretization is derived from an energy inequality that controls the $ H^1 $-norm of the solution, leading to uniform bounds in the $ L^rown $-norm.