Efficient Space-Time Finite Element Solvers for Hyperbolic Optimal Control Problems with L2 and Energy Regularization

Core Concepts

The authors propose and analyze robust iterative solvers for systems of linear algebraic equations arising from the space-time finite element discretization of reduced optimality systems defining the approximate solution of hyperbolic distributed, tracking-type optimal control problems with both the standard L2 and the more general energy regularizations.

Abstract

The content presents the following key highlights and insights: The authors consider abstract optimal control problems (OCPs) of the form: Find the state yϱ and the control uϱ minimizing the cost functional J(yϱ, uϱ) subject to the state equation Byϱ = uϱ. They focus on tracking-type, distributed hyperbolic OCPs represented by the wave operator B = □. For the L2 regularization with U = L2(Q), the authors derive error estimates for the deviation of the exact state yϱ from the desired state yd, and show that the optimal choice of the regularization parameter ϱ is linked to the space-time finite element mesh-size h by the relation ϱ = h4. For the energy regularization with U = P∗ = [H1,1 0;,0(Q)]∗, the authors establish that the optimal choice of the regularization parameter is ϱ = h2, independent of the regularity of the desired state yd. The authors construct robust (parallel) iterative solvers for the reduced finite element optimality systems, exploiting the spectral equivalence between the Schur complement and the mass matrix. This allows for efficient solution methods, such as the Schur-Complement Preconditioned Conjugate Gradient (SC-PCG) method. The results are generalized to variable regularization parameters adapted to the local behavior of the mesh-size that can heavily change in the case of adaptive mesh refinements. Numerical results are presented to illustrate the theoretical findings.

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Key Insights Distilled From

Robust finite element solvers for distributed hyperbolic optimal control problems

by Ulri... at arxiv.org 04-08-2024

https://arxiv.org/pdf/2404.03756.pdf

Deeper Inquiries

How can the proposed methods be extended to handle additional control and/or state constraints in the optimal control problems

The proposed methods can be extended to handle additional control and/or state constraints in the optimal control problems by incorporating them into the variational formulations and discretization processes. For control constraints, penalty methods or augmented Lagrangian approaches can be used to enforce the constraints within the optimization problem. This involves adding penalty terms to the cost functional or introducing Lagrange multipliers to the optimality system. State constraints can be handled similarly by incorporating them into the state equation or the adjoint equation. The finite element discretization would need to be adjusted to accommodate these additional constraints, ensuring that the resulting system of equations reflects the constraints accurately. Iterative solvers would also need to be adapted to solve the constrained optimization problems efficiently, possibly requiring the development of specialized algorithms tailored to the specific constraints involved.

What are the implications of the loss of regularity discussed in Remark 4, and how can it be addressed in practice

The loss of regularity discussed in Remark 4 can have significant implications for the convergence and accuracy of the numerical methods. When the regularity of the target function decreases, the convergence rates of the numerical schemes may deteriorate, leading to slower convergence or reduced accuracy of the solutions. In practice, this loss of regularity can be addressed by refining the mesh or increasing the polynomial degree of the finite element basis functions to capture the irregularities in the solution more effectively. Adaptive mesh refinement techniques can also be employed to concentrate computational resources in regions where the solution exhibits high oscillations or irregular behavior. Additionally, specialized numerical methods designed to handle functions with lower regularity, such as discontinuous Galerkin methods or adaptive algorithms tailored to handle irregular solutions, can be utilized to improve the accuracy and convergence of the computations.

Can the space-time finite element discretization and the iterative solvers be adapted to handle more general hyperbolic operators beyond the wave equation

The space-time finite element discretization and the iterative solvers can be adapted to handle more general hyperbolic operators beyond the wave equation by modifying the variational formulations and the discretization schemes to accommodate the specific properties of the new hyperbolic operators. This may involve adjusting the basis functions, the mesh refinement strategies, and the numerical algorithms to account for the characteristics of the new operators, such as different wave speeds, boundary conditions, or source terms. The iterative solvers can be tailored to exploit the structure and properties of the new hyperbolic operators, potentially requiring the development of new preconditioning techniques or solution strategies optimized for the specific operator equations. By customizing the finite element methods and solvers to the particular hyperbolic operators under consideration, the numerical simulations can accurately capture the behavior of the systems and provide reliable solutions for a broader class of problems.