Robust finite element solvers for distributed hyperbolic optimal control problems

TL;DR

This work develops robust, parallel solvers for space-time finite element discretizations of reduced optimality systems arising from hyperbolic distributed tracking-type OCPs with both $L^2$ and energy regularizations. By tying the regularization parameter $\varrho$ to the mesh size $h$ ($\varrho= h^4$ for $L^2$, $\varrho= h^2$ for energy), the authors prove spectral equivalences that enable efficient Schur-complement and saddle-point solvers, including Bramble–Pasciak PCG and mass-lumped preconditioning. They extend the framework to adaptive meshes with variable $\varrho$ and demonstrate, through extensive 2D/3D numerical experiments with smooth, continuous, and discontinuous targets, that the solvers exhibit near mesh-independent convergence and scalability, with nested iterations further improving efficiency. The results pave the way for scalable, high-fidelity control of hyperbolic PDEs on unstructured space-time meshes and suggest broad applicability to other hyperbolic constraints and adaptive strategies.

Abstract

We propose, analyze, and test new 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 $L^2$ and the more general energy regularizations. In contrast to the usual time-stepping approach, we discretize the optimality system by space-time continuous piecewise-linear finite element basis functions which are defined on fully unstructured simplicial meshes. If we aim at the asymptotically best approximation of the given desired state $y_d$ by the computed finite element state $y_{\varrho h}$, then the optimal choice of the regularization parameter $\varrho$ is linked to the space-time finite element mesh-size $h$ by the relations $\varrho=h^4$ and $\varrho=h^2$ for the $L^2$ and the energy regularization, respectively. For this setting, we can construct robust (parallel) iterative solvers for the reduced finite element optimality systems. These results can be 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. The numerical results illustrate the theoretical findings firmly.

Figures (5)

• Figure 1: Convergence history for all targets when solving the mass-lumped SC system: $d=2$ (left) and $d=3$ (right).
• Figure 2: Convergence history for all the targets (\ref{['Sec:NumericalResults:Eqn:Example1:SmoothTarget']})-(\ref{['Sec:NumericalResults:Eqn:Example3:DiscontinuousTarget']}) and for energy regularization: $d=2$ (left) and $d=3$ (right).
• Figure 3: Targets $y_{d,i}\in H_{0;0,}^{1,1}(Q)\cap H^2(Q)$, $i=1,2,3$.
• Figure 4: Convergence plots for the targets $y_{d,i}$, $i=1,2,3$, choosing $\varrho_j =2^{-j}$, $j=14,\ldots,23$ for the $L^2$-regularization where the reference solution $y_{\varrho_j}=y_{\varrho_j h}\in Y_h$ is computed via a finite element method on a uniform mesh with $n_h=131072$ simplicial elements and $m_h = 65280$ DoFs with mesh size $h=2.7621$e$-3$.
• Figure 5: Convergence plots for the targets $y_{d,i}$, $i=1,2,3$, choosing $\varrho_j =10^{-j}$, $j=2,\ldots,11$ for the $L^2$-regularization where the reference solution $y_{\varrho_j}=y_{\varrho_j h}\in Y_h$ is computed via a finite element method on a uniform mesh with $n_h=131072$ simplicial elements and $m_h = 65280$ DoFs with mesh size $h=2.7621$e$-3$.

Theorems & Definitions (27)

• Remark 1
• Remark 2
• Lemma 1
• proof
• Corollary 1
• Proposition 1
• Remark 3
• Lemma 2: LLSY:LoescherSteinbach:2024SINUM
• Remark 4
• Remark 5