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Optimal complexity solution of space-time finite element systems for state-based parabolic distributed optimal control problems

Richard Löscher, Michael Reichelt, Olaf Steinbach

TL;DR

The work addresses distributed optimal control of the parabolic heat equation by employing an energy norm defined on the anisotropic Sobolev space $H^{1,1/2}_{0;0,}(Q)$, realized computably via a modified Hilbert transform ${\mathcal{H}}_T$. This enables a reduced formulation with a regularization term $\varrho\langle D u_\varrho,u_\varrho\rangle_Q$ and a space-time tensor-product finite element discretization that yields anisotropic error estimates and stable, efficient solvers. A key contribution is an optimal-complexity solver based on diagonalizing the temporal operator and decoupling into independent spatial problems, solvable with conjugate gradients and accelerated by a fast discrete sine transform, achieving $\mathcal{O}(M_x N_t\log N_t)$ complexity. Numerical experiments across smooth, anisotropic, and discontinuous targets, as well as nonlinear and constrained variants, validate the theory and demonstrate robust convergence and scalability. The framework offers a scalable approach for large-scale parabolic control problems and can be extended to adaptive schemes and more general state or control constraints.

Abstract

We consider a distributed optimal control problem subject to a parabolic evolution equation as constraint. The control will be considered in the energy norm of the anisotropic Sobolev space $[H_{0;,0}^{1,1/2}(Q)]^\ast$, such that the state equation of the partial differential equation defines an isomorphism onto $H^{1,1/2}_{0;0,}(Q)$. Thus, we can eliminate the control from the tracking type functional to be minimized, to derive the optimality system in order to determine the state. Since the appearing operator induces an equivalent norm in $H_{0;0,}^{1,1/2}(Q)$, we will replace it by a computable realization of the anisotropic Sobolev norm, using a modified Hilbert transformation. We are then able to link the cost or regularization parameter $\varrho>0$ to the distance of the state and the desired target, solely depending on the regularity of the target. For a conforming space-time finite element discretization, this behavior carries over to the discrete setting, leading to an optimal choice $\varrho = h_x^2$ of the regularization parameter $\varrho$ to the spatial finite element mesh size $h_x$. Using a space-time tensor product mesh, error estimates for the distance of the computable state to the desired target are derived. The main advantage of this new approach is, that applying sparse factorization techniques, a solver of optimal, i.e., almost linear, complexity is proposed and analyzed. The theoretical results are complemented by numerical examples, including discontinuous and less regular targets. Moreover, this approach can be applied also to optimal control problems subject to non-linear state equations.

Optimal complexity solution of space-time finite element systems for state-based parabolic distributed optimal control problems

TL;DR

The work addresses distributed optimal control of the parabolic heat equation by employing an energy norm defined on the anisotropic Sobolev space , realized computably via a modified Hilbert transform . This enables a reduced formulation with a regularization term and a space-time tensor-product finite element discretization that yields anisotropic error estimates and stable, efficient solvers. A key contribution is an optimal-complexity solver based on diagonalizing the temporal operator and decoupling into independent spatial problems, solvable with conjugate gradients and accelerated by a fast discrete sine transform, achieving complexity. Numerical experiments across smooth, anisotropic, and discontinuous targets, as well as nonlinear and constrained variants, validate the theory and demonstrate robust convergence and scalability. The framework offers a scalable approach for large-scale parabolic control problems and can be extended to adaptive schemes and more general state or control constraints.

Abstract

We consider a distributed optimal control problem subject to a parabolic evolution equation as constraint. The control will be considered in the energy norm of the anisotropic Sobolev space , such that the state equation of the partial differential equation defines an isomorphism onto . Thus, we can eliminate the control from the tracking type functional to be minimized, to derive the optimality system in order to determine the state. Since the appearing operator induces an equivalent norm in , we will replace it by a computable realization of the anisotropic Sobolev norm, using a modified Hilbert transformation. We are then able to link the cost or regularization parameter to the distance of the state and the desired target, solely depending on the regularity of the target. For a conforming space-time finite element discretization, this behavior carries over to the discrete setting, leading to an optimal choice of the regularization parameter to the spatial finite element mesh size . Using a space-time tensor product mesh, error estimates for the distance of the computable state to the desired target are derived. The main advantage of this new approach is, that applying sparse factorization techniques, a solver of optimal, i.e., almost linear, complexity is proposed and analyzed. The theoretical results are complemented by numerical examples, including discontinuous and less regular targets. Moreover, this approach can be applied also to optimal control problems subject to non-linear state equations.
Paper Structure (6 sections, 8 theorems, 105 equations, 3 figures, 5 tables)

This paper contains 6 sections, 8 theorems, 105 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. A parabolic distributed optimal control problem
  3. Optimal control in anisotropic Sobolev spaces
  4. Optimal solution strategies
  5. Numerical results
  6. Conclusions

Key Result

Lemma 3.1

Let $u_\varrho \in H^{1,1/2}_{0;0,}(Q)$ be the unique solution of the variational formulation VF aniso. For $\overline{u} \in L^2(Q)$ there holds while for $\overline{u} \in H^{1,1/2}_{0;0,}(Q)$ the following error estimates hold true: If in addition $D \overline{u} \in L^2(Q)$ is satisfied for $\overline{u} \in H^{1,1/2}_{0;0,}(Q)$, as well as follow. Moreover, when choosing $v = u_\varrho$ i

Figures (3)

  • Figure 1: Simulation results for the nonlinear turning wave example with $n_t=n_x=128$. Target (left), state (middle), and control (right) at $t=0.25$ (top), $t=0.5$ (middle), and $t=0.75$ (bottom).
  • Figure 2: Simulation times for all cases.
  • Figure 3: Gallbladder with discontinuous target function, left column: state $u_{\varrho,h}$, right column: target $\overline{u}$ for times $t=0.24$ (top), $t=0.3$ (middle), and $t=0.5$ (bottom).

Theorems & Definitions (9)

  • Lemma 3.1
  • Remark 3.1
  • Corollary 3.2
  • Lemma 3.3
  • Theorem 3.4
  • Lemma 3.5
  • Corollary 3.6
  • Lemma 4.1
  • Lemma 4.2