Searcharxiv⌕ Search

arXiv subjects

Ulrich Langer

Publications and source records attributed to Ulrich Langer.

At least 19 recordsLinked to original sources
arXiv2025

Numerical solution of elliptic distributed optimal control problems with boundary value tracking

We consider some boundary value tracking optimal control problem constrained by a Neumann boundary value problem for some elliptic partial differential equation where the control acts as right-hand side. This optimal control problem can be reformulated asa state-based variational problem that is the starting point for the finite element discretizion. In this paper, we only consider atensor-product finite element discretizion for which optimal discretization error estimates and fast solvers can be derived.Numerical experiments illustrate the theoretical results quantitatively.

math.NA↗
arXiv2025

State-based approach to the numerical solution of Dirichlet boundary optimal control problems for the Laplace equation

We investigate the Dirichlet boundary control of the Laplace equation, considering the control in $H^{1/2}(\partial \Omega)$, which is the natural space for Dirichlet data when the state belongs to $H^1(\Omega)$. The cost of the control is measured in the $H^{1/2}(\partial \Omega)$ norm that also plays the role of the regularization term. We discuss regularization and finite element error estimates enabling us to derive an optimal relation between the finite element mesh size $h$ and the regularization parameter $\varrho$, balancing the energy cost for the control and the accuracy of the approximation of the desired state. This relationship is also crucial in designing efficient solvers. We also discuss additional box constraints imposed on the control and the state. Our theoretical findings are complemented by numerical examples, including one example with box constraints.

math.NA↗
arXiv2025

State-based nested iteration solution of optimal control problems with PDE constraints

We consider an abstract framework for the numerical solution of optimal control problems (OCPs) subject to partial differential equations (PDEs). Examples include not only the distributed control of elliptic PDEs such as the Poisson equation discussed in this paper in detail but also parabolic and hyperbolic equations. The approach covers the standard $L^2$ setting as well as the more recent energy regularization, also including state and control constraints. We discretize OCPs subject to parabolic or hyperbolic PDEs by means of space-time finite elements similar as in the elliptic case. We discuss regularization and finite element error estimates, and derive an optimal relation between the regularization parameter and the finite element mesh size in order to balance the accuracy, and the energy costs for the corresponding control. Finally, we also discuss the efficient solution of the resulting systems of algebraic equations, and their use in a state-based nested iteration procedure that allows us to compute finite element approximations to the state and the control in asymptotically optimal complexity. The numerical results illustrate the theoretical findings quantitatively.

math.NA↗
arXiv2024

Robust finite element solvers for distributed hyperbolic optimal control problems

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.

math.NA↗
arXiv2024

A Posteriori Single- and Multi-Goal Error Control and Adaptivity for Partial Differential Equations

This work reviews goal-oriented a posteriori error control, adaptivity and solver control for finite element approximations to boundary and initial-boundary value problems for stationary and non-stationary partial differential equations, respectively. In particular, coupled field problems with different physics may require simultaneously the accurate evaluation of several quantities of interest, which is achieved with multi-goal oriented error control. Sensitivity measures are obtained by solving an adjoint problem. Error localization is achieved with the help of a partition-of-unity. We also review and extend theoretical results for efficiency and reliability by employing a saturation assumption. The resulting adaptive algorithms allow to balance discretization and non-linear iteration errors, and are demonstrated for four applications: Poisson's problem, non-linear elliptic boundary value problems, stationary incompressible Navier-Stokes equations, and regularized parabolic $p$-Laplace initial-boundary value problems. Therein, different finite element discretizations in two different software libraries are utilized, which are partially accompanied with open-source implementations on GitHub.

math.NA↗
arXiv2023

Parallel iterative solvers for discretized reduced optimality systems

We propose, analyze, and test new iterative solvers for large-scale systems of linear algebraic equations arising from the finite element discretization of reduced optimality systems defining the finite element approximations to the solution of elliptic tracking-type distributed optimal control problems with both the standard $L_2$ and the more general energy regularizations. If we aim at an approximation of the given desired state $y_d$ by the computed finite element state $y_h$ that asymptotically differs from $y_d$ in the order of the best $L_2$ approximation under acceptable costs for the control, then the optimal choice of the regularization parameter $\varrho$ is linked to the 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 efficient 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 case of adaptive mesh refinement. Similar results can be obtained for the space-time finite element discretization of the corresponding parabolic and hyperbolic optimal control problems.

math.NA↗
arXiv2023

Mass-lumping discretization and solvers for distributed elliptic optimal control problems

The purpose of this paper is to investigate the effects of the use of mass-lumping in the finite element discretization of the reduced first-order optimality system arising from a standard tracking-type, distributed elliptic optimal control problem with $L_2$ regularization. We show that mass-lumping will not affect the $L_2$ error between the desired state and the computed state, but will lead to a Schur-complement system that allows for a fast matrix-by-vector multiplication. We show that the use of the Schur-Complement Preconditioned Conjugate Gradient method in a nested iteration setting leads to an asymptotically optimal solver with respect to the complexity.

math.NA↗
arXiv2022

An adaptive finite element method for distributed elliptic optimal control problems with variable energy regularization

We analyze the finite element discretization of distributed elliptic optimal control problems with variable energy regularization, where the usual $L^2(\Omega)$ norm regularization term with a constant regularization parameter $\varrho$ is replaced by a suitable representation of the energy norm in $H^{-1}(\Omega)$ involving a variable, mesh-dependent regularization parameter $\varrho(x)$. It turns out that the error between the computed finite element state $\widetilde{u}_{\varrho h}$ and the desired state $\overline{u}$ (target) is optimal in the $L^2(\Omega)$ norm provided that $\varrho(x)$ behaves like the local mesh size squared. This is especially important when adaptive meshes are used in order to approximate discontinuous target functions. The adaptive scheme can be driven by the computable and localizable error norm $\| \widetilde{u}_{\varrho h} - \overline{u}\|_{L^2(\Omega)}$ between the finite element state $\widetilde{u}_{\varrho h}$ and the target $\overline{u}$. The numerical results not only illustrate our theoretical findings, but also show that the iterative solvers for the discretized reduced optimality system are very efficient and robust.

math.NA↗
arXiv2022

Robust finite element discretization and solvers for distributed elliptic optimal control problems

We consider standard tracking-type, distributed elliptic optimal control problems with $L^2$ regularization, and their finite element discretization. We are investigating the $L^2$ error between the finite element approximation $u_{\varrho h}$ of the state $u_\varrho$ and the desired state (target) $\bar{u}$ in terms of the regularization parameter $\varrho$ and the mesh size $h$ that leads to the optimal choice $\varrho = h^4$. It turns out that, for this choice of the regularization parameter, we can devise simple Jacobi-like preconditioned MINRES or Bramble-Pasciak CG methods that allow us to solve the reduced discrete optimality system in asymptotically optimal complexity with respect to the arithmetical operations and memory demand. The theoretical results are confirmed by several benchmark problems with targets of various regularities including discontinuous targets.

math.NA↗
arXiv2022

Robust space-time finite element error estimates for parabolic distributed optimal control problems with energy regularization

We consider space-time tracking optimal control problems for linear para\-bo\-lic initial boundary value problems that are given in the space-time cylinder $Q = Ω\times (0,T)$, and that are controlled by the right-hand side $z_\varrho$ from the Bochner space $L^2(0,T;H^{-1}(Ω))$. So it is natural to replace the usual $L^2(Q)$ norm regularization by the energy regularization in the $L^2(0,T;H^{-1}(Ω))$ norm. We derive a priori estimates for the error $\|\widetilde{u}_{\varrho h} - \bar{u}\|_{L^2(Q)}$ between the computed state $\widetilde{u}_{\varrho h}$ and the desired state $\bar{u}$ in terms of the regularization parameter $\varrho$ and the space-time finite element mesh-size $h$, and depending on the regularity of the desired state $\bar{u}$. These estimates lead to the optimal choice $\varrho = h^2$. The approximate state $\widetilde{u}_{\varrho h}$ is computed by means of a space-time finite element method using piecewise linear and continuous basis functions on completely unstructured simplicial meshes for $Q$. The theoretical results are quantitatively illustrated by a series of numerical examples in two and three space dimensions.

math.NA↗
arXiv2022

Matrix-free Monolithic Multigrid Methods for Stokes and Generalized Stokes Problems

We consider the widely used continuous $\mathcal{Q}_{k}$-$\mathcal{Q}_{k-1}$ quadrilateral or hexahedral Taylor-Hood elements for the finite element discretization of the Stokes and generalized Stokes systems in two and three spatial dimensions. For the fast solution of the corresponding symmetric, but indefinite system of finite element equations, we propose and analyze matrix-free monolithic geometric multigrid solvers that are based on appropriately scaled Chebyshev-Jacobi smoothers. The analysis is based on results by Schöberl and Zulehner (2003). We present and discuss several numerical results for typical benchmark problems.

math.NA↗
arXiv2021

Space-time finite element methods for the initial temperature reconstruction

This work is devoted to the reconstruction of the initial temperature in the backward heat equation using the space-time finite element method on fully unstructured space-time simplicial meshes proposed by Steinbach (2015). Such a severely ill-posed problem is tackled by the standard Tikhonov regularization method. This leads to a related optimal control for an parabolic equation in the space-time domain. In this setting, the control is taken as initial condition, whereas the terminal observation data serve as target. The objective becomes a standard terminal observation functional combined with the Tikhonov regularization. The space-time finite element method is applied to the space-time optimality system that is well-posed for a fixed regularization parameter.

math.NA↗
arXiv2021

Space-time hexahedral finite element methods for parabolic evolution problems

We present locally stabilized, conforming space-time finite element methods for parabolic evolution equations on hexahedral decompositions of the space-time cylinder. Tensor-product decompositions allow for anisotropic a priori error estimates, that are explicit in spatial and temporal meshsizes. Moreover, tensor-product finite elements are suitable for anisotropic adaptive mesh refinement strategies provided that an appropriate a posteriori discretization error estimator is available. We present such anisotropic adaptive strategies together with numerical experiments.

math.NA↗
arXiv2021

Robust discretization and solvers for elliptic optimal control problems with energy regularization

We consider the finite element discretization and the iterative solution of singularly perturbed elliptic reaction-diffusion equations in three-dimensional computational domains. These equations arise from the optimality conditions for elliptic distributed optimal control problems with energy regularization that were recently studied by M.~Neumüller and O.~Steinbach (2020). We provide quasi-optimal a priori finite element error estimates which depend both on the mesh size $h$ and on the regularization parameter $\varrho$. The choice $\varrho = h^2$ ensures optimal convergence which only depends on the regularity of the target function. For the iterative solution, we employ an algebraic multigrid preconditioner and a balancing domain decomposition by constraints (BDDC) preconditioner. We numerically study robustness and efficiency of the proposed algebraic preconditioners with respect to the mesh size $h$, the regularization parameter $\varrho$, and the number of subdomains (cores) $p$. Furthermore, we investigate the parallel performance of the BDDC preconditioned conjugate gradient solver.

math.NA↗
arXiv2020

Efficient Direct Space-Time Finite Element Solvers for Parabolic Initial-Boundary Value Problems in Anisotropic Sobolev Spaces

We consider a space-time variational formulation of parabolic initial-boundary value problems in anisotropic Sobolev spaces in combination with a Hilbert-type transformation. This variational setting is the starting point for the space-time Galerkin finite element discretization that leads to a large global linear system of algebraic equations. We propose and investigate new efficient direct solvers for this system. In particular, we use a tensor-product approach with piecewise polynomial, globally continuous ansatz and test functions. The developed solvers are based on the Bartels-Stewart method and on the Fast Diagonalization method, which result in solving a sequence of spatial subproblems. The solver based on the Fast Diagonalization method allows to solve these spatial subproblems in parallel leading to a full parallelization in time. We analyze the complexity of the proposed algorithms, and give numerical examples for a two-dimensional spatial domain, where sparse direct solvers for the spatial subproblems are used.

math.NA↗
arXiv2020

Parallel matrix-free higher-order finite element solvers for phase-field fracture problems

Phase-field fracture models lead to variational problems that can be written as a coupled variational equality and inequality system. Numerically, such problems can be treated with Galerkin finite elements and primal-dual active set methods. Specifically, low-order and high-order finite elements may be employed, where, for the latter, only few studies exist to date. The most time-consuming part in the discrete version of the primal-dual active set (semi-smooth Newton) algorithm consists in the solutions of changing linear systems arising at each semi-smooth Newton step. We propose a new parallel matrix-free monolithic multigrid preconditioner for these systems. We provide two numerical tests, and discuss the performance of the parallel solver proposed in the paper. Furthermore, we compare our new preconditioner with a block-AMG preconditioner available in the literature.

math.NA↗
arXiv2020

Space-time finite element discretization of parabolic optimal control problems with energy regularization

We analyze space-time finite element methods for the numerical solution of distributed parabolic optimal control problems with energy regularization in the Bochner space $L^2(0,T;H^{-1}(Ω))$. By duality, the related norm can be evaluated by means of the solution of an elliptic quasi-stationary boundary value problem. When eliminating the control, we end up with the reduced optimality system that is nothing but the variational formulation of the coupled forward-backward primal and adjoint equations. Using Babuška's theorem, we prove unique solvability in the continuous case. Furthermore, we establish the discrete inf-sup condition for any conforming space-time finite element discretization yielding quasi-optimal discretization error estimates. Various numerical examples confirm the theoretical findings. We emphasize that the energy regularization results in a more localized control with sharper contours for discontinuous target functions, which is demonstrated by a comparison with an $L^2$ regularization and with a sparse optimal control approach.

math.NA↗