arXiv subjects

Ira Neitzel

Publications and source records attributed to Ira Neitzel.

9 recordsLinked to original sources

arXiv2025

Optimal control of quasilinear parabolic PDEs with gradient terms and pointwise constraints on the gradient of the state

We derive existence results and first order necessary optimality conditions for optimal control problems governed by quasilinear parabolic PDEs with a class of first order nonlinearities that include for instance quadratic gradient terms. Pointwise in space and time or averaged in space and pointwise in time constraints on the gradient of the state control the growth of the nonlinear terms. We rely on and extend the improved regularity analysis for quasilinear parabolic PDEs on a whole scale of function spaces from [Hoppe et al, 2023]. In case of integral in space gradient-constraints we derive first-order optimality conditions under rather general regularity assumptions for domain, coefficients, and boundary conditions, similar to e.g. [Bonifacius and Neitzel, 2018]. In the case of pointwise in time and space gradient-constraints we use slightly stronger regularity assumptions leading to a classical smoother $W^{2,p}$-setting similar to [Casas and Chrysafinos, 2018].

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arXiv2023

Coefficient Control of Variational Inequalities

Within this chapter, we discuss control in the coefficients of an obstacle problem. Utilizing tools from H-convergence, we show existence of optimal solutions. First order necessary optimality conditions are obtained after deriving directional differentiability of the coefficient to solution mapping for the obstacle problem. Further, considering a regularized obstacle problem as a constraint yields a limiting optimality system after proving, strong, convergence of the regularized control and state variables. Numerical examples underline convergence with respect to the regularization. Finally, some numerical experiments highlight the possible extension of the results to coefficient control in phase-field fracture.

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arXiv2023

Global-in-time solutions for quasilinear parabolic PDEs with mixed boundary conditions in the Bessel dual scale

We prove existence and uniqueness of global-in-time solutions in the $W^{-1,p}_D$-$W^{1,p}_D$-setting for abstract quasilinear parabolic PDEs with nonsmooth data and mixed boundary conditions, including a nonlinear source term with at most linear growth. Subsequently, we use a bootstrapping argument to achieve improved regularity of these global-in-time solutions within the functional-analytic setting of the interpolation scale of Bessel-potential dual spaces $H^{\theta-1,p}_D = [W^{-1,p}_D,L^p]_\theta$ with $\theta \in [0,1]$ for the abstract equation under suitable additional assumptions. This is done by means of new nonautonomous maximal parabolic regularity results for nonautonomous differential operators operators with H\"older-continuous coefficients on Bessel-potential spaces. The upper limit for $\theta$ is derived from the maximum degree of H\"older continuity for solutions to an elliptic mixed boundary value problem in $L^p$.

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arXiv2023

Analysis of Four-Dimensional Variational Data Assimilation Problems in Low Regularity Spaces

We carry out a rigorous analysis of four-dimensional variational data assimilation ($4D$-VAR) problems for linear and semilinear parabolic partial differential equations. Continuity of the state with respect to the spatial variable is required since pointwise observations of the state variable appear in the cost functional. Using maximal parabolic regularity tools, we prove this regularity for initial conditions with $L^\beta$-regularity guaranteed by control constraints, rather than Sobolev regularity of the controls ensured by artificial cost terms. We obtain existence of optimal controls and first order necessary optimality conditions for both the convex and nonconvex problem with spatial dimension $d=2,3$, as well as second order sufficient optimality conditions for the nonconvex problem for $d=2$.

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arXiv2020

First-order conditions for the optimal control of the obstacle problem with state constraints

We consider an optimal control problem in which the state is governed by an unilateral obstacle problem (with obstacle from below) and restricted by a pointwise state constraint (from above). In the presence of control constraints, we prove, via regularization of the state constraints, that a system of C-stationarity is necessary for optimality. In the absence of control constraints, we show that local minimizers are even strongly stationary by a careful discussion of the primal first-order conditions of B-stationary type.

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arXiv2019

Finite element error estimates for one-dimensional elliptic optimal control by BV functions

We consider an optimal control problem governed by a one-dimensional elliptic equation that involves univariate functions of bounded variation as controls. For the discretization of the state equation we use linear finite elements and for the control discretization we analyze two strategies. First, we use variational discretization of the control and show that the $L^2$- and $L^\infty$-error for the state and the adjoint state are of order ${\mathcal O}(h^2)$ and that the $L^1$-error of the control behaves like ${\mathcal O}(h^2)$, too. These results rely on a structural assumption that implies that the optimal control of the original problem is piecewise constant and that the adjoint state has nonvanishing first derivative at the jump points of the control. If, second, piecewise constant control discretization is used, we obtain $L^2$-error estimates of order $\mathcal{O}(h)$ for the state and $W^{1,\infty}$-error estimates of order $\mathcal{O}(h)$ for the adjoint state. Under the same structural assumption as before we derive an $L^1$-error estimate of order $\mathcal{O}(h)$ for the control. We discuss optimization algorithms and provide numerical results for both discretization schemes indicating that the error estimates are optimal.

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arXiv2019

A sparse control approach to optimal sensor placement in PDE-constrained parameter estimation problems

We present a systematic approach to the optimal placement of finitely many sensors in order to infer a finite-dimensional parameter from point evaluations of the solution of an associated parameter-dependent elliptic PDE. The quality of the corresponding least squares estimator is quantified by properties of the asymptotic covariance matrix depending on the distribution of the measurement sensors. We formulate a design problem where we minimize functionals related to the size of the corresponding confidence regions with respect to the position and number of pointwise measurements. The measurement setup is modeled by a positive Borel measure on the spatial experimental domain resulting in a convex optimization problem. For the algorithmic solution a class of accelerated conditional gradient methods in measure space is derived, which exploits the structural properties of the design problem to ensure convergence towards sparse solutions. Convergence properties are presented and the presented results are illustrated by numerical experiments.

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arXiv2019

Multigoal-oriented optimal control problems with nonlinear PDE constraints

In this work, we consider an optimal control problem subject to a nonlinear PDE constraint and apply it to the regularized $p$-Laplace equation. To this end, a reduced unconstrained optimization problem in terms of the control variable is formulated. Based on the reduced approach, we then derive an a posteriori error representation and mesh adaptivity for multiple quantities of interest. All quantities are combined to one, and then the dual-weighted residual (DWR) method is applied to this combined functional. Furthermore, the estimator allows for balancing the discretization error and the nonlinear iteration error. These developments allow us to formulate an adaptive solution strategy, which is finally substantiated via several numerical examples.

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arXiv2018

A Lagrange Multiplier Method for Semilinear Elliptic State Constrained Optimal Control Problems

In this paper we apply an augmented Lagrange method to a class of semilinear elliptic optimal control problems with pointwise state constraints. We show strong convergence of subsequences of the primal variables to a local solution of the original problem as well as weak convergence of the adjoint states and weak* convergence of the multipliers associated to the state constraint. Moreover, we show existence of stationary points in arbitrary small neighborhoods of local solutions of the original problem. Additionally, various numerical results are presented.

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