Michael Hinze

Optimization with PDE constraints

This project is concerned with the development, analysis and application of new, innovative mathematical techniques for the solution of constrained optimization problems where a partial differential equation (PDE) or a system of PDEs appears as an essential part of the constraints. Such optimization problems arise in a wide variety of important applications in the form of, e.g., parameter identification problems, optimal design problems, or optimal control problems. The efficient and robust solution of PDE constrained optimization problems has a strong impact on more traditional applications in, e.g., automotive and aerospace industries and chemical processing, as well as on applications in recently emerging technologies in materials and life sciences including environmental protection, bioand nanotechnology, pharmacology, and medicine. The appropriate mathematical treatment of PDE constrained optimization problems requires the integrated use of advanced methodologies from the theory of optimization and optimal control in a functional analytic setting, the theory of PDEs as well as the development and implementation of powerful algorithmic tools from numerical mathematics and scientific computing. Experience has clearly shown that the design of efficient and reliable numerical solution methods requires a fundamental understanding of the subtle interplay between optimization in function spaces and numerical discretization techniques which can only be achieved by a close cooperation between researchers from the above mentioned fields.

A variational discretization concept in control constrained optimization: the linear-quadratic case

A new discretization concept for optimal control problems with control constraints is introduced which utilizes for the discretization of the control variable the relation between adjoint state and control. Its key feature is not to discretize the space of admissible controls but to implicitly utilize the first order optimality conditions and the discretization of the state and adjoint equations for the discretization of the control. For discrete controls obtained in this way an optimal error estimate is proved. The application to control of elliptic equations is discussed. Finally it is shown that the new concept is numerically implementable with only slight increase in program management. A numerical test confirms the theoretical investigations.

Proper Orthogonal Decomposition Surrogate Models for Nonlinear Dynamical Systems: Error Estimates and Suboptimal Control

Nonlinear Dynamical System Let V and H be real separable Hilbert spaces and suppose that V is dense in H with compact embedding. By 〈· , ·〉H we denote the inner product in H. The inner product in V is given by a symmetric bounded, coercive, bilinear form a : V × V → IR: 〈φ,ψ〉V = a(φ,ψ) for all φ,ψ ∈ V (10.16) with associated norm given by ‖ · ‖V = √ a(· , ·). Since V is continuously injected into H, there exists a constant cV > 0 such that ‖φ‖H ≤ cV ‖φ‖V for all φ ∈ V. (10.17) We associate with a the linear operator A: 〈Aφ,ψ〉V ′,V = a(φ,ψ) for all φ,ψ ∈ V, where 〈· , ·〉V ′,V denotes the duality pairing between V and its dual. Then, by the Lax-Milgram lemma, A is an isomorphism from V onto V ′. Alternatively, A can be considered as a linear unbounded self-adjoint operator in H with domain D(A) = {φ ∈ V : Aφ ∈ H}. By identifying H and its dual H ′ it follows that 10 POD: Error Estimates and Suboptimal Control 269 D(A) ↪→ V ↪→ H = H ′ ↪→ V ′, each embedding being continuous and dense, when D(A) is endowed with the graph norm of A. Moreover, let F : V × V → V ′ be a bilinear continuous operator mapping D(A) × D(A) into H. To simplify the notation we set F (φ) = F (φ,φ) for φ ∈ V . For given f ∈ C([0, T ];H) and y0 ∈ V we consider the nonlinear evolution problem d dt 〈y(t), φ〉H + a(y(t), φ) + 〈F (y(t)), φ〉V ′,V = 〈f(t), φ〉H (10.18a) for all φ ∈ V and t ∈ (0, T ] a.e. and y(0) = y0 in H. (10.18b) Assumption (A1). For every f ∈ C([0, T ];H) and y0 ∈ V there exists a unique solution of (10.18) satisfying y ∈ C([0, T ];V ) ∩ L(0, T ;D(A)) ∩H(0, T ;H). (10.19) Computation of the POD Basis Throughout we assume that Assumption (A1) holds and we denote by y the unique solution to (10.18) satisfying (10.19). For given n ∈ IN let 0 = t0 < t1 < . . . < tn ≤ T (10.20) denote a grid in the interval [0, T ] and set δtj = tj − tj−1, j = 1, . . . , n. Define ∆t = max (δt1, . . . , δtn) and δt = min (δt1, . . . , δtn). (10.21) Suppose that the snapshots y(tj) of (10.18) at the given time instances tj , j = 0, . . . , n, are known. We set V = span {y0, . . . , y2n}, where yj = y(tj) for j = 0, . . . , n, yj = ∂ty(tj−n) for j = n + 1, . . . , 2n with ∂ty(tj) = (y(tj)−y(tj−1))/δtj , and refer to V as the ensemble consisting of the snapshots {yj} j=0, at least one of which is assumed to be nonzero. Furthermore, we call {tj}j=0 the snapshot grid. Notice that V ⊂ V by construction. Throughout the remainder of this section we let X denote either the space V or H. 270 Michael Hinze and Stefan Volkwein Remark 10.2.1 (compare [KV01, Remark 1]). It may come as a surprise at first that the finite difference quotients ∂ty(tj) are included into the set V of snapshots. To motivate this choice let us point out that while the finite difference quotients are contained in the span of {yj} j=0, the POD bases differ depending on whether {∂ty(tj)}j=1 are included or not. The linear dependence does not constitute a difficulty for the singular value decomposition which is required to compute the POD basis. In fact, the snapshots themselves can be linearly dependent. The resulting POD basis is, in any case, maximally linearly independent in the sense expressed in (P ) and Proposition 10.2.5. Secondly, in anticipation of the rate of convergence results that will be presented in Section 10.3.3 we note that the time derivative of y in (10.18) must be approximated by the Galerkin POD based scheme. In case the terms {∂ty(tj)}j=1 are included in the snapshot ensemble, we are able to utilize the estimate

Second Order Methods for Optimal Control of Time-Dependent Fluid Flow

Second order methods for open loop optimal control problems governed by the two-dimensional instationary Navier--Stokes equations are investigated. Optimality systems based on a Lagrangian formulation and adjoint equations are derived. The Newton and quasi-Newton methods as well as various variants of SQP methods are developed for applications to optimal flow control, and their complexity in terms of system solves is discussed. Local convergence and rate of convergence are proved. A numerical example illustrates the feasibility of solving optimal control problems for two-dimensional instationary Navier--Stokes equations by second order numerical methods in a standard workstation environment.

Convergence of a Finite Element Approximation to a State-Constrained Elliptic Control Problem

We consider an elliptic optimal control problem with pointwise state constraints. The cost functional is approximated by a sequence of functionals which are obtained by discretizing the state equation with the help of linear finite elements and enforcing the state constraints in the nodes of the triangulation. The corresponding minima are shown to converge in

Instantaneous control of backward-facing step flows

L^2

Error estimates for abstract linear-quadratic optimal control problems using proper orthogonal decomposition

to the exact control as the discretization parameter tends to zero. Furthermore, error bounds for the control and the state are obtained in both two and three space dimensions. Finally, we present numerical examples which confirm our analytical findings.

Finite Element Approximation of Dirichlet Boundary Control for Elliptic PDEs on Two- and Three-Dimensional Curved Domains

In the present paper suboptimal boundary control strategies for the time-dependent, incompressible flow over the backward-facing step are considered. The objective consists in the reduction of the recirculation bubble of the flow behind the step. Several cost functionals are suggested and a frame for the derivation of the optimality systems for a general class of cost functionals is presented. Numerical examples are given.

Optimal control of the free boundary in a two-phase Stefan problem

Abstract In this paper we investigate POD discretizations of abstract linear–quadratic optimal control problems with control constraints. We apply the discrete technique developed by Hinze (Comput. Optim. Appl. 30:45–61, 2005) and prove error estimates for the corresponding discrete controls, where we combine error estimates for the state and the adjoint system from Kunisch and Volkwein (Numer. Math. 90:117–148, 2001; SIAM J. Numer. Anal. 40:492–515, 2002). Finally, we present numerical examples that illustrate the theoretical results.

Model reduction for circuit simulation

We consider the variational discretization of elliptic Dirichlet optimal control problems with constraints on the control. The underlying state equation, which is considered on smooth two- and three-dimensional domains, is discretized by linear finite elements taking into account domain approximation. The control variable is not discretized. We obtain optimal error bounds for the optimal control in two and three space dimensions and prove a superconvergence result in two dimensions, provided that the underlying mesh satisfies some additional condition. We confirm our analytical findings by numerical experiments.