49M15 Newton-type methods
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For the solution of optimal control problems governed by nonlinear parabolic PDEs, methods working on the reduced objective functional are often employed to avoid a full spatio-temporal discretization of the problem. The evaluation of the reduced gradient requires one solve of
the state equation forward in time, and one backward solve of the ad-joint equation. The state enters into the adjoint equation, requiring the storage of a full 4D data set. If Newton-CG methods are used, two additional trajectories have to be stored. To get numerical results which are accurate enough, in many case very fine discretizations in time and space are necessary, which leads to a significant amount of data to be stored and transmitted to mass storage. Lossy compression methods were
developed to overcome the storage problem by reducing the accuracy of the stored trajectories. The inexact data induces errors in the reduced gradient and reduced Hessian. In this paper, we analyze the influence of such a lossy trajectory compression method on Newton-CG methods for optimal control of parabolic PDEs and design an adaptive strategy for choosing appropriate quantization tolerances.
This paper presents efficient computational techniques for solving an optimization problem in cardiac defibrillation governed by the monodomain equations. Time-dependent electrical currents injected at different spatial positions act as the control. Inexact Newton-CG methods are used, with reduced gradient computation by adjoint solves. In order to reduce the computational complexity, adaptive mesh refinement for state and adjoint equations is performed. To reduce the high storage and bandwidth demand imposed by adjoint gradient and Hessian-vector evaluations, a lossy compression technique for storing trajectory data is applied. An adaptive choice of quantization tolerance based on error estimates is developed in order to ensure convergence. The efficiency of the proposed approach is demonstrated on numerical examples.
Fast nonlinear programming methods following the all-at-once approach usually employ Newton's method for solving linearized Karush-Kuhn-Tucker (KKT) systems. In nonconvex problems, the Newton direction is only guaranteed to be a descent direction if the Hessian of the Lagrange function is positive definite on the nullspace of the active constraints, otherwise some modifications to Newton's method are necessary. This condition can be verified using the signs of the KKT's eigenvalues (inertia), which are usually available from direct solvers for the arising linear saddle point problems. Iterative solvers are mandatory for very large-scale problems, but in general do not provide the inertia. Here we present a preconditioner based on a multilevel incomplete $LBL^T$ factorization, from which an approximation of the inertia can be obtained. The suitability of the heuristics for application in optimization methods is verified on an interior point method applied to the CUTE and COPS test problems, on large-scale 3D PDE-constrained optimal control problems, as well as 3D PDE-constrained optimization in biomedical cancer hyperthermia treatment planning. The efficiency of the preconditioner is demonstrated on convex and nonconvex problems with $150^3$ state variables and $150^2$ control variables, both subject to bound constraints.
A continuity result for Nemyckii Operators and some applications in PDE constrained optimal control
(2006)
This work explores two applications of a classical result on the continuity of Nemyckii operators to optimal control with PDEs. First, we present an alternative approach to the analysis of Newton's method for function space problems involving semi-smooth Nemyckii operators. A concise proof for superlinear convergence is presented, and sharpened bounds on the rate of convergence are derived. Second, we derive second order sufficient conditions for problems, where the underlying PDE has poor regularity properties. We point out that the analytical structure in both topics is essentially the same.
A thorough convergence analysis of the Control Reduced Interior Point Method in function space is performed. This recently proposed method is a primal interior point pathfollowing scheme with the special feature, that the control variable is eliminated from the optimality system. Apart from global linear convergence we show, that this method converges locally almost quadratically, if the optimal solution satisfies a function space analogue to a non-degeneracy condition. In numerical experiments we observe, that a prototype implementation of our method behaves in compliance with our theoretical results.
A primal interior point method for control constrained optimal control problems with PDE constraints is considered. Pointwise elimination of the control leads to a homotopy in the remaining state and dual variables, which is addressed by a short step pathfollowing method. The algorithm is applied to the continuous, infinite dimensional problem, where discretization is performed only in the innermost loop when solving linear equations. The a priori elimination of the least regular control permits to obtain the required accuracy with comparable coarse meshes. Convergence of the method and discretization errors are studied, and the method is illustrated at two numerical examples.
A primal-dual interior point method for optimal control problems with PDE constraints is considered. The algorithm is directly applied to the infinite dimensional problem. Existence and convergence of the central path are analyzed. Numerical results from an inexact continuation method applied to a model problem are shown.
The paper extends affine conjugate Newton methods from convex to nonconvex minimization, with particular emphasis on PDE problems originating from compressible hyperelasticity. Based on well-known schemes from finite dimensional nonlinear optimization, three different algorithmic variants are worked out in a function space setting, which permits an adaptive multilevel finite element implementation. These algorithms are tested on two well-known 3D test problems and a real-life example from surgical operation planning.
A new approach to the numerical solution of optimal control problems including control and state constraints is presented. Like hybrid methods, the approach aims at combining the advantages of direct and indirect methods. Unlike hybrid methods, however, our method is directly based on interior-point concepts in function space --- realized via an adaptive multilevel scheme applied to the complementarity formulation and numerical continuation along the central path. Existence of the central path and its continuation towards the solution point is analyzed in some theoretical detail. An adaptive stepsize control with respect to the duality gap parameter is worked out in the framework of affine invariant inexact Newton methods. Finally, the performance of a first version of our new type of algorithm is documented by the successful treatment of the well-known intricate windshear problem.
An affine invariant convergence analysis for inexact augmented Lagrangian-SQP methods is presented. The theory is used for the construction of an accuracy matching between iteration errors and truncation errors, which arise from the inexact linear system solves. The theoretical investigations are illustrated numerically by an optimal control problem for the Burgers equation.
A path following algorithm for linear complementarity problems is presented. Given a point $z$ that approximates a point $z(\tau)$ on the central path with complementarity gap $\tau$, one determines a parameter $\theta\in (0,1)$ so that this point satisfies the hypothesis of the affine invariant Kantorovich Theorem for the equation defining $z((1-\theta)\tau)$. It is shown that $\theta$ is bounded below by a multiple of $n^{-1/2}$, where $n$ is the dimension of the problem. Since the hypothesis of of the Kantorovich Theorem is satisfied the sequence generated by Newton's method, or by the simplified Newton method, will converge to $z((1-\theta)\tau)$. We show that the number of steps required to obtain an acceptable approximation of $z((1-\theta)\tau)$ is bounded above by a number independent of $n$. Therefore the algorithm has $O(\sqrt{n}L)$-iteration complexity. The parameters of the algorithm can be determined in such a way that only one Newton step is needed each time the complementarity gap is decreased.
A wide range of free boundary problems occurring in engineering andindustry can be rewritten as a minimization problem for astrictly convex, piecewise smooth but non--differentiable energy functional.The fast solution of related discretized problemsis a very delicate question, because usual Newton techniquescannot be applied. We propose a new approach based on convex minimization and constrained Newton type linearization. While convex minimization provides global convergence of the overall iteration, the subsequent constrained Newton type linearization is intended to accelerate the convergence speed. We present a general convergence theory and discuss several applications.