49M15 Newton-type methods
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Keywords
- optimal control (3)
- Newton-CG (2)
- trajectory storage (2)
- Cahn-Hilliard system (1)
- Elasto-plastic contact, Variational inequality of the 2nd kind, Fenchel duality, Moreau-Yosida/Tichonov regularization, Path-following, Semismooth Newton (1)
- Lavrentiev type regularization (1)
- Newton methods in function space (1)
- PDE Optimization, Conditional Value-At-Risk, Uncertainty Quantification (1)
- Parabolic optimal control (1)
- Schur complement (1)
Uncertainty is inevitable when solving science and engineering application problems. In the face of
uncertainty, it is essential to determine robust and risk-averse solutions. In this work,
we consider a class of PDE-constrained optimization problems in which the PDE coefficients
and inputs may be uncertain. We introduce two approximations for minimizing the
conditional value-at-risk for such PDE-constrained optimization problems. These approximations are based
on the primal and dual formulations of the conditional value-at-risk. For the primal problem,
we introduce a smooth approximation of the conditional value-at-risk in order to utilize
derivative-based optimization algorithms and to take advantage of the convergence properties
of quadrature-based discretizations. For this smoothed conditional value-at-risk, we prove
differentiability as well as consistency of our approximation. For the dual problem, we
regularize the inner maximization problem, rigorously derive optimality conditions, and demonstrate
the consistency of our approximation. Furthermore, we propose a fixed-point iteration that takes
advantage of the structure of the regularized optimality conditions and provides a means of calculating
worst-case probability distributions based on the given probability level. We conclude with numerical
results.
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 adjoint 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.
A Fenchel dualization scheme for the one-step time-discretized contact problem of quasi-static
elasto-plasticity with kinematic or isotropic hardening is considered. The associated path is induced
by a combined Moreau-Yosida / Tichonov regularization of the dual problem.
The sequence of solutions to the regularized problems is shown to converge strongly to the optimal
displacement-stress-strain triple of the original elasto-plastic contact problem in the space-continuous setting.
This property relies on the density of the intersection of certain convex sets which is shown as well.
It is also argued that the mappings associated with the resulting problems are Newton- or slantly differentiable.
Consequently, each regularized subsystem can be solved mesh-independently at a local
superlinear rate of convergence. For efficiency purposes, an inexact path-following approach is proposed and
a numerical validation of the theoretical results is given.
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.
We introduce and analyze nonsmooth Schur-Newton methods for a class of nonsmooth saddle point problems. The method is able to solve problems where the primal energy decomposes into a convex smooth part and a convex separable but nonsmooth part. The method is based on nonsmooth Newton techniques for an equivalent unconstrained dual problem. Using this we show that it is globally convergent even for inexact evaluation of the linear subproblems.
An adaptive finite element semi-smooth Newton solver for
the Cahn-Hilliard model with double obstacle free energy is proposed. For this purpose, the governing system is discretised in time using a semi-implicit scheme, and the resulting time-discrete system is formulated as an optimal control problem with pointwise constraints on the control. For the numerical solution of the optimal control problem, we propose a function space based algorithm which combines a Moreau-Yosida regularization technique for handling the control constraints with a semi-smooth-Newton method for solving the optimality systems of the resulting sub-problems. Further, for the discretization in space and in connection with the proposed algorithm, an adaptive finite element method is considered. The performance of the overall algorithm is illustrated by numerical experiments.
A continuity result for Nemyckii Operators and some applications in PDE constrained optimal control
(2008)
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 primal-dual interior point method for state-constrained parabolic optimal control problems is considered. By a Lavrentiev type regularization, the state constraints are transformed to mixed control-state constraints which, after a simple transformation, can be handled as control constraints. Existence and convergence of the central path are shown. Moreover, the convergence of a short step interior point algorithm is proven in a function space setting. The theoretical properties of the algorithm are confirmed by numerical examples.