European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
10:30   MS30: Numerical methods for PDE-constrained and controlled optimization problems with appplications (Part 1)
Chair: Maya Neytcheva
25 mins
Applications of the PRESB preconditioning method for OPT-PDE problems
Owe Axelsson
Abstract: The abstract is enclosed in the attachment.
25 mins
Fast Interior Point Solvers and Preconditioning for PDE-Constrained Optimization
John Pearson
Abstract: In this talk we consider the effective numerical solution of PDE-constrained optimization problems with additional box constraints on the state and control variables. Upon discretization, these may give rise to problems of quadratic or nonlinear programming form: a sensible solution strategy is to apply an interior point method, provided one can solve the large and structured matrix systems that arise at each Newton step. We therefore consider fast and robust preconditioned iterative methods for these systems, examining two cases: (i) where L^2 norms measure the misfit between state and desired state, as well as the control; (ii) with an additional L^1 norm term promoting sparsity in the control. Having motivated and derived our recommended preconditioners, and shown some theoretical results on saddle point systems, we present numerical results demonstrating the potency of our solvers. This talk is based on work with Jacek Gondzio, Margherita Porcelli, and Martin Stoll.
25 mins
Spectral Analysis of Saddle-point Matrices from Optimal Control PDE Problems
Fabio Durastante, Isabella Furci
Abstract: Optimization problems with constraints given in terms of Partial Differential Equations are a ubiquitous problem of the applied mathematics. We consider here the distributed optimal control for the Poisson equation~[3], and focus on the sequences of saddle-point linear systems stemming from its Finite Element approximation $\mathcal{A}}_N$. Our main objective is then devising an efficient solution strategy for them by proving that the matrix sequence $\{\mathcal{A}_N\}_N$ belongs to the class of \emph{Generalized Locally Toeplitz} sequences~[2] in their most general block multilevel setting. This framework permits describing the spectrum of the matrix sequence in a compact form, i.e., it ensures the existence of a measurable matrix--valued function $\kappa$, called symbol, correlated with the sequence. The knowledge of $\kappa$ enables the detailed study of the distribution and the localization of the spectrum of the sequence together with the behavior of its extremal and outlier eigenvalues, and thus of its conditioning. By exploiting these results, we propose a specific solution strategy based on a preconditioned Krylov method. The study of the complexity, and of the convergence speed of the proposed method is then carried out by means of the information acquired from the symbol of the preconditioned sequence. [1] Durastante, F., and I. Furci. Spectral Analysis of Saddle-point Matrices from Optimization problems with Elliptic PDE Constraints. 2019, arXiv e-prints , arXiv:1903.01869. [2] Garoni, C., and S. Serra-Capizzano. Generalized Locally Toeplitz Sequences: Theory and Applications. Vol. 1. Springer, 2017. [3] Tröltzsch, F. Optimal control of partial differential equations: theory, methods, and applications. Vol. 112. American Mathematical Soc., 2010.
25 mins
PDE-constrained optimization: Matrix structures and preconditioners
Ivo Dravins, Maya Neytcheva
Abstract: \documentclass[12pt,a4paper]{article} \title{PDE-constrained optimization: \\ Matrix structures and preconditioners} \author{Ivo Dravins, Maya Neytcheva } \begin{document} \maketitle Optimization and control of processes governed by partial differential equations are central to many important applications. The importance of their numerical simulation steadily grows, together with the need for robust and efficient numerical techniques to solve the arising large-scale algebraic problems. In this presentation we the minimization of a cost functional of the form $$ J(y, u) = \frac{1}{2} \|y-y_d\|^2_{L^2 (\Omega_0)}+\frac{\alpha}{2}\|u\|^2_{L^2(\Omega)} + \beta\|u\|^2_{L^1(\Omega)}, $$ optimizing for both a state and a control variable, $y$ and $u$, correspondingly. In addition we include constraints on $u$ and or $y$ to be within certain bounds and on $u$ - to be sparse, which is achieved by including the $L^1$ norm term in $J$. Here $y_0$ is some desired state, $0<\alpha<1$ and $0<\beta<1$ are regularization parameters. The arising algebraic problems in the above setting are nonlinear, solved using the so-called semi-smooth Newton method. The focus in this presentation is on the structure of the matrices in the arising linear systems to be solved at each nonlinear iteration and approaches to construct numerically and computationally efficient preconditioners. We illustrate the performance of the involved nonlinear and linear solvers with some numerical experiments. The implementation is done in \texttt{Julia}. \end{document}