Publications

2008

• A characterization of singular electromagnetic fields by an inductive approach
  
  • Assous F.
  • Ciarlet Patrick
  • Garcia E.
  International Journal of Numerical Analysis and Modeling, Institute for Scientific Computing and Information, 2008, 5 (3), pp.491-515. In this article, we are interested in the mathematical modeling of singular electromagnetic fields, in a non-convex polyhedral domain. We first describe the local trace (i. e. defined on a face) of the normal derivative of an L2 function, with L2 Laplacian. Among other things, this allows us to describe dual singularities of the Laplace problem with homogeneous Neumann boundary condition. We then provide generalized integration by parts formulae for the Laplace, divergence and curl operators. With the help of these results, one can split electromagnetic fields into regular and singular parts, which are then characterized. We also study the particular case of divergence-free and curl-free fields, and provide non-orthogonal decompositions that are numerically computable. © 2008 Institute for Scientific Computing and Information.
• Numerical solution of large-scale Lyapunov equations, Riccati equations, and linear-quadratic optimal control problems
  
  • Benner Peter
  • Li Jing-Rebecca
  • Penzl Thilo
  Numerical Linear Algebra with Applications, Wiley, 2008, 15 (9), pp.755-777. We study large-scale, continuous-time linear time-invariant control systems with a sparse or structured state matrix and a relatively small number of inputs and outputs. The main contributions of this paper are numerical algorithms for the solution of large algebraic Lyapunov and Riccati equations and linearquadratic optimal control problems, which arise from such systems. First, we review an alternating direction implicit iteration-based method to compute approximate low-rank Cholesky factors of the solution matrix of large-scale Lyapunov equations, and we propose a refined version of this algorithm. Second, a combination of this method with a variant of Newton's method (in this context also called Kleinman iteration) results in an algorithm for the solution of large-scale Riccati equations. Third, we describe an implicit version of this algorithm for the solution of linear-quadratic optimal control problems, which computes the feedback directly without solving the underlying algebraic Riccati equation explicitly. Our algorithms are efficient with respect to both memory and computation. In particular, they can be applied to problems of very large scale, where square, dense matrices of the system order cannot be stored in the computer memory. We study the performance of our algorithms in numerical experiments. (10.1002/nla.622)
  DOI : 10.1002/nla.622