48 research outputs found
A Preconditioned Iteration Method for Solving Sylvester Equations
A preconditioned gradient-based iterative method is derived by judicious selection of two auxil- iary matrices. The strategy is based on the Newton’s iteration method and can be regarded as a
generalization of the splitting iterative method for system of linear equations. We analyze the convergence of the method and illustrate that the approach is able to considerably accelerate the convergence of the gradient-based iterative method
Iterative and doubling algorithms for Riccati-type matrix equations: a comparative introduction
We review a family of algorithms for Lyapunov- and Riccati-type equations
which are all related to each other by the idea of \emph{doubling}: they
construct the iterate of another naturally-arising fixed-point
iteration via a sort of repeated squaring.
The equations we consider are Stein equations , Lyapunov
equations , discrete-time algebraic Riccati equations
, continuous-time algebraic Riccati equations
, palindromic quadratic matrix equations , and
nonlinear matrix equations . We draw comparisons among these
algorithms, highlight the connections between them and to other algorithms such
as subspace iteration, and discuss open issues in their theory.Comment: Review article for GAMM Mitteilunge
On dynamical low-rank integrators for matrix differential equations
This thesis is concerned with dynamical low-rank integrators for matrix differential equations, typically stemming from space discretizations of partial differential equations. We first construct and analyze a dynamical low-rank integrator for second-order matrix differential equations, which is based on a Strang splitting and the projector-splitting integrator, a dynamical low-rank integrator for first-order matrix
differential equations proposed by Lubich and Osedelets in 2014. For the analysis, we derive coupled recursive inequalities, where we express the global error of the scheme in terms of a time-discretization error and a low-rank error contribution. The first can be treated with Taylor series expansion of the exact solution. For the latter, we make use of an induction argument and the convergence result derived by Kieri, Lubich, and Walach in 2016 for the projector-splitting integrator.
From the original method, several variants are derived which are tailored to, e.g., stiff or highly oscillatory second-order problems. After discussing details on the implementation of dynamical low-rank schemes, we turn towards rank-adaptivity. For the projector-splitting integrator we derive both a technique to realize changes in the approximation ranks efficiently and a heuristic to choose the rank appropriately over time. The core idea is to determine the rank such that the error of the low-rank
approximation does not spoil the time-discretization error. Based on the rank-adaptive pendant of the projector-splitting integrator, rank-adaptive dynamical low-rank integrators for (stiff and non-stiff) first-order and second-order matrix differential equations are derived. The thesis is concluded with numerical experiments to confirm our theoretical findings
Structured condition number for multiple right-hand side linear systems with parameterized quasiseparable coefficient matrix
In this paper, we consider the structured perturbation analysis for multiple
right-hand side linear systems with parameterized coefficient matrix.
Especially, we present the explicit expressions for structured condition
numbers for multiple right-hand sides linear systems with {1;1}-quasiseparable
coefficient matrix in the quasiseparable and the Givens-vector representations.
In addition, the comparisons of these two condition numbers between themselves,
and with respect to unstructured condition number are investigated. Moreover,
the effective structured condition number for multiple right-hand sides linear
systems with {1;1}-quasiseparable coefficient matrix is proposed. The
relationships between the effective structured condition number and structured
condition numbers with respect to the quasiseparable and the Givens-vector
representations are also studied. Numerical experiments show that there are
situations in which the effective structured condition number can be much
smaller than the unstructured ones
5 Post-processing methods for passivity enforcement
Many physical systems are passive (or dissipative): they are unable to generate energy on their own, but they can store energy in some form while exchanging power with the surrounding environment. This chapter describes the most prominent approaches for ensuring that Reduced Order Models are passive, so that their math- ematical representation satisfies an appropriate dissipativity condition. The main focus is on Linear and Time-Invariant (LTI) systems in state-space form. Different conditions for testing passivity of a given LTI model are discussed, including Linear Matrix Inequalities (LMIs), Frequency-Domain Inequalities, and spectral conditions on associated Hamiltonian matrices. Then we describe common approaches for perturbing a given non-passive system to enforce its passivity. Various examples from electronic applications are used to demonstrate both theory and algorithm performance