Odjel za matematiku Sveučilišta Josipa Jurja Strossmayera u Osijeku organizira predavanje Low rank approximations of infinite-dimensional Lyapunov equations, na engleskom jeziku
NAJAVA – Predavanje će se održati u četvrtak, 6. studenoga 2014. godine s početkom u 17 sati u predavaonici broj 36 na Odjelu za matematiku Sveučilišta Josipa Jurja Strossmayera u Osijeku (Trg Ljudevita Gaja 6).
Predavanje će održati:
- Izv. prof. dr. sc. Luka Grubišić, Department of Mathematics, University of Zagreb, Zagreb, Croatia
S obzirom da je riječ o predavanju na engleskom jeziku, na tom jeziku objavljujemo i kratak opis predavanja:
We analyze the convergence properties of an explicitly constructed sequence of low rank approximations to the solution of an infinite dimensional operator Lyapunov equation. As an application of our abstract theory we consider approximations of linear control system in the context of model order reduction by balanced truncation.
In particular we are interested in systems governed by the heat equation with both distributed as well as boundary control. In a recent study by Opmeer et.al the authors have presented an ADI-based algorithm in infinite dimensional setting which was used to construct low rank approximations of the solutions of a Lyapunov equation.
Let now A and B be unbounded operators. The formal expression AX+XA’=-BB’, where A’ and B’ are appropriate operator duals, is called an abstract Lyapunov equation. We analyze the approximation properties of solutions of abstract Lyapunov equations in the setting of a scale of Hilbert spaces associated to an unbounded diagonalizable operator which satisfies the Kato’s square root theorem. We call an (unbounded) operator A diagonalizable if there exists a bounded operator Q, with a bounded inverse, such that the (unbounded) operator Q^{-1}AQ is a normal operator with a compact resolvent.
We assume that A generates an exponentially stable analytic semigroup and we construct—using sinc-quadrature techniques—a rank (2k+1) Ran(B) approximation X_{2k} to the operator X.
In the case of a (more strongly) unbounded control operator B, e.g. an operator only bounded in a weighted Hilbert space, we obtain the same type of convergence estimates in an associated weighted norm on (the subspace of) the space of compact operators.
Based on our convergence estimates we also discuss ramifications of this analysis for the design of adaptive finite element methods including the analysis of the influence of linear algebra approximations on the overall process.
