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NEW DOCTORAL DEGREES Optimization of the solution of parameter depending Sylvester equation and applications
Ivana Kuzmanović
; Department of Mathematics, University of Osijek, Osijek, Croatia
Abstract
In dissertation the problem of solving and optimizing the solution of structured Sylvester (i.e. Lyapunov) and $T$-Sylvester matrix equations is considered, with special focus on the parameter dependent Sylvester and $T$-Sylvester equations.
It has been shown that the use of a structure can significantly contribute to acceleration of the process of solving the Sylvester as well as the $T$-Sylvester equation, and especially some related sequences of equations that arise e.g. in the process of optimization of the solution of parameter-dependent Sylvester and $T$-Sylvester equations.
Due to the appropriate structure, Sylvester equations in which matrices are the sum of simple matrices (e.g. diagonal or block-diagonal)
with small-rank matrices are considered in dissertation, i.e. Sylvester equations of the form
\[(A_0+U_1V_1)X+X(B_0+U_2V_2)=E,\] where $A_0,B_0$ are simple matrices and $U_1,V_1,U_2,V_2$ are matrices with a low rank $r$.
By using the standard Sherman-Morrison-Woodbury formula it is possible to obtain the so-called Sherman-Morrison-Woodbury formula for the solution of the previous equation. The obtained formula can be used for the construction of an algorithm that solves the equation of the given form much more efficiently than standard algorithms.
An algorithm based on the Sherman-Morrison-Woodbury formula is especially efficient for computing the solution of the parameter dependent Sylvester equation
\[(A_0-vU_1V_1)X(v)+X(v)(B_0-vU_2V_2)=E\]
for many different values of parameter $v$. While the standard methods need $ O (n ^ 3) $ elementary operations for each different value of parameter $v$, the method based on the Sherman-Morrison-Woodbury formula has complexity $\mathcal{O}(rkn^2)$, where $r,k\ll n$ for the first value of $v$, while each of the following solving processes with the other value of $v$ needs only $\mathcal{O}(rn^2)$ operations, where $k$ is the dimension of the corresponding Krylov subspace. In addition, this approach also allows computation of derivatives of $ X (v) $ in $ O (rn ^ 2) $ elementary operations, which enables efficient optimization of the solution $ X (v) $ with respect to parameter $ v$.
A special case, a parameter-dependent Lyapunov equation of the form \[(A_0-vUU^T)X(v)+X(v)(A_0-vUU^T)^T=E,\]
occurs during calculation and optimization of dampers' viscosity in mechanical systems with respect to the criterion of minimizing the average total unit energy.
Keywords
Hrčak ID:
93303
URI
Publication date:
5.12.2012.
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