A memory-efficient model order reduction for time-delay systems. (English) Zbl 1305.65166
The authors consider a system of linear delay differential equations with time-invariant coefficient matrices. W. Michiels et al. [SIAM J. Matrix Anal. Appl. 32, No. 4, 1399–1421 (2011; Zbl 1247.65086)] constructed a larger approximating linear system without delay using approximations by the Chebyshev polynomials. Consequently, a model-order reduction of the larger system is feasible, where a moment matching approach based on Krylov subspaces results in a small linear system. The Arnoldi process yields an orthonormal basis in the required orthogonalisation.
Now the authors derive a two-level Arnoldi process for the orthogonalisation. This construction applies the sparse and banded structure of the matrix and a specific assumption on the initial vector in the definition of the Krylov subspaces. It follows that the memory requirements of the two-level Arnoldi algorithm are significantly lower in comparison to the classical Arnoldi process if the dimension of the original system of delay differential equations is large. Yet the computational effort coincides for both techniques. The stability of the two-level Arnoldi algorithm is also indicated.
The two-level Arnoldi process yields an orthogonal matrix, which can be used for the reduction of the approximating system without delay as well as the original system with delay. Concerning the direct reduction of the delay differential equations, the authors prove a moment matching property based on the transfer functions of the systems.
Finally, numerical simulations of a test example are presented, where a semidiscretisation in space of a delay partial differential equation yields a linear system with delay in time of the considered form. Differences between the transfer functions of the exact systems and reduced systems are illustrated. On the one hand, the results indicate that the classical Arnoldi process and the two-level approach yield the same accuracy in the reduction of the system without delay. On the other hand, the direct reduction of the delay differential equations is more accurate in comparison to the reduction of the approximating system without delay.
Now the authors derive a two-level Arnoldi process for the orthogonalisation. This construction applies the sparse and banded structure of the matrix and a specific assumption on the initial vector in the definition of the Krylov subspaces. It follows that the memory requirements of the two-level Arnoldi algorithm are significantly lower in comparison to the classical Arnoldi process if the dimension of the original system of delay differential equations is large. Yet the computational effort coincides for both techniques. The stability of the two-level Arnoldi algorithm is also indicated.
The two-level Arnoldi process yields an orthogonal matrix, which can be used for the reduction of the approximating system without delay as well as the original system with delay. Concerning the direct reduction of the delay differential equations, the authors prove a moment matching property based on the transfer functions of the systems.
Finally, numerical simulations of a test example are presented, where a semidiscretisation in space of a delay partial differential equation yields a linear system with delay in time of the considered form. Differences between the transfer functions of the exact systems and reduced systems are illustrated. On the one hand, the results indicate that the classical Arnoldi process and the two-level approach yield the same accuracy in the reduction of the system without delay. On the other hand, the direct reduction of the delay differential equations is more accurate in comparison to the reduction of the approximating system without delay.
Reviewer: Roland Pulch (Greifswald)
MSC:
| 65L03 | Numerical methods for functional-differential equations |
| 65F25 | Orthogonalization in numerical linear algebra |
| 34K06 | Linear functional-differential equations |
| 34K28 | Numerical approximation of solutions of functional-differential equations (MSC2010) |
| 65L20 | Stability and convergence of numerical methods for ordinary differential equations |
| 65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |
| 35R10 | Partial functional-differential equations |
Keywords:
orthogonalisation; Arnoldi process; model-order reduction; moment matching; Krylov subspace; memory saving algorithm; numerical example; stability; semidiscretization; delay partial differential equationCitations:
Zbl 1247.65086Software:
SOARReferences:
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