A POD Projection Method for Large-Scale Algebraic Riccati Equations
The solution of large-scale matrix algebraic Riccati equations is important for instance in control design and model reduction and remains an active area of research. We consider a class of matrix algebraic Riccati equations (AREs) arising from a linear system along with a weighted inner product. This problem class often arises from a spatial discretization of a partial differential equation system. We propose a projection method to obtain low rank solutions of AREs based on simulations of linear systems coupled with proper orthogonal decomposition. The method can take advantage of existing (black box) simulation code to generate the projection matrices. We also develop new weighted norm residual computations and error bounds. We present numerical results demonstrating that the proposed approach can produce highly accurate approximate solutions. We also brie y discuss making the proposed approach completely data-based so that one can use existing simulation codes without accessing system matrices.
B. Kramer and J. R. Singler, "A POD Projection Method for Large-Scale Algebraic Riccati Equations," Numerical Algebra, Control and Optimization, vol. 6, no. 4, pp. 413-435, American Institute of Mathematical Sciences, Dec 2016.
The definitive version is available at https://doi.org/10.3934/naco.2016018
Mathematics and Statistics
Center for High Performance Computing Research
Keywords and Phrases
Algebraic Riccati Equations; Control Theory; Large-Scale; Proper Orthogonal Decomposition; Reduced-Order Modeling
International Standard Serial Number (ISSN)
Article - Journal
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