Balanced Pod for Model Reduction of Linear PDE Systems: Convergence Theory
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We consider convergence analysis for a model reduction algorithm for a class of linear infinite dimensional systems. The algorithm computes an approximate balanced truncation of the system using solution snapshots of specific linear infinite dimensional differential equations. The algorithm is related to the proper orthogonal decomposition, and it was first proposed for systems of ordinary differential equations by Rowley (Int. J. Bifurc. Chaos Appl. Sci. Eng. 15(3), 997-1013, 2005). For the convergence analysis, we consider the algorithm in terms of the Hankel operator of the system, rather than the product of the system Gramians as originally proposed by Rowley. For exponentially stable systems with bounded finite rank input and output operators, we prove that the balanced realization can be expressed in terms of balancing modes, which are related to the Hankel operator. The balancing modes are required to be smooth, and this can cause computational difficulties for PDE systems. We show how this smoothness requirement can be lessened for parabolic systems, and we also propose a variation of the algorithm that avoids the smoothness requirement for general systems. We prove entrywise convergence of the matrices in the approximate reduced order models in both cases, and present numerical results for two example PDE systems.