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CROSSGRAMIAN BASED MODEL REDUCTION FOR DATASPARSE SYSTEMS ∗
"... Abstract. Model order reduction (MOR) is common in simulation, control and optimization of complex dynamical systems arising in modeling of physical processes and in the spatial discretization of parabolic partial differential equations (PDEs) in two or more dimensions. Typically, after a semidiscre ..."
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Abstract. Model order reduction (MOR) is common in simulation, control and optimization of complex dynamical systems arising in modeling of physical processes and in the spatial discretization of parabolic partial differential equations (PDEs) in two or more dimensions. Typically, after a semidiscretization of the differential operator by the finite element method (FEM) or by the boundary element method, we have a large statespace dimension n = O(10 4). It is assumed that the number of inputs and outputs is equal and much smaller than n. We show how to compute an approximate reducedorder system of order r ≪ n with a balancingrelated model reduction method. The method is based on the computation of the crossGramian (CG) X, which is the solution of one Sylvester equation. As standard algorithms for the solution of Sylvester equations are of limited use for largescale systems, we investigate approaches based on the sign function method. To make this iterative method applicable in the largescale setting, we use a modified iteration scheme for computing lowrank factors of the solution X and we incorporate structural information from the underlying PDE model into the approach. By using datasparse matrix approximations, hierarchical matrix formats, and the corresponding formatted arithmetic we obtain an efficient solver having linearpolylogarithmic complexity. We show that the reducedorder model can then be computed from the lowrank factors directly. Numerical experiments demonstrate the efficiency of our approach.
SYSTEMTHEORETIC METHODS FOR MODEL REDUCTION OF LARGESCALE SYSTEMS: SIMULATION, CONTROL, AND INVERSE PROBLEMS
"... Abstract. Model (order) reduction, MOR for short, is an ubiquitous tool in the analysis and simulation of dynamical systems, control design, circuit simulation, structural dynamics, CFD, etc. In systems and control, MOR methods based on balanced truncation (BT) and its relatives have been widely use ..."
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Abstract. Model (order) reduction, MOR for short, is an ubiquitous tool in the analysis and simulation of dynamical systems, control design, circuit simulation, structural dynamics, CFD, etc. In systems and control, MOR methods based on balanced truncation (BT) and its relatives have been widely used. In other areas, they have been less successful as it is common belief that their computational complexity is too high to apply them to largescale problems involving sparse matrices. We will review the recent development of efficient algorithms for solving matrix equations that make balancingrelated model reduction methods competitive to other MOR approaches these new implementations fall into the same complexity class as the omnipresent Krylov subspace methods. As balancingrelated methods offer the advantage of computable error bounds that allow for an adaptive choice of the order of the reduced model and moreover, they can be shown to preserve certain system properties like stability, passivity, dissipativity, etc., these new BT implementations become attractive in various application areas. These include • nanoelectronics/VLSI design, where MOR is inevitable for circuit simulation, • (optimal) control of physical processes described by partial differential equations (PDEs), • inverse problems related to the identification of input signals, e.g., for tracking control. We will discuss some particular aspects arising in these areas when applying BTrelated MOR techniques. The performance of several BTrelated approaches will be demonstrated using examples from a variety of application areas. 1