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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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Cited by 3 (2 self)
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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
BALANCINGRELATED MODEL REDUCTION FOR DATASPARSE SYSTEMS
"... truncation Model reduction is an ubiquitous tool to facilitate or even enable the simulation, optimization and control of largescale dynamical systems. Application areas range from microelectronics (device and circuit simulation) over computational biology, control of mechanical and electrical syst ..."
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truncation Model reduction is an ubiquitous tool to facilitate or even enable the simulation, optimization and control of largescale dynamical systems. Application areas range from microelectronics (device and circuit simulation) over computational biology, control of mechanical and electrical systems to flow control and PDEconstrained optimization. In this talk, we will focus on problems from the latter application areas. In particular, we will discuss model reduction techniques based on system balancing for (optimal) control of parabolic partial differential equations. After discretization of the elliptic (spatial) differential operator by FEM or BEM methods, largescale, sparse (in case of FEM) or datasparse (in case of BEM) linear control systems are obtained. Due to the cubic complexity of standard implementations of balanced truncation and relatives, it is not possible to apply these methods directly to such systems. Here, we will discuss how the use of hierarchical matrices and the corresponding formatted arithmetic enables us to implement balanced truncation and related algorithms for model reduction in almost linear complexity. Our approach is based on the sign function method for solving Lyapunov equations, where matrix inversions, additions, and multiplications are replaced by the corresponding operations for hierarchical matrices [1]. Numerical experiments will demonstrate the applicability of this approach to control problems for several types of parabolic PDEs.
On the QR Decomposition of HMatrices
"... The hierarchical (H) matrix format allows storing a variety of dense matrices from certain applications in a special datasparse way with linearpolylogarithmic complexity. Many operations from linear algebra like matrixmatrix and matrixvector products, matrix inversion and LU decomposition can be ..."
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The hierarchical (H) matrix format allows storing a variety of dense matrices from certain applications in a special datasparse way with linearpolylogarithmic complexity. Many operations from linear algebra like matrixmatrix and matrixvector products, matrix inversion and LU decomposition can be implemented efficiently using the Hmatrix format. Due to its importance in solving many problems in numerical linear algebra like leastsquares problems, it is also desirable to have an efficient QR decomposition of Hmatrices. In the past, two different approaches for this task have been suggested in [Lin02] and [Beb08]. We will review the resulting methods and suggest a new algorithm to compute the QR decomposition of an Hmatrix. Like other Harithmetic operations the HQR decomposition is of linearpolylogarithmic complexity. We will compare our new algorithm with the older ones by using two series of test examples and