698 research outputs found
Peer Methods for the Solution of Large-Scale Differential Matrix Equations
We consider the application of implicit and linearly implicit
(Rosenbrock-type) peer methods to matrix-valued ordinary differential
equations. In particular the differential Riccati equation (DRE) is
investigated. For the Rosenbrock-type schemes, a reformulation capable of
avoiding a number of Jacobian applications is developed that, in the autonomous
case, reduces the computational complexity of the algorithms. Dealing with
large-scale problems, an efficient implementation based on low-rank symmetric
indefinite factorizations is presented. The performance of both peer approaches
up to order 4 is compared to existing implicit time integration schemes for
matrix-valued differential equations.Comment: 29 pages, 2 figures (including 6 subfigures each), 3 tables,
Corrected typo
On a family of low-rank algorithms for large-scale algebraic Riccati equations
In [3] it was shown that four seemingly different algorithms for computing
low-rank approximate solutions to the solution of large-scale
continuous-time algebraic Riccati equations (CAREs) generate the same sequence when used with the same
parameters. The Hermitian low-rank approximations are of the form where is a matrix with only few columns and is a
small square Hermitian matrix. Each generates a low-rank Riccati residual
such that the norm of the residual can be evaluated easily
allowing for an efficient termination criterion. Here a new family of methods
to generate such low-rank approximate solutions of CAREs is proposed.
Each member of this family of algorithms proposed here generates the same
sequence of as the four previously known algorithms. The approach is
based on a block rational Arnoldi decomposition and an associated block
rational Krylov subspace spanned by and Two specific versions of
the general algorithm will be considered; one will turn out to be a rediscovery
of the RADI algorithm, the other one allows for a slightly more efficient
implementation compared to the RADI algorithm. Moreover, our approach allows
for adding more than one shift at a time
Adaptive high-order splitting schemes for large-scale differential Riccati equations
We consider high-order splitting schemes for large-scale differential Riccati
equations. Such equations arise in many different areas and are especially
important within the field of optimal control. In the large-scale case, it is
critical to employ structural properties of the matrix-valued solution, or the
computational cost and storage requirements become infeasible. Our main
contribution is therefore to formulate these high-order splitting schemes in a
efficient way by utilizing a low-rank factorization. Previous results indicated
that this was impossible for methods of order higher than 2, but our new
approach overcomes these difficulties. In addition, we demonstrate that the
proposed methods contain natural embedded error estimates. These may be used
e.g. for time step adaptivity, and our numerical experiments in this direction
show promising results.Comment: 23 pages, 7 figure
Effizientes Lösen von groĂskaligen Riccati-Gleichungen und ein ODE-Framework fĂŒr lineare Matrixgleichungen
This work considers the iterative solution of large-scale matrix equations. Due to the size of the system matrices in large-scale Riccati equations the solution can not be calculated directly but is approximated by a low rank matrix ZYZ^*. Herein Z is a basis of a low-dimensional rational Krylov subspace. The inner matrix Y is a small square matrix. Two ways to choose this inner matrix are examined: By imposing a rank condition on the Riccati residual and by projecting the Riccati residual onto the Krylov subspace generated by Z. The rank condition is motivated by the well-known ADI iteration. The ADI solutions span a rational Krylov subspace and yield a rank-p residual. It is proven that the rank-p condition guarantees existence and uniqueness of such an approximate solution. Known projection methods are generalized to oblique projections and a new formulation of the Riccati residual is derived, which allows for an efficient evaluation of the residual norm. Further a truncated approximate solution is characterized as the solution of a Riccati equation, which is projected to a subspace of the Krylov subspace generated by Z. For the approximate solution of Lyapunov equations a system of ordinary differential equations (ODEs) is solved via Runge-Kutta methods. It is shown that the space spanned by the approximate solution is a rational Krylov subspace with poles determined by the time step sizes and the eigenvalues of the matrices of the Butcher tableau of the used Runge-Kutta method. The method is applied to a model order reduction problem. The analytical solution of the system of ODEs satisfies an algebraic invariant. Those Runge-Kutta methods which preserve this algebraic invariant are characterized by a simple condition on the corresponding Butcher tableau. It is proven that these methods are equivalent to the ADI iteration. The invariance approach is transferred to Sylvester equations.Diese Arbeit befasst sich mit der numerischen Lösung hochdimensionaler Matrixgleichungen mittels iterativer Verfahren. Aufgrund der GröĂe der Systemmatrizen in groĂskaligen algebraischen Riccati-Gleichung kann die Lösung nicht direkt bestimmt werden, sondern wird durch eine approximative Lösung ZYZ^* von geringem Rang angenĂ€hert. Hierbei wird Z als Basis eines rationalen Krylovraums gewĂ€hlt und enthĂ€lt nur wenige Spalten. Die innere Matrix Y ist klein und quadratisch. Es werden zwei Wege untersucht, die Matrix Y zu wĂ€hlen: Durch eine Rang-Bedingung an das Riccati-Residuum und durch Projektion des Riccati-Residuums auf den von Z erzeugten Krylovraum. Die Rang-Bedingung wird durch die wohlbekannten ADI-Verfahren motiviert. Die approximativen ADI-Lösungen spannen einen Krylovraum auf und fĂŒhren zu einem Riccati-Residuum vom Rang p. Es wird bewiesen, dass die Rang-p-Bedingung Existenz und Eindeutigkeit einer solchen approximativen Lösung impliziert. Aus diesem Ergebnis werden effiziente iterative Verfahren abgeleitet, die eine solche approximative Lösung erzeugen. Bisher bekannte Projektionsverfahren werden auf schiefe Projektionen erweitert und es wird eine neue Formulierung des Riccati-Residuums hergeleitet, die eine effiziente Berechnung der Norm erlaubt. Weiter wird eine abgeschnittene approximative Lösung als Lösung einer Riccati-Gleichung charakterisiert, die auf einen Unterraum des von Z erzeugten Krylovraums projiziert wird. Um die Lösung der Lyapunov-Gleichung zu approximieren wird ein System gewöhnlicher Differentialgleichungen mittels Runge-Kutta-Verfahren numerisch gelöst. Es wird gezeigt, dass der von der approximativen Lösung aufgespannte Raum ein rationaler Krylovraum ist, dessen Pole von den Zeitschrittweiten der Integration und den Eigenwerten der Koeffizientenmatrix aus dem Butcher-Tableau des verwendeten Runge-Kutta-Verfahrens abhĂ€ngen. Das Verfahren wird auf ein Problem der Modellreduktion angewendet. Die analytische Lösung des Differentialgleichungssystems erfĂŒllt eine algebraische Invariante. Diejenigen Runge-Kutta-Verfahren, die diese Invariante erhalten, werden durch eine Bedingung an die zugehörigen Butcher-Tableaus charakterisiert. Es wird gezeigt, dass diese speziellen Verfahren Ă€quivalent zur ADI-Iteration sind. Der Invarianten-Ansatz wird auf Sylvester-Gleichungen ĂŒbertragen
Low-rank updates and a divide-and-conquer method for linear matrix equations
Linear matrix equations, such as the Sylvester and Lyapunov equations, play
an important role in various applications, including the stability analysis and
dimensionality reduction of linear dynamical control systems and the solution
of partial differential equations. In this work, we present and analyze a new
algorithm, based on tensorized Krylov subspaces, for quickly updating the
solution of such a matrix equation when its coefficients undergo low-rank
changes. We demonstrate how our algorithm can be utilized to accelerate the
Newton method for solving continuous-time algebraic Riccati equations. Our
algorithm also forms the basis of a new divide-and-conquer approach for linear
matrix equations with coefficients that feature hierarchical low-rank
structure, such as HODLR, HSS, and banded matrices. Numerical experiments
demonstrate the advantages of divide-and-conquer over existing approaches, in
terms of computational time and memory consumption
Efficient Numerical Solution of Large Scale Algebraic Matrix Equations in PDE Control and Model Order Reduction
Matrix Lyapunov and Riccati equations are an important tool in mathematical systems theory. They are the key ingredients in balancing based model order reduction techniques and linear quadratic regulator problems. For small and moderately sized problems these equations are solved by techniques with at least cubic complexity which prohibits their usage in large scale applications.
Around the year 2000 solvers for large scale problems have been introduced. The basic idea there is to compute a low rank decomposition of the quadratic and dense solution matrix and in turn reduce the memory and computational complexity of the algorithms. In this thesis efficiency enhancing techniques for the low rank alternating directions implicit iteration based solution of large scale matrix equations are introduced and discussed. Also the applicability in the context of real world systems is demonstrated.
The thesis is structured in seven central chapters. After the introduction chapter 2 introduces the basic concepts and notations needed as fundamental tools for the remainder of the thesis. The next chapter then introduces a collection of test examples spanning from easily scalable academic test systems to badly conditioned technical applications which are used to demonstrate the features of the solvers. Chapter four and five describe the basic solvers and the modifications taken to make them applicable to an even larger class of problems. The following two chapters treat the application of the solvers in the context of model order reduction and linear quadratic optimal control of PDEs. The final chapter then presents the extensive numerical testing undertaken with the solvers proposed in the prior chapters.
Some conclusions and an appendix complete the thesis
Two Iterative algorithms for the matrix sign function based on the adaptive filtering technology
In this paper, two new efficient algorithms for calculating the sign function
of the large-scale sparse matrix are proposed by combining filtering algorithm
with Newton method and Newton Schultz method respectively. Through the
theoretical analysis of the error diffusion in the iterative process, we
designed an adaptive filtering threshold, which can ensure that the filtering
has little impact on the iterative process and the calculation result.
Numerical experiments are consistent with our theoretical analysis, which shows
that the computational efficiency of our method is much better than that of
Newton method and Newton Schultz method, and the computational error is of the
same order of magnitude as that of the two methods.Comment: 18 pages,12 figure
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