30 research outputs found
Efficient Numerical Algorithms for Balanced Stochastic Truncation
We propose an efficient numerical algorithm for relative error model reduction based on balanced stochastic truncation. The method uses full-rank factors of the Gramians to be balanced versus each other and exploits the fact that for large-scale systems these Gramians are often of low numerical rank. We use the easy-to-parallelize sign function method as the major computational tool in determining these full-rank factors and demonstrate the numerical performance of the suggested implementation of balanced stochastic truncation model reduction
From Algebraic Riccati equations to unilateral quadratic matrix equations: old and new algorithms
The problem of reducing an algebraic Riccati equation to a unilateral quadratic matrix equation (UQME) of the
kind is analyzed. New reductions are introduced
which enable one to prove some theoretical and computational properties.
In particular we show that the structure preserving doubling algorithm
of B.D.O. Anderson [Internat. J. Control, 1978] is nothing else but the
cyclic reduction algorithm applied to a suitable UQME. A new algorithm
obtained by complementing our reductions with the shrink-and-shift tech-
nique of Ramaswami is presented. Finally, faster algorithms which require
some non-singularity conditions, are designed. The non-singularity re-
striction is relaxed by introducing a suitable similarity transformation of
the Hamiltonian
Institute for Computer Applications in Science and Engineering (ICASE)
Research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, numerical analysis and computer science during the period April 1, 1983 through September 30, 1983 is summarized
Approximate Analytical Relationships for Linear Optimal Aeroelastic Flight Control Laws
This dissertation introduces new methods to uncover functional relationships between design parameters of a contemporary control design technique and the resulting closed-loop properties. Three new methods are developed for generating such relationships through analytical expressions: the Direct Eigen-Based Technique, the Order of Magnitude Technique, and the Cost Function Imbedding Technique. Efforts concentrated on the linear-quadratic state-feedback control-design technique applied to an aeroelastic flight control task. For this specific application, simple and accurate analytical expressions for the closed-loop eigenvalues and zeros in terms of basic parameters such as stability and control derivatives, structural vibration damping and natural frequency, and cost function weights are generated. These expressions explicitly indicate how the weights augment the short period and aeroelastic modes, as well as the closed-loop zeros, and by what physical mechanism. The analytical expressions are used to address topics such as damping, nonminimum phase behavior, stability, and performance with robustness considerations, and design modifications. This type of knowledge is invaluable to the flight control designer and would be more difficult to formulate when obtained from numerical-based sensitivity analysis
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SciCADE 95: International conference on scientific computation and differential equations
This report consists of abstracts from the conference. Topics include algorithms, computer codes, and numerical solutions for differential equations. Linear and nonlinear as well as boundary-value and initial-value problems are covered. Various applications of these problems are also included
Multimodeling, Singular Perturbations and Chained Aggregation of Large Scale Systems
Coordinated Science Laboratory was formerly known as Control Systems LaboratoryDepartment of Energy / US ERDA EX-76-C-01-208
H2, fixed architecture, control design for large scale systems
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 1990.Includes bibliographical references (p. 227-234).by Mathieu Mercadal.Ph.D
Polynomial Eigenproblems: a Root-Finding Approach
A matrix polynomial, also known as a polynomial matrix,
is a polynomial whose coefficients are matrices; or, equivalently, a matrix whose elements are polynomials.
If the matrix polynomial P(x) is regular, that is if p(x):=det(P(x)) is not identically zero,
the polynomial eigenvalue problem associated with P(x) is equivalent to the
computation of the roots of the polynomial p(x); such roots are called the
eigenvalues of the regular matrix polynomial P(x). Sometimes, one is also interested in computing
the corresponding (left and right) eigenvectors.
Recently, much literature has been addressed to the polynomial
eigenvalue problem. This line of research is currently very active: the theoretical properties of PEPs are studied,
and fast and numerically stable methods are sought
for their numerical solution. The most commonly encountered
case is the one of degree 2 polynomials,
but there exist applications where higher degree polynomials appear. More generally, PEPs are special
cases belonging to the wider class of nonlinear eigenvalue problems. Amongst nonlinear eigenvalue problems, rational eigenvalue
problems
can be immediately brought to polynomial form, multiplying them by their least common denominator; truly
nonlinear eigenvalue problems may be approximated with PEPs, truncating some matrix power series, or with
rational eigenproblems, using rational approximants such as Padé approximants.
To approximate numerically the solutions of PEPs, several
algorithms have been introduced based on the technique of
linearization where the polynomial problem is replaced by a linear
pencil with larger size and the customary methods for the generalised
eigenvalue problem, like for instance the QZ algorithm, are applied.
This thesis is addressed to the design and analysis of
algorithms for the polynomial eigenvalue problem based on a root-finding approach.
A root-finder is applied to the
characteristic equation p(x)=0. In particular, we discuss algorithms based on the
Ehrlich-Aberth iteration.
The Ehrlich-Aberth iteration (EAI) is a method that simultaneously approximates all
the roots of a (scalar) polynomial.
In order to adapt the EAI to the numerical solution of a PEP,
we propose a method based on the Jacobi formula; two implementation of the EAI are
discussed, of which one uses a linearization and the other works directly on the matrix polynomial.
The algorithm that we propose has quadratic computational complexity with respect to
the degree k of the matrix polynomial. This leads to computational advantage when the ratio k^2/n, where
n is the dimension of the matrix coefficients, is large.
Cases of this kind can be encountered, for instance,
in the truncation of matrix power series. If k^2/n is small, the EAI can be implemented
in such a way that its asymptotic complexity is cubic (or slightly supercubic) in nk, but QZ-based methods appear to be
faster in this case. Nevertheless, experiments suggest that the EAI can improve the approximations of the QZ in terms of forward
error, so that even when it is not as fast as other algorithms it is still suitable as a refinement method.
The EAI does not compute the eigenvectors. If they are needed, the EAI
can be combined with other methods such as the SVD or the inverse iteration. In the
experiments we performed, eigenvectors were computed in this way, and
they were approximated with higher accuracy with respect to the QZ.
Another root-finding approach to PEPs, similar to the EAI, is to
apply in sequence the Newton method to each single eigenvalue, using an implicit deflation of the previously
computed roots of the determinant
in order to avoid to approximate twice the same eigenvalue. Our numerical experience
suggests that in terms of efficiency the EAI is superior with respect to the sequential Newton method with deflation.
Specific attention concerns structured problems where the matrix
coefficients have some additional feature which is reflected on
structural properties of the roots. For instance, in the case of
T-palindromic polynomials, the roots are encountered in pairs (x,1/x).
In this case the goal is to design
algorithms which take advantage of this additional information about
the eigenvalues and deliver approximations to the eigenvalues which
respect these symmetries independently of the rounding
errors.
Within this setting, we study the case of polynomials endowed with specific properties
like, for instance, palindromic, T-palindromic, Hamiltonian, symplectic, even/odd, etc.,
whose eigenvalues have special symmetries in the complex plane.
In general, we
may consider the case of structures where the roots can be grouped in
pairs as (x,f(x)), where f(x) is any self-inverse analytic function such that.
We propose a unifying treatment of structured polynomials belonging to this class
and show how the EAI can be adapted to deal with them in a
very effective way. Several structured variants of the EAI are available to this goal:
they are described in this thesis.
All of such variants enable one to
compute only a subset of eigenvalues and to recover the remaining part
of the spectrum by means of the symmetries satisfied by the
eigenvalues. By exploiting the structure of the problem, this
approach leads to a saving on the number of floating point operations and provides
algorithms which yield numerical approximations fulfilling
the symmetry properties. Our research on the structured EAI can of course be applied also to scalar polynomials: in the
next future, we plan to exploit our results and design new features for the software MPSolve.
When studying the theoretical properties of the change of variable, useful to design one of the structured EAI methods, we had the
chance to discover some theorems on the behaviour of the complete eigenstructure of a matrix polynomial under a rational change of
variable. Such results are discussed in this thesis.
Some, but not all, of the different structured versions of the EAI algorithm have a drawback: accuracy is lost for eigenvalues that
are close to a finite number of critical values, called exceptional eigenvalues. On the other
hand, it turns out that at least for some specific structures
the versions that suffer from this problem are also the most efficient ones: thus, it is desirable to circumvent the
loss of accuracy. This can
be done by the design of a structured refinement Newton algorithm. Besides
its application to structured PEPs, this algorithm can have further application to the computation of the
roots of scalar polynomials whose roots appear in pairs.
In this thesis, we also present the results of several
numerical experiments performed in order to test the effectiveness of
our approach in terms of speed and of accuracy. We have compared the
Ehrlich-Aberth iteration with the Matlab functions polyeig and quadeig. In the structured case, we have also considered, when
available, other structured methods, say, the URV algorithm by
Schroeder . Moreover, the different versions of our algorithm are compared one with another