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We give an algorithm to compute N steps of a convolution quadrature approximation
to a continuous temporal convolution using only O(N logN) multiplications and O(logN) active
memory. The method does not require evaluations of the convolution kernel, but instead O(logN)
evaluations of its Laplace transform, which is assumed sectorial. The algorithm can be used for the
stable numerical solution with quasi-optimal complexity of linear and nonlinear integral and integrodifferential
equations of convolution type. In a numerical example we apply it to solve a subdiffusion
equation with transparent boundary conditions.
In this paper we propose a new finite element realization of the Perfectly Matched
Layer method (PML-method). Our approach allows to deal with a wide class of
polygonal domains and with certain types of inhomogeneous exterior domains.
Among the covered inhomogeneities are open waveguide structures playing an essential
role in integrated optics. We give a detailed insight into implementation
aspects. Numerical examples show exponential convergence behavior to the exact
solution with the thickness of the PML sponge layer.
Lyapunov and exponential dichotomy spectral theory is extended
from ordinary differential equations (ODEs) to nonautonomous
differential-algebraic equations (DAEs). By using orthogonal
changes of variables, the original DAE system is transformed into
appropriate condensed forms, for which concepts such as Lyapunov
exponents, Bohl exponents, exponential dichotomy and spectral
intervals of various kinds can be analyzed via the resulting
underlying ODE. Some essential differences between the spectral
theory for ODEs and that for DAEs are pointed out. Numerical
methods for computing the spectral intervals associated with
Lyapunov and Sacker-Sell (exponential dichotomy) spectra are
derived by modifying and extending those methods proposed for ODEs. Perturbation theory and error analysis are discussed, as
well. Finally, some numerical examples are presented to illustrate
the theoretical results and the properties of the numerical
methods.
A Generic Grid Interface for Parallel and Adaptive Scientific Computing. Part I: Abstract Framework
(2007)
We give a mathematically rigorous definition of a grid for algorithms solving
partial differential equations. Unlike previous approaches, our grids have a
hierarchical structure. This makes them suitable for geometric multigrid
algorithms and hierarchical local grid refinement. The description is also
general enough to include geometrically nonconforming grids. The definitions
in this article serve as the basis for an implementation of an abstract grid
interface as C++ classes in the DUNE.
In a companion paper [Matheon-Preprint 403] we introduced an abstract definition of a parallel and adaptive hierarchical grid for scientific computing. Based on this
definition we derive an efficient interface specification as a set of C++ classes.
This interface separates the applications from the grid data structures.
Thus, user implementations become independent of the underlying grid
implementation. Modern C++ template techniques are used to provide an
interface implementation without big performance losses.
The implementation is realized as part of the
software environment DUNE.
Numerical tests demonstrate the flexibility and the efficiency of our approach.
In many applications such as data compression, imaging or
genomic data analysis,
it is important to approximate a given $m\times n$ matrix $A$
by a matrix $B$ of rank at most $k$ which is much smaller than $m$ and $n$.
The best rank $k$ approximation can be determined via
the singular value decomposition
which, however, has prohibitively
high computational complexity and storage requirements
for very large $m$ and $n$.
We present an optimal least squares algorithm for computing a rank $k$
approximation to an $m\times n$ matrix $A$ by reading
only a limited number of rows and columns of $A$.
The algorithm has complexity $\mathcal O(k^2\max(m,n))$ and
allows to iteratively improve given rank $k$
approximations by reading additional rows and
columns of $A$. We also show how this approach can be extended
to tensors and present numerical results.
In this paper, we discuss stability properties of positive descriptor systems in the continuous-time as well as in the discrete-time case. We present different characterisations of positivity and establish generalised stability criteria for the case of positive descriptor systems. We show that if the spectral projector onto the right finite deflating subspace of the matrix pair $(E,A)$ is non-negative, then all stability criteria for standard positive systems take a comparably simple form in the positive descriptor case. Furthermore, we provide sufficient conditions that guarantee entry-wise non-negativity along with positive semi-definiteness of solutions of generalised projected Lyapunov equations. As an application of the framework established throughout this paper, we exemplarily generalise two criteria for the stability of two switched standard positive systems under arbitrary switching to the descriptor case.
Adjoint Broyden a la GMRES
(2007)
It is shown here that a compact storage implementation of a quasi-Newton
method based on the adjoint Broyden update reduces in the affine
case exactly to the well established GMRES procedure. Generally,
storage and linear algebra effort per step are small multiples of $n\cdot k$,
where $n$ is the number of variables and $k$ the number of steps taken
in the current cycle. In the affine case the storage is exactly $(n+k)\cdot k$
and in the nonlinear case the same bound can be achieved if adjoints,
i.e. transposed Jacobian-vector products are available. A
transposed-free variant that relies exclusively on Jacobian-vector
products (or possibly their approximation by divided differences)
requires roughly twice the storage and turns out to be somewhat slower
in our numerical experiments reported at the end.
Structured eigenvalue conditioning and backward error of a class of polynomial eigenvalue problems
(2007)
Characterisations of simple eigenvalues of complex matrix polynomials with *-even/odd and *-palindromic/antipalindromic structures that have the same normwise condition number with respect to structure preserving and arbitrary perturbations are obtained. Here * denotes either the transpose T or the conjugate tranpose *. In the process we obtain formulae for the normwise structured condition number of simple eigenvalues of T-palindromic/antipalindromic and *-even/odd polynomials. Moreover, conditions under which the normwise structured backward error of approximate eigenvalues of such polynomials is equal to the unstructured error are also derived. These lead to complete characterisations of approximate eigenvalues that have the same structured and unstructured backward errors for the *-even/odd and T-even/odd polynomials.