For linear differential-algebraic equations (DAEs) with properly
stated leading terms the property of being numerically qualified
guarantees that qualitative properties of DAE solutions are reflected
by the numerical approximations. In this case BDF and Runge-Kutta
methods integrate the inherent regular ODE.
Here, we extend these results to general linear methods. We show
how general linear methods having stiff accuracy can be applied to
linear DAEs of index 1 and 2. In addition to the order conditions for
ODEs, general linear methods for DAEs have to satisfy additional
conditions.
As general linear methods require a starting procedure to start the
integration we put special emphasis on finding suitable starting
methods for index-2 DAEs.
Balancing a matrix by a simple and accurate similarity transformation can improve
the speed and accuracy of numerical methods for computing eigenvalues. We describe
balancing strategies for a large and sparse Hamiltonian matrix H. It is first shown how
to permute H to irreducible form while retaining its structure. This form can be used to
decompose the Hamiltonian eigenproblem into smaller-sized problems. Next, we discuss
the computation of a symplectic scaling matrix D so that the norm of D 1 HD is reduced.
The considered scaling algorithm is solely based on matrix-vector products and thus particularly
suitable if the elements of H are not explicitly given. The merits of balancing
for eigenvalue computations are illustrated by several practically relevant examples.
The periodic QR algorithm is a strongly backward stable method for computing the
eigenvalues of products of matrices, or equivalently for computing the eigenvalues of
block cyclic matrices. The main purpose of this paper is to show that this algorithm
is numerically equivalent to the standard QR algorithm. It will be demonstrated
how this connection may be used to develop a better understanding of the periodic
QR algorithm.
Two algorithms for the solution of discrete-time periodic Lyapunov
equations are presented. The first one is a variant of the
squared Smith iteration, which is solely based on matrix multiplications
and thus attractive to parallel computing environments.
The second algorithm is based on Krylov subspaces and
employs a recently developed variant of the block Arnoldi algorithm.
It is particularly suited for periodic Lyapunov equations
with large and sparse coefficient matrices. We also demonstrate
how these methods can be applied to balanced truncation model
reduction of periodic discrete-time systems and the solution of
periodic Riccati equations.
Uniform lower and upper bounds for positive finite-element approximations
to semilinear elliptic equations in several space dimensions subject to
mixed Dirichlet-Neumann boundary conditions are derived. The main feature is
that the non-linearity may be non-monotone and unbounded. The discrete minimum
principle provides a positivity-preserving approximation if the discretization
parameter is small enough and if some structure conditions on the non-linearity
and the triangulation are assumed. The discrete maximum principle also holds
for degenerate diffusion coefficients. The proofs are based on Stampacchia's truncation
technique and on a variational formulation. Both methods are settled on
careful estimates on the truncation operator.
A strategy for controlling the stepsize in the numerical integration of stochastic
differential equations (SDEs) is presented. It is based on estimating the p-th mean of
local errors. The strategy leads to deterministic stepsize sequences that are identical
for all paths. For the family of Euler schemes for SDEs with small noise we derive
computable estimates for the dominating term of the p-th mean of local errors
and show that the strategy becomes efficient for reasonable stepsizes. Numerical
experience is reported for test examples including scalar SDEs and a stochastic
circuit model.
This paper demonstrates simulation tools for edge-emitting multi quantum well (MQW) lasers.
Properties of the strained MQW active region are simulated by eight-band kp calculations. Then, a 2D
simulation along the transverse cross section of the device is performed based on a drift-diffusion model,
which is self-consistently coupled to heat transport and equations for the optical field. Furthermore, a
method is described, which allows for an efficient quasi 3D simulation of dynamic properties of multisection
edge-emitting lasers.
We consider the linear-quadratic optimal control problem for a controlled differential
algebraic equation (DAE). Under minimal assumptions on the DAE concerning index and regularity
it will be possible to prove that the sufficient optimality condition given in papers of G. Kurina
and R. März is also a necessary condition. This condition includes the solution of an appropriate
boundary value problem and, in the special case of an explicit ordinary differential equation, the
condition is equal to the well known necessary and sufficient condition of the classical linear-quadratic
optimal control problem.
A Structure-Preserving Method for Generalized Algebraic RiccatiEquations Based on Pencil Arithmetic
(2004)
This paper describes a numerical method for extracting the stable
right deflating subspace of a matrix pencil Z Y using
a spectral projection method. It has several advantages compared
to other spectral projection methods like the sign function
method. In particular it avoids the rounding error induced
loss of accuracy associated with matrix inversions. The new algorithm
is particularly well adapted to solving continuous-time
algebraic Riccati equations. In numerical examples, it solves
Riccati equations to high accuracy.