An important issue for solving multistage stochastic programs consists in the approximate representation of the (multivariate) stochastic input process in the form of a scenario tree. In this paper, forward and backward approaches are developed for generating scenario trees out of an initial fan of individual scenarios. Both approaches are motivated by the recent stability result in [15] for optimal values of multistage stochastic programs. They are based on upper bounds for the two relevant ingredients of the stability estimate, namely, the probabilistic and the filtration distance, respectively. These bounds allow to control the process of recursive scenario reduction [13] and branching. Numerical experience is reported for constructing multivariate scenario trees in electricity portfolio management.
Descriptor systems present a general
mathematical framework for the modelling, simulation and control of complex dynamical systems arising in many areas of mechanical,
electrical and chemical engineering. This
paper presents a survey of the current theory of descriptor systems,concerning
solvability, stability, model reduction, controllability, observability and optimal control.
In this work we derive an exact discrete artificial boundary condition
for the Crank-Nicolson scheme for solving the Black-Scholes
equation for the valuation of American options.
To ensure stability and to avoid any numerical reflections
we derive the artificial boundary
condition on a purely discrete level.
Since the exact discrete artificial boundary condition
includes a convolution with respect to time
with a weakly decaying kernel, its numerical evaluation
becomes very costly for large-time simulations.
As a remedy we construct approximate artificial boundary conditions
with a kernel having the form of a
finite sum-of-exponentials, which can be evaluated in a very
efficient recursion. We prove a simple stability criteria
for the approximated artificial boundary conditions.
Finally we illustrate the
efficiency of the proposed method on several examples
and compare it to previously obtained discretized artificial boundary conditions.
Classical stability properties of solutions
that are well-known for ordinary differential
equations (ODEs) are generalized to differential-algebraic equations (DAEs).
A new test equation is derived for the analysis of numerical methods applied
to DAEs with respect to the stability of the numerical approximations.
Morevover, a stabilization technique is developed to improve the stability of classical DAE integration methods. The stability regions for these stabilized discretization methods are determined and it is shown that they much better reproduce the stability properties known for the ODE case
than in the unstabilized form.
Movies that depict the stability regions for several methods are included for interactive use.
By extending the stability analysis of [17] for multistage stochastic programs we show that their solution sets behave stable with respect to the sum of an Lr-distance and a filtration distance. Based on such stability results we suggest a scenario tree generation method for the (multivariate) stochastic input process. It starts with a fan of individual scenarios and consists of a recursive deletion and branching procedure which is controlled by bounding the approximation error. Some numerical experience for generating scenario trees in electricity portfolio management is reported.
Mixed-integer two-stage stochastic programs with fixed recourse matrix, random recourse costs, technology matrix, and right-hand sides are considered. Quantitative continuity properties of its optimal value and solution set are derived when the underlying probability distribution is perturbed with respect to an appropriate probability metric.
Lagrangian invariant subspaces for symplectic matrices play an important role in the numerical solution of discrete time, robust and optimal control problems. The sensitivity (perturbation) analysis of these subspaces, however, is a difficult problem, in particular, when the eigenvalues are on or close to some critical regions in the complex plane, such as the unit circle.
We present a detailed perturbation analysis for several different cases of real and complex symplectic matrices. We analyze stability and conditional stability
as well as the index of stability for these subspaces.
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.
This paper is intended to be a first step towards the continuous dependence of dynamical contact problems on the initial data as well as the uniqueness of a solution. Moreover, it provides the basis for a proof of the convergence of popular time integration schemes as the Newmark method.
We study a frictionless dynamical contact problem between both linearly elastic and viscoelastic bodies which is formulated via the Signorini contact conditions. For viscoelastic materials fulfilling the Kelvin-Voigt constitutive law, we find a characterization of the class of problems which satisfy a perturbation result in a non-trivial mix of norms in function space. This characterization is given in the form of a stability condition on the contact stresses at the contact boundaries.
Furthermore, we present perturbation results for two well-established approximations of the classical Signorini condition: The Signorini condition formulated in velocities and the model of normal compliance, both satisfying even a sharper version of our stability condition.
The Lang-Kobayashi model is a system of delay differential equations (DDEs) describing the dynamics of a semiconductor laser under delayed optical feedback. In this paper, we study the stability of so called external cavity modes (ECMs), which are harmonic oscillations corresponding to stationary lasing states. We focus on experimentally relevant situations, when the delay is large compared to the internal time scales of the laser. In this case, both the number of ECMs and the number of critical eigenvalues grows to infinity. Applying a newly developed asymptotic description for the spectrum of linearized DDEs with long delay, we are able to overcome this difficulty and to give a complete description of the stability properties of all ECMs. In particular, we distinguish between different types of weak and strong instabilities and calculate bifurcation diagrams that indicate the regions with different stability properties and the transitions between them.