Refine
Year of publication
- 2010 (70) (remove)
Language
- English (70)
Keywords
- dissipativity (3)
- optimal control (3)
- backward error (2)
- behavior approach (2)
- even pencil (2)
- even polynomial (2)
- finite elements (2)
- linear quadratic optimal control (2)
- problem (2)
- structured eigenvalue problem (2)
Project
We study the class of linear differential-algebraic m-input m-output systems
which have a transfer function with proper inverse.
A sufficient condition for the transfer function to have proper inverse
it that the system has 'strict and non-positive relative degree'.
We present two main results:
First, a so called 'zero dynamics form' is derived: this form is - within the class of system equivalence - a simple "almost normal" form of the DAE; it is a counterpart to the well-known Byrnes-Isidori form for
ODE systems with strictly proper transfer function.
The 'zero dynamics form' is exploited to characterize structural properties such as
asymptotically stable zero dynamics,
minimum phase, and high-gain stabilizability.
The zero dynamics are characterized by (A,E,B)-invariant subspaces.
Secondly, it is shown that the 'funnel controller' (that is a static nonlinear output error feedback) achieves, for all DAE systems with asymptotically stable zero dynamics and transfer function with proper inverse, tracking of a reference signal by the output signal within a pre-specified funnel. This funnel determines the transient behaviour.
Under market frictions like illiquidity or transaction costs, contingent claims
can incorporate some inevitable intrinsic risk that cannot be completely hedged
away but remains with the holder. In general, they cannot be synthesized by
dynamical trading in liquid assets and hence not be priced by no-arbitrage arguments alone. Still, an agent can determine a valuation with respect to her
preferences towards risk. The utility indifference value for a variation in the
quantity of illiquid assets held by the agent is defined as the compensating variation
of wealth, under which her maximal expected utility remains unchanged.
We study discrete curvatures computed from nets of curvature lines on a given smooth surface and prove their uniform convergence to smooth principal curvatures. We provide explicit error bounds, with constants depending only on properties of the smooth limit surface and the shape regularity of the discrete net.
The weak Stratonovich integral with respect to fractional Brownian motion with Hurst parameter 1/6
(2010)
Let $B$ be a fractional Brownian motion with Hurst parameter
$H=1/6$. It is known that the symmetric Stratonovich-style Riemann sums
for $\int g(B(s))\,dB(s)$ do not, in general, converge in probability.
We show, however, that they do converge in law in the Skorohod space of
c\`adl\`ag functions. Moreover, we show that the resulting stochastic
integral satisfies a change of variable formula with a correction term
that is an ordinary It\^o integral with respect to a Brownian motion
that is independent of $B$.
The PSurface Library
(2010)
We describe psurface, a C++ library that allows to store and access piecewise linear mappings between simplicial surfaces in $\R^2$ and $\R^3$. These mappings are stored in a graph data structure and can be constructed explicitly, by projection, or by surface simplification. Piecewise linear maps can be used, e.g., to construct boundary
approximations for finite element grids, and grid intersections for domain decomposition methods. In computer graphics the mappings allow to build level-of-detail representations as well as texture- and bump maps. We document the data structures and algorithms used and show how \psurface is used in the numerical analysis framework Dune
and the visualization software Amira.
We consider simple models of financial markets with less and better
informed investors described by a smaller and a larger filtration on a
general stochastic basis that describes the market dynamics, including
continuous and jump components. We study the relation between different forms of non existance of arbitrage and the characteristics of the stochastic basis under the different filtrations. This is achieved through the analysis of the properties of the numéraire portfolio. Furthermore, we focus on the problem of calculating the additional logarithmic utility of the better informed investor in terms of the Shannon antropy of is additional information. The information drift, i.e. the drift to eliminate in order to preserved the martingale property in the larger filtration terms out to be the crucial quantity needed to tackle these problems. We show that the expected
ed logarithmic utility increment due to better information equals its Shannon
entropy also in case of a pure jump basis with jumps that are quadratically
hedgeable, and so extend a similar result known for bases consisting of
continuous semimartingales. An example illustrates that the equality may
not persist if both continuous and jump components are present in the
underlying.
In this paper we consider the first exit problem of an overdamped
Lévy driven particle in a confining potential. We survey results
obtained in recent years from our work on the Kramers' times for
dynamical systems of this type with Lévy perturbations containing
heavy, and exponentially light jumps, and compare them to the well
known case of dynamical systems with Gaussian perturbations. It
turns out that exits induced by Lévy processes with jumps are
always essentially faster than Gaussian exits.
Supporting Global Numerical Optimization of Rational Functions by Generic Symbolic Convexity Tests
(2010)
Convexity is an important property in nonlinear optimization since it allows to apply efficient local methods for finding global solutions. We propose to apply symbolic methods to prove or disprove convexity of rational functions over a polyhedral domain. Our algorithms reduce convexity questions to real quantifier elimination problems. Our methods are implemented and publicly available in the open source computer algebra system REDUCE. Our long term goal is to integrate REDUCE as a ``workhorse'' for symbolic computations into a numerical solver.
Functional Magnetic Resonance Imaging inherently involves noisy measurements and a severe multiple test
problem. Smoothing is usually used to reduce the effective number of multiple
comparisons and to locally integrate the signal and hence increase the
signal-to-noise ratio. Here, we provide a new structural adaptive segmentation
algorithm (AS)
that naturally combines the signal detection with noise reduction in one procedure.
Moreover, the new method
is closely related to a recently proposed structural adaptive smoothing
algorithm and preserves shape and spatial extent of activation areas without
blurring the borders.