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We characterise the asymptotic smile and term structure of implied volatility in the Heston model at small maturities and all strikes. Using saddlepoint methods we derive a small-maturity expansion formula for call option prices, which we then transform into a closed-form expansion (including the leading-order and correction terms) for implied volatility. This refined expansion reveals the relationship between the small-expiry smile and all Heston parameters (including the pair in the volatility drift coefficient), sharpening the leading-order result of~\cite{FJ09I} which found the relationship between the zero-expiry smile and the diffusion coefficients. We solve for in/out-of-the-money and at-the-money cases; in the latter case our proof involves subleading-order saddlepoint approximation along a suitable path of integration.
We investigate the worst-case behavior of the simplex algorithm on linear programs
with 3 variables, that is, on 3-dimensional simple polytopes. Among the
pivot rules that we consider, the “random edge” rule yields the best asymptotic
behavior as well as the most complicated analysis. All other rules turn out to be
much easier to study, but also produce worse results: Most of them show essentially
worst-possible behavior; this includes both Kalai’s “random-facet” rule, which is
known to be subexponential without dimension restriction, as well as Zadeh’s deterministic
history-dependent rule, for which no non-polynomial instances in general
dimensions have been found so far.
The background for the general mathematical link between utility and information
theory investigated in this paper is a simple financial market model with two kinds of small
traders: less informed traders and insiders whose extra information is represented by an
enlargement of the other agents' filtration. The expected logarithmic utility increment,
i.e. the difference of the insider's and the less informed trader's expected logarithmic
utility is described in terms of the information drift, i.e. the drift one has to eliminate
in order to perceive the price dynamics as a martingale from the insider's perspective.
On the one hand, we describe the information drift in a very general setting by natural
quantities expressing the probabilistic better informed view of the world. This on the
other hand allows us to identify the additional utility by entropy related quantities known
from information theory. In particular in a complete market in which the insider has some
fixed additional information during the entire trading interval, its utility increment can
be represented by the Shannon information of his extra knowledge. For general markets,
and in some particular examples, we provide estimates of maximal utility by information
inequalities.
The Real Multiple Dual
(2009)
In this paper we present a dual representation for the multiple stopping
problem, hence multiple exercise options. As such it is a natural generalization of the
method in Rogers (2002) and Haugh and Kogan (2004) for the standard stopping
problem for American options. We consider this representation as the real dual as it is
solely expressed in terms of an infimum over martingales rather than an infimum over
martingales and stopping times as in Meinshausen and Hambly (2004). For the multiple
dual representation we present three Monte Carlo simulation algorithms which require
only one degree of nesting.
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.
Relaying is a protocol extension for cellular wireless computer networks; in order to utilize radio resources more efficiently, several hops are allowed within one cell. This paper investigates the principle potential of relaying by casting transmission scheduling as a mathematical optimization problem, namely, a linear program. We analyze the throughput gains showing that, irrespective of the concrete scheduling algorithm, performance gains of up to 30\% on average for concrete example networks are achievable.
When simulating isolated resonators, the application of transparent boundary conditions causes the approximated spectrum to be polluted with spurious solutions. Distinguishing these artificial solutions from solutions with a physical meaning is often difficult and requires a priori knowledge of the spectrum or the expected field distribution of resonant states. We present an implementation of the pole condition that distinguishes between incoming and outgoing waves by the location of the poles of their Laplace transform as transparent boundary condition. This implementation depends on one tuning parameter. We will use the sensitivity of the computed solutions to perturbations of this parameter as a means to identify spurious solutions. To obtain global statements, we will combine this technique with a convergence monitor for the boundary condition.
The perturbation and ADAE index of a degenerated hyperbolic system modelling a heat exchanger
(2007)
The heat exchanger in a heat pump can be modelled by the zero Mach-number limit of the Euler equations of compressible fluid flow. This system turns out to be a coupled hyperbolic/parabolic equation with coupled, time-dependent boundary conditions. Using the theory of abstract differential-algebraic equations it is shown that the frozen coefficient system has ADAE index 1. Moreover, the much stronger result is proven that the system has time-perturbation index one and space-perturbation index two even in the case of time-dependent boundary conditions. The results are stated in terms of the original physical variables. The estimates agree well with numerical experiments.
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.
We propose a class of Markovian agent based models for the time evolution of a share price in an interactive market. The models rely on a microscopic description of a market of buyers and sellers who change their opinion about the stock value in a stochastic way. The actual price is determined in realistic way by matching (clearing) offers until no further transactions can be performed. Some analytic results for a non-interacting model are presented. We also propose basic interaction mechanisms and show in simulations that these already reproduce certain particular features of prices in real stock markets.
Many online problems encountered in real-life involve a two-stage decision process: upon arrival of a new request, an irrevocable
first-stage decision (the assignment of a specific resource to the request) must be made immediately, while in a second stage process, certain ``subinstances'' (that is, the instances of all requests assigned to a particular resource) can be solved to optimality (offline) later.
We introduce the novel concept of an Online Target Date Assignment Problem (OnlineTDAP) as a general framework for online problems with this nature. Requests for the OnlineTDAP become known at certain dates. An online algorithm has to assign a target date to each request, specifying on which date the request should be processed (e.g., an appointment with a customer for a
washing machine repair). The cost at a target date is given by the downstream cost, the optimal cost of processing all requests
at that date w.r.t. some fixed downstream offline optimization problem (e.g., the cost of an optimal dispatch for service
technicians). We provide general competitive algorithms for the OnlineTDAP independently of the particular downstream problem,
when the overall objective is to minimize either the sum or the maximum of all downstream costs. As the first basic examples, we analyze the competitive ratios of our algorithms for the particular academic downstream problems of bin-packing, nonpreemptive scheduling on identical parallel machines, and routing a traveling salesman.
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 the planning process of railway companies, we propose to integrate important
decisions of network planning, line planning, and vehicle scheduling into the task of periodic
timetabling. From such an integration, we expect to achieve an additional potential for
optimization.
Models for periodic timetabling are commonly based on the Periodic Event Scheduling
Problem (PESP). We show that, for our purpose of this integration, the PESP has to be extended
by only two features, namely a linear objective function and a symmetry requirement.
These extensions of the PESP do not really impose new types of constraints, because practitioners
have already required them even when only planning timetables autonomously without
interaction with other planning steps.
The mixed regularity of electronic wave functions in fractional order and weighted Sobolev spaces
(2012)
The paper continues the study of the regularity of electronic wave functions in Hilbert spaces of mixed derivatives. It is shown that the eigenfunctions of electronic Schr\"odinger operators and their
exponentially weighted counterparts possess, roughly speaking, square integrable mixed weak derivatives of fractional order $\vartheta$ for $\vartheta<3/4$. The bound $3/4$ is best possible and can neither be reached nor surpassed. Such results are important for the study
of sparse grid-like expansions of the wave functions and show that their asymptotic convergence rate measured in terms of the number of ansatz functions involved does not deteriorate with the number of electrons.
The minimization of an L^{\infty}-functional subject to an elliptic PDE and state constraints
(2008)
We study the optimal control of a maximum-norm objective functional subjected to an elliptic-type PDE and pointwise state constraints. The problem is transformed into a problem where the non-differentiable L^{\infty}-norm in the functional will be replaced by a scalar variable and additional state constraints. This problem is solved by barrier methods. We will show the existence and convergence of the central path for a class of barrier functions. Numerical experiments complete the presentation.
Given a general mixed integer program (MIP), we automatically detect block structures in the constraint matrix together with the coupling by capacity constraints arising from multi-commodity flow formulations. We
identify the underlying graph and generate cutting planes based on cuts in the detected network. Our implementation adds a separator to the branch-and-cut libraries of SCIP and CPLEX. We make use of the complemented mixed integer rounding framework (cMIR) but provide a special purpose aggregation heuristic that exploits the network structure. Our separation scheme speeds-up the computation for a large set of MIPs coming from network design problems
by a factor of two on average.
The well-known Kalman-Yakubovich-Popov Lemma establishes an equivalence between dissipativity and the solvability of a linear matrix inequality. In this paper we strengthen this result by showing the equivalence of dissipativity to the solvability of a so-called Lur'e equation, which mainly is a linear matrix inequality with a rank minimizing condition. Finally, we apply the result to standard systems to obtain the well-known result about the solvability of the algebraic Riccati equation.
This paper introduces the line connectivity
problem, a generalization of the Steiner tree problem and a
special case of the line planning problem. We study its complexity and
give an IP formulation in terms of an exponential number of
constraints associated with "line cut constraints". These inequalities
can be separated in polynomial time. We also generalize the Steiner
partition inequalities.
We derive a full asymptotic expansion for call option prices and a third order approximation for implied volatility in the large-time, large log-moneyness regime for a general exponential Levy model, by extending the saddlepoint argument used in Forde,Jacquier & Mijatovic for the Heston model. As for the Heston model, there are two special log-moneyness values where the call option asymptotics are qualitatively different, and we use an Edgeworth expansion to deal with these cases. We also characterise the behaviour of the implied volatility skew at large-maturities; in particular we show that the derivative of the dimensionless implied variance with respect to log-moneyness exists and is less than or equal to 4 in the large-maturity limit, which is consistent with the bound on the right and left-side derivative given in Rogers&Tehranchi.
We investigate properties of a certain countably infinite graph called the
infinite locally random graph, written R_N. The graph R_N arises in the study
of models for massive, self-organizing networks like the web-graph. We
characterize the isomorphism type of R_N as the limit of a random process, and
via a domination elimination ordering. We prove that R_N satisfies vertex
deletion properties generalizing inexhaustibility. As is the case for the
infinite random graph R, R_N has a universal automorphism group and
endomorphism monoid. Unlike R, R_N isometrically embeds all finite graphs.
The hypergraph assignment problem (HAP) is the generalization of assignments
from directed graphs to directed hypergraphs. It serves, in particular,
as a universal tool to model several train composition rules in vehicle rotation
planning for long distance passenger railways. We prove that even for problems
with a small hyperarc size and hypergraphs with a special partitioned structure
the HAP is NP-hard and APX-hard. Further, we present an extended integer
linear programming formulation which implies, e. g., all clique inequalities.
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.
The goal of this paper is to show how the heat treatment of steel can be modelled in terms of a mathematical optimal
control problem. The approach is applied to laser surface hardening and the cooling of a steel slab including
mechanical effects. Finally, it is shown how the results can be utilized in industrial practice by a coupling with
machine-based control.
We present an extension module for the Dune system. This module, called dune-subgrid, allows to mark elements of another Dune hierarchical grid. The set of marked elements can then be accessed as a Dune grid in its own right. dune-subgrid is free software and is available for download. We describe the functionality and use of dune-subgrid, comment on its implementation, and give two example applications.
First, we show how dune-subgrid can be used for micro-FE simulations of trabecular bone. Then we present an algorithm that allows to use exact residuals for the adaptive solution of the spatial problems of time-discretized evolution equations.
The Stefan problem is coupled with a spatially inhomogeneous and anisotropic Gibbs-Thomson condition at the phase boundary. We show the long-time existence of weak solutions for the non-degenerate Stefan problem with a
spatially inhomogeneous and anisotropic Gibbs-Thomson law and a conditional existence result for the corresponding degenerate Stefan problem. To this end, approximate solutions are constructed by means of variational problems
for energy functionals with spatially inhomogeneous and anisotropic interfacial energy. By passing to the limit, we establish solutions of the Stefan problem with a spatially inhomogeneous and anisotropic Gibbs--Thomson law in a weak generalized BV-formulation.
The Stefan problem is coupled with a spatially inhomogeneous and anisotropic Gibbs-Thomson condition at the phase boundary. We show the long-time existence of weak solutions for the non-degenerate Stefan problem with a spatially inhomogeneous and anisotropic Gibbs-Thomson law and a conditional existence result for the corresponding degenerate Stefan problem. To this end, approximate solutions are constructed by means of variational problems
for energy functionals with spatially inhomogeneous and anisotropic interfacial energy. By passing to the limit, we establish solutions of the Stefan problem with a spatially inhomogeneous and anisotropic Gibbs-Thomson law in a weak generalized BV-formulation.
A relational structure is a core, if all its endomorphisms are embeddings. This notion is important for the classification for the computational complexity of constraint satisfaction problems. It is a fundamental fact that every finite structure S has a core, i.e., S has an endomorphism e such that the structure induced by e(S) is a core; moreover, the core is unique up to isomorphism.
We prove that this result remains valid for countably categorical structures, and prove that every countably categorical structure has a core, which is unique up to isomorphism, and which is again countably categorical. We thus reduced the classification for the complexity of constraint satisfaction problems with countably categorical templates to the classifiaction for constraint satisfaction problems where the templates are countably categorical cores. We also show that a core of a countably categorical structure Gamma is model complete, and therefore universal-existential axiomatizable. If Gamma contains all primitive positive definable relations, then the core of Gamma admits quantifier elimination. We discuss consequences for constraint satisfaction with countably categorical templates.
The paper addresses primal interior point method for state constrained PDE optimal
control problems. By a Lavrentiev regularization, the state constraint is transformed to a mixed
control-state constraint with bounded Lagrange multiplier. Existence and convergence of the central
path are established, and linear convergence of a short-step pathfollowing method is shown. The
behaviour of the regularizations are demonstrated by numerical examples.
In this paper, mean curvature type equations with general potentials and contact angle boundary conditions are considered. We extend the ideas of Ural'tseva, formulating sharper hypotheses for the existence of a classical solution.
Corner stone for these results is a method to estimate quantities on the boundary of the free surface. We moreover provide alternative proofs for the higher-order estimates, and for the existence result.
Single-hop WDM networks with a central Passive Star Coupler (PSC), as well as single-hop networks with
a central Arrayed-Waveguide Grating (AWG) and a single transceiver at each node, have been extensively
studied as solutions for the quickly increasing amounts of unicast and multicast traffic in the metropolitan
area. The main bottlenecks of these networks are the lack of spatial wavelength reuse in the studied PSC
based networks and the single transceiver in the studied AWG based metro WDM networks. In this paper
we develop and evaluate the FT EE ???? FREE AWG network, which is based on a central AWG and has arrays
of fixed-tuned transmitters and receivers at each node. Transceiver arrays are a mature technology, making
the proposed network practical. In addition, the transmitter arrays allow for high speed signaling over the
AWG while the receiver arrays relieve the receiver bottleneck arising from multicasting in conjunction with
spatial wavelength reuse on the AWG. Our results from probabilistic analysis and simulation indicate that
the FTEE ???? FREE AWG network gives particularly good throughput-delay performance for a mix of unicast
and multicast traffic.
Single–hop WDM networks with a central Passive
Star Coupler (PSC), as well as single–hop networks with a central
Arrayed–Waveguide Grating (AWG) and a single transceiver at
each node, have been extensively studied as solutions for the
quickly increasing amounts of unicast and multicast traffic in
the metropolitan area. The main bottlenecks of these networks
are the lack of spatial wavelength reuse in the studied PSC based
networks and the single transceiver in the studied AWG based
metro WDM networks. In this paper we develop and evaluate
the FT? ? FR? AWG network, which is based on a central
AWG and has arrays of fixed–tuned transmitters and receivers
at each node. Transceiver arrays are a mature technology, making
the proposed network practical. In addition, the transmitter
arrays allow for high speed signaling over the AWG while
the receiver arrays relieve the receiver bottleneck arising from
multicasting in conjunction with spatial wavelength reuse on the
AWG. Our results from probabilistic analysis and simulation
indicate that the FT? ? FR? AWG network gives particularly
good throughput–delay performance for multicast traffic with
small multicast group sizes or localized destination nodes, as
well as for a mix of unicast and multicast traffic.
Starting from the logical description of gene regulatory networks developed by R.~Thomas, we introduce an enhanced modelling approach
based on timed automata. We obtain a refined qualitative description of the dynamical behaviour by exploiting not only information on ratios of kinetic parameters related to synthesis and decay, but also constraints on the time delays associated with the operations of the system. We develop a formal framework for handling such temporal constraints using timed automata, discuss the relationship with the original Thomas formalism, and demonstrate the potential of our approach by analysing an illustrative gene regulatory network of bacteriophage~$\lambda$.
We consider a coupled system of two elliptic PDEs, where the elliptic term in the first equation shares the properties of the $p(x)$-Laplacian with discontinuous exponent, while in the second equation we have to deal with an a~priori $L^1$ term on the right hand side. Such a system of equations is suitable for the description of various electrothermal effects, in particular those, where the non-Ohmic behavior can change dramatically with respect to the spatial variable. We prove the existence of a weak solution under very weak assumptions on the data and also under general structural assumptions on the constitutive equations of the model. The main difficulty consists in the fact that we have to overcome simultaneously two obstacles - the discontinuous variable exponent (which limits the use of standard methods) and the $L^1$ right hand side of the heat equation. Our existence proof based on Galerkin approximation is highly constructive and therefore seems to be suitable also for numerical purposes.
Periodic timetabling for railway networks is usually modeled by the Periodic Event Scheduling
Problem (PESP). This model permits to express many requirements that practitioners impose
on periodic railway timetables. We discuss a requirement practitioners are asking for, but which,
so far, has not been the topic of mathematical studies: the concept of symmetry.
Several motivations why symmetric timetables might seem promising will be given. However,
we provide examples showing that symmetry leads to suboptimality.
To integrate symmetry into the graph model of the PESP, there are many obstacles to overcome.
Nevertheless, adding symmetry requirements to mixed-integer programming formulations
explicitly, enables MIP solvers such as CPLEX
to terminate earlier with good solutions.
We consider the design of a logical network topology, together with node hardware, link capacities, and a survivable routing of demands. In addition to all single node failures, the routing must also survive multiple logical link failures caused by single failures in the underlying physical network. Furthermore, the number of logical links supported by a physical link is bounded. We propose an integer linear programming model for this design problem, together with a branch-and-cut based solution approach combined with column generation. The model and algorithm are tested on three real-world test instances, and preliminary results are given.
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.
A thorough convergence analysis of the Control Reduced Interior Point
Method in function space is performed. This recently proposed method is a
primal interior point pathfollowing scheme with the special feature, that the
control variable is eliminated from the optimality system. Apart from global
linear convergence we show, that this method converges locally almost quadratically,
if the optimal solution satisfies a certain non-degeneracy condition. In
numerical experiments we observe, that a prototype implementation of our
method behaves as predicted by our theoretical results.
An optimal control problem for a 2-d elliptic equation is investigated with pointwise control constraints.
This paper is concerned with discretization of the control by piecewise constant functions. The state and
the adjoint state are discretized by linear finite elements. Approximations of the optimal solution of the continuous
optimal control problem will be constructed by a projection of the discrete adjoint state. It is proved that these
approximations have convergence order h2.
Second-order sufficient optimality conditions are established for the optimal control
of semilinear elliptic and parabolic equations with pointwise constraints on the control and the state. In
contrast to former publications on this subject, the cone of critical directions is the smallest possible in the
sense that the second-order sufficient conditions are the closest to the associated necessary ones. The theory
is developed for elliptic distributed controls in domains up to dimension three. Moreover, problems of elliptic
boundary control and parabolic distributed control are discussed in spatial domains of dimension two and one,
respectively.
In this paper we study the shape and growth of structured pseudospectra
for small matrix perturbations of the form $A \leadsto
A_\Delta=A+B\Delta C$, $\Delta \in \DD$, $\|\Delta\|\leq \delta$.
It is shown that the properly scaled pseudospectra components converge
to non-trivial limit sets as $\delta$ tends to 0.
We discuss the relationship of these limit sets with $\mu$-values and
structured eigenvalue condition numbers for multiple eigenvalues.
Let $\lambda$ be a nonderogatory eigenvalue of $A \in \C^{n \times
n}$. The sensitivity of $\lambda$ with respect to matrix
perturbations
$A \leadsto A+\Delta,\Delta \in \DD$, is measured by the structured
condition number $\kappa_\DD(A,\lambda)$. Here $\DD$ denotes the set
of admissible perturbations. However, if $\DD$ is not a vector space
over $\C$ then $\kappa_\DD(A,\lambda)$ provides only incomplete
information about the mobility of $\lambda$ under small
perturbations from $\DD$. The full
information is then given by a certain set $K_\DD(x,y)\subset \C$
which depends on $\DD$ and
a pair of normalized right and left eigenvectors $x,y$. In this paper
we study the sets $K_\DD(x,y)$ and obtain methods for computing
them.
In particular we show that $K_\DD(x,y)$ is an ellipse in some
important cases.
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.
This paper investigates the effect of structure-preserving perturbations on the eigenvalues
of linearly and nonlinearly structured eigenvalue problems. Particular attention is paid to
structures that form Jordan algebras, Lie algebras, and automorphism groups of a scalar product.
Bounds and computable expressions for structured eigenvalue condition numbers are derived for
these classes of matrices, which include complex symmetric, pseudo symmetric, persymmetric, skewsymmetric,
Hamiltonian, symplectic, and orthogonal matrices. In particular we show that under mild
assumptions on the scalar product, the structured and unstructured eigenvalue condition numbers
are equal for structures in Jordan algebras. For Lie algebras, the effect on the condition number of
incorporating structure varies greatly with the structure. We identify Lie algebras for which structure
does not affect the eigenvalue condition number.
Structured eigenvalue backward errors of matrix pencils and polynomials with palindromic structures
(2014)
We derive formulas for the backward error of an approximate eigenvalue of a *-palindromic
matrix polynomial with respect to *-palindromic perturbations. Such formulas are also obtained
for complex T-palindromic pencils and quadratic
polynomials. When the T-palindromic polynomial is real, then we derive the backward error
of a real number considered as an approximate eigenvalue of the matrix polynomial with
respect to real T-palindromic perturbations.
In all cases the corresponding minimal structure preserving perturbations are obtained as well.
The results are illustrated by numerical experiments. These show that there is
significant difference between the backward errors with respect to structure
preserving and arbitrary perturbations in many cases.
We derive a formula for the backward error of a complex number $\lambda$ when considered as an approximate eigenvalue
of a Hermitian matrix pencil or polynomial with respect to Hermitian perturbations. The same are also obtained for approximate
eigenvalues of matrix pencils and polynomials with related structures like skew-Hermitian, $*$-even and $*$-odd.
Numerical experiments suggest that in many cases there is a significant difference between the backward
errors with respect to perturbations that preserve structure and those with respect to arbitrary perturbations.
Dissipative Hamiltonian (DH) systems are an important concept in energy based modeling of dynamical
systems. One of the major advantages of the DH formulation is that the system encodes system
properties in an algebraic way in the system. Making use of the structure,
it is easy to see that DH systems are stable. In this paper
the question is discussed when a linear constant coefficient DH system is on the boundary
of the region of asymptotic stability, i.e., when it has purely imaginary eigenvalues,
or how much it has to be perturbed to be on this boundary. For unstructured systems this distance to instability (stability radius) is well-understood. In this paper,
explicit formulas for this distance under structure preserving perturbations are determined.
It is also shown (via numerical examples) that under structured perturbations the asymptotical
stability of a DH system is much more robust than for unstructured perturbations, since the
distance can be much larger.