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This paper proposes a new mathematical model for the open pit mine planning problem,
based on continuous functional analysis. The traditional models for this problem have been
constructed by using discrete 0-1 decision variables, giving rise to large-scale combinatorial
and Mixed Integer Programming (MIP) problems. Instead, we use a continuous approach
which allows for a refined imposition of slope constraints associated with geotechnical stability.
The model introduced here is posed in a suitable functional space, essentially the
real-valued functions that are Lipschitz continuous on a given two dimensional bounded region.
We derive existence results and investigate some qualitative properties of the solutions
A duality based semismooth Newton framework for solving variational inequalities of the second kind
(2010)
In an appropriate function space setting, semismooth Newton methods are proposed
for iteratively computing the solution of a rather general class of variational inequalities (VIs) of the
second kind. The Newton scheme is based on the Fenchel dual of the original VI problem which
is regularized if necessary. In the latter case, consistency of the regularization with respect to the
original problem is studied. The application of the general framework to specific model problems
including Bingham flows, simplified friction, or total variation regularization in mathematical imaging
is described in detail. Finally, numerical experiments are presented in order to verify the theoretical
results.
We propose a generalization of the Structured Doubling Algorithm (SDA) to compute invariant subspaces
of structured matrix pencils
that arise in the context of solving linear quadratic optimal control problems.
The new algorithm is
designed to attain better accuracy when the classical Riccati equation approach for the solution of the optimal control problem is not well suited because
the stable and unstable invariant subspaces are not well separated (due to eigenvalues near or on the imaginary
axis) or in the case when the Riccati solution does not exist at all. We analyze the convergence
of the method and compare the new method with the classical SDA algorithm as well as some recent structured QR-methods.
Bovine fertility is the subject of extensive research in animal sciences,
especially because fertility of dairy cows has declined during the last
decades. The regulation of estrus is controlled by the complex interplay
of various organs and hormones. Mathematical modeling of the bovine
estrous cycle could help in understanding the dynamics of this complex
biological system. In this paper we present a mechanistic mathematical
model of the bovine estrous cycle that includes the processes of follicle
and corpus luteum development and the key hormones that interact to
control these processes. The model generates successive estrous cycles of
21 days, with three waves of follicle growth per cycle. The model contains
12 differential equations and 54 parameters. Focus in this paper is on
development of the model, but also some simulation results are presented,
showing that a set of equations and parameters is obtained that describes
the system consistent with empirical knowledge. Even though the majority
of the mechanisms that are included in the model are based on relations
that in literature have only been described qualitatively (i.e. stimulation
and inhibition), the output of the model is surprisingly well in line with
empirical data. This model of the bovine estrous cycle could be used
as a basis for more elaborate models with the ability to study effects of
external manipulations and genetic differences.
We propose a robust and efficient numerical discretization scheme for the infinitesimal generator of a diffusion process based on a finite volume approximation. The resulting discrete-space operator can be interpreted as a jump process on the mesh whose invariant measure is precisely the cell approximation of the Boltzmann distribution of the original process. Moreover the resulting jump process preserves the detailed balance property of the original stochastic process.
The aim of this paper is to devise an adaptive timestep control in the contact--stabilized Newmark method (CONTACX) for dynamical contact problems between two viscoelastic bodies in the framework of Signorini's condition. In order to construct a comparative scheme of higher order accuracy, we extend extrapolation techniques. This approach demands a subtle theoretical investigation of an asymptotic error expansion of the contact--stabilized Newmark scheme. On the basis of theoretical insight and numerical observations, we suggest an error estimator and a timestep selection which also cover the presence of contact. Finally, we give a numerical example.
This paper presents three different adaptive algorithms for eigenvalue problems
associated with non-selfadjoint partial differential
operators. The basis for the developed algorithms is a homotopy method.
The homotopy method starts from a well-understood selfadjoint problem,
for which well-established adaptive methods are available.
Apart from the adaptive grid refinement, the progress of the homotopy as
well as the solution of the iterative method are adapted to balance the contributions
of the different error sources.
The first algorithm balances the homotopy, discretization and approximation errors with respect
to a fixed step-size $\tau$ in the homotopy.
The second algorithm combines the adaptive step-size control for the homotopy
with an adaptation in space that ensures an error below a fixed tolerance $\varepsilon$.
The third algorithm allows the complete adaptivity in space,
homotopy step-size as well as the iterative algebraic eigenvalue solver.
All three algorithms are compared in numerical examples.
We introduce an electronic model for solar cells including energy resolved defect
densities. The resulting drift-diffusion model corresponds to a generalized
van Roosbroeck system with additional source terms coupled with ODEs containing space and
energy as parameters for all defect densities. The system has to be considered in
heterostructures and with mixed boundary conditions from device simulation.
We give a weak formulation of the problem. If the boundary data and the sources
are compatible with thermodynamic equilibrium the free energy along solutions
decays monotonously. In other cases it may be increasing, but we estimate its growth.
We establish boundedness and uniqueness results and prove the existence of a
weak solution. This is done by considering a regularized problem, showing its
solvability and the boundedness of its solutions independent of the regularization level.
We present and compare two different approaches to conditional
risk measures. One approach draws from vector space based convex analysis
and presents risk measures as functions on L^p spaces while the other approach
utilizes module based convex analysis where conditional risk measures are defined on L^p type modules. Both approaches utilize general duality theory for
vector valued convex functions in contrast to the current literature in which
we fi nd ad hoc dual representations. By presenting several applications such
as monotone and sub(cash) invariant hulls with corresponding examples we
illustrate that module based convex analysis is well suited to the concept of
conditional risk measures.
This paper is devoted to the numerical approximation of Lyapunov and Sacker-Sell spectral intervals for linear differential-algebraic equations (DAEs). The spectral analysis for DAEs is improved and the concepts of leading directions and solution subspaces associated with spectral intervals are extended to DAEs. Numerical methods
based on smooth singular value decompositions are introduced for computing all or only some spectral intervals and their associated leading directions. The numerical algorithms as well as implementation issues are discussed in detail and numerical examples are presented to illustrate the theoretical results.
Arrow Debreu Prices
(2010)
Arrow Debreu prices are the prices of ‘atomic’ time and state contingent
claims which deliver one unit of a specific consumption good if a specific uncertain
state realizes at a specific future date. For instance, claims on the good
‘ice cream tomorrow’ are split into different commodities depending whether the
weather will be good or bad, so that good-weather and bad-weather ice cream
tomorrow can be traded separately. Such claims were introduced by K.J. Arrow
and G. Debreu in their work on general equilibrium theory under uncertainty,
to allow agents to exchange state and time contingent claims on goods. Thereby
the general equilibrium problem with uncertainly can be reduced to a conventional
one without uncertainty. In finite state financial models, Arrow-Debreu
securities delivering one unit of the numeraire good can be viewed as natural
atomic building blocks for all other state-time contingent financial claims; their
prices determine a unique arbitrage-free price system.
Cross–derivatives are mixed partial derivatives that are obtained by differentiating at most
once in every coordinate direction. They are a computational tool in combinatorics and high–
dimensional integration. Here we present two methods of computing exact values of all cross–
derivatives at a given point both following the general philosophy of automatic differentiation.
Implementation details are discussed and numerical results given.
Riemann surfaces naturally appear in the analysis of complex functions that are branched over the complex plane. However, they usually possess a complicated topology and are thus hard to understand. We present an algorithm for constructing Riemann surfaces as meshes in R3 from explicitly given branch points with corresponding branch indices. The constructed surfaces cover the complex plane by the canonical
projection onto R2 and can therefore be considered as multivalued graphs
over the plane – hence they provide a comprehensible visualization of the
topological structure. Complex functions are elegantly visualized using domain coloring on
a subset of C. By applying domain coloring to the automatically constructed Riemann surface models, we generalize this approach to deal with functions which cannot be entirely visualized in the complex plane.
We study balanced model reduction for stable bilinear systems in the limit of partly vanishing Hankel singular values. We show that the dynamics admit a splitting into fast and slow subspaces and prove an averaging principle for the slow dynamics. We illustrate our method with an example from stochastic control (density evolution of a dragged Brownian particle) and discuss issues of structure preservation and positivity.
We study balanced model reduction of partially-observed linear stochastic differential equa- tions of Langevin type. Balancing the equations of motion gives rise to a singularly perturbed system of equations with slow and fast degrees of freedom, and we prove that in the limit of the fast variables becoming infinitely fast, the solutions converge to the solution of a reduced-order Langevin equation. We illustrate the method with several numerical examples and discuss the relation to model reduction of deterministic control systems that have an underlying Hamiltonian structure.
Diffusion weighted imaging is a magnetic resonance based method to investigate
tissue micro-structure especially in the human brain via water diffusion.
Since the standard diffusion tensor model for the acquired data failes in
large portion of the brain voxel more sophisticated models have bee developed.
Here, we report on the package dti and how some of these models
can be used with the package.
The behavior approach and the problem of dissipativity have both been introduced and studied extensively by Willems et al. However, a computationally feasible method to check dissipativity is missing. Current methods will mostly rely on symbolic representations of rational functions. We will discuss a new characterization for linear systems in behavior form that allows to check dissipativity via the solution of a para-Hermitian, polynomial eigenvalue problem. Thus, we can employ standard methods of cubic complexity.
We derive a Crooks-Jarzynski-type identity for computing free energy differences between metastable states that is based on nonequilibrium diffusion processes. Furthermore we outline a brief derivation of an infinite-dimensional stochastic partial differential equation that can be used to efficiently generate the ensemble of trajectories connecting the metastable states.