Refine
Year of publication
Language
- English (1103) (remove)
Keywords
- optimal control (27)
- stability (14)
- integer programming (11)
- Stochastic programming (9)
- finite elements (9)
- mixed integer programming (9)
- Hamiltonian matrix (8)
- finite element method (8)
- model reduction (8)
- state constraints (8)
We consider Large Deformation Diffeomorphic Metric Mapping of general $m$-currents. After stating an optimization algorithm in the function space of admissable morph generating velocity fields, two innovative aspects in this framework are presented and numerically investigated: First, we spatially discretize the velocity field with conforming adaptive finite elements and discuss advantages of this new approach. Second, we directly compute the temporal evolution of discrete $m$-current attributes.
We introduce hexagonal global parameterizations, a new
type of surface parameterizations in which parameter lines respect six-fold rotational symmetries (6-RoSy). Such parameterizations enable the tiling of surfaces with nearly regular hexagonal or triangular patterns, and can be used for triangular remeshing.
To construct a hexagonal parameterization on a surface, we provide an automatic algorithm to generate a 6-RoSy field that respects directional and singularity features of the surface. This field is then used to direct a hexagonal global parameterization. The framework, called HexCover, extends the QuadCover algorithm and formulates necessary conditions for hexagonal parameterization.
We demonstrate the usefulness of our geometry-aware global parameterization with applications such as surface tiling with nearly regular textures and geometry patterns, as well as triangular and hexagonal remeshing.
Despite the success of quad-based 2D surface parameterization methods, effective parameterization algorithms for 3D volumes with cubes, i.e. hexahedral elements, are still missing. CubeCover is a first approach which provides both, a consistent theoretical framework for volume parameterization plus a full pipeline for generating
a hexahedral tessellation of a given volume with boundary aligned cubes which are guided by a frame field.
The input of CubeCover is a tetrahedral volume mesh. First, a frame field is designed with manual input from
the designer. It guides the interior and boundary layout of the parameterization. Then, the parameterization and
the hexahedral mesh are computed so as to align with the given frame field.
CubeCover has similarities to the QuadCover algorithm and extends it from 2D surfaces to 3D volumes. The
paper also provides theoretical results for 3D hexahedral parameterizations and analyses topological properties
of the appropriate function space.
We derive gradient-flow formulations for systems describing drift-diffusion processes of a finite number of species which undergo mass-action type reversible reactions. Our investigations cover heterostructures, where material parameter may depend in a nonsmooth way on the space variable. The main results concern a gradient flow formulation for electro-reaction-diffusion systems
with active interfaces permitting drift-diffusion processes and reactions of species living on the interface and transfer mechanisms allowing bulk species to jump into an interface or to pass through interfaces.
The gradient flows are formulated in terms of two functionals: the free energy and the dissipation potential. Both functionals consist of a bulk and an interface integral. The interface integrals determine the interface dynamics as well as the self-consistent coupling to the model in the bulk. The advantage of the gradient structure is that it automatically generates thermodynamically consistent models.
The ability of velocity methods to describe changes
of topology by creating defects like holes is investigated. For the shape optimization energy-type objective functions are considered, which depend on the geometry by means of state variables. The state system is represented by abstract, quadratic, constrained minimization problems stated over domains with defects. The velocity
method provides the shape derivative of the objective function due to finite variations of a defect. Suffcient conditions are
derived which allow us to pass the shape derivative to the limit
with respect to diminishing defect, thus, to obtain the "topological derivative" of the objective function due to a topology change.
An illustrative example is presented for a circular hole bored at
the tip of a crack.
Chimera states are particular trajectories
in systems of phase oscillators with non-local coupling
that display a spatio-temporal pattern of coherent and incoherent motion.
We present here a detailed analysis
of the spectral properties for such trajectories.
First, we study numerically their Lyapunov spectrum
and its behavior for an increasing number of oscillators.
The spectra demonstrate the hyperchaotic nature of the chimera states
and show a correspondence of the Lyapunov dimension
with the number of incoherent oscillators.
Then, we pass to the thermodynamic limit equation
and present an analytic approach
to the spectrum of a corresponding linearized evolution operator.
We show that in this setting, the chimera state is neutrally stable
and that the continuous spectrum coincides with the limit
of the hyperchaotic Lyapunov spectrum obtained for the finite size systems.
For a system of globally pulse-coupled phase-oscillators,
we derive conditions for stability of the completely synchronous
state and all possible two-cluster states and explain
how the different states are naturally connected via
bifurcations. The coupling is modeled using the phaseresponse-
curve (PRC), which measures the sensitivity of
each oscillator’s phase to perturbations. For large systems
with a PRC, which turns to zero at the spiking threshold,
we are able to find the parameter regions where multiple
stable two-cluster states coexist and illustrate this by an
example. In addition, we explain how a locally unstable
one-cluster state may form an attractor together will its
homoclinic connections. This leads to the phenomenon
of intermittent, asymptotic synchronization with abating
beats away from the perfect synchrony.
We prove a necessary and sufficient criterion for the exponential
stability of periodic solutions of delay differential equations with
large delay. We show that for sufficiently large delay the Floquet
spectrum near criticality is characterized by a set of curves, which we
call asymptotic continuous spectrum, that is independent on the
delay.
In this paper we study the destabilization mechanism in a ring of unidirectionally coupled oscillators. We derive an amplitude equation of Ginzburg-Landau type that describes the destabilization of the stationary state for systems with a large number of oscillators. Based on this amplitude equation, we are able to provide an explanation for the fast transition to chaos (or hyperchaos)
that can be observed in such systems. We show that the parameter interval, where the transition from a stable periodic state to chaos occurs, scales like the inverse
square of the number of oscillators in the ring. In particular, for a sufficiently large
number of oscillators a practically immediate transition to chaos can be observed.
The results are illustrated by a numerical study of a system of unidirectionally
coupled Duffing oscillators.
Three families of transmission conditions of different order are proposed for thin conducting sheets in the eddy current model. Resolving the thin sheet by a finite element mesh is often not possible. With these transmission conditions only the middle curve, but not the thin sheet itself, has not to be resolved by a finite element mesh. The families of transmission conditions are derived by an asymptotic expansion for small sheet thicknesses $\eps$, where each family results from a different asymptotic framework. In the first asymptotic framework the conductivity remains constant, scales with $1/\eps$ in the second and with $1/\eps^2$ in the third. The different asymptotics lead to different limit conditions, namely the vanishing sheet, a non-trivial borderline case, and the impermeable sheet, as well as different transmission conditions of higher orders. We investigated the stability, the convergence of the transmission conditions as well as their robustness. We call transmission conditions robust, if they provide accurate approximation for a wide range of sheet thicknesses and conductivities. We introduce an ordering of transmission conditions for the same sheet with respect to the robustness, and observe that the condition derived for the $1/\eps$ asymptotics is the most robust limit condition, contrary to order 1 and higher, where the transmission conditions derived for the $1/\eps^2$ asymptotics turn out to be most robust.
Network flows over time form a fascinating area of research. They model the temporal
dynamics of network flow problems occurring in a wide variety of
applications. Research in this area has been pursued in two different and mainly independent
directions with respect to time modeling: discrete and continuous time models.
In this paper we deploy measure theory in order to introduce a general model of network flows over time combining both discrete and continuous aspects into a single model. Here, the flow on each arc is modeled as a Borel measure on the real line (time axis) which assigns to each
suitable subset a real value, interpreted as the amount of
flow entering the arc over the subset. We focus on the maximum flow problem formulated in a network where capacities on arcs are also given as Borel measures and storage might be allowed at the nodes of the network. We generalize the concept of cuts to the case of these Borel Flows and extend the famous MaxFlow-MinCut Theorem.
We study a diffusion model of phase field type, consisting of
a system of two partial differential equations encoding the balances
of microforces and microenergy; the two unknowns are the order parameter
and the chemical potential. By a careful development of uniform estimates and
the deduction of certain useful boundedness properties, we
prove existence and uniqueness of a global-in-time
smooth solution to the associated initial/boundary-value problem;
moreover, we give a description of the relative $\omega$-limit set.
Dynamic network flow problems model the temporal evolution of flows over time and also consider changes of network parameters such as capacities, costs, supplies, and demands over time. These problems have been extensively studied in the past because of their important role in real world applications such as transport, traffic, and logistics. This has led to many results, but the more challenging continuous time model still lacks some of the key features such as network related optimality conditions and algorithms that are available in the static case.
The aim of this paper is to advance the state of the art for dynamic network flows by developing the continuous time analogues of several well-known optimality conditions for static network flows. Specifically, we establish a reduced cost optimality condition, a negative cycle optimality condition, and a strong duality result for a very general class of dynamic
network flows. The underlying idea is to construct a dual feasible solution that proves optimality when the residual network (with respect to a given flow) contains no dynamic cycles with negative cost. We also discuss a generic negative cycle-canceling algorithm resulting from the corresponding optimality criterion and point out promising directions for future research.
This paper concerns the shortest path problem for a network in which arc costs can vary with
time, each arc has a transit time, parking with a corresponding
time-varying cost is allowed at the nodes, and time is modeled as a continuum. The resulting problem is called the {\em continuous-time dynamic shortest path problem}, which is well studied in the literature. However, the problem appears as a subproblem when one wishes to test, via an algorithm for dynamic shortest paths, the presence of negative cycles in the residual network in order to develop continuous-time analogues of several well-known optimality conditions for continuous-time dynamic network flow problems. But, in general, the residual network contains arcs with negative transit times and hence the results in the literature are useless for these purposes since all results are based on the assumption of positive transit times.
In this paper, we relax this condition to allow negative transit times. We study a corresponding linear program in space of measures and prove the existence of an optimal extreme point solution. Moreover, we define a dual problem and establish a strong duality result that shows under certain assumptions that the value of the linear program equals the value of the dual problem and both values are attained. We also present counterexamples to show that strong duality only holds under these assumptions.
We consider the dynamic shortest path problem in the continuous-time model because of its importance. This problem has been extensively studied in the literature. But so far, all contributions to this problem are based on the assumption that all transit times are strictly positive. However, in order to study dynamic network flows it is essential to support negative transit times since they occur quite naturally in residual networks.
In this paper we extend the work of Philpott [SIAM Control Opt.,~1994, pp.~538--552] to the case of arbitrary (also negative and irrational) transit times. We study a corresponding linear program in a space of measures and give a full characterization of its extreme points. In particular, we show a one-to-one correspondence between extreme points and dynamic paths.
Research on flows over time has been conducted mainly in two separate and mainly independent approaches, namely \emph{discrete} and \emph{continuous} models, depending on whether a discrete or continuous representation of time is used. Recently, Borel flows have been introduced to build a bridge between these two models.
In this paper, we consider the maximum Borel flow problem formulated in a network where capacities on arcs are given as Borel measures and storage might be allowed at the nodes of the network. This problem is formulated as a linear program in a space of measures. We define a dual problem and prove a strong duality result. We show that strong duality is closely related to a MaxFlow-MinCut Theorem.
We investigate optimal elliptic
regularity (within the scale of Sobolev spaces) of anisotropic
div--grad operators in three dimensions at a multi-material vertex on
the Neumann boundary part of a polyhedral spatial domain. The
gradient of a solution to the corresponding elliptic PDE (in a
neighbourhood of the vertex) is integrable to an index greater than
three.
One of the most challenging problems in dynamic concurrent multiscale
simulations is the reflectionless transfer of physical quantities between the
different scales. In particular, when coupling molecular dynamics and finite
element discretizations in solid body mechanics, often spurious wave reflections
are introduced by the applied coupling technique. The reflected waves are
typically of high frequency and are arguably of little importance in the domain
where the finite element discretization drives the simulation.
In this work, we provide an analysis of this phenomenon.
Based on the gained
insight, we derive a new coupling approach, which neatly separates high and low
frequency waves. Whereas low frequency waves are permitted to
bridge the scales, high frequency waves can be removed by applying damping techniques without affecting the coupled share of the solution. As a consequence, our new method almost completely eliminates unphysical wave reflections and deals in a consistent way with waves of arbitrary frequencies. The separation of
wavelengths is achieved by employing a discrete $L^2$-projection, which acts as a
low pass filter. Our coupling constraints enforce matching in the range of this projection. With respect to the numerical realization this approach
has the advantage of a small number of constraints, which is computationally
efficient. Numerical results in one and two dimensions confirm our theoretical
findings and illustrate the performance of our new weak coupling approach.
We investigate a control problem for the heat equation. The goal is to find an optimal heat transfer coefficient in the Robin boundary condition such that a desired temperature distribution at the boundary is adhered. To this end we consider a function space setting in which the heat flux across the boundary is forced to be an Lp function with respect to the surface measure, which in turn implies higher regularity for the time derivative of temperature. We show that the corresponding elliptic operator generates a strongly continuous semigroup of contractions and apply the concept of maximal parabolic regularity. This allows to show the existence of an optimal control and the derivation of necessary and sufficient optimality conditions.
We consider an inverse problem arising in laser-induced thermotherapy, a minimally invasive method for cancer treatment, in which cancer tissue is destroyed by coagulation. For the dosage planning numerical simulation plays an important role. To this end a crucial problem is to identify the thermal growth kinetics of the coagulated zone. Mathematically, this problem is a nonlinear and nonlocal parabolic heat source inverse problem. The solution to this inverse problem is defined as the minimizer of a non-convex cost functional. The existence of the minimizer is proven. We derive the Gateaux derivative of the cost functional, which is based on the adjoint system, and use it for a numerical approximation of the optimal coefficient.
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.
We consider the mechanical coupling of a geometrically exact Cosserat rod to a linear elastic continuum. The coupling conditions are formulated in the nonlinear rod configuration space. We describe a Dirichlet--Neumann algorithm for the coupled system, and use it to simulate the static stresses in a human knee joint, where the Cosserat rods are models for the ligaments.
Systems of rail-mounted vehicles play a key role in many logistics applications, and the efficiency of their operation frequently has a significant impact on the overall performance of the surrounding production environment. In theory, assigning transport requests to the vehicles of such systems and scheduling their execution amounts to finding k tours on a common line, where tours may never cross each other in time--dynamic collision constraints need to be respected. The goal is to minimize the makespan for a given set of transport requests.
We establish a model capturing the core challenges in transport planning problems of this type and relate it to other models in literature. After proving NP-hardness for a basic version of the problem, the large part of the paper is dedicated to devising various fast heuristic algorithms suitable for practice. We present computational results regarding the performance of the algorithms proposed for several classes of problem instances.
In this paper we study the distributed optimal control for the Cahn-Hilliard system. A general class of free energy potentials is allowed which, in particular, includes the double-obstacle potential. The latter potential yields an optimal control problem of a parabolic variational inequality which is of fourth-order in space. We show the existence of optimal controls to approximating problems where the potential is replaced by a mollified version of its Moreau-Yosida approximation. Corresponding first order optimality conditions for the mollified problems are given. For this purpose a new result on the continuous Fr\'echet differentiability of superposition operators with values in Sobolev spaces is established. Besides the convergence of optimal controls of the mollified problems to an optimal control of the original problem, we also derive first order optimality conditions for the original problem by a limit process. The newly derived stationarity system corresponds to a function space version of C-stationarity.
We discuss the numerical solution of large scale nonlinear eigenvalue problems and frequency
response problems that arise in the analysis, simulation and optimization of acoustic fields.
We report about the cooperation with the company SFE in Berlin. We present the challenges
in the current industrial problems and the state-of-the-art of current methods. The difficulties
that arise with current off-the-shelf methods are discussed and several industrial examples are presented. It is documented that industrial cooperation is by no means a one-way street
of transfer from academia to industry but the challenges arising in industrial practice also lead to new mathematical questions which actually change the mathematical theory and methods.
In optimal control problems with nonlinear time-dependent 3D PDEs, full 4D discretizations are usually prohibitive due to the storage requirement. For this reason gradient and Newton type methods working on the reduced functional are often employed. The computation of the reduced gradient requires one solve of the state equation forward in time, and one backward solve of the adjoint equation. The state enters into the adjoint equation, again requiring the storage of a full 4D data set. We propose a lossy compression algorithm using an inexact but cheap predictor for the state data, with additional entropy coding of prediction errors. As the data is used inside a discretized, iterative algorithm, lossy compression
maintaining a certain error bound turns out to be sufficient.
The ability of velocity methods to describe changes
of topology by creating defects like holes is investigated. For the
shape optimization energy-type objective functions are considered,
which depend on the geometry by means of state variables. The
state system is represented by abstract, quadratic, constrained
minimization problems stated over domains with defects. The velocity
method provides the shape derivative of the objective function
due to finite variations of a defect. Sufficient conditions are
derived which allow us to pass the shape derivative to the limit
with respect to diminishing defect, thus, to obtain the topological
derivative" of the objective function due to a topology change.
An illustrative example is presented for a circular hole bored at
the tip of a crack
This paper deals with error estimates for space-time
discretizations of 3D model for isothermal stress-induced
transformations in shape-memory materials. After recalling
existence and uniqueness results, a fully-discrete approximation is presented and an explicit space-time
convergence rate of order is derived.
This note addresses a three-dimensional model for isothermal stress-induced transformation in shape-memory polycrystalline materials. We treat the problem within the framework of the energetic formulation of rate-independent processes and investigate existence and continuous dependence issues at both the constitutive relation and quasi-static evolution level. Moreover, we focus on time and space approximation as well as on regularization and parameter asymptotics.
This paper discusses the stability of quasi-static paths for a continuous elastic-plastic system with hardening in a one-dimensional (bar) domain. Mathematical formulations, as well as existence and uniqueness results for dynamic and quasi-static problems involving elastic-plastic systems with linear kinematic hardening are recalled in the paper. The concept of stability of quasi-static paths used here is essentially a continuity property of the system dynamic solutions relatively to the quasi-static ones, when (as in Lyapunov stability) the size of initial perturbations is decreased and the rate of application of the forces (which plays the role of the small parameter in singular perturbation problems) is also decreased to zero. The stability of the quasi-static paths of these elastic-plastic systems is the main result proved in the paper.
Mathematical results on existence for viscoelastodynamic problems with unilateral constraints
(2011)
We study a damped wave equation and the evolution of a Kelvin-Voigt material, both problems have unilateral boundary conditions. Under appropriate regularity assumptions on the initial data, both problems possess a weak solution which is obtained as the limit of a sequence of penalized problems; the functional properties of all the traces are precisely identified through Fourier analysis, and this enables us to infer the existence of a strong solution.
Energetic solutions to rate-independent processes are usually constructed via time-incremental minimization problems. In this work we show that all energetic solutions can be approximated by incremental problems if we allow approximate minimizers, where the error in minimization has to be of the order of the time step. Moreover, we study sequences of problems where the energy functionals have a Gamma limit.
A model for the evolution of damage that allows for complete disintegration is addressed. Small strains and a linear response function are assumed. The ``flow rule'' for the damage parameter is rate-independent. The stored energy involves the gradient of the damage variable, which determines an internal length-scale. Quasi-static fully rate-independent evolution is considered as well as rate-dependent evolution including viscous/inertial effects. Illustrative 2-dimensional computer simulations are presented, too.
Quasistatic small-strain plasticity in the limit of small hardening and its numerical approximation
(2011)
The quasistatic rate-independent evolution of the Prager-Ziegler-type model of linearized plasticity with hardening is shown to converge to the rate-independent evolution of the Prandtl-Reuss elastic/perfectly plastic model. Based on the concept of energetic solutions we study the convergence of the solutions in the limit for hardening coefficients converging to 0 by using the abstract method of Gamma-convergence for rate-independent systems. An unconditionally convergent numerical scheme is devised and 2D and 3D numerical experiments are presented. A two-sided energy inequality is a posteriori verified to document experimental convergence rates.
Deflated and augmented Krylov subspace methods: Basic Facts and a Breakdown-free deflated MINRES
(2011)
In this paper we consider deflation and augmentation techniques for accelerating
the convergence of Krylov subspace methods for the solution of nonsingular linear
algebraic systems. The two techniques are conceptually different from
preconditioning. Deflation "removes" certain parts from the operator, while
augmentation adds a subspace to the Krylov subspace. Both approaches have been
used in a variety of methods and settings. For Krylov subspace methods that
satisfy a (Petrov-) Galerkin condition we show that augmentation can in general
be achieved implicitly by projecting the residuals appropriately and correcting
the approximate solutions in a final step. In this context, we analyze known
methods to deflate CG, GMRes and MinRes. Our analysis reveals that the recently
proposed RMinRes method can break down. We show how such breakdowns can be
avoided by choosing a special initial guess, and we derive a breakdown-free
deflated MinRes method. In numerical experiments we study the properties of
different variants of MinRes analyzed in this paper.
We introduce a~numerical method for the numerical solution of the Lur'e matrix equations that arise, for instance, in linear-quadratic infinite time horizon optimal control. The method is based on the characterization of the solutions in terms of deflating subspaces of a suitable even matrix pencil. Via a Cayley transformation, the problem is transformed to the discrete-time case. This leaves us with a symplectic problem with several Jordan blocks of eigenvalue 1 and even size, which arise from the remaining eigenvalues at infinity of the original problem. For the solution of this modified problem, we use the {\em structure-preserving doubling algorithm} (SDA), an iterative scheme for the solution of dense continuous- and discrete-time algebraic Riccati equations. Unlike other iterative schemes, this algorithm converges also when the pencil has eigenvalues on the unit circle, as is the case in our problem. Implementation issues such as the choice of the parameter $\gamma$ in the Cayley transform are discussed. The numerical examples presented confirm the effectiveness of this method.
This paper presents concepts and implementation of the finite element toolbox Kaskade 7, a flexible C++ code for solving elliptic and parabolic PDE systems. Issues such as problem formulation, assembly and adaptivity are discussed at the example of optimal control problems. Trajectory compression for parabolic optimization problems is considered as a case study.
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.
Recently, the format of TT tensors
\cite{hackbuschHT,osele1,tyrtosele2,tyrtosele3} has turned out to be
a promising new format for the approximation of solutions of high
dimensional problems. In this paper, we prove some new results for
the TT representation of a tensor $U \in \R^{n_1\times \ldots\times
n_d}$ and for the manifold of tensors of TT-rank $\underline{r}$.\As a first result, we prove that the TT (or compression) ranks $r_i$
of a tensor $U$ are unique and equal to the respective separation
ranks of $U$ if the components of the TT decomposition are required to
fulfil a certain maximal rank condition. We then show that the set
$\mathcal{T}$ of TT tensors of fixed rank $\underline{r}$ forms an embedded
manifold in $\R^{n^d}$, therefore preserving the essential theoretical
properties of the Tucker format, but often showing an improved scaling
behaviour. Extending a similar approach for matrices \cite{conte_lub},
we introduce certain gauge conditions to obtain a unique
representation of the tangent space $\cT_U\mathcal{T}$ of $\mathcal{T}$
and deduce a
local parametrization of the TT manifold. The parametrisation of
$\cT_{U}\mathcal{T}$ is often crucial for an algorithmic treatment of
high-dimensional time-dependent PDEs and minimisation problems
\cite{lubuch_blau}. We conclude with remarks on those applications and
present some numerical examples.
We investigate the regularity of the weak solution to elliptic transmission problems that involve two layered anisotropic materials separated by a boundary intersecting interface. Under a compatibility condition for the angle of contact of the two surfaces and the boundary data, we prove the existence of square-integrable second derivatives, and the global Lipschitz continuity of the solution. We show that the second weak derivatives remain integrable to a certain power less than two if the compatibility condition is violated.
While it is well-known that the standard integral operator K of (stationary) diffuse-gray radiation, as it occurs in the radiosity equation, is compact if the domain of radiative interaction is sufficiently regular, we show noncompactness of the operator if the domain is polyhedral. We also show that a stationary operator is never compact when reinterpreted in a transient setting. Moreover, we provide new proofs, which do not use the compactness of K, for 1 being a simple eigenvalue of K for connected enclosures, and for I-(1-e)K being invertible, provided the emissivity e does not vanish identically.
In this paper we will consider elliptic boundary value problems with
oscillatory diffusion coefficient, say A. We will derive regularity
estimates in Sobolev norms which are weighted by certain derivatives of A.
The constants in the regularity estimates then turn out to be independent of
the variations in A.
These regularity results will be employed for the derivation of error
estimates for hp-finite element discretizations which are explicit with
respect to the local variations of the diffusion coefficient.
We consider the numerical solution of projected Lyapunov equations using Krylov subspace iterative methods. Such equations
play a fundamental role in balanced truncation model reduction of descriptor systems. We present generalizations of
the extended block and global Arnoldi methods to projected Lyapunov equations and compare these methods with the alternating direction implicit method with respect to performance on different examples.
A deflation strategy is also proposed to overcome possible breakdown in the
recurrence.
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 describe the appearance and stability of spatio-temporal periodic
patterns (rotating waves) in unidirectional rings of coupled oscillators
with delayed couplings. We show how delays in the coupling lead
to a splitting of each rotating wave into several new ones. The appearance
of rotating waves is mediated by Hopf bifurcations of the symmetric
equilibrium.
We also conclude that the coupling delays can be effectively
replaced by increasing the number of oscillators in the chain.
The phenomena are shown for Stuart-Landau
oscillators as well as for coupled FitzHugh-Nagumo systems interacting
via excitatory chemical synapses.
Classical surface parameterization algorithms often place singularities
in order to enhance the quality of the resulting parameter map. Unfortunately, singularities of positive integral index (as the north pole of a sphere) were not handled since they cannot be described with piecewise linear parameter functions on a triangle mesh. Preprocessing is needed to adapt the mesh connectivity. We present an extension to the QuadCover parameterization algorithm [KNP07], which allows to handle those singularities. A singularity of positive integral index can be resolved using bilinear parameter functions on quadrilateral elements. This generalization
of piecewise linear functions for quadrilaterals enriches the space of parameterizations. The resulting parameter map can be visualized by textures using a rendering system which supports quadrilateral elements, or it can be used for remeshing into a pure quad mesh.
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.
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.
We present a novel algorithm for automatic parameterization of tube-like surfaces of arbitrary genus such as the surfaces of knots, trees, blood vessels, neurons, or any tubular graph with a globally consistent stripe texture. We use the principal curvature frame field of the underlying tube-like surface to guide the creation of a global, topologically consistent stripe parameterization of the surface. Our algorithm extends the QuadCover algorithm and is based, first, on the use of so-called projective vector fields instead of frame fields, and second, on different types of branch points. That does not only simplify the mathematical theory, but also reduces computation time by the decomposition of the underlying stiffness matrices.
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.
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 package fmri is provided for analysis of single run functional
Magnetic Resonance Imaging data. It implements structural adaptive smoothing
methods with signal detection for adaptive noise reduction which avoids blurring
of edges of activation areas. fmri provides fmri analysis from time series
modeling to signal detection and publication-ready images.
Modeling the orientation distribution function by mixtures of angular central Gaussian distributions
(2010)
In this paper we develop a tensor mixture model for diffusion weighted imaging
data using an automatic model selection criterion for the order of tensor
components in a voxel. We show that the weighted orientation distribution
function for this model can be expanded into a mixture of angular central
Gaussian distributions. We show properties of this model in extensive
simulations and in a high angular resolution experimental data set. The results
suggest that the model may improve imaging of cerebral fiber tracts. We
demonstrate how inference on canonical model parameters may give rise to new
clinical applications.
In this work, we study the spectra and eigenmodes of the Hessian of various discrete surface energies and discuss applications to shape analysis. In particular, we consider a physical model that describes the vibration modes and frequencies of a surface through the eigenfunctions and eigenvalues of the Hessian of a deformation energy, and we
derive a closed form representation for the Hessian (at the rest state of the energy) for a general class of deformation energies. Furthermore, we design a quadratic energy, such that the eigenmodes of the Hessian of
this energy are sensitive to the extrinsic curvature of the surface. Based on these spectra and eigenmodes, we derive two shape signatures. One that measures the similarity of points on a surface, and another that
can be used to identify features of the surface. In addition, we discuss a
spectral quadrangulation scheme for surfaces.
We discuss the perturbation analysis for
eigenvalues and eigenvectors of structured homogeneous matrix polynomials with
Hermitian, skew-Hermitian, H-even and H-odd structure.
We construct minimal structured perturbations (structured backward errors) such that an
approximate eigenpair is an exact eigenpair of an appropriate perturbed structured matrix
polynomial. We present various comparisons with unstructured backward
errors and previous error bounds derived for the non-homogeneous case
and show that our bounds present a significant improvement.
Many applications give rise to matrix polynomials whose coefficients have
a kind of reversal symmetry, a structure we call palindromic.
Several properties of scalar palindromic polynomials are derived,
and together with properties of compound matrices, used to
establish the Smith form of regular and singular T-palindromic matrix polynomials,
over arbitrary fields.
The invariant polynomials are shown to
inherit palindromicity,
and their structure is described in detail.
Jordan structures of palindromic matrix polynomials are characterized,
and necessary conditions for the
existence of structured linearizations established.
In the odd degree case, a constructive procedure for building
palindromic linearizations shows that the necessary conditions are sufficient as well.
The Smith form for *-palindromic polynomials is also analyzed. Finally, results for palindromic matrix polynomials over fields of
characteristic two are presented.
We define a risk averse nonanticipative feasible policy for multistage stochastic programs and propose a methodology to implement it. The approach is based on dynamic programming equations written for a risk averse formulation of the problem.
This formulation relies on a new class of multiperiod risk functionals called extended polyhedral risk measures. Dual representations of such risk functionals are given and used to derive conditions of coherence. In the one-period case, conditions for convexity and consistency with second order stochastic dominance are also provided. The risk averse dynamic programming equations are specialized considering convex combinations of one-period extended polyhedral risk measures such as spectral risk measures.
To implement the proposed policy, the approximation of the risk averse recourse functions for stochastic linear programs is discussed. In this context, we detail a stochastic dual dynamic programming algorithm which converges to the optimal value of the risk averse problem.
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.
To address the plurality of interpretations of the subjective notion of risk, we describe it by means of a risk order and concentrate on the context invariant features of diversification and monotonicity. Our main results are uniquely characterized robust representations of lower semicontinuous risk orders on vector spaces and convex sets. This representation covers most instruments related to risk and allow for a differentiated interpretation depending on the underlying context which is illustrated in different settings: For random variables, risk perception can be interpreted as model risk, and we compute among others the robust representation of the economic index of riskiness. For lotteries, risk perception can be viewed as distributional risk and we study the "Value at Risk". For consumption patterns, which excerpt an intertemporality dimension in risk perception, we provide an interpretation in terms of discounting risk and discuss some examples.
We study the risk assessment of uncertain cash flows in terms of dynamic convex risk measures for processes as introduced in Cheridito, Delbaen, and Kupper (2006). These risk measures take into account not only the amounts but also the timing of a cash flow. We discuss their robust representation in terms of suitably penalized probability measures on the optional $\sigma$-field. This yields an explicit analysis both of model and discounting ambiguity. We focus on supermartingale criteria for time consistency. In particular we show how ``bubbles'' may appear in the dynamic penalization, and how they cause a breakdown of asymptotic safety of the risk assessment procedure.
Dynamic risk measures
(2010)
This paper gives an overview of the theory of dynamic convex risk measures for random variables in discrete time setting. We summarize robust representation results of conditional convex risk measures, and we characterize various time consistency properties of dynamic risk measures in terms of acceptance sets, penalty functions, and by supermartingale properties of risk processes and penalty functions.
The classical valuation of an uncertain cash flow in discrete time consists in taking the expectation of the sum of the discounted future payoffs under a fixed probability measure, which is assumed to be known. Here we discuss the valuation problem in the context of Knightian uncertainty. Using results from the theory of convex risk measures, but without assuming the existence of a global reference measure, we derive a robust representation of concave valuations with an infinite time horizon, which specifies the interplay between model uncertainty and uncertainty about the time value of money.
Recently, there is a growing trend to offer guarantee products where the investor is allowed to shift her account/investment value between multiple funds. The switching right is granted a finite number per year, i.e. it is American style with multiple exercise possibilities. In consequence, the pricing and the risk management is based on the switching strategy which maximizes the value of the guarantee put option. We analyze the optimal stopping problem in the case of one switching right within different model classes and compare the exact price with the lower price bound implied by the optimal deterministic switching time. We show that, within the class of log-price processes with independent increments, the stopping problem is solved by a deterministic stopping time if (and only if) the price process is in addition continuous. Thus, in a sense, the Black & Scholes model is the only (meaningful) pricing model where the lower price bound gives the exact price. It turns out that even moderate deviations from the Black & Scholes model assumptions give a lower price bound which is really below the exact price. This is illustrated by means of a stylized stochastic volatility model setup.
In this paper we consider the optimal stopping problem for general dynamic monetary utility functionals. Sufficient conditions for the Bellman principle and the existence of optimal stopping times are provided. Particular attention is payed to representations which allow for a numerical treatment in real situations. To this aim, generalizations of standard evaluation methods like policy iteration, dual and consumption based approaches are developed in the context of general dynamic monetary utility functionals. As a result, it turns out that the possibility of a particular generalization depends on specific properties of the utility functional under consideration.
Boolean modeling frameworks have long since proved their worth for capturing and analyzing essential characteristics of complex systems.
Hybrid approaches aim at exploiting the advantages of Boolean formalisms while refining expressiveness. In this paper, we present a formalism that augments Boolean models with stochastic aspects. More specifically, biological reactions effecting a system in a given state are associated
with probabilities, resulting in dynamical behavior represented as a Markov chain. Using this approach, we model and analyze the cytokinin
response network of Arabidopsis thaliana with a focus on clarifying the character of an important feedback mechanism.
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 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.
Motivated by the analysis of passive control systems, we undertake a detailed perturbation analysis of Hamiltonian matrices that have eigenvalues on the imaginary axis. We construct minimal Hamiltonian perturbations that move and coalesce eigenvalues of opposite sign characteristic to form multiple eigenvalues with mixed sign characteristics, which are then moved from the imaginary axis to specific locations in the complex plane by small Hamiltonian perturbations. We also present a numerical method to compute upper bounds for the minimal perturbations that move all eigenvalues of a given Hamiltonian matrix outside a vertical strip along the imaginary axis.
In this work we propose a general framework for the structured perturbation
analysis of several classes of structured matrix polynomials in homogeneous
form, including complex symmetric, skew-symmetric, even and odd matrix polynomials. We introduce structured backward errors for approximate eigenvalues and eigenvectors and we construct minimal structured perturbations such that an approximate eigenpair is an exact eigenpair of an appropriately perturbed matrix polynomial. This work extends previous work for the non-homogeneous case (we include infinite eigenvalues) and we show that the structured backward errors improve the known unstructured backward errors.
Mathematical programs in which the constraint set is partially defined by the solutions of an elliptic variational inequality, so-called ``elliptic MPECs'', are formulated in reflexive Banach spaces. With the goal of deriving explicit first order optimality conditions amenable to the development of numerical procedures, variational analytic concepts are both applied and further developed. The paper is split into two main parts. The first part concerns the derivation of conditions in which the state constraints are assumed to be polyhedric sets. This part is then completed by two examples, the latter of which involves pointwise bilateral bounds on the gradient of the state. The second part begins with the derivation of a formula for the second order (Mosco) epiderivative of the indicator function of a general convex set. This result is then used to derive analogous conditions to those which are presented in the first part. Finally, an elliptic MPEC is considered important to the study of elasto-plasticity in which the pointwise Euclidean norm of the gradient of the state is bounded. Explicit strong stationarity conditions are provided for this problem.
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.
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
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.
We consider the behavior of a modulated wave solution to
an $\mathbb{S}^1$-equivariant autonomous system of differential equations under an external
forcing of modulated wave type. The modulation frequency of the forcing is assumed to be close to the
modulation frequency of the modulated wave solution, while the wave frequency of the forcing is supposed to be far
from that of the modulated wave solution. We describe the domain in the three-dimensional
control parameter space (of frequencies and amplitude of the forcing)
where stable locking of the modulation frequencies of the forcing and the modulated wave solution
occurs.
Our system is a simplest case scenario for the behavior of self-pulsating lasers under the influence of external
periodically modulated
optical signals.
We show that the coupled balance equations for a large class of dissipative materials
can be cast in the form of GENERIC (General Equations for Non-Equilibrium
Reversible Irreversible Coupling). In dissipative solids, also called generalized standard
materials, the state of a material point is described by dissipative internal variables in addition to the elastic deformation and the temperature. The framework GENERIC allows
for an efficient derivation of thermodynamically consistent coupled field equations,
while revealing additional underlying physical structures, like the role of the free energy
as the driving potential for reversible effects and the role of the free entropy (Massieu potential) as the driving potential for dissipative effects.
Applications to large and small-strain thermoplasticity is given. Moreover, for the
quasistatic case, where the deformation can be statically eliminated, we derive a generalized
gradient structure for the internal variable and the temperature with a reduced
entropy as driving functional.
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.
In this paper we study BSDEs arising from a special class of backward stochastic partial differential equations (BSPDEs) that is intimately related to utility maximization problems with respect to arbitrary utility functions. After providing existence and uniqueness we discuss the numerical realizability. Then we study utility maximization problems on incomplete financial markets whose dynamics are governed by continuous semimartingales. Adapting standard methods that solve the utility maximization problem using BSDEs, we give solutions for the portfolio optimization problem which involve the delivery of a liability at maturity. We illustrate our study by numerical simulations for selected examples. As a byproduct we prove existence of a solution to a very particular quadratic growth BSDE with unbounded terminal condition. This complements results on this topic obtained in [6,7,8].
We prove central and non-central limit theorems for the
Hermite variations of the anisotropic fractional Brownian sheet
$W^{\alpha, \beta}$
with Hurst parameter $(\alpha, \beta) \in (0,1)2$. When $0<\alpha \leq
1-\frac{1}{2q}$ or $0<\beta \leq 1-\frac{1}{2q}$ a central limit theorem
holds for the renormalized Hermite variations of order $q\geq 2$, while
for $1-\frac{1}{2q}<\alpha, \beta < 1$ we prove that these variations
satisfy a non-central limit theorem. In fact, they converge to a random
variable which is the value of a two-parameter Hermite process at time
$(1,1)$.
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 $B$.
In this Note we consider a Lipschitz backward stochastic
differential equation (BSDE) driven by a continuous martingale $M$. We
prove (in Theorem \ref{theorem:main}) that if $M$ is a strong Markov
process and if the BSDE has regular data then the unique solution
$(Y,Z,N)$ of the BSDE is reduced to $(Y,Z)$, \textit{i.e.} the
orthogonal martingale $N$ is equal to zero, showing that in a Markovian
setting the "usual" solution $(Y,Z)$ (of a BSDE with regular data) has
not to be completed by a strongly orthogonal component even if $M$ does
not enjoy the martingale representation property.
We extend some recent works by Delong and Imkeller concerning Backward
stochastic differential equations with time delayed generators (delay
BSDE). We provide sharper a priori estimates and show that the solution
of a delay BSDE is in $L^p$. We introduce decoupled systems of SDE and
delay BSDE (which we term delay FBSDE) and give sufficient conditions
for the variational differentiability of their solutions. We connect
these derivatives to the Malliavin derivatives of such delay FBSDE via
the usual representation formulas which in turn give access to several
path regularity results. In particular we prove an extension of the
$L2$-path regularity result for delay FBSDE.
Model Reduction for a Class of Nonlinear Electrical Circuits by Reduction of Linear Subcircuits
(2010)
We analyze a model reduction approach for a class of electrical circuits containing nonlinear resistance.
In our approach
the linear subcircuits are extracted and replaced with linear passive reduced-order models. The resulting nonlinear
reduced-order model preserves admissibility and passivity. Moreover, we derive a-priori bounds for the error between
the input-output map of the original circuit equations and that of our reduced-order model. These bounds
are valid for all inputs. Since only linear subcircuits are reduced, our approach is effective when the number of nonlinear resistances is relatively small. The performance of our approach is illustrated numerically.
This paper concerns $n\times n$ linear one-dimensional hyperbolic systems of the type
$$
\om\partial_tu_j + a_j(x)\partial_xu_j + \sum\limits_{k=1}^nb_{jk}(x)u_k = f_j(x,t),\; j=1,\ldots,n,
$$
with periodicity conditions in time and reflection boundary conditions in space.
We state sufficient conditions on the data $\om, a_j$ and $b_{jk}$
such that the system has a Fredholm like solvability behavior.
Moreover, we state sufficient conditions on the data such that for any right hand side there exists exactly
one solution, that the solution survives under small perturbations of the data, and that the corresponding
data-to-solution-map
is smooth with respect to appropriate function space norms. In particular, those sufficient conditions
imply that no small denominator effects occur. Moreover, we show that perturbations of the coefficients $\om$ and $a_j$ lead to
essentially different results than perturbations of the coefficients $b_{jk}$, in general.
In this article we compare two different calmness conditions which are
widely used in the literature on bilevel programming and on mathematical
programs with equilibrium constraints. In order to do so, we consider convex
bilevel programming as a kind of intersection between both research areas.
The so-called partial calmness concept is based on the function value
approach for describing the lower level solution set. Alternatively,
calmness in the sense of multifunctions may be considered for perturbations
of the generalized equation representing the same lower level solution set.
Both concepts allow to derive first order necessary optimality conditions
via tools of generalized differentiation introduced by Mordukhovich. They
are very different, however, concerning their range of applicability and the
form of optimality conditions obtained. The results of this paper seem to
suggest that partial calmness is considerably more restrictive than calmness
of the perturbed generalized equation. This fact is also illustrated by
means of a dicretized obstacle control problem.
Based on a thermodynamically consistent model for precipitation in gallium arsenide crystals including surface tension and bulk stresses by Dreyer and Duderstadt, we propose different mathematical models to describe the size evolution of liquid droplets in a crystalline solid. The first class of models treats the diffusion-controlled regime of interface motion, while the second class is concerned with the interface-controlled regime of interface motion. Our models take care of conservation of mass and substance. We consider homogenised models, where different length scales of the experimental situation have been exploited in order to simplify the equations. These homogenised models generalise the well-known Lifshitz-Slyozov-Wagner model for Ostwald ripening. Mean field models capture the main properties of our system and are well adapted for numerics and further analysis. Numerical evidence suggests in which case which one of the two regimes might be appropriate to the experimental situation.
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 prove global convergence of an inexact polyhedral Gau\ss--Seidel method for the minimization of strictly convex functionals that are continuously differentiable on each polyhedron of a polyhedral decomposition of
their domains of definition. While being known to be very slow by themselves, such methods are a cornerstone for fast, globally convergent multigrid methods. Our result generalizes the proof of Kornhuber and Krause [2006] for differentiable functionals on the Gibbs simplex. Example applications are given that require the generality of our approach.
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.
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.
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.
We revisit the problem of the linear response of a constrained mechanical system. In doing so we show that the standard expressions of Green and Kubo carry over to the constrained case without any alteration. The argument is based on the appropriate definition of constrained expectations by means of which Liouville’s theorem and the Green-Kubo relations naturally follow.
We propose a nonequilibrium sampling method for computing free energy profiles along a given reaction coordinate. The method consists of two parts: a controlled Langevin sampler that generates nonequilibrium bridge paths conditioned by the reaction coordinate, and Jarzynski’s formula for reweighting the paths. Our derivation of the equa- tions of motion of the sampler is based on stochastic perturbation of a controlled dissipative Hamiltonian system, for which we prove Jarzynski’s identity as a special case of the Feynman-Kac formula. We illustrate our method by means of a suitable numerical example and briefly discuss issues of optimally choosing the control protocol for the reaction coordinate.
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.
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.