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Group-velocity matched cross-phase modulation between a fundamental soliton and a dispersive wave-packet has been previously suggested for optical switching applications similar to an optical transistor. Moreover, the nonlinear interaction in the resulting group-velocity horizon can be exploited for adiabatic compression of the soliton down into the few-cycle regime. Here we show that both mechanisms can be combined. In such a transient compressor, parameters of the dispersive wave may then serve to actively control the soliton compression and adjust the pulse duration in the presence of disturbances. While a certain amount of control is already enabled by the delay between soliton and dispersive wave, the means of controlling the compression process are substantially enhanced by additionally manipulating the chirp of the dispersive wave. Moreover, controlling the chirp of the dispersive wave also enables correction for limitations of the compression scheme due to a self-frequency shift of the soliton or for uncompensated dispersion in the scheme. This substantially widens the practicality of the compression scheme and other applications of the highly efficient nonlinear interaction at the group-velocity horizon.
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.
In this paper, we introduce and study analytically a vectorial Cahn-Hilliard reaction model coupled with rate-dependent damage processes. The recently proposed Cahn-Hilliard reaction model can e.g. be used to describe the behavior of electrodes of lithium-ion batteries as it includes both the intercalation reactions at the surfaces and the separation into different phases. The coupling with the damage process allows considering simultaneously the evolution of a damage field, a second important physical effect occurring during the charging or discharging of lithium-ion batteries.
Mathematically, this is realized by a Cahn-Larché system with a non-linear Newton boundary condition for the chemical potential and a doubly non-linear differential inclusion for the damage evolution. We show that this system possesses an underlying generalized gradient structure which incorporates the non-linear Newton boundary condition. Using this gradient structure and techniques from the field of convex analysis we are able to prove constructively the existence of weak solutions of the
coupled PDE system.
We discuss analytical and numerical methods for the optimization of optoelectronic devices by performing optimal control of the PDE governing the carrier transport with respect to the doping profile. First, we provide a cost functional that is a sum of a regularization and a contribution, which is motivated by the modal net gain that appears in optoelectronic models of bulk or quantum- well lasers. Then, we state a numerical discretization, for which we study optimized solutions for different regularizations and for vanishing weights.
We derive thermodynamically consistent models of reaction-diffusion equations
coupled to a heat equation. While the total energy is conserved, the total
entropy serves as a driving functional such that the full coupled system is a
gradient flow. The novelty of the approach is the Onsager structure, which is
the dual form of a gradient system, and the formulation in terms of the
densities and the internal energy. In these variables it is possible to assume
that the entropy density is strictly concave such that there is a unique
maximizer (thermodynamical equilibrium) given linear constraints on the total
energy and suitable density constraints.
We consider two particular systems of this type, namely, a
diffusion-reaction bipolar energy transport system, and a
drift-diffusion-reaction energy
transport system with confining potential. We prove corresponding
entropy-entropy production inequalities with explicitly calculable constants
and establish the convergence to thermodynamical equilibrium, at first in
entropy and further in $L^1$ using Cziszar-Kullback-Pinsker type inequalities.
We consider a non-isothermal multi-phase field model.
We subsequently discretize implicitly in time and with
linear finite elements. The arising algebraic problem is
formulated in two variables where one is the multi-phase
field, and the other contains the inverse temperature field.
We solve this saddle point problem numerically by a
non-smooth Schur-Newton approach using truncated
non-smooth Newton multigrid methods. An application in
grain growth as occurring in liquid phase crystallization
of silicon is considered.
In these notes we discuss two approaches to evolutionary Γ- convergence of gradient systems in Hilbert spaces. The formulation of the gradient system is based on two functionals, namely the energy functional and the dissipation potential, which allows us to employ Γ- convergence methods. In the first approach we consider families of uni- formly convex energy functionals such that the limit passage of the time-dependent problems can be based on the theory of evolutionary variational inequalities as developed by Daneri and Savar ́e 2010. The second approach uses the equivalent formulation of the gradient system via the energy-dissipation principle and follows the ideas of Sandier and Serfaty 2004.
We apply both approaches to rigorously derive homogenization limits for Cahn–Hilliard-type equations. Using the method of weak and strong two-scale convergence via periodic unfolding, we show that the energy and dissipation functionals Γ-converge. In conclusion, we will give specific examples for the applicability of each of the two approaches.
Structure-preserving generic low-rank perturbations are studied for classes of structured matrix pencils, including real symmetric, complex symmetric, and complex Hermitian pencils. For singular pencils it is analyzed which characteristic quantities stay invariant in the perturbed canonical form, and it is shown that the regular part of a structured matrix pencil is not affected by generic perturbations of rank one. When the rank one perturbations involve a scaling parameter, the behavior of the canonical forms in dependence of this parameter is analyzed as well.
We study a stationary thermistor model describing the electrothermal behavior of organic semiconductor devices featuring non-Ohmic current-voltage laws and self-heating effects. The coupled system consists of the current-flow equation for the electrostatic potential and the heat equation with Joule heating term as source. The self-heating in the device is modeled by an Arrhenius-like temperature dependency of the electrical conductivity. Moreover, the non-Ohmic electrical behavior is modeled by a power law such that the electrical conductivity depends nonlinearly on the electric field. Notably, we allow for functional substructures with different power laws, which gives rise to a $p(x)$-Laplace-type problem with piecewise constant exponent.
We prove the existence and boundedness of solutions in the two-dimensional case. The crucial point is to establish the higher integrability of the gradient of the electrostatic potential to tackle the Joule heating term. The proof of the improved regularity is based on Caccioppoli-type estimates, Poincar\'e inequalities, and a Gehring-type Lemma for the $p(x)$-Laplacian. Finally, Schauder's fixed-point theorem is used to show the existence of solutions.
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.
Trajectory- or mesh-based methods for analyzing the dynamical behavior of large
molecules tend to be impractical due to the curse of dimensionality - their computational cost increases
exponentially with the size of the molecule. We propose a method to break the curse by a
novel square root approximation of transition rates, Monte Carlo quadrature and a discretization
approach based on solving linear programs. With randomly sampled points on the molecular energy
landscape and randomly generated discretizations of the molecular conguration space as our initial
data, we construct a matrix describing the transition rates between adjacent discretization regions.
This transition rate matrix yields a Markov State Model of the molecular dynamics. We use Perron
cluster analysis and coarse-graining techniques in order to identify metastable sets in conguration
space and approximate the transition rates between the metastable sets. Application of our method
to a simple energy landscape on a two-dimensional conguration space provides proof of concept and
an example for which we compare the performance of dierent discretizations. We show that the
computational cost of our method grows only polynomially with the size of the molecule. However,
nding discretizations of higher-dimensional conguration spaces in which metastable sets can be
identied remains a challenge.
The chemical master equation is a fundamental equation in chemical kinetics. It underlies the classical reaction-rate equations and takes stochastic effects into account. In this paper we give a simple argument showing that the solutions of a large class of chemical master equations are bounded in weighted $\ell_1$-spaces and possess high-order moments. This class includes all equations in which no reactions between two or more already present molecules and further external reactants occur that add mass to the system. As an illustration for the implications of this kind of regularity, we analyze the effect of truncating the state space. This leads to an error analysis for the finite state projections of the chemical master equation, an approximation that forms the basis of many numerical methods.
The present work deals with the resolution of the Poisson equation in a bounded domain made of a thin and periodic layer of finite length placed into a homogeneous medium.
We provide and justify a high order asymptotic expansion which takes into account the boundary layer effect occurring in the vicinity of the periodic layer as well as the corner singularities appearing in the neighborhood of the extremities of the layer. Our approach combines mixes
the method of matched asymptotic expansions and the method of periodic surface homogenization.
We study non-isothermal nucleation and growth phase transformations, which are described by a generalized Avrami model for the phase transition coupled with an energy balance to account for recalescence
effects. The main novelty of our work is the identification of temperature dependent nucleation rates. We prove that such rates can be uniquely identified from measurements in a subdomain and apply an optimal control approach to develop a numerical strategy for its computation.
Let $\sigma_t(x)$ denote the implied volatility at maturity t for a strike $K = S_0 e^{x t}$, where $x \in R$ and $S_0$ is the current value of the underlying. We show that $\sigma_t(x)$ has a uniform (in $x$) limit as maturity t tends to infinity, given by the formula \sigma_{\infty}(x) = \sqrt2 (h^*(x)^{1/2} + (h^*(x) − x)^{1/2}, for $x$ in some compact neighbourhood of zero in the class of affine stochastic volatility models. The function $h^*$ is the convex dual of the limiting cumulant generating function $h$ of the scaled log-spot process. We express $h$ in terms of the functional characteristics of the underlying model. The proof of the limiting formula rests on the large deviation behaviour of the scaled log-spot process as time tends to infinity. We apply our results to obtain the limiting smile for several classes of stochastic volatility models with jumps used in applications (e.g. Heston with state-independent jumps, Bates with state-dependent jumps and Barndorff-Nielsen-Shephard model).
In this article we show, that the binding kinetics of a molecular system can be
identied by a projection of a continuous process onto a nite number of macro states. We thus
interpret binding kinetics as a projection. When projecting onto non-overlapping macro states the
Markovianity is spoiled. As a consequence, the description of e.g. a receptor-ligand system by a two
state kinetics is not accurate. By assigning a degree of membership to each state, we abandon the
non-overlapping approach. This overlap is crucial for a correct mapping of binding eects by Markov
State Models with regard to their long time behavior. It enables us to describe the highly discussed
rebinding eect, where the spatial arrangement of the system has the be included. By introducing
a \degree of fuzziness" we have an indicator for the strength of the rebinding eect, such that the
minimal rebinding eect can be derived from an optimization problem. The fuzziness also includes
some new paradigms for molecular kinetics. These new model paradigms show good agreement with
experimental data.
Markov State Models (MSMs) are widely used to represent molecular
conformational changes as jump-like transitions between subsets of the conformational
state space. However, the simulation of peptide folding in explicit water is
usually said to be unsuitable for the MSM framework. In this article, we summarize
the theoretical background of MSMs and indicate that explicit water simulations do
not contradict these principles. The algorithmic framework of a meshless conformational
space discretization is applied to an explicit water system and the sampling
results are compared to a long-term molecular dynamics trajectory. The meshless
discretization approach is based on spectral clustering of stochastic matrices (MSMs)
and allows for a parallelization of MD simulations. In our example of Trialanine we
were able to compute the same distribution of a long term simulation in less computing
time.
This paper is concerned with the distributed optimal control of a
time-discrete Cahn--Hilliard/Navier--Stokes system with variable
densities.
It focuses on the double-obstacle potential which yields an optimal
control problem for a family of coupled systems in each time instance of a
variational inequality of fourth order and the Navier--Stokes equation.
By proposing a suitable time-discretization, energy estimates are proved
and the existence of solutions to the primal system and of optimal
controls is established for the original problem as well as for a family
of regularized problems. The latter correspond to Moreau--Yosida type
approximations of the double-obstacle potential. The consistency of these
approximations is shown and first order optimality conditions for the
regularized problems are derived. Through a limit process, a stationarity
system for the original problem is established which is related to a
function space version of C-stationarity.
We propose a composite step method, designed for equality constrained optimization with partial differential equations. Focus is laid on the construction of a globalization scheme, which is based on cubic regularization of the objective and an affine covariant damped Newton method for feasibility. We show finite termination of the inner loop and fast local convergence of the algorithm. We discuss preconditioning strategies for the iterative solution of the arising linear systems with projected conjugate gradient. Numerical results are shown for optimal control problems subject to a nonlinear heat equation and subject to nonlinear elastic equations arising from an implant design problem in craniofacial surgery.
In a strong constant electric field, a dielectric particle
immersed in a weakly conducting fluid exhibits spontaneous rotations. This phenomenon is known under the name of the Quincke effect. In the original setup the particle was suspended on a silk thread and performed torsional oscillations of remarkably high amplitude.
We derive the governing equations for this experiment, and
ascertain that onset of oscillations from the quiescent state corresponds to the supercritical Hopf bifurcation.
For the case of a soft thread, we characterize the regime of large-scale torsional relaxation oscillations: explicit estimates are derived for their period and amplitude,
effects of bifurcation delay are described. In a stronger electric field, these relaxation oscillations yield to small-scale erratic rotations of the pendulum.
We study a mechanical equilibrium problem for a material consisting of two components with different densities, which allows to change the outer shape by changing the interface between the subdomains. We formulate the shape design problem of compensating unwanted workpiece
changes by controlling the interface, employ regularity results for transmission problems for a rigorous derivation of optimality conditions based on the speed method, and conclude with some numerical results based on a spline approximation of the interface.
In this paper we propose a time discretization of a system of two parabolic equations describing diffusion-driven atom rearrangement in crystalline matter. The equations express the balances of microforces and microenergy; the two phase fields are the order parameter and the chemical potential. The initial and boundary-value problem for the evolutionary system is known to be well posed. Convergence of the discrete scheme to the solution of the continuous problem is proved by a careful development of uniform estimates, by weak compactness and a suitable treatment of
nonlinearities. Moreover, for the difference of discrete
and continuous solutions we prove an error estimate of
order one with respect to the time step.
This note is concerned with a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial conditions. The system arises from a model of two-species phase segregation
on an atomic lattice [22]; it consists of the balance equations of microforces and microenergy; the two unknowns are the order parameter rho and the chemical potential mu. Some recent results obtained for this class of problems is reviewed and, in the case of a nonconstant and nonlinear atom mobility, uniqueness and continuous dependence on the initial data are shown with the help of a new line of
argumentation developed in [12].
We consider viscoelastic solids undergoing thermal expansion and exhibiting hysteresis effects due to plasticity or phase transformations. Within the framework of generalized standard solids, the problem is described in a 3D setting by the momentum equilibrium equation, the flow rule describing the dependence of the stress on the strain history, and the heat transfer equation. Under appropriate regularity assumptions on the data, a local existence result for this thermodynamically consistent system is established, by combining existence results for ordinary differential equations in Banach spaces with a fixed-point argument. Then global estimates are obtained by using both the classical energy estimate and more specific techniques for the heat equation introduced by Boccardo and Gallouet. Finally a global existence result is derived.
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.
Uncertainty is inevitable when solving science and engineering application problems. In the face of
uncertainty, it is essential to determine robust and risk-averse solutions. In this work,
we consider a class of PDE-constrained optimization problems in which the PDE coefficients
and inputs may be uncertain. We introduce two approximations for minimizing the
conditional value-at-risk for such PDE-constrained optimization problems. These approximations are based
on the primal and dual formulations of the conditional value-at-risk. For the primal problem,
we introduce a smooth approximation of the conditional value-at-risk in order to utilize
derivative-based optimization algorithms and to take advantage of the convergence properties
of quadrature-based discretizations. For this smoothed conditional value-at-risk, we prove
differentiability as well as consistency of our approximation. For the dual problem, we
regularize the inner maximization problem, rigorously derive optimality conditions, and demonstrate
the consistency of our approximation. Furthermore, we propose a fixed-point iteration that takes
advantage of the structure of the regularized optimality conditions and provides a means of calculating
worst-case probability distributions based on the given probability level. We conclude with numerical
results.
This paper is concerned with the distributed optimal control of a
time-discrete Cahn--Hilliard/Navier--Stokes system with variable
densities.
It focuses on the double-obstacle potential which yields an optimal
control problem for a family of coupled systems in each time instance of a
variational inequality of fourth order and the Navier--Stokes equation.
By proposing a suitable time-discretization, energy estimates are proved
and the existence of solutions to the primal system and of optimal
controls is established for the original problem as well as for a family
of regularized problems. The latter correspond to Moreau--Yosida type
approximations of the double-obstacle potential. The consistency of these
approximations is shown and first order optimality conditions for the
regularized problems are derived. Through a limit process, a stationarity
system for the original problem is established which is related to a
function space version of C-stationarity.
With symmetric local absorbing boundary conditions for the Helmholtz equation scattering problems can be solved on a truncated domain, where the outgoing radiation condition is approximated by a Dirichlet-to-Neumann map with higher tangential derivatives on its outer boundary. Feng's conditions are symmetric local absorbing boundary conditions, which are based on an asymptotic expansion of the coefficients of the exact Dirichlet-to-Neumann map for large radia of the circular outer boundary. In this article we analyse the well-posedness of variational formulations with symmetric local absorbing boundary conditions in general and show how the modelling error introduced by Feng's conditions depends on the radius of the truncated domain.
Adaptive Numerical Solution of Eigenvalue Problems arising from Finite Element Models. AMLS vs. AFEM
(2015)
We discuss adaptive numerical methods for the solution of eigenvalue problems arising either
from the finite element discretization of a partial differential equation (PDE) or from discrete finite element modeling.
When a model is described by a partial differential equation, the adaptive finite element method
starts from a coarse finite element mesh which, based on a posteriori error estimators, is adaptively refined
to obtain eigenvalue/eigenfunction approximations of prescribed accuracy. This method is well established for classes of elliptic PDEs,
but is still in its infancy for more complicated PDE models.
For complex technical systems, the typical approach is to directly derive finite element models
that are discrete in space and are combined with macroscopic models to describe certain phenomena like damping or friction.
In this case one typically starts with a fine uniform mesh and computes eigenvalues and eigenfunctions using
projection methods from numerical linear algebra that are often combined with the algebraic multilevel
substructuring to achieve an adequate performance.
These methods work well in practice but their convergence and error analysis is rather difficult.
We analyze the relationship between these two extreme approaches. Both approaches have their pros and cons which are discussed in detail.
Our observations are demonstrated with several numerical examples.
This paper deals with the effect of generic but structured low rank perturbations on the Jordan structure and sign
characteristic of matrices that have structure in an indefinite inner product space.
The paper is a follow-up of earlier papers in which the effect of rank one perturbations
was considered. Several results that are in contrast to the case of unstructured low rank
perturbations of general matrices are presented here.
In large-area Organic Light-Emitting Diodes (OLEDs) spatially inhomogeneous luminance at high power due to inhomogeneous current flow and electrothermal feedback can be observed. To describe these self-heating effects in organic semiconductors we present a stationary thermistor model based on the heat equation for the temperature coupled to a p-Laplace-type equation for the electrostatic potential with mixed boundary conditions. The p-Laplacian describes the non-Ohmic electrical behavior of the organic material. Moreover, an Arrhenius-like temperature dependency of the electrical conductivity is considered.
We introduce a finite-volume scheme for the system and discuss its relation to recent network models for OLEDs. In two spatial dimensions we derive a priori estimates for the temperature and the electrostatic potential and prove the existence of a weak solution by Schauder's fixed point theorem.
The numerical simulation of the band structure of three-dimensional dispersive metallic photonic crystals with face-centered cubic lattices leads to large-scale nonlinear eigenvalue problems, which are very challenging due to a high dimensional subspace associated with the eigenvalue zero and the fact that the desired eigenvalues (with smallest real part) cluster near the zero eigenvalues. For
the solution of the eigenvalue problem, a Newton-type iterative method is proposed and the nullspace-free method is applied to exclude the zero eigenvalues from the associated generalized eigenvalue problem. To find the successive eigenvalue/eigenvector pairs, we propose a new non-equivalence deflation method to transform converged eigenvalues to infinity, while all other eigenvalues remain unchanged. The deflated problem is then solved by the same Newton-type method, which uses a hybrid method that combines the Jacobi-Davidson, the shift-invert residual Arnoldi and nonlinear Arnoldi methods to compute the clustered eigenvalues. Numerical results illustrate that the method is robust even for the case of computing many eigenvalues in very large problems.
Several classes of optimal control of electromagnetic fields are considered. Special emphasis is
laid on a non-standard $H$-based formulation of the equations of electromagnetism in multiply connected conductors. By this technique, the Maxwell equations can be solved with reduced computational complexity. While the magnetic field $H$ in the conductor is obtained from an elliptic equation
with the $\curl \sigma^{-1} \curl$ operator, an elliptic equation with the $\div \mu \nabla$ operator is set up for a potential $\psi$ in the isolator.
Both equations are coupled by appropriate interface conditions. In all problems, the
electrical current is controlled in the conducting domain. Several types of control functions are discussed. In particular, the problem of sparse optimal control is investigated in a package of electrical wires. For all problems, the associated sensitivity
analysis is performed.
A survey of methods from numerical linear algebra for linear constant coefficient differential-algebraic equations (DAEs) and descriptor control systems is presented. We discuss numerical methods to check the solvability properties of DAEs as well as index reduction and regularization techniques. For descriptor systems we discuss controllability and observability properties and how these can be checked numerically. These methods are based on staircase forms and derivative arrays, transformed with real orthogonal transformations that are discussed in detail. Then we use the reformulated problems in several control applications for differential-algebraic equations ranging from regular and singular linear-quadratic optimal and robust control to dissipativity checking. We discuss these applications and give a systematic overview over the theory and the numerical solution methods. In particular, we show that all these applications can be treated with a common approach that is based on the computation of eigenvalues and deflating subspaces of even matrix pencils. The unified approach allows to generalize and improve several techniques that are currently in use in systems and control.
The efficient and reliable computation of guided modes in photonic crystal wave-guides is of great importance for designing optical devices. Transparent boundary conditions based on Dirichlet-to-Neumann operators allow for an exact computation of well-confined modes and modes close to the band edge in the sense that no modelling error is introduced. The well-known super-cell method, on the other hand, introduces a modelling error which may become prohibitively large for guided modes that are not well-confined. The Dirichlet-to-Neumann transparent boundary conditions are, however, not applicable for all frequencies as they are not uniquely defined and their computation is unstable for a countable set of frequencies that correspond to so called Dirichlet eigenvalues. In this work we describe how to overcome this theoretical difficulty introducing Robin-to-Robin transparent boundary conditions whose construction do not exhibit those forbidden frequencies. They seem, hence, well suited for an exact and reliable computation of guided modes in photonic crystal wave-guides.
Regularization and Numerical Solution of the Inverse Scattering Problem using Shearlet Frames
(2014)
Regularization techniques for the numerical solution of nonlinear inverse scattering
problems in two space dimensions are discussed. Assuming that the boundary of a scatterer is its most prominent feature, we exploit as model the class of cartoon-like functions.
Since functions in this class are asymptotically optimally sparsely approximated by shearlet frames, we consider shearlets as a means for the regularization in a Tikhonov method.
We examine both directly the nonlinear problem and a linearized problem obtained by
the Born approximation technique. As problem classes we study the acoustic inverse
scattering problem and the electromagnetic inverse scattering problem. We show that
this approach introduces a sparse regularization for the nonlinear setting and we present
a result describing the behavior of the local regularity of a scatterer under linearization,
which shows that the linearization does not affect the sparsity of the problem. The analytical results are illustrated by numerical examples for the acoustic inverse scattering problem that highlight the effectiveness of this approach.
An eigenvalue perturbation theory under rank-one perturbations is developed for classes
of real matrices that are symmetric with respect to a non-degenerate bilinear form,
or Hamiltonian with respect to a non-degenerate skew-symmetric form.
In contrast to the case of complex matrices, the sign characteristic is a crucial feature
of matrices in these classes. The behavior of the sign characteristic under generic
rank-one perturbations is analyzed in each of these two classes of matrices.
Partial results are presented, but some questions remain open. Applications
include boundedness and robust boundedness for solutions of structured systems
of linear differential equations with respect to general perturbations as well
as with respect to structured rank perturbations of the coefficients.
Using a standard first-order optimality condition for nonsmooth optimization problems, a general framework for a descent method is developed. This setting is applied to a class of mathematical programs with equilibrium constraints in function space from which a new algorithm is derived. Global convergence of the algorithm is demonstrated in function space and the results are then illustrated by numerical experiments.
We derive formulae for the second-order subdifferential of polyhedral norms. These formulae are fully explicit in terms of initial data. In a first step we rely on the explicit formula for the coderivative of normal cone mapping to polyhedra. Though being explicit, this formula is quite involved and difficult to apply. Therefore, we derive simple formulae for the 1-norm and - making use of a recently obtained formula for the second-order subdifferential of the maximum function - for the maximum norm.
In this paper we present an efficient algorithm for the calculation of photonic crystal band structures and band structures of photonic crystal waveguides. Our method relies on the fact that the dispersion curves of the band structure are smooth functions of the quasi-momentum in the one-dimensional Brillouin zone. We show the derivation and computation of the group velocity, the group velocity dispersion, and any higher derivative of the dispersion curves. These derivatives are then employed in a Taylor expansion of the dispersion curves. We control the error of the Taylor expansion with the help of a residual estimate and introduce an adaptive scheme for the selection of nodes in the one-dimensional Brillouin zone at which we solve the underlying eigenvalue problem and compute the derivatives of the dispersion curves. The proposed algorithm is not only advantageous as it decreases the computational effort to compute the band structure but also because it allows
for the identification of crossings and anti-crossings of dispersion curves, respectively. This identification is not possible with the standard approach of solving the underlying eigenvalue problem at a discrete set of values of the quasi-momentum without taking the mode parity into account.
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.
A mechanical equilibrium problem for a material consisting of two components with dierent densities is considered. Due to the heterogeneous material densities, the
outer shape of the underlying workpiece can be changed by shifting the interface between
the subdomains. In this paper, the problem is modeled as a shape design problem for optimally compensating unwanted workpiece changes. The associated control variable is
the interface. Regularity results for transmission problems are employed for a rigorous
derivation of suitable first-order optimality conditions based on the speed method. The paper concludes with several numerical results based on a spline approximation of the
interface.
Multi-valued network models can be described by their topology and a set of parameters capturing the effects of the regulators for each component. Dynamics can then be derived and represented as state transition systems.
Different network models may lead to the same transition system, meaning dynamics analysis of a representative model covers a larger class of models. While rather clear in the Boolean case, the properties contributing to this effect become more involved for multi-valued models. We analyse these properties and present a mathematical description of the resulting model equivalence classes.
In this paper a new theorem is formulated
which allows a rigorous proof of the shape differentiability without the usage of the material derivative; the domain expression is automatically obtained and the boundary expression is easy to derive.
Furthermore, the theorem is applied to a cost function which depends on a quasi-linear transmission
problem. Using a Gagliardo penalization the existence of optimal shapes is established.
A coupling of discrete and continuous optimization to solve kinodynamic motion planning problems
(2014)
This paper studies the relationship between the material derivative method, the shape derivative method, the min-max formulation of Correa and Seeger, and the Lagrange method introduced by Cea. A theorem is formulated
which allows a rigorous proof of the shape differentiability without the usage of material derivative;
the domain expression is automatically obtained and the boundary expression is easy to derive.
Furthermore, the theorem is applied to a cost function which depends on a quasi-linear transmission
problem. Using a Gagliardo penalization the existence of optimal shapes is established.
An algorithm to detect collisions between robots moving along given trajectories is presented. The method is a combination of the adaptive
dynamic collision checking developed by Schwarzer et al. and Lin and Canny's algorithm, which computes efficiently the distance between two polyhedra. The resulting algorithm is part of a global model that computes the optimal task assignment, sequencing and kinodynamic
motion planning in a robotic work-cell.
Two optimal control problems for instationary magnetization
processes are considered in 3D spatial domains that
include electrically conducting and nonconducting regions. The magnetic
fields are generated by induction coils. In the first model, the induction coil
is considered as part of the conducting region and the electrical current is taken
as control. In the second, the coil is viewed as part of the nonconducting region and the
electrical voltage is the control. Here, an integro-differential equation accounts
for the magnetic induction law that couples the given
electrical voltage with the induced electrical current in the
induction coil.
We derive first-order necessary
optimality condition for the optimal controls of both problems. Based on them,
numerical methods of gradient type are applied. Moreover, we report on the application
of model reduction by POD that lead to tremendous savings. Numerical tests are
presented for academic 3D geometries but also for a real-world application.
We study the fundamental problem of scheduling bidirectional traffic across machines arranged on a path. The main feature of the problem is that jobs traveling in the same direction can be scheduled in quick succession on a machine, while jobs in the other direction have to wait for an additional transit time. We show that this tradeoff makes the problem significantly harder than the related flow shop problem, by showing that it is NP-hard even for jobs with identical processing and transit times. We give polynomial algorithms for a single machine and any constant number of machines. In contrast, we show the problem to be NP-hard on a single machine and with identical processing and transit times if some pairs of jobs in different directions are allowed to run on the machine concurrently. We generalize a PTAS of Afrati et al. [1999] for one direction and a single machine to the bidirectional case on any constant number of machines.
The analysis of adaptive finite element methods in practice immediately leads to eigenvalue clusters which requires the simultaneous marking in adaptive finite element methods. A first analysis for multiple eigenvalues of the recent work [Dai, He, Zhou, arXiv Preprint 1210.1846v2] introduces an adaptive method whose marking strategy is based on the element-wise sum of local error estimator contributions for multiple eigenvalues. This paper proves optimality of a practical adaptive algorithm for eigenvalue clusters for the eigenvalues of the Laplace operator in terms of nonlinear approximation classes. All estimates are explicit in the initial mesh-size, the eigenvalues and the cluster width to clarify the dependence of the involved constants.
For regular matrix pencils the distance in norm to the nearest singular pencil
under low rank perturbation is studied. Characterizations of this distance are derived via the Weyl function of the perturbation. Special attention is paid to the Hermitian pencil case.
Estimates for the distance of a given pencil to the set of singular pencils are obtained.