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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].