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We consider convex optimization problems with $k$th order stochastic dominance constraints for $k\ge 2$. We discuss distances of random variables that are relevant for the dominance relation and establish quantitative stability results for optimal values and solution sets in terms of a suitably selected probability metrics.Moreover, we provide conditions ensuring that the optimal value function is Hadamard directionally differentiable. Finally, we discuss some implications of the results for empirical (Monte Carlo,
sample average) approximations of dominance constrained optimization models.
Some mathematical problems related to the 2nd order optimal shape of a crystallization interface
(2012)
We consider the problem to optimize the stationary temperature distribution and the equilibrium shape of the solid-liquid interface in a two-phase system subject to a temperature gradient. The interface satisfies the minimization principle of the free energy, while the temperature is solving the heat equation with a radiation boundary conditions at the outer wall. Under the condition that the temperature gradient is uniformly negative in the direction of crystallization, the interface is expected to have a global graph representation. We reformulate this condition as a pointwise constraint on the gradient of the state, and we derive the first order optimality system for a class of objective functionals that account for the second surface derivatives, and for the surface temperature gradient.
We characterize the Smith form of skew-symmetric matrix polynomials
over an arbitrary field $\F$,
showing that all elementary divisors occur with even multiplicity.
Restricting the class of equivalence transformations to unimodular congruences,
a Smith-like skew-symmetric canonical form
for skew-symmetric matrix polynomials is also obtained.
These results are used to analyze the eigenvalue and elementary divisor structure
of matrices expressible as products of two skew-symmetric matrices,
as well as the existence of structured linearizations
for skew-symmetric matrix polynomials.
By contrast with other classes of structured matrix polynomials
(e.g., alternating or palindromic polynomials),
every regular skew-symmetric matrix polynomial
is shown to have a structured strong linearization.
While there are singular skew-symmetric polynomials of even degree
for which a structured linearization is impossible,
for each odd degree we develop a skew-symmetric companion form
that uniformly provides a structured linearization
for every regular and singular skew-symmetric polynomial
of that degree.
Finally, the results are applied to the construction of minimal
symmetric factorizations of skew-symmetric rational matrices.
We consider risk-averse formulations of multistage stochastic linear programs. For these formulations, based on convex combinations of spectral risk measures, risk-averse dynamic programming equations can be written. As a result, the Stochastic Dual Dynamic Programming
(SDDP) algorithm can be used to obtain approximations of
the corresponding risk-averse recourse functions. This allows us to define a risk-averse nonanticipative feasible policy for thestochastic linear program. Formulas for the cuts that approximate the recourse functions are given.
We consider the solution of a system of stochastic generalized equations (SGE) where the underlying functions are mathematical expectation of random set-valued mappings. SGE has many applications such as characterizing optimality conditions of a nonsmooth stochastic optimization problem and a stochastic equilibrium problem. We derive quantitative continuity of expected value of the set-valued mapping with respect to the variation of the underlying
probability measure in a metric space. This leads to the subsequent qualitative and quantitative stability analysis of solution set mappings of the SGE. Under some metric regularity conditions, we derive Aubin's property of the solution set mapping with respect to the change of probability measure. The established results are
applied to stability analysis of stationary points of classical one stage and two stage stochastic minimization problems, two stage stochastic mathematical programs with equilibrium constraints and stochastic programs with second order dominance constraints.
This paper is concerned with a PDE-constrained optimization problem of induction heating, where the state equations consist of 3D time--dependent heat equations coupled with 3D time--harmonic eddy current equations. The control parameters are given by finite real numbers representing applied alternating voltages which enter the eddy current equations via impressed current. The optimization problem is to find optimal voltages so that, under certain constraints on the voltages and the temperature, a desired temperature can be optimally achieved. As there are finitely many control parameters but the state constraint has to be satisfied in an infinite number of points, the problem belongs to a class of semi--infinite programming problems. We present a rigorous analysis of the optimization problem and a numerical strategy based on our theoretical result.
We discuss shape optimization problems for cylindrical tubes that are loaded by time-dependent applied force. This is a problem of shape optimization that leads to optimal control in linear elasticity theory. We determine the optimal thickness of a cylindrical tube minimizing the deformation of the tube under the influence of the external force. The main difficulty is that the state equation is a hyperbolic partial differential equation of 4th order. First order necessary conditions for the optimal solution are derived. Based on them, a numerical method is set up and numerical examples are presented.
Optimal Thickness of a Cylindrical Shell -- An Optimal Control Problem in Linear Elasticity Theory
(2012)
In this paper we discuss optimization problems for cylindrical tubes which are loaded by an applied force. This is a problem of optimal control in linear elasticity theory (shape optimization). We are looking for an optimal thickness minimizing the deflection (deformation) of the tube under the influence of an external force.
From basic equations of mechanics, we derive the equation of deformation. We apply the displacement approach from shell theory and make use of the hypotheses of
Mindlin and Reissner. A corresponding optimal control problem is formulated and first order necessary conditions for the optimal solution (optimal thickness) are derived.
We present numerical examples which were solved by the finite element method.
This paper is devoted to an optimal control problem of Maxwell's equations in the presence of pointwise state constraints. The control is given by a divergence--free three--dimensional vector function representing an applied current density. To cope with the divergence--free constraint on the control, we consider a vector potential ansatz. Due to the lack of regularity of the control--to--state mapping, existence of Lagrange multipliers cannot be guaranteed. We regularize the optimal control problem by penalizing the pointwise state constraints. Optimality conditions for the regularized problem can be derived straightforwardly. It also turns out that the solution of the regularized problem enjoys higher regularity which then allows us to establish its convergence towards the solution of the unregularized problem. The second part of the paper focuses on the numerical analysis of the regularized optimal control problem. Here the state and the control are discretized by N\'ed\'elec's curl--conforming edge elements. Employing the higher regularity property of the optimal control, we establish an a priori error estimate for the discretization error in the $\boldsymbol{H}(\bold{curl})$--norm. The paper ends by numerical results including a numerical verification of our theoretical results.
An optimal control problem arising in the context of 3D electromagnetic induction heating is investigated. The state equation is given by a quasilinear stationary heat equation coupled with a semilinear time-harmonic eddy current equation. The temperature-dependent electrical conductivity and the presence of pointwise inequality state-constraints represent the main challenge of the paper. In the first part of the paper, the existence and regularity of the state are addressed. The second part of the paper deals with the analysis of the corresponding linearized equation. Some sufficient conditions are presented which guarantee the solvability of the linearized system. The final part of the paper is concerned with the optimal control. The aim of the optimization is to find the optimal voltage such that a desired temperature can be achieved optimally. The corresponding first-order necessary optimality condition is presented.
Some optimal control problems for linear and nonlinear ordinary differential equations related to the optimal switching between
different magnetic fields are considered. The main aim is to move an electrical initial current by a controllable
voltage in shortest time to a desired terminal current and to hold it afterwards. Necessary optimality conditions are derived by
Pontryagin's principle and a Lagrange technique. In the case of a linear system, the principal structure of time-optimal controls is
discussed. The associated optimality systems are solved by a one-shot strategy
using a multigrid software package. Various numerical examples are discussed.
This paper deals with the computation of regular coderivatives of
solution maps associated with a frequently arising class of generalized equations.
The constraint sets are given by (not necessarily convex) inequalities,
and we do not assume linear independence of gradients to active constraints.
The achieved results enable us to state several versions of sharp necessary optimality
conditions in optimization problems with equilibria governed by such
generalized equations. The advantages are illustrated by means of examples.
We introduce nonsmooth Schur-Newton methods for the solution of the nonlinear discrete saddle-point problems arising from discretized vector-valued Cahn-Hilliard equations with logarithmic and obstacle potentials. The discrete problems are obtained by semi-implicit discretization in time and a first order finite element discretization in space. We incorporate the linear constraints that enforce solutions to stay on the Gibbs simplex using Lagrangian multipliers and prove existence of these multipliers under the assumption of a non-trivial initial condition for the order parameters.
The authors propose a recycling MINRES scheme for a solution of subsequent self-adjoint linear systems as appearing, for example, in the Newton process for solving nonlinear equations. Ritz vectors are automatically extracted from one MINRES run and then used for self-adjoint deflation in the next. The method is designed to work with a preconditioner and arbitrary inner products. Numerical experiments with nonlinear Schrödinger equations indicate a substantial decrease in computation time when recycling is used.
Mathematical modeling of Czochralski type growth processes for semiconductor bulk single crystals
(2012)
This paper deals with the mathematical modeling and simulation of
crystal growth processes by the so-called Czochralski method and related methods,
which are important industrial processes
to grow large
bulk single crystals of semiconductor materials such as, e.g., gallium arsenide
(GaAs) or silicon (Si) from the melt.
In particular, we investigate a recently developed
technology in which traveling magnetic fields are applied in order to
control
the behavior of the turbulent melt flow. Since numerous different physical effects
like electromagnetic fields, turbulent melt flows, high temperatures, heat transfer via
radiation, etc., play an important role in the process, the corresponding mathematical
model leads to an extremely difficult system of initial-boundary value problems for
nonlinearly coupled partial differential equations. In this paper, we describe a mathematical
model that is under use for the simulation of real-life growth scenarios, and we give an overview
of mathematical results and numerical simulations that have been obtained for it in recent years.
In this paper, we deal with a hydraulic reservoir optimization problem with uncertainty on
inflows in a joint chance constrained programming setting. In particular, we will consider inflows with
a persistency effect, following a causal time series model, and examine the impact of the ”Gaussian”
assumption for such inflows. We present an iterative algorithm for solving similarly structured joint
chance constrained programming problems that requires a Slater point and the computation of gradients.
Several alternatives to the joint chance constraint problem are presented. In particular, we present an
individual chance constraint problem and a robust model. We illustrate the interest of joint chance
constrained programming by comparing results obtained on a realistic hydro-valley with those obtained
from the alternative models. Despite the fact that the alternative models often require less hypothesis
on the law of the inflows, we show that they yield conservative and costly solutions. The simpler models,
such as the individual chance constraint one, are shown to yield insufficient robustness and are therefore
not useful. We therefore conclude that Joint Chance Constrained programming appears as a technique
offering a good trade-off between cost and robustness and can be tractable for complex realistic models.
We provide lower estimates for the norm of gradients of Gaussian
distribution functions and apply the results obtained to a special class of
probabilistically constrained optimization problems. In particular, it is shown
how the precision of computing gradients in such problems can be controlled
by the precision of function values for Gaussian distribution functions. Moreover,
a sensitivity result for optimal values with respect to perturbations of the
underlying random vector is derived. It is shown that the so-called maximal
increasing slope of the optimal value with respect to the Kolmogorov distance
between original and perturbed distribution can be estimated explicitly from
the input data of the problem.
Global spatial regularity for elasticity models with cracks, contact and other nonsmooth constraints
(2012)
A global higher differentiability result in Besov
spaces is proved for the displacement fields of linear elastic models
with self contact.
Domains with cracks are studied, where nonpenetration
conditions/Signorini conditions are imposed on the crack faces.
It is shown that
in a neighborhood of crack tips (in 2D) or
crack fronts (3D) the displacement fields are
$B^{3/2}_{2,\infty}$ regular.
The proof relies on a difference
quotient argument for the directions tangential to the crack. In order
to obtain the regularity estimates also in the normal direction, an
argument due to
Ebmeyer/Frehse/Kassmann is modified.
The methods are then applied to further examples like
contact problems with nonsmooth rigid foundations, to a model with
Tresca friction and
to minimization problems with
nonsmooth energies and constraints as they occur for instance in the modeling of
shape memory alloys.
Based on Falk's approximation Theorem for variational
inequalities, convergence rates for FE-discretizations of contact
problems are derived relying on the proven regularity properties.
Several numerical examples illustrate the theoretical results.
Global higher integrability of minimizers of variational problems with mixed boundary conditions
(2012)
We consider integral functionals with densities of p-growth, with respect to gradients, on a Lipschitz domain with mixed boundary conditions. The aim of this paper is to prove that, under uniform estimates within certain classes of p-growth and coercivity assumptions on the density, the minimizers are of higher integrability order, meaning that they belong to the space of first order Sobolev functions with an integrability of order $p+\epsilon$ for a uniform $\epsilon >0$. The results are applied to a model describing damage evolution in a nonlinear elastic body and to a model for shape memory alloys.
Existence result for a class of generalized standard materials with thermomechanical coupling
(2012)
This paper deals with the study of a three-dimensional model of thermomechanical coupling for viscous solids exhibiting hysteresis effects. This model is written in accordance with the formalism of generalized standard materials. It is composed by the momentum equilibrium equation combined with the flow rule, which describes some stress-strain dependance, and the heat-transfer equation. An existence result for this thermodynamically consistent problem is obtained by using a fixed-point argument and some qualitative properties of the solutions are established.
We derive and study a dynamical model for suspensions of negatively buoyant particles on an incline. Our theoretical model includes the settling/sedimentation due to gravity as well as the resuspension of particles induced by shear-induced migration, leading to disaggregation of the dense sediment layer. Out of the three different regimes observed in the experiments, we focus on the so-called settled case, where the particles settle out of the flow, and two distinct fronts, liquid and particle, form. Using an approach relying on asymptotics, we systematically connect our dynamic model with the previously developed equilibrium theory for particle-laden flows. We show that the resulting transport equations for the liquid and the particles are of hyperbolic type, and study the dilute limit, for which we derive the analytic solution. We also carry out a systematic experimental study of the settled regime, focusing on the motion of the liquid and the particle fronts. Finally, we carry out numerical simulations of our transport equations. We show that the model predictions for small to moderate values of the particle volume fraction and the inclination angle of the solid substrate agree well with the experimental data.
We investigate a distributed optimal control problem for a phase field
model of Cahn-Hilliard type. The model describes two-species phase segregation
on an atomic lattice under the presence of diffusion; it has been introduced recently in
[4], on the basis of the theory developed in [15], and consists of a system of two
highly nonlinearly coupled PDEs. For this reason, standard arguments of optimal control theory do not apply
directly, although the control constraints and the cost functional are of standard type.
We show that the problem admits a solution, and we derive the first-order
necessary conditions of optimality.
We discuss the possibility of computing eigenpairs of some prototypical linear second-order self-adjoint elliptic partial differential operator (or its high-resolution finite element discretization) by numerical upscaling techniques. We compute a low-dimensional generalized finite element space that preserves small eigenvalues in a superconvergent way. The approximate eigenpairs are then obtained by solving the corresponding low-dimensional algebraic eigenvalue problem. The rigorous error bounds are based on two-scale decompositions of H1 by means of a certain Clement-type quasi-interpolation operator.
Quasi-Monte Carlo algorithms are studied for designing discrete approximations of two-stage linear stochastic programs. Their integrands are piecewise linear, but neither smooth nor of bounded variation in the sense of Hardy and Krause. We show that under some weak geometric condition on the two-stage model all terms of their
ANOVA decomposition, except the one of highest order, are smooth and, hence, certain Quasi-Monte Carlo algorithms may achieve the optimal rate of convergence $O(n^{-1+\delta})$ with $\delta\in(0,\frac{1}{2})$ and a constant not depending on the dimension if the integrands belong to weighted tensor product Sobolev spaces with properly selected weights. The geometric condition is generically (i.e., almost everywhere) satisfied if the underlying distribution is normal. We also discuss sensitivity
indices and efficient dimensions of two-stage integrands, and suggest a dimension reduction heuristic for such integrands.
Piecewise linear convex functions arise as integrands in stochastic programs. They are Lipschitz continuous on their domain, but do not belong to tensor product Sobolev spaces. Motivated by applying Quasi-Monte Carlo methods we show that all terms of their ANOVA decomposition, except the one of highest order, are smooth if the underlying densities are smooth and certain geometric condition is satisfied. The latter condition is generically satisfied in the normal case.
In this paper we revisit models for the description of the evolution of crystalline films with anisotropic surface energies.
We prove equivalences of symmetry properties of anisotropic surface energy models commonly used in the literature.
Then we systematically develop a framework for the derivation of surface diffusion models for the self-assembly of quantum dots during Stranski-Krastanov
growth that include surface energies also with large anisotropy as well as the effect of wetting energy,
elastic energy and a randomly perturbed atomic deposition flux.
A linear stability analysis for the resulting sixth-order semilinear evolution equation for the thin film surface shows that that the new model allows
for large anisotropy and gives rise to the formation of anisotropic quantum dots. The nonlinear three-dimensional evolution is investigated via numerical solutions.
These suggest that increasing anisotropy stabilizes the faceted surfaces and may lead to a dramatic slow-down of the coarsening of the dots.
Analysis of the second phase of the GMRES convergence for a convection-diffusion model problem
(2012)
It is well konwn that GMRES applied to linear algebraic systems arising from a convection-diffusion model problem that has been discretized by the streamline upwind Petrov-Galerkin (SUPG) method, typically displays two distinct phases of convergence: a slow initial phase followed by a convergence acceleration in the second phase. This paper complements the known results on the length of the initial phase by analyzing how the acceleration in the second phase of convergence is related to the mesh Peclet number and the choice of the stabilization parameter in the SUPG discretization. The analysis is based on some new expressions and bounds for the GMRES residuals, which can be of general interest.
We investigate a nonstandard phase field
model of Cahn-Hilliard type. The model, which was introduced in
[16], describes two-species phase segregation and consists of a
system of two highly nonlinearly coupled PDEs. It has been studied
recently in
[5], [6] for the case of homogeneous Neumann
boundary conditions. In this paper, we investigate the case that the
boundary condition for one of the unknowns of the system is of third
kind and nonhomogeneous. For the resulting system, we show
well-posedness, and we study optimal boundary control
problems. Existence of optimal controls is shown, and the first-order
necessary optimality conditions are derived. Owing to the strong
nonlinear couplings in the PDE system, standard arguments of optimal
control theory do not apply directly, although the control constraints
and the cost functional will be of standard type.
This paper is concerned with a diffusion model of phase-field type, consisting
of a {parabolic} system of two partial differential equations{,} interpreted as balances
of microforces and microenergy{, for two unknowns: the problem's order parameter $\rho$}
and the chemical potential $\mu$; each equation includes a viscosity term -- respectively, $\varepsilon \,\partial_t\mu$ and $\delta\,\partial_t\rho$ -- with $\varepsilon$ and $\delta$ two positive parameters; the field equations are complemented by Neumann homogeneous boundary conditions and suitable initial conditions. In a recent paper \cite{CGPS3}, we proved that this problem is \wepo\ and investigated the \loti\ \bhv\ of its $(\varepsilon,\delta)-$solutions. Here we discuss the asymptotic limit of the system as $\eps$
tends to $0$. We prove convergence of
$(\varepsilon,\delta)-$solutions to the corresponding solutions for
the case $\eps =0$, whose long-time behavior we characterize; in the
proofs, we employ compactness and monotonicity arguments.
This paper discusses adaptive finite element methods (AFEMs) for the solution of elliptic eigenvalue problems associated with partial differential operators. An adaptive method based on nodal-patch refinement leads to an asymptotic error reduction property for the computed sequence of simple eigenvalues and eigenfunctions. This justifies the use of the proven saturation property for a class of reliable and efficient hierarchical a posteriori error estimators. Numerical experiments confirm that the saturation property is present even for very coarse meshes for many examples; in other cases the smallness assumption on the initial mesh may be severe.
In this paper we propose and analyze a new Multiscale Method for solving semi-linear elliptic problems with heterogeneous and highly variable coeffcient functions. For this purpose we construct a generalized finite element basis that spans a low dimensional multiscale space. The basis is assembled by performing localized linear finescale computations in small patches that have a diameter of order H |log(H)| where H is the coarse mesh size. Without any assumptions on the type of the oscillations in the coeffcients, we give a rigorous proof for a linear convergence of the H1-error with respect to the coarse mesh
size. To solve the arising equations, we propose an algorithm that is based on a damped Newton scheme in the multiscale space.
We consider a semilinear parabolic equation subject to a nonlinear dynamical boundary condition that is related to the so-calles Wentzell boundary condition. First, we prove the existence and uniqueness of global solutions as well as the existence of a global attractor. Then we derive a suitable Lojasiewicz-Simon-type inequality to show the convergence of global solutions to single steady states as time tends to infinity under the assumption that the nonlinear terms $f$, $g$ are real analytic. Moreover, we provide an estimate for the convergence rate.
A nonlocal quasilinear multi-phase system with nonconstant specific heat and heat conductivity
(2012)
In this paper, we prove the existence
and global boundedness from above for a solution to an
integrodifferential model for nonisothermal multi-phase
transitions under nonhomogeneous third type boundary conditions.
The system couples a quasilinear internal energy balance
ruling the evolution of the absolute temperature with a vectorial
integro-differential inclusion governing the vectorial
phase-parameter dynamics. The specific heat and the heat
conductivity k are allowed to depend both on the order parameter
$\chi$ and on the absolute temperature $\teta$ of the system, and
the convex component of the free energy may or may not be
singular. Uniqueness and continuous data dependence are
also proved under additional assumptions.
A novel Finite Element Method (FEM) for the computational simulation in particle reinforced composite materials with many inclusions is presented. It is based on an adapted mesh which consists of triangles and parametric quadrilaterals in 2D. The number of elements and, hence, the number of degrees of freedom are proportional to the number of inclusions. The error of the method is independent of the distance of the neighboring inclusions. While being related to network methods, the approach can tackle more general settings. We present an efficient residual a posteriori error estimator which enables to compute reliable upper and lower error bounds. Several numerical examples illustrate the performance of the method and the error estimator. Moreover, it is demonstrated that the assumption of a lattice structure of inclusions can easily lead to incorrect predictions about material properties.
A theorem on error estimates for smooth nonlinear programming
problems in Banach spaces is proved that can be used to derive
optimal error estimates for optimal control problems. This theorem is applied
to a class of optimal control problems for quasilinear elliptic equations.
The state equation is approximated by a finite element scheme, while different
discretization methods are used for the control functions. The distance of
locally optimal controls to their discrete approximations is estimated.
A mathematical model for instationary magnetization
processes is considered, where the underlying spatial domain
includes electrically conducting and nonconducting regions. The
model accounts for the magnetic induction law that couples the given
electrical voltage with the induced electrical current in the
induction coil. By a theorem of Showalter on degenerate parabolic
equations, theorems on existence, uniqueness, and regularity of the
solution to the associated Maxwell integrodifferential system are
proved.