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The goal of this paper is to show how the heat treatment of steel can be modelled in terms of a mathematical optimal
control problem. The approach is applied to laser surface hardening and the cooling of a steel slab including
mechanical effects. Finally, it is shown how the results can be utilized in industrial practice by a coupling with
machine-based control.
In this paper we investigate the performance of several out-of-the box solvers for mixed-integer quadratically constrained programmes (MIQCPs) on an open pit mine production scheduling problem with mixing constraints. We compare the solvers BARON, Couenne, SBB, and SCIP to a problem-specific algorithm on two different MIQCP formulations. The computational results presented show that general-purpose solvers with no particular knowledge of problem structure are able to nearly match the performance of a hand-crafted algorithm.
Stability-based methods for scenario generation in stochastic programming are reviewed. In particular, we briefly discuss Monte Carlo sampling, Quasi-Monte Carlo methods, quadrature rules based on sparse grids and optimal quantization. In addition, we provide some convergence results based on recent developments in multivariate integration. The method of optimal scenario reduction and techniques for scenario trees generation are also reviewed.
An adaptive finite element semi-smooth Newton solver for
the Cahn-Hilliard model with double obstacle free energy is proposed. For this purpose, the governing system is discretised in time using a semi-implicit scheme, and the resulting time-discrete system is formulated as an optimal control problem with pointwise constraints on the control. For the numerical solution of the optimal control problem, we propose a function space based algorithm which combines a Moreau-Yosida regularization technique for handling the control constraints with a semi-smooth-Newton method for solving the optimality systems of the resulting sub-problems. Further, for the discretization in space and in connection with the proposed algorithm, an adaptive finite element method is considered. The performance of the overall algorithm is illustrated by numerical experiments.
We study the dynamics of a ring of unidirectionally coupled autonomous
Duffing oscillators. Starting from a situation where the individual
oscillator without coupling has only trivial equilibrium dynamics,
the coupling induces complicated transitions to periodic, quasiperiodic,
chaotic, and hyperchaotic behavior. We study these transitions in
detail for small and large numbers of oscillators. Particular attention
is paid to the role of unstable periodic solutions for the appearance
of chaotic rotating waves, spatiotemporal structures and the Eckhaus
effect for a large number of oscillators. Our analytical and numerical
results are confirmed by a simple experiment based on the electronic
implementation of coupled Duffing oscillators.
Abstract—We present a novel coder for lossless compression
of adaptive multiresolution meshes that exploits their special
hierarchical structure. The heart of our method is a new
progressive connectivity coder that can be combined with
leading geometry encoding techniques. The compressor uses
the parent/child relationships inherent to the hierarchical mesh.
We use the rules that accord to the refinement scheme and
store bits only where it leaves freedom of choice, leading to
compact codes that are free of redundancy. To illustrate our
scheme we chose the widespread red-green refinement, but
the underlying concepts can be directly transferred to other
adaptive refinement schemes as well. The compression ratio of
our method exceeds that of state-of-the-art coders by a factor
of 2 to 3 on most of our benchmark models.
{We suggest hierarchical a posteriori error estimators for time-discretized
Allen-Cahn and Cahn-Hilliard equations with logarithmic potential and investigate
their robustness numerically.
We observe that the associated effectivity ratios seem to saturate for decreasing mesh size
and are almost independent of the temperature.
For selfadjoint matrices in an indefinite inner product, possible canonical forms are identified that arise when the matrix is subjected to a selfadjoint generic rank one perturbation. Genericity is understood in
the sense of algebraic geometry. Special attention is paid to the perturbation
behavior of the sign characteristic. Typically, under such a perturbation,
for every given eigenvalue, the largest Jordan block of the eigenvalue is
destroyed and (in case the eigenvalue is real) all other Jordan blocks
keep their sign characteristic. The new eigenvalues, i.e., those eigenvalues of
the perturbed matrix that are not eigenvalues of the original matrix,
are typically simple, and in some cases information is provided about their sign
characteristic (if the new eigenvalue is real). The main results are proved by using
the well known canonical forms of selfadjoint matrices in an indefinite inner product, a version of the Brunovsky
canonical form and on general results concerning rank one perturbations.
We study two related problems in non-preemptive scheduling and packing of malleable tasks with precedence constraints to minimize the makespan. We distinguish the scheduling variant, in which we allow the free choice of processors, and the packing variant, in which a task must be assigned to a contiguous subset of processors.
For precedence constraints of bounded width, we completely resolve the complexity status for any particular problem setting concerning width bound and number of processors, and give polynomial-time algorithms with best possible performance. For both, scheduling and packing malleable tasks, we present an FPTAS for the NP-hard problem variants and exact algorithms for all remaining special cases. To obtain the positive results, we do not require the common monotonous penalty assumption on processing times, whereas our hardness results hold even when assuming this restriction.
With the close relation between contiguous scheduling and strip packing, our FPTAS
is the first (and best possible) constant factor approximation for (malleable) strip packing under special precedence constraints.
This paper is intended to be a first step towards the continuous dependence of dynamical contact problems on the initial data as well as the uniqueness of a solution. Moreover, it provides the basis for a proof of the convergence of popular time integration schemes as the Newmark method.
We study a frictionless dynamical contact problem between both linearly elastic and viscoelastic bodies which is formulated via the Signorini contact conditions. For viscoelastic materials fulfilling the Kelvin-Voigt constitutive law, we find a characterization of the class of problems which satisfy a perturbation result in a non-trivial mix of norms in function space. This characterization is given in the form of a stability condition on the contact stresses at the contact boundaries.
Furthermore, we present perturbation results for two well-established approximations of the classical Signorini condition: The Signorini condition formulated in velocities and the model of normal compliance, both satisfying even a sharper version of our stability condition.
This paper deals with error estimates for space-time discretizations in the context of evolutionary variational inequalities of rate-independent type. After introducing a general abstract evolution problem, we address a fully-discrete approximation and provide a priori error estimates. The application of the abstract theory to a semilinear case is detailed. In particular, we provide explicit space-time convergence rates for the isothermal Souza-Auricchio model for shape-memory alloys.
This paper presents a combined adaptive finite element method with an iterative algebraic eigenvalue solver for the Laplace eigenvalue problem of quasi-optimal computational complexity. The analysis is based on a direct approach for eigenvalue problems and allows the use of higher order conforming finite element spaces with fixed polynomial degree k>0. The optimal adaptive finite element eigenvalue solver (AFEMES) involves a proper termination criterion for the algebraic eigenvalue solver and does not need any coarsening. Numerical evidence illustrates the optimal computational complexity.
On probabilistic constraints induced by rectangular sets and multivariate normal distributions
(2009)
In this paper, we consider optimization problems under probabilistic constraints which are defined by two-sided
inequalities for the underlying normally distributed random vector. As a main step
for an algorithmic solution of such problems, we derive a derivative formula for (normal) probabilities
of rectangles as functions of their lower or upper bounds. This formula allows to reduce the calculus
of such derivatives to the calculus of (normal) probabilities of rectangles themselves thus generalizing a
similar well-known statement for multivariate normal distribution functions. As an application, we consider
a problem from water reservoir management. One of the outcomes of the problem solution is that the
(still frequently encountered) use of simple individual probabilistic can completely fail. In contrast, the
(more difficult) use of joint probabilistic constraints which heavily depends on the derivative formula mentioned
before yields very reasonable and robust solutions over the whole time horizon considered.
Alternating matrix polynomials, that is, polynomials whose coefficients
alternate between symmetric and skew-symmetric matrices,
generalize the notions of even and odd scalar polynomials.
We investigate the Smith forms of alternating matrix polynomials,
showing that each invariant factor is an even or odd scalar polynomial.
Necessary and sufficient conditions
are derived for a given Smith form to be that of an alternating matrix polynomial.
These conditions allow a characterization of the possible Jordan structures
of alternating matrix polynomials,
and also lead to necessary and sufficient conditions
for the existence of structure-preserving strong linearizations.
Most of the results are applicable to singular as well
as regular matrix polynomials.
The paper proposes goal-oriented error estimation and mesh refinement
for optimal control problems with elliptic PDE constraints using the value
of the reduced cost functional as quantity of interest. Error representation,
hierarchical error estimators, and greedy-style error indicators are derived and
compared to their counterparts when using the all-at-once cost functional as
quantity of interest. Finally, the efficiency of the error estimator and generated
meshes are demonstrated on numerical examples.
We introduce geodesic finite elements as a new way to discretize
the nonlinear configuration space of a geometrically exact Cosserat rod.
These geodesic finite elements naturally generalize standard one-dimensional
finite elements to spaces of functions with values in a Riemannian manifold.
For the special orthogonal group, our approach reproduces the
interpolation formulas of [Crisfield/Jelenic:1999].
Geodesic finite elements are
conforming and lead to objective and path-independent problem formulations.
We introduce geodesic finite elements for general Riemannian manifolds,
discuss the relationship between geodesic finite elements and
coefficient vectors, and estimate the interpolation error.
Then we use them to find static equilibria of hyperelastic Cosserat rods.
Using the Riemannian trust-region algorithm of [Absil/Mahony/Sepulchre:2008]
we show numerically that the discretization error depends optimally on
the mesh size.
he generalized Langevin equation is useful for modeling a wide
range of physical processes. Unfortunately its parameters,
especially the memory function, are difficult to determine for
nontrivial processes. In this paper, relations between a
time-discrete generalized Langevin model and discrete multivariate
autoregressive (AR) or autoregressive moving average models (ARMA)
are established. This allows a wide range of discrete linear
methods known from time series analysis to be applied. In
particular, the determination of the memory function {\it via} the
order of the respective AR or ARMA model is addressed. The method
is illustrated on a one-dimensional test system and subsequently
applied to the molecular dynamics of a biomolecule which exhibits
an interesting relationship between the solvent method used, the
molecular conformation and the depth of the memory.
e propose an algorithm for the fast and efficient simulation of polymers represented by
chains of hard spheres. The particles are linked by holonomic bond constraints.
While the motion of the polymers is free (i.e., no collisions occur) the equations
of motion can be easily integrated using a collocation-based partitioned Gauss-Runge-Kutta method.
The method is reversible, symplectic and preserves energy. Moreover the numerical scheme allows the integration using much longer time steps than any explicit integrator such as the popular Verlet method. If polymers collide the point of impact can be determined to arbitrary precision by simple nested intervals. Once the collision point is known the impulsive contribution can be computed analytically. We illustrate our approach by means of a suitable numerical example.
We study Balanced Truncation for stochastic differential equations. In doing so, we adopt ideas from large deviations theory and discuss notions of controllability and obervability for dissipative Hamiltonian systems with degenerate noise term, also known as Langevin equations. For partially-observed Langevin equations, we illustrate model reduction by balanced truncation with an example from molecular dynamics and discuss aspects of structure-preservation.
or stable linear input-output systems, the method of balanced truncation (B.C.~Moore, {\em IEEE Trans. Auto. Contr.} {\bf AC-26}, 17--32, 1981) consists in finding a coordinate transformation such that modes which are least sensitive to the external input (controllability) also give the least output (observability) and therefore can be neglected. A drawback is that projecting the original equations of motion onto the subspace of interest typically fails to preserve the problem's physical structure, e.g., if the original equations are of second-order form or Hamiltonian. For Hamiltonian systems, a natural way of restricting a system to a subspace is by means of constraints, and we show, employing singular perturbation arguments, that balanced truncation can be done in a structure-preserving fashion. The thus obtained reduced Hamiltonian system preserves stability and passivity and satisfies the usual balanced truncation error bound.