Humboldt-Universität zu Berlin
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Keywords
- Gas networks (2)
- Mixed-integer nonlinear optimization (2)
- spheric-radial decomposition (2)
- Abs-smooth Algorithmic Differentiation (1)
- Active Signature Method (1)
- Closed-loop stability (1)
- Controlling Regulators and Compressors (1)
- Convergence (1)
- DAE (1)
- Euler-Gleichungen, isotherme Euler-Gleichungen, Modellhierarchie, Netzelemente (1)
The directional differentiability of the solution mapping for a class of variational inequali-
ties of the second kind inspired by applications in fluid mechanics and moving free boundary
problems is investigated. The result is particularly relevant for the model predictive control
or optimal control of such variational inequalities in that it can be used to derive stationarity
conditions and efficient numerical methods.
Using a standard first-order optimality condition for nonsmooth optimization prob-
lems, 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 address the problem of optimally placing sensor networks for convection-diffusion
processes where the convective part is perturbed. The problem is formulated as an optimal control
problem where the integral Riccati equation is a constraint and the design variables are sensor
locations. The objective functional involves a term associated to the trace of the solution to the
Riccati equation and a term given by a constrained optimization problem for the directional derivative
of the previous quantity over a set of admissible perturbations. The paper addresses the existence
of the derivative with respect to the convective part of the solution to the Riccati equation, the
well-posedness of the optimization problem and finalizes with a range of numerical tests.
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 instant 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 with respect to the regularization parameter, a
stationarity system for the original problem is established. The resulting system corresponds to a
function space version of C-stationarity which is a special notion of stationarity for MPECs.
In this paper we study the structure of solutions of the one dimensional weighted total
variation regularisation problem, motivated by its application in signal recovery tasks. We study
in depth the relationship between the weight function and the creation of new discontinuities in
the solution. A partial semigroup property relating the weight function and the solution is shown
and analytic solutions for simply data functions are computed. We prove that the weighted total
variation minimisation problem is well-posed even in the case of vanishing weight function, despite
the lack of coercivity. This is based on the fact that the total variation of the solution is bounded
by the total variation of the data, a result that it also shown here. Finally the relationship to the
corresponding weighted fidelity problem is explored, showing that the two problems can produce
completely different solutions even for very simple data functions.
In this paper, optimal control problems subject to a nonlinear scalar conservation law are
studied. Such optimal control problems are challenging both at the continuous and at the discrete
level since the control-to-state operator poses difficulties as it is, e.g., not differentiable. Therefore
discretization of the underlying optimal control problem should be designed with care. Here the
discretize-then-optimize approach is employed where first the full discretization of the objective
function as well as the underlying PDE is considered. Then, the derivative of the reduced objective
is obtained by using an adjoint calculus. In this paper total variation diminishing Runge-Kutta
(TVD-RK) methods for the time discretization of such problems are studied. TVD-RK methods,
also called strong stability preserving (SSP), are originally designed to preserve total variation of
the discrete solution. It is proven in this paper that providing an SSP state scheme, is enough to
ensure stability of the discrete adjoint. However requiring SSP for both discrete state and adjoint is
too strong. Also approximation properties that the discrete adjoint inherits from the discretization
of the state equation are studied. Moreover order conditions are derived. In addition, optimal
choices with respect to CFL constant are discussed and numerical experiments are presented.
We present a novel mathematical algorithm to assist gas network operators in managing uncertainty,
while increasing reliability of transmission and supply. As a result, we solve an optimization problem
with a joint probabilistic constraint over an infinite system of random inequalities. Such models arise
in the presence of uncertain parameters having partially stochastic and partially non-stochastic character.
The application that drives this new approach is a stationary network with uncertain demand
(which are stochastic due to the possibility of fitting statistical distributions based on historical measurements)
and with uncertain roughness coefficients in the pipes (which are uncertain but non-stochastic due to a lack of
attainable measurements).
We study the sensitivity of local uncertainties in the roughness coefficients and their impact on a highly reliable
network operation. In particular, we are going to answer the question, what is the maximum uncertainty that is
allowed (shaping a 'maximal' uncertainty set) around nominal roughness coefficients, such that random demands in
a stationary gas network can be satisfied at given high probability level for no matter which realization of
true roughness coefficients within the uncertainty set.
One ends up with a constraint, which is probabilistic with respect to the load of gas
and robust with respect to the roughness coefficients. We demonstrate how such constraints can be dealt with in
the framework of the so-called spheric-radial decomposition of multivariate Gaussian distributions.
The numerical solution of a corresponding optimization problem is illustrated.
The results might assist the network operator with the implementation
of cost-intensive roughness measurements.
Quasi-Monte Carlo algorithms are studied for designing discrete approximations of two-stage linear stochastic programs with random right-hand side and
continuous probability distribution. The latter should allow for a transformation to a
distribution with independent marginals. The two-stage integrands are piecewise linear,
but neither smooth nor lie in the function spaces considered for QMC error analysis.
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 continuously differentiable and that first and second order ANOVA terms have mixed first order partial
derivatives and belong to L2 . Hence, randomly shifted lattice rules (SLR) may achieve
the optimal rate of convergence O(n−1+δ ) with δ ∈ (0, 12 ] and a constant not depending
on the dimension if the effective superposition dimension is at most two. We discuss
effective dimensions and dimension reduction for two-stage integrands. The geometric
condition is shown to be satisfied almost everywhere if the underlying probability distribution is normal and principal component analysis (PCA) is used for transforming
the covariance matrix. Numerical experiments for a large scale two-stage stochastic
production planning model with normal demand show that indeed convergence rates
close to the optimal are achieved when using SLR and randomly scrambled Sobol’ point
sets accompanied with PCA for dimension reduction.