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A new method for noise removal of arbitrary surfaces
meshes is presented which focuses on the preservation
and sharpening of non-linear geometric features such
as curved surface regions and feature lines. Our method
uses a prescribed mean curvature flow (PMC) for simplicial
surfaces which is based on three new contributions:
1. the definition and efficient calculation of a
discrete shape operator and principal curvature properties
on simplicial surfaces that is fully consistent with
the well-known discrete mean curvature formula, 2. an
anisotropic discrete mean curvature vector that combines
the advantages of the mean curvature normal with
the special anisotropic behaviour along feature lines of
a surface, and 3. an anisotropic prescribed mean curvature
flow which converges to surfaces with an estimated
mean curvature distribution and with preserved nonlinear
features. Additionally, the PMC flow prevents
boundary shrinkage at constrained and free boundary
segments.
We study perturbations of a stochastic program with a probabilistic constraint and r-concave original probability distribution. First we improve our earlier results substantially and provide conditions implying Hölder continuity properties of the solution sets w.r.t. the Kolmogorov distance of probability distributions. Secondly, we derive an upper Lipschitz continuity property for solution sets under more restrictive conditions on the original program and on the perturbed probability measures. The latter analysis applies to linear-quadratic models and is based on work by Bonnans and Shapiro. The stability results are illustrated by numerical tests showing the different asymptotic behaviour of parametric and nonparametric estimates in a program with a normal probabilistic constraint.
We consider stochastic programs with risk measures in the objective and study
stability properties as well as decomposition structures. Thereby we place emphasis on dynamic
models, i.e., multistage stochastic programs with multiperiod risk measures. In this context, we
define the class of polyhedral risk measures such that stochastic programs with risk measures taken
from this class have favorable properties. Polyhedral risk measures are defined as optimal values of
certain linear stochastic programs where the arguments of the risk measure appear on the right-hand
side of the dynamic constraints. Dual representations for polyhedral risk measures are derived and
used to deduce criteria for convexity and coherence. As examples of polyhedral risk measures we
propose multiperiod extensions of the Conditional-Value-at-Risk.
We consider multistage stochastic optimization models containing nonconvex constraints, e.g.,
due to logical or integrality requirements. We study three variants of Lagrangian relaxations and of the corresponding
decomposition schemes, namely, scenario, nodal and geographical decomposition. Based on
convex equivalents for the Lagrangian duals, we compare the duality gaps for these decomposition schemes.
The first main result states that scenario decomposition provides a smaller or equal duality gap than nodal
decomposition. The second group of results concerns large stochastic optimization models with loosely coupled
components. The results provide conditions implying relations between the duality gaps of geographical
decomposition and the duality gaps for scenario and nodal decomposition, respectively.
Portfolio and risk management problems of power
utilities may be modeled by multistage stochastic programs. These
models use a set of scenarios and corresponding probabilities
to model the multivariate random data process (electrical load,
stream flows to hydro units, and fuel and electricity prices). For
most practical problems the optimization problem that contains
all possible scenarios is too large. Due to computational complexity
and to time limitations this program is often approximated by
a model involving a (much) smaller number of scenarios. The proposed
reduction algorithms determine a subset of the initial scenario
set and assign new probabilities to the preserved scenarios.
The scenario tree construction algorithms successively reduce the
number of nodes of a fan of individual scenarios by modifying the
tree structure and by bundling similar scenarios. Numerical experience
is reported for constructing scenario trees for the load
and spot market prices entering a stochastic portfolio management
model of a German utility
We present a mixed-integer multistage stochastic programming model for the short term unit commitment of a hydro-thermal power system under uncertainty in load, inflow to reservoirs, and prices for fuel and delivery contracts. The model is implemented for uncertain load and tested on realistic data from a German power utility. Load scenario trees are generated by a procedure consisting of two steps: (i) Simulation of load scenarios using an explicit respresentation of the load distribution and (ii) construction of a tree out of these scenarios. The dimension of the corresponding mixed-integer programs ranges up to 200,000 binary and 350,000 continuous variables. The model is solved by a Lagrangian-based decomposition strategy exploiting the loose coupling structure. Solving the Lagrangian dual by a proximal bundle method leads to a successive decomposition into single unit subproblems, which are solved by specific algorithms. Finally, Lagrangian heuristics are used to construct nearly optimal first stage decisions.
Mathematical models for the electricity portfolio
management of a utility that owns a hydro-thermal generation system
and trades on the power market often lead to complex stochastic
optimization problems. We present a new approach to solving
stochastic hydro-storage subproblems that occur when stochastic
Lagrangian relaxation is applied to solving such models. The special
structure of such hydro-storage subproblems allows the design
of a stochastic network flow algorithm. The algorithm represents
a stochastic extension of a relaxation method, that algorithmically
solves the linear minimum cost flow problem. It is based on the
iterative improvement of dual costs. Numerical experience of the
new algorithm is reported and its performance is compared with
that of standard LP software .
This work is concerned with transparent boundary conditions (TBCs) for systems of Schrödinger type equations, namely the time-dependent kp-Schrödinger equations. These TBCs
have to be constructed for the discrete scheme, in order to maintain stability and to avoid
numerical re
ections. The discrete transparent boundary conditions (DTBCs) are constructed
using the solution of the exterior problem with Laplace and Z-transformation respectively.
Hence we will analyse the numerical error caused by the inverse Z-transformation. Since
these DTBCs are non-local in time and thus very costly, we present approximate DTBCs,
that allow a fast calculation of the boundary terms.
It is known that for each combinatorial type of convex 3-dimensional
polyhedra, there is a representative with edges tangent to the unit sphere.
This representative is unique up to projective transformations that fix the unit
sphere. We show that there is a unique representative (up to congruence) with
edges tangent to the unit sphere such that the origin is the barycenter of the
points where the edges touch the sphere.