The sink location problem is a combination of network flow and location problems: From a given set of nodes in a flow network a minimum cost subset \(W\) has to be selected such that given supplies can be transported to the nodes in \(W\). In contrast to its counterpart, the source location problem which has already been studied in the literature, sinks have, in general, a limited capacity. Sink location has a decisive application in evacuation planning, where the supplies correspond to the number of evacuees and the sinks to emergency shelters.
We classify sink location problems according to capacities on shelter nodes, simultaneous or non-simultaneous flows, and single or multiple assignments of evacuee groups to shelters. Resulting combinations are interpreted in the evacuation context and analyzed with respect to their worst-case complexity status.
There are several approaches to tackle these problems: Generic solution methods for uncapacitated problems are based on source location and modifications of the network. In the capacitated case, for which source location cannot be applied, we suggest alternative approaches which work in the original network. It turns out that latter class algorithms are superior to the former ones. This is established in numerical tests including random data as well as real world data from the city of Kaiserslautern, Germany.
Building interoperation among separately developed software units requires checking their conceptual assumptions and constraints. However, eliciting such assumptions and constraints is time consuming and is a challenging task as it requires analyzing each of the interoperating software units. To address this issue we proposed a new conceptual interoperability analysis approach which aims at decreasing the analysis cost and the conceptual mismatches between the interoperating software units. In this report we present the design of a planned controlled experiment for evaluating the effectiveness, efficiency, and acceptance of our proposed conceptual interoperability analysis approach. The design includes the study objectives, research questions, statistical hypotheses, and experimental design. It also provides the materials that will be used in the execution phase of the planned experiment.
A new algorithm for optimization problems with three objective functions is presented which computes a representation for the set of nondominated points. This representation is guaranteed to have a desired coverage error and a bound on the number of iterations needed by the algorithm to meet this coverage error is derived. Since the representation does not necessarily contain nondominated points only, ideas to calculate bounds for the representation error are given. Moreover, the incorporation of domination during the algorithm and other quality measures are discussed.
The ordered weighted averaging objective (OWA) is an aggregate function over multiple optimization criteria which received increasing attention by the research community over the last decade. Different to the ordered weighted sum, weights are attached to ordered objective functions (i.e., a weight for the largest value, a weight for the second-largest value and so on). As this contains max-min or worst-case optimization as a special case, OWA can also be considered as an alternative approach to robust optimization.
For linear programs with OWA objective, compact reformulations exist, which result in extended linear programs. We present new such reformulation models with reduced size. A computational comparison indicates that these formulations improve solution times.
We consider an uncertain traveling salesman problem, where distances between nodes are not known exactly, but may stem from an uncertainty set of possible scenarios. This uncertainty set is given as intervals with an additional bound on the number of distances that may deviate from their expected, nominal value.
A recoverable robust model is proposed, that allows a tour to change a bounded number of edges once a scenario becomes known. As the model contains an exponential number of constraints and variables, an iterative algorithm is proposed, in which tours and scenarios are computed alternately.
While this approach is able to find a provably optimal solution to the robust model, it also needs to solve increasingly complex subproblems. Therefore, we also consider heuristic solution procedures based on local search moves using a heuristic estimate of the actual objective function. In computational experiments, these approaches are compared.
Finally, an alternative recovery model is discussed, where a second-stage recovery tour is not required to visit all nodes of the graph. We show that the previously NP-hard evaluation of a fixed solution now becomes solvable in polynomial time.
We argue that the concepts of resilience in engineering science and robustness in mathematical optimization are strongly related. Using evacuation planning as an example application, we demonstrate optimization techniques to improve solution resilience. These include a direct modelling of the uncertainty for stochastic or robust optimization, as well as taking multiple objective functions into account.
Glioma is a common type of primary brain tumor, with a strongly invasive potential, often exhibiting nonuniform, highly irregular growth. This makes it difficult to assess
the degree of extent of the tumor, hence bringing about a supplementary challenge for the treatment. It is therefore necessary to understand the
migratory behavior of glioma in greater detail.
In this paper we propose a multiscale model for glioma growth and migration. Our model couples the microscale dynamics (reduced to the binding of surface receptors to the
surrounding tissue) with a kinetic transport equation for the cell density on the mesoscopic level of individual cells. On the latter scale we also include the
proliferation of tumor cells via effects of interaction with the tissue. An adequate parabolic scaling yields a convection-diffusion-reaction equation, for which the coefficients
can be explicitly determined from the information about the tissue obtained by diffusion tensor imaging. Numerical simulations relying on DTI measurements confirm the biological
findings that glioma spreads
along white matter tracts.
In this paper we propose a procedure to extend classical numerical schemes for
hyperbolic conservation laws to networks of hyperbolic conservation laws. At the
junctions of the network we solve the given coupling conditions and minimize the
contributions of the outgoing numerical waves. This flexible procedure allows
us to also use central schemes at the junctions. Several numerical examples are
considered to investigate the performance of this new approach compared to the
common Godunov solver and exact solutions.
Starting from the two-scale model for pH-taxis of cancer cells introduced in , we consider here an extension accounting for tumor heterogeneity w.r.t. treatment sensitivity and a treatment approach including chemo- and radiotherapy. The effect of peritumoral region alkalinization on such therapeutic combination is investigated with the aid of numerical simulations.
Variational methods in imaging are nowadays developing towards a quite universal and
tool, allowing for highly successful approaches on tasks like denoising, deblurring, inpainting,
segmentation, super-resolution, disparity, and optical flow estimation. The overall structure of such approaches is of the form
D(Ku) + alpha R(u) to min_u
where the functional D is a data fidelity term also depending on some input data f and
measuring the deviation of Ku from such and R is a regularization functional. Moreover
K is a (often linear) forward operator modeling the dependence of data on an underlying
image, and alpha is a positive regularization parameter. While D is often smooth and (strictly)
convex, the current practice almost exclusively uses nonsmooth regularization functionals.
The majority of successful techniques is using nonsmooth and convex functionals like the total variation and generalizations thereof, cf. [28, 31, 40], or l_1-norms of coeefficients arising
from scalar products with some frame system, cf.  and references therein.
The efficient solution of such variational problems in imaging demands for appropriate algorithms.
Taking into account the specific structure as a sum of two very different terms
to be minimized, splitting algorithms are a quite canonical choice. Consequently this field
has revived the interest in techniques like operator splittings or augmented Lagrangians. In
this chapter we shall provide an overview of methods currently developed and recent results
as well as some computational studies providing a comparison of different methods and also
illustrating their success in applications.
We start with a very general viewpoint in the first sections, discussing basic notations, properties
of proximal maps, firmly non-expansive and averaging operators, which form the basis
of further convergence arguments. Then we proceed to a discussion of several state-of-the
art algorithms and their (theoretical) convergence properties. After a section discussing issues
related to the use of analogous iterative schemes for ill-posed problems, we present some practical convergence studies in numerical examples related to PET and spectral CT reconstruction.