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Network flows over time form a fascinating area of research. They model the temporal
dynamics of network flow problems occurring in a wide variety of
applications. Research in this area has been pursued in two different and mainly independent
directions with respect to time modeling: discrete and continuous time models.
In this paper we deploy measure theory in order to introduce a general model of network flows over time combining both discrete and continuous aspects into a single model. Here, the flow on each arc is modeled as a Borel measure on the real line (time axis) which assigns to each
suitable subset a real value, interpreted as the amount of
flow entering the arc over the subset. We focus on the maximum flow problem formulated in a network where capacities on arcs are also given as Borel measures and storage might be allowed at the nodes of the network. We generalize the concept of cuts to the case of these Borel Flows and extend the famous MaxFlow-MinCut Theorem.
We study a diffusion model of phase field type, consisting of
a system of two partial differential equations encoding the balances
of microforces and microenergy; the two unknowns are the order parameter
and the chemical potential. By a careful development of uniform estimates and
the deduction of certain useful boundedness properties, we
prove existence and uniqueness of a global-in-time
smooth solution to the associated initial/boundary-value problem;
moreover, we give a description of the relative $\omega$-limit set.
Dynamic network flow problems model the temporal evolution of flows over time and also consider changes of network parameters such as capacities, costs, supplies, and demands over time. These problems have been extensively studied in the past because of their important role in real world applications such as transport, traffic, and logistics. This has led to many results, but the more challenging continuous time model still lacks some of the key features such as network related optimality conditions and algorithms that are available in the static case.
The aim of this paper is to advance the state of the art for dynamic network flows by developing the continuous time analogues of several well-known optimality conditions for static network flows. Specifically, we establish a reduced cost optimality condition, a negative cycle optimality condition, and a strong duality result for a very general class of dynamic
network flows. The underlying idea is to construct a dual feasible solution that proves optimality when the residual network (with respect to a given flow) contains no dynamic cycles with negative cost. We also discuss a generic negative cycle-canceling algorithm resulting from the corresponding optimality criterion and point out promising directions for future research.
This paper concerns the shortest path problem for a network in which arc costs can vary with
time, each arc has a transit time, parking with a corresponding
time-varying cost is allowed at the nodes, and time is modeled as a continuum. The resulting problem is called the {\em continuous-time dynamic shortest path problem}, which is well studied in the literature. However, the problem appears as a subproblem when one wishes to test, via an algorithm for dynamic shortest paths, the presence of negative cycles in the residual network in order to develop continuous-time analogues of several well-known optimality conditions for continuous-time dynamic network flow problems. But, in general, the residual network contains arcs with negative transit times and hence the results in the literature are useless for these purposes since all results are based on the assumption of positive transit times.
In this paper, we relax this condition to allow negative transit times. We study a corresponding linear program in space of measures and prove the existence of an optimal extreme point solution. Moreover, we define a dual problem and establish a strong duality result that shows under certain assumptions that the value of the linear program equals the value of the dual problem and both values are attained. We also present counterexamples to show that strong duality only holds under these assumptions.
We consider the dynamic shortest path problem in the continuous-time model because of its importance. This problem has been extensively studied in the literature. But so far, all contributions to this problem are based on the assumption that all transit times are strictly positive. However, in order to study dynamic network flows it is essential to support negative transit times since they occur quite naturally in residual networks.
In this paper we extend the work of Philpott [SIAM Control Opt.,~1994, pp.~538--552] to the case of arbitrary (also negative and irrational) transit times. We study a corresponding linear program in a space of measures and give a full characterization of its extreme points. In particular, we show a one-to-one correspondence between extreme points and dynamic paths.
Research on flows over time has been conducted mainly in two separate and mainly independent approaches, namely \emph{discrete} and \emph{continuous} models, depending on whether a discrete or continuous representation of time is used. Recently, Borel flows have been introduced to build a bridge between these two models.
In this paper, we consider the maximum Borel flow problem formulated in a network where capacities on arcs are given as Borel measures and storage might be allowed at the nodes of the network. This problem is formulated as a linear program in a space of measures. We define a dual problem and prove a strong duality result. We show that strong duality is closely related to a MaxFlow-MinCut Theorem.
We investigate optimal elliptic
regularity (within the scale of Sobolev spaces) of anisotropic
div--grad operators in three dimensions at a multi-material vertex on
the Neumann boundary part of a polyhedral spatial domain. The
gradient of a solution to the corresponding elliptic PDE (in a
neighbourhood of the vertex) is integrable to an index greater than
three.
One of the most challenging problems in dynamic concurrent multiscale
simulations is the reflectionless transfer of physical quantities between the
different scales. In particular, when coupling molecular dynamics and finite
element discretizations in solid body mechanics, often spurious wave reflections
are introduced by the applied coupling technique. The reflected waves are
typically of high frequency and are arguably of little importance in the domain
where the finite element discretization drives the simulation.
In this work, we provide an analysis of this phenomenon.
Based on the gained
insight, we derive a new coupling approach, which neatly separates high and low
frequency waves. Whereas low frequency waves are permitted to
bridge the scales, high frequency waves can be removed by applying damping techniques without affecting the coupled share of the solution. As a consequence, our new method almost completely eliminates unphysical wave reflections and deals in a consistent way with waves of arbitrary frequencies. The separation of
wavelengths is achieved by employing a discrete $L^2$-projection, which acts as a
low pass filter. Our coupling constraints enforce matching in the range of this projection. With respect to the numerical realization this approach
has the advantage of a small number of constraints, which is computationally
efficient. Numerical results in one and two dimensions confirm our theoretical
findings and illustrate the performance of our new weak coupling approach.
We investigate a control problem for the heat equation. The goal is to find an optimal heat transfer coefficient in the Robin boundary condition such that a desired temperature distribution at the boundary is adhered. To this end we consider a function space setting in which the heat flux across the boundary is forced to be an Lp function with respect to the surface measure, which in turn implies higher regularity for the time derivative of temperature. We show that the corresponding elliptic operator generates a strongly continuous semigroup of contractions and apply the concept of maximal parabolic regularity. This allows to show the existence of an optimal control and the derivation of necessary and sufficient optimality conditions.
We consider an inverse problem arising in laser-induced thermotherapy, a minimally invasive method for cancer treatment, in which cancer tissue is destroyed by coagulation. For the dosage planning numerical simulation plays an important role. To this end a crucial problem is to identify the thermal growth kinetics of the coagulated zone. Mathematically, this problem is a nonlinear and nonlocal parabolic heat source inverse problem. The solution to this inverse problem is defined as the minimizer of a non-convex cost functional. The existence of the minimizer is proven. We derive the Gateaux derivative of the cost functional, which is based on the adjoint system, and use it for a numerical approximation of the optimal coefficient.
In this paper, mean curvature type equations with general potentials and contact angle boundary conditions are considered. We extend the ideas of Ural'tseva, formulating sharper hypotheses for the existence of a classical solution.
Corner stone for these results is a method to estimate quantities on the boundary of the free surface. We moreover provide alternative proofs for the higher-order estimates, and for the existence result.
We consider the mechanical coupling of a geometrically exact Cosserat rod to a linear elastic continuum. The coupling conditions are formulated in the nonlinear rod configuration space. We describe a Dirichlet--Neumann algorithm for the coupled system, and use it to simulate the static stresses in a human knee joint, where the Cosserat rods are models for the ligaments.
Systems of rail-mounted vehicles play a key role in many logistics applications, and the efficiency of their operation frequently has a significant impact on the overall performance of the surrounding production environment. In theory, assigning transport requests to the vehicles of such systems and scheduling their execution amounts to finding k tours on a common line, where tours may never cross each other in time--dynamic collision constraints need to be respected. The goal is to minimize the makespan for a given set of transport requests.
We establish a model capturing the core challenges in transport planning problems of this type and relate it to other models in literature. After proving NP-hardness for a basic version of the problem, the large part of the paper is dedicated to devising various fast heuristic algorithms suitable for practice. We present computational results regarding the performance of the algorithms proposed for several classes of problem instances.
In this paper we study the distributed optimal control for the Cahn-Hilliard system. A general class of free energy potentials is allowed which, in particular, includes the double-obstacle potential. The latter potential yields an optimal control problem of a parabolic variational inequality which is of fourth-order in space. We show the existence of optimal controls to approximating problems where the potential is replaced by a mollified version of its Moreau-Yosida approximation. Corresponding first order optimality conditions for the mollified problems are given. For this purpose a new result on the continuous Fr\'echet differentiability of superposition operators with values in Sobolev spaces is established. Besides the convergence of optimal controls of the mollified problems to an optimal control of the original problem, we also derive first order optimality conditions for the original problem by a limit process. The newly derived stationarity system corresponds to a function space version of C-stationarity.
We discuss the numerical solution of large scale nonlinear eigenvalue problems and frequency
response problems that arise in the analysis, simulation and optimization of acoustic fields.
We report about the cooperation with the company SFE in Berlin. We present the challenges
in the current industrial problems and the state-of-the-art of current methods. The difficulties
that arise with current off-the-shelf methods are discussed and several industrial examples are presented. It is documented that industrial cooperation is by no means a one-way street
of transfer from academia to industry but the challenges arising in industrial practice also lead to new mathematical questions which actually change the mathematical theory and methods.
In optimal control problems with nonlinear time-dependent 3D PDEs, full 4D discretizations are usually prohibitive due to the storage requirement. For this reason gradient and Newton type methods working on the reduced functional are often employed. The computation of the reduced gradient requires one solve of the state equation forward in time, and one backward solve of the adjoint equation. The state enters into the adjoint equation, again requiring the storage of a full 4D data set. We propose a lossy compression algorithm using an inexact but cheap predictor for the state data, with additional entropy coding of prediction errors. As the data is used inside a discretized, iterative algorithm, lossy compression
maintaining a certain error bound turns out to be sufficient.
The ability of velocity methods to describe changes
of topology by creating defects like holes is investigated. For the
shape optimization energy-type objective functions are considered,
which depend on the geometry by means of state variables. The
state system is represented by abstract, quadratic, constrained
minimization problems stated over domains with defects. The velocity
method provides the shape derivative of the objective function
due to finite variations of a defect. Sufficient conditions are
derived which allow us to pass the shape derivative to the limit
with respect to diminishing defect, thus, to obtain the topological
derivative" of the objective function due to a topology change.
An illustrative example is presented for a circular hole bored at
the tip of a crack
This paper deals with error estimates for space-time
discretizations of 3D model for isothermal stress-induced
transformations in shape-memory materials. After recalling
existence and uniqueness results, a fully-discrete approximation is presented and an explicit space-time
convergence rate of order is derived.
This note addresses a three-dimensional model for isothermal stress-induced transformation in shape-memory polycrystalline materials. We treat the problem within the framework of the energetic formulation of rate-independent processes and investigate existence and continuous dependence issues at both the constitutive relation and quasi-static evolution level. Moreover, we focus on time and space approximation as well as on regularization and parameter asymptotics.