We investigate the inverse scale space flow as a decomposition method for decomposing data into generalised singular vectors. We show that the inverse scale space flow, based on convex and absolutely one-homogeneous regularisation functionals, can decompose data represented by the application of a forward operator to a linear combination of generalised singular vectors into its individual singular vectors. We verify that for this decomposition to hold true, two additional conditions on the singular vectors are sufficient: orthogonality in the data space and inclusion of partial sums of the subgradients of the singular vectors in the subdifferential of the regularisation functional at zero.We also address the converse question of when the inverse scale space flow returns a generalised singular vector given that the initial data is arbitrary (and therefore not necessarily in the range of the forward operator). We prove that the inverse scale space flow is guaranteed to return a singular vector if the data satisfies a novel dual singular vector condition.We conclude the paper with numerical results that validate the theoretical results and that demonstrate the importance of the additional conditions required to guarantee the decomposition result.
In research and practice, public transportation planning is executed in a series of steps, which are often divided into the strategic, the tactical, and the operational planning phase. Timetables are normally designed in the tactical phase, taking into account a given line plan, safety restrictions arising from infrastructural constraints, as well as regularity requirements and bounds on transfer times. In this paper, however, we propose a timetabling approach that is aimed at decision making in the strategic phase of public transportation planning and to determine an outline of a timetable that is good from the passengers' perspective. Instead of including explicit synchronization constraints between train runs (as most timetabling models do), we include the adaption time (waiting time at the origin station) in the objective function to ensure regular connections between passengers' origins and destinations. We model the problem as a mixed integer quadratic program and linearise it. Furthermore we propose a heuristic to generate starting solutions. We illustrate the type of solutions found by our approach on two case studies based on the Dutch railway network and analyse trade-offs that are made to balance dwell times and regularity of trains.
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