779 research outputs found
Minimizing movement: Fixed-parameter tractability
We study an extensive class of movement minimization problems that arise from many practical scenarios but so far have little theoretical study. In general, these problems involve planning the coordinated motion of a collection of agents (representing robots, people, map labels, network messages, etc.) to achieve a global property in the network while minimizing the maximum or average movement (expended energy). The only previous theoretical results about this class of problems are about approximation and are mainly negative: many movement problems of interest have polynomial inapproximability. Given that the number of mobile agents is typically much smaller than the complexity of the environment, we turn to fixed-parameter tractability. We characterize the boundary between tractable and intractable movement problems in a very general setup: it turns out the complexity of the problem fundamentally depends on the treewidth of the minimal configurations. Thus, the complexity of a particular problem can be determined by answering a purely combinatorial question. Using our general tools, we determine the complexity of several concrete problems and fortunately show that many movement problems of interest can be solved efficiently.</jats:p
Multiscale analysis of singularly perturbed finite dimensional gradient flows: the minimizing movement approach
We perform a convergence analysis of a discrete-in-time minimization scheme
approximating a finite dimensional singularly perturbed gradient flow. We allow
for different scalings between the viscosity parameter and the
time scale . When the ratio diverges, we
rigorously prove the convergence of this scheme to a (discontinuous) Balanced
Viscosity solution of the quasistatic evolution problem obtained as formal
limit, when , of the gradient flow. We also characterize the
limit evolution corresponding to an asymptotically finite ratio between the
scales, which is of a different kind. In this case, a discrete interfacial
energy is optimized at jump times
Motion of three-dimensional elastic films by anisotropic surface diffusion with curvature regularization
Short time existence for a surface diffusion evolution equation with
curvature regularization is proved in the context of epitaxially strained
three-dimensional films. This is achieved by implementing a minimizing movement
scheme, which is hinged on the -gradient flow structure underpinning
the evolution law. Long-time behavior and Liapunov stability in the case of
initial data close to a flat configuration are also addressed.Comment: 44 page
- …