研究会 (2019 年 06 月 08 日)

概要

1.Single particle tracking (SPT) is a powerful class of methods for
  studying the dynamics of biomolecules inside living cells. The
  techniques reveal both trajectories of individual particles, with a
  resolution well below the diffraction limit of light, and the
  parameters defining the motion model, such as diffusion coefficients
  and confinement lengths.  Existing algorithms assume these
  parameters are constant throughout an experiment. However, it has
  been demonstrated that they often vary with time as the tracked
  particles move through different regions in the cell or as
  conditions inside the cell change in response to stimuli. In this
  work we apply the method of local Maximum Likelihood (ML) estimation
  to the SPT application combined with change detection. Local ML uses
  a sliding window over the data, estimating the model parameters in
  each window. Once we have found the values for the parameters before
  and after the change, we apply offline change detection to know the
  exact time of the change. Then, we reestimate these parameters and
  show that there is an improvement in the estimation of key
  parameters found in SPT. Preliminary results using simulated data
  with a basic diffusion model with additive Gaussian noise show that
  our proposed algorithm is able to track abrupt changes in the
  parameters as they evolve during a trajectory.

2.Optimization problems in which control variables are penalized via
  the L0 norm have attracted a renewed interest due to its connection to
  sparsity. Since the penalty cost is defined as the length of the
  support of controls, the optimization tends to make the control input
  identically zero on a set with positive measures and the optimal
  control is switched off completely on parts of the time domain. This
  is why the problem is referred to as sparse optimal control. In this
  talk, I introduce some characterizations of the sparse optimal
  control.
    Firstly, we investigate the sparse optimal control for linear systems
  with a final state constraint. In the light of the fact that the Lp
  optimization problems with p ∈ (0, 1] have been seen in the context of
  the sparse optimization, we show the relationship among the Lp
  optimizations with p ∈ [0,1]. A sufficient condition under which the
  sparse optimization problem is exactly solved by the associated L1
  optimization problem is obtained. In addition, an equivalence theorem
  among the Lp optimal solutions with p ∈ [0, 1) is shown.
    Secondary, we investigate the sparse optimal control for nonlinear
  systems with a terminal cost. In contrast to the linear case, I
  directly deal with the underlying non-smooth and non-convex L0 cost
  without the aid of any Lp relaxations and derive the sparse optimal
  feedback map. For this purpose, the dynamic programming approach is
  adopted and the value function is analyzed. Due to the non-smoothness
  of the L0 cost functional, in general, the value function is not
  differentiable in the domain. Then, we characterize the value function
  as a viscosity solution to the associated Hamilton-Jacobi-Bellman
  (HJB) equation. Based on the result, we derive a sufficient and
  necessary condition for the L0 optimality, which immediately gives the
  optimal feedback map. In addition, we show an equivalence with the
  associated L1 optimal control problem.