The method of using periodic approximations to compute the spectral decomposition of the Koopman operator is generalized to the class of measure-preserving flows on compact metric spaces. It is shown that the spectral decomposition of the continuous one-parameter unitary group can be approximated from an intermediate time discretization of the flow. A sufficient condition is established between the time-discretization of the flow and the spatial discretization of the periodic approximation, so that weak convergence of spectra will occur in the limit. This condition effectively translates to the requirement that the spatial refinements must occur at a faster pace than the temporal refinements. This result is contrasted with the well-known CLF condition of finite difference schemes for advection equations. Numerical results of spectral computations are shown for some benchmark examples of volume-preserving flows.
a b s t r a c tThis paper presents a sparse collocation method for solving the time-dependent Hamilton-Jacobi-Bellman (HJB) equation associated with the continuous-time optimal control problem on a fixed, finite timehorizon with integral cost functional. Through casting the problem in a recursive framework using the value-iteration procedure, the value functions of every iteration step is approximated with a time-varying multivariate simplex B-spline on a certain state domain of interest. In the collocation scheme, the timedependent coefficients of the spline function are further approximated with ordinary univariate B-splines to yield a discretization for the value function fully in terms of piece-wise polynomials. The B-spline coefficients are determined by solving a sequence of highly sparse quadratic programming problems. The proposed algorithm is demonstrated on a pair of benchmark example problems. Simulation results indicate that the method can yield increasingly more accurate approximations of the value function by refinement of the triangulation.
An algorithm for the detection and delineation of breaths is described. The proposed algorithm takes into account the different, common modes of ventilation like the pressure controlled, volume controlled and patient triggered modes of ventilation. Airway flow curve is used as the basic delineator and the airway pressure and the Co2 concentration curves are used to confirm the delineation. A flow chart is also included to explain the algorithm. The detailed explanation and modifications, for additional confirmation and for the selections of constants, to check for the rise or fall of the pressure and Co2 curves, are also included.
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