Analysis and extraction of strongly frequency modulated signals have been a challenging problem for Adaptive Data Analysis methods; e.g., Empirical Mode Decomposition (EMD) [13]. In fact, many of the Newtonian dynamical systems, including conservative mechanical systems, are sources of signals with low to strong levels of frequency modulation. Analysis of such signals is an important issue in system identification problems. In this paper, we present a novel method to accurately extract Intrawave Signals. This method is a descendant of Sparse Time-Frequency Representation (STFR) methods [8, 7]. We will present numerical examples to show the performance of this new algorithm. Theoretical analysis of convergence of the algorithm is also presented as a support for the method. We will show that the algorithm is stable to noise perturbation, as well.

Based on the recently developed data-driven time-frequency analysis [16], we propose a two-level method to look for the sparse time-frequency decomposition of multiscale data. In the two-level method, we rst run a local algorithm to get a good approximation of the instantaneous frequency. We then pass this instantaneous frequency to the global algorithm to get an accurate global Intrinsic Mode Function (IMF) and instantaneous frequency. The two-level method alleviates the diculty of the mode mixing to some extent. We also present a method to reduce the end eects.

We propose and study a new parallel one-shot Lagrange--Newton--Krylov--Schwarz (LNKSz) algorithm for shape optimization problems constrained by steady incompressible Navier--Stokes equations discretized by finite element methods on unstructured moving meshes. Most existing algorithms for shape optimization problems iteratively solve the three components of the optimality system: the state equations for the constraints, the adjoint equations for the Lagrange multipliers, and the design equations for the shape parameters. Such approaches are relatively easy to implement, but generally are not easy to converge as they are basically nonlinear Gauss--Seidel algorithms with three large blocks. In this paper, we introduce a fully coupled, or the so-called one-shot, approach which solves the three components simultaneously. First, we introduce a moving mesh finite element method for the shape optimization

In this paper, we present entropy satisfying schemes for solving an integro-differential equation that describes the evolution of a population structured with respect to a continuous trait. In [P.-E. Jabin and G. Raoul, J. Math. Biol., 63 (2011), pp. 493–517] solutions are shown to converge toward the so-called evolutionary stable distribution (ESD) as time becomes large, using the relative entropy. At the discrete level, the ESD is shown to be the solution to a quadratic programming problem and can be computed by any well-established nonlinear programing algorithm. The schemes are then shown to satisfy the entropy dissipation inequality on the set where initial data are positive and the numerical solutions tend toward the discrete ESD in time. An alternative algorithm is presented to capture the global ESD for nonnegative initial data, which is made possible due to the mutation mechanism built into the modified scheme. A series of numerical tests are given to confirm both accuracy and the entropy satisfying property and to underline the efficiency of capturing the large time asymptotic behavior of numerical solutions in various settings.

In this paper, we develop high-order bound-preserving (BP) discontinuous Galerkin (DG) methods for the coupled system of compressible miscible displacements on triangular meshes. We consider the problem with multi-component fluid mixture and the (volumetric) concentration of the $j$th component, $c_j$, should be between 0 and 1. There are three main difficulties. Firstly, $c_j$ does not satisfy a maximum-principle. Therefore, the numerical techniques introduced in (X. Zhang and C.-W. Shu, Journal of Computational Physics, 229 (2010), 3091-3120) cannot be applied directly. The main idea is to apply the positivity-preserving techniques to all $c_j's$ and enforce $\sum_jc_j=1$ simultaneously to obtain physically relevant approximations. By doing so, we have to treat the time derivative of the pressure $dp/dt$ as a source in the concentration equation and choose suitable fluxes in the pressure and concentration equations. Secondly, it is not easy to construct first-order numerical fluxes for interior penalty DG methods on triangular meshes. One of the key points in the high-order BP technique applied in this paper is the combination of high-order and lower-order numerical fluxes. We will construct second-order BP schemes and use the second-order numerical fluxes as the lower-order one. Finally, the classical slope limiter cannot be applied to $c_j$. To construct the BP technique, we will not approximate $c_j$ directly. Therefore, a new limiter will be introduced. Numerical experiments will be given to demonstrate the high-order accuracy and good performance of the numerical technique.