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In this paper, we introduce a new adaptive data analysis method to study trend and instantaneous frequency of nonlinear and nonstationary data. This method is inspired by the Empirical Mode Decomposition method (EMD) and the recently developed compressed (compressive) sensing theory. The main idea is to look for the sparsest representation of multiscale data within the largest possible dictionary consisting of intrinsic mode functions of the form {a(t) cos(θ(t))}, where a ∈ V (θ), V (θ) consists of the functions smoother than cos(θ(t)) and θ   0. This problem can be formulated as a nonlinear l0 optimization problem. In order to solve this optimization problem, we propose a nonlinear matching pursuit method by generalizing the classical matching pursuit for the l0 optimization problem. One important advantage of this nonlinear matching pursuit method is it can be implemented very efficiently and is very stable to noise. Further, we provide an error analysis of our nonlinear matching pursuit method under certain scale separation assumptions. Extensive numerical examples will be given to demonstrate the robustness of our method and comparison will be made with the state-of-the-art methods. We also apply our method to study data without scale separation, and data with incomplete or under-sampled data.

A popular and classical method for finding the best rank one approximation of a real tensor is the higher order power method (HOPM). It is known in the literature that the iterative sequence generated by HOPM converges globally, while the convergence rate can be superlinear, linear or sublinear. In this paper, we examine the convergence rate of HOPM in solving the best rank one approximation problem of real tensors. We first show that the iterative sequence of HOPM always converges globally and provide an explicit eventual sublinear convergence rate. The sublinear convergence rate estimate is in terms of the dimension and the order of the underlying tensor space. Then, we examine the concept of nondegenerate singular vector tuples and show that, if the sequence of HOPM converges to a nondegenerate singular vector tuple, then the global convergence rate is R-linear. We show that, for almost all tensors (in the sense of Lebesgue measure), all the singular vector tuples are nondegenerate, and so, the HOPM “typically” exhibits global R-linear convergence rate. Moreover, without any regularity assumption, we establish that the sequence generated by HOPM always converges globally and R-linearly for orthogonally decomposable tensors with order at least 3. We achieved this by showing that each nonzero singular vector tuple of an orthogonally decomposable tensor with order at least 3 is nondegenerate.

The main purpose of this paper is to analyze the stability and error estimates of the local discontinuous Galerkin (LDG) methods coupled with implicit-explicit (IMEX) time discretization schemes, for solving multi-dimensional convection-diffusion equations with nonlinear convection. By establishing the important relationship between the gradient and the interface jump of the numerical solution with the independent numerical solution of the gradient in the LDG method, on both rectangular and triangular elements, we can obtain the same stability results as in the one-dimensional case, i.e, the IMEX LDG schemes are unconditionally stable for the multi-dimensional convection-diffusion problems, in the sense that the time-step $\dt$ is only required to be upper-bounded by a positive constant independent of the spatial mesh size $h$. Furthermore, by the aid of the so-called elliptic projection and the adjoint argument, we can also obtain optimal error estimates in both space and time, for the corresponding fully discrete IMEX LDG schemes, under the same condition, i.e, if piecewise polynomial of degree $k$ is adopted on either rectangular or triangular meshes, we can show the convergence accuracy is of order $\Ocal(h^{k+1}+\dt^s)$ for the $s$-th order IMEX LDG scheme ($s=1,2,3$) under consideration. Numerical experiments are also given to verify our main results.

Gibbs phenomenon is the particular manner how a global spectral approximation of a piecewise analytic function behaves at the jump discontinuity. The truncated spectral series has large oscillations near the jump, and the overshoot does not decay as the number of terms in the truncated series increases. There is therefore no convergence in the maximum norm, and convergence in smooth regions away from the discontinuity is also slow. In earlier work, a methodology is proposed to completely overcome this difficulty in the context of spectral collocation methods, resulting in the recovery of exponential accuracy from collocation point values of a piecewise analytic function. In this paper, we extend this methodology to handle spectral collocation methods for functions which are analytic in the open interval but have singularities at end-points. With this extension, we are able to obtain exponential accuracy from collocation point values of such functions. Similar to earlier work, the proof is constructive and uses the Gegenbauer polynomials $C^\lambda_n(x)$. The result implies that the Gibbs phenomenon can be overcome for smooth functions with endpoint singularities.