In this paper, we establish a connection between the recently developed data-driven timefrequency analysis [10, 11] and the classical second order differential equations. The main idea of the data-driven time-frequency analysis is to decompose a multiscale signal into the sparsest collection of Intrinsic Mode Functions (IMFs) over the largest possible dictionary via nonlinear optimization. These IMFs are of the form a(t) cos(θ(t)) where the amplitude a(t) is positive and slowly varying. The non-decreasing phase function θ(t) is determined by the data and in general depends on the signal in a nonlinear fashion. One of the main results of this paper is that we show that each IMF can be associated with a solution of a second order ordinary differential equation of the form x¨ + p(x, t)x˙ + q(x, t) = 0. Further, we propose a localized variational formulation for this problem and develop an effective l1-based optimization method to recover p(x, t) and q(x, t) by looking for a sparse representation of p and q in terms of the polynomial basis. Depending on the form of nonlinearity in p(x, t) and q(x, t), we can define the order of nonlinearity for the associated IMF. This generalizes a concept recently introduced by Prof. N. E. Huang et al. [15]. Numerical examples will be provided to illustrate the robustness and stability of the proposed method for data with or without noise. This manuscript should be considered as a proof of concept.

The problem is concerned about computing virtual coordinates for greedy routing in a wireless ad hoc network. Consider a set of wireless nodes S densely deployed inside a geometric domain R2. Nodes within communication range can directly communicate with each other. We ask whether one can compute a set of virtual coordinates for S such that greedy routing has guaranteed delivery. In particular, each node forwards the message to the neighbor whose distance to the destination, computed under the virtual coordinates and some metric function d, is the smallest. If such a neighbor can always be found, greedy routing successfully delivers the message to the destination. The problem can be phrased as finding a greedy embedding of S in some geometric space, such that greedy routing always succeeds. In the setting of this entry, we assume that the nodes are a dense sample of the domain such that

The goal of smooth function reconstruction on a 2D or 3D manifold is to obtain a smooth function on surfaces or higher dimensional manifolds. It is a common problem in computer graphics and computational mathematics, especially in civil engineering including structural analysis of solid objects. In this chapter, we introduce a new method using harmonic functions for solving this problem. This method contains the following steps: (1) Partition the boundary surfaces of the 3D manifold based on sample points so that each sample point is on the edge of the partition. (2) Use gradually varied interpolation on the edges so that each point on the edge will be assigned a value. In addition, all values on the edge are gradually varied. (3) Use discrete harmonic functions to fit the unknown points, i.e. the points inside each partition patch. This solution of the fitting becomes the piecewise harmonic function.

Recent technical advances lead to the coupling of PET and MRI scanners, enabling one to acquire functional and anatomical data simultaneously. In this paper, we propose a tight frame based PET-MRI joint reconstruction model via the joint sparsity of tight frame coefficients. In addition, a nonconvex balanced approach is adopted to take the different regularities of PET and MRI images into account. To solve the nonconvex and nonsmooth model, a proximal alternating minimization algorithm is proposed, and the global convergence is present based on the Kurdyka--Łojasiewicz property. Finally, the numerical experiments show that our proposed models achieve better performance over the existing PET-MRI joint reconstruction models.

We begin a systematical study on the effect of numerical viscosities. In this paper we investigate the behavior of shock-capturing methods for slowly moving shocks. It is known that for slowly moving shocks even a first-order scheme, such as the Godunov or Roe type methods, will generate downstream oscillatory wave patterns that cannot be effectively damped by the dissipation of these first-order schemes. The purpose of this paper is to understand the formation and behavior of these downstream patterns. Our study shows that the downstream errors are generated by the unsteady nature of the viscous shock profiles and behave diffusively. The scenario is as follows. When solving the compressible Euler equations by shock capturing methods, the smeared density profile introduces a momentum spike at the shock location if the shock moves slowly. Downstream waves will necessarily emerge in order to balance the momentum mass carried by the spike for the momentum conservation. Although each family of waves decays inl∞andl2while they preserve the same mass, the perturbing nature of the viscous or spike profile is a constant source for the generation of new downstream waves, causing spurious solutions foralltime. Higher order TVD or ENO type interpolations accentuate this problem.