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.

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.

In this paper, we study the problem of computing the effective diffusivity for a particle moving in chaotic and stochastic flows. In addition, we numerically investigate the residual diffusion phenomenon in chaotic advection. The residual diffusion refers to the nonzero effective (homogenized) diffusion in the limit of zero molecular diffusion as a result of chaotic mixing of the streamlines. In this limit, traditional numerical methods typically fail since the solutions of the advection-diffusion equations develop sharp gradients. Instead of solving the Fokker--Planck equation in the Eulerian formulation, we compute the motion of particles in the Lagrangian formulation, which is modeled by stochastic differential equations (SDEs). We propose an effective numerical integrator based on a splitting method to solve the corresponding SDEs in which the deterministic subproblem is symplectic preserving while the random subproblem

Quantitative susceptibility mapping (QSM) uses the phase data in magnetic resonance signals to visualize a three-dimensional susceptibility distribution by solving the magnetic field to susceptibility inverse problem. Due to the presence of zeros of the integration kernel in the frequency domain, QSM is an ill-posed inverse problem. Although numerous regularization-based models have been proposed to overcome this problem, incompatibility in the field data, which leads to deterioration of the recovery, has not received enough attention. In this paper, we show that the data acquisition process of QSM inherently generates a harmonic incompatibility in the measured local field. Based on this discovery, we propose a novel regularization-based susceptibility reconstruction model with an additional sparsity-based regularization term on the harmonic incompatibility. Numerical experiments show that the proposed method achieves better performance than existing approaches.

We develop Random Batch Methods for interacting particle systems with large number of particles. These methods use small but random batches for particle interactions, thus the computational cost is reduced from $O(N^2)$ per time step to $O(N)$, for a system with $N$ particles with binary interactions. On one hand, these methods are efficient Asymptotic-Preserving schemes for the underlying particle systems, allowing $N$-independent time steps and also capture, in the $N \to \infty$ limit, the solution of the mean field limit which are nonlinear Fokker-Planck equations; on the other hand, the stochastic processes generated by the algorithms can also be regarded as new models for the underlying problems. For one of the methods, we give a particle number independent error estimate under some special interactions. Then, we apply these methods to some representative problems in mathematics, physics, social and data sciences, including the Dyson Brownian motion from random matrix theory, Thomson's problem, distribution of wealth, opinion dynamics and clustering. Numerical results show that the methods can capture both the transient solutions and the global equilibrium in these problems.