We study front speeds of curvature and strain G-equations arising in turbulent combustion. These G-equations are HamiltonJacobi type level set partial differential equations (PDEs) with non-coercive Hamiltonians and degenerate nonlinear second order diffusion. The Hamiltonian of a strain G-equation is also non-convex. Numerical computation is performed based on monotone discretization and weighted essentially nonoscillatory (WENO) approximation of transformed G-equations on a fixed periodic domain. The advection field in the computation is a two dimensional Hamiltonian flow consisting of a periodic array of counter-rotating vortices, or cellular flows. Depending on whether the evolution is predominantly in the hyperbolic or parabolic regimes, suitable explicit and semi-implicit time stepping methods are chosen. The turbulent flame speeds are computed as the linear growth rates of large time solutions. A

We study homogenization of the G-equation with a flow straining term (or the strain G-equation) in two dimensional periodic cellular flow. The strain G-equation is a highly non-coercive and non-convex level set HamiltonJacobi equation. The main objective is to investigate how the flow induced straining (the nonconvex term) influences front propagation as the flow intensity <i>A</i> increases. Three distinct regimes are identified. When <i>A</i> is below the critical level, homogenization holds and the turbulent flame speed <i>s</i> _T (effective Hamiltonian) is well-defined for any periodic flow with small divergence and is enhanced by the cellular flow as <i>s</i> _T <i>O</i>(<i>A</i>/log <i>A</i>). In the second regime where <i>A</i> is slightly above the critical value, homogenization breaks down, and <i>s</i> _T is not well-defined along any direction. Solutions become a mixture of a fast moving part and a stagnant part. When <i>A</i> is

In this paper, we prove that the celebrated Arnold--Beltrami--Childress (ABC) flow with parameters A=B=C=1, given by $ \dot x =\sin z+\cos y,\ \dot y = \sin x+\cos z,\ \dot z =\sin y + \cos x $, has periodic orbits on (2\pi\mathbb T)^3 with rotation vectors parallel to (1,0,0), (0,1,0), and (0,0,1). Despite ABC flows being studied since the 1960s, this seems to be the first time that existence of nonperturbative periodic orbits has been established for them. The main difficulty here is the lack of a variational structure for these flows, and our proof instead relies on their symmetry properties. As an application of our result, we show that the well-known G-equation model of turbulent combustion with this ABC flow on (2\pi\mathbb T)^3 has a linear (i.e., maximal possible) flame speed enhancement rate as the flow amplitude grows to infinity. To the best of our knowledge, this is the first time an asymptotic flame speed growth law has been established for a

We study the residual diffusion phenomenon in chaotic advection computationally via adaptive orthogonal basis. The chaotic advection is generated by a class of time periodic cellular flows arising in modeling transition to turbulence in Rayleigh-Bnard experiments. The residual diffusion refers to the non-zero effective (homogenized) diffusion in the limit of zero molecular diffusion as a result of chaotic mixing of the streamlines. In this limit, the solutions of the advection-diffusion equation develop sharp gradients, and demand a large number of Fourier modes to resolve, rendering computation expensive. We construct adaptive orthogonal basis (training) with built-in sharp gradient structures from fully resolved spectral solutions at few sampled molecular diffusivities. This is done by taking snapshots of solutions in time, and performing singular value decomposition of the matrix consisting of these

Given a set of mixtures, blind source separation attempts to retrieve the source signals without or with very little information of the mixing process. We present a geometric approach for blind separation of nonnegative linear mixtures termed <i>facet component analysis</i>. The approach is based on facet identification of the underlying cone structure of the data. Earlier works focus on recovering the cone by locating its vertices (vertex component analysis) based on a mutual sparsity condition which requires each source signal to possess a stand-alone peak in its spectrum. We formulate alternative conditions so that enough data points fall on the facets of a cone instead of accumulating around the vertices. To find a regime of unique solvability, we make use of both geometric and density properties of the data points and develop an efficient facet identification method by combining data classification and linear