We propose to incorporate a weighted difference of anisotropic and isotropic total variation (TV) norms into a relaxed formulation of the two phase Mumford-Shah (MS) model for image segmentation. We show results exceeding those obtained by the MS model when using the standard TV norm to regularize partition boundaries. In particular, examples illustrating the qualitative differences between the proposed model and the standard MS one are shown. A fast numerical method is introduced to minimize the proposed model utilizing the difference-of-convex algorithm (DCA) and the primal dual hybrid gradient (PDHG) method.

Variational principle of Kolmogorov-Petrovsky-Piskunov (KPP) minimal front speeds provides a fast and accurate way for speed calculations. A variational principle based computation is carried out on a large ensemble of KPP random speeds through spatial, mean zero, stationary, Gaussian random shear flows inside two dimensional channel domains. In the regime of small root mean square (rms) shear amplitude, the enhancement of the ensemble averaged KPP front speed obeys the quadratic law. In the large rms amplitude regime, the enhancement follows the linear law. An asymptotic ensemble averaged speed formula is derived and agrees well with the numerics. Related theoretical results are presented with a brief outline of the ideas in the proofs. The ensemble averaged speed is found to increase sublinearly with enlarging channel widths, while the speed variance decreases. Direct simulations in the

An orthogonal discrete auditory transform (ODAT) from sound signal to spectrum is constructed by combining the auditory spreading matrix of Schroeder et. al. and the time one map of a discrete nonlocal Schrodinger equation. Thanks to the dispersive smoothing property of the Schrodinger evolution, ODAT spectrum is a smoother than that of the discrete Fourier transform (DFT) consistent with human audition. ODAT and DFT are compared in signal denoising tests with spectral thresholding method. The signals are noisy speech segments. ODAT outperforms DFT in signal to noise ratio (SNR) when the noise level is relatively high.

Differential optical absorption spectroscopy (DOAS) is a powerful tool for detecting and quantifying trace gases in atmospheric chemistry\cite {Platt_Stutz08}. DOAS spectra consist of a linear combination of complex multi-peak multi-scale structures. Most DOAS analysis routines in use today are based on least squares techniques, for example, the approach developed in the 1970s uses polynomial fits to remove a slowly varying background, and known reference spectra to retrieve the identity and concentrations of reference gases. An open problem is to identify unknown gases in the fitting residuals for complex atmospheric mixtures.

We present a novel modification to the well-known infomax algorithm of blind source separation. Under natural gradient descent, the infomax algorithm converges to a stationary point of a limiting ordinary differential equation. However, due to the presence of saddle points or local minima of the corresponding likelihood function, the algorithm may be trapped around these bad stationary points for a long time, especially if the initial data are near them. To speed up convergence, we propose to add a sequence of random perturbations to the infomax algorithm to shake the iterating sequence so that it is captured by a path descending to a more stable stationary point. We analyze the convergence of the randomly perturbed algorithm, and illustrate its fast convergence through numerical examples on blind demixing of stochastic signals. The examples have analytical structures so that saddle points or local minima of