We present a hybrid asymptotic/numerical method for the accurate computation of single- and double-layer heat potentials in two dimensions. It has been shown in previous work that simple quadrature schemes suffer from a phenomenon called “geometrically induced stiffness,” meaning that formally high-order accurate methods require excessively small time steps before the rapid convergence rate is observed. This can be overcome by analytic integration in time, requiring the evaluation of a collection of spatial boundary integral operators with non-physical, weakly singular kernels. In our hybrid scheme, we combine a local asymptotic approximation with the evaluation of a few boundary integral operators involving only Gaussian kernels, which are easily accelerated by a new version of the fast Gauss transform. This new scheme is robust, avoids geometrically induced stiffness, and is easy to use in the presence of moving geometries. Its extension to three dimensions is natural and straightforward, and should permit layer heat potentials to become flexible and powerful tools for modeling diffusion processes.

Sparse modeling/approximation of images plays an important role in image restoration. Instead of using a fixed system to sparsely model any input image, a more promising approach is using a system that is adaptive to the input image. A non-convex variational model is proposed in [1] for constructing a tight frame that is optimized for the input image, and an alternating scheme is used to solve the resulting non-convex optimization problem. Although it showed good empirical performance in image denoising, the proposed alternating iteration lacks convergence analysis. This paper aims at providing the convergence analysis of the method proposed in [1]. We first established the sub-sequence convergence property of the iteration scheme proposed in [1], i.e., there exists at least one convergent sub-sequence and any convergent sub-sequence converges to a stationary point of the minimization problem. Moreover, we showed that the original method can be modified to have sequence convergence, i.e., the modified algorithm generates a sequence that converges to a stationary point of the minimization problem.

Hoogendoorn and Bovy (Transportation Research Part B, 2004, 38(7), 571--592) developed an approach for a pedestrian user-optimal dynamic assignment in continuous time and space. Although their model was proposed for pedestrian traffic, it can also be applied to urban cities. The model is very general, and consists of a conservation law (CL) and a Hamilton-Jacobi-Bellman (HJB) equation that contains a minimum value problem. However, only an isotropic application example was given in their paper. We claim that the HJB equation is difficult to compute numerically in an anisotropic case. To overcome this, we reformulate their model for a dense urban city that is arbitrary in shape and has a single central business district (CBD). In our model, the minimum value problem is only used in the CL portion, and the HJB equation reduces to a Hamilton-Jacobi (HJ) equation for easier computation. The dynamic path equilibrium of our model is proven in a different way from theirs, and a numerical algorithm is also provided to solve the model. Finally, we show a numerical example under the anisotropic case and compare the results with those of the isotropic case.

In this paper we introduce a high-order discontinuous Galerkin method for two-dimensional incompressible flow in the vorticity stream-function formulation. The momentum equation is treated explicitly, utilizing the efficiency of the discontinuous Galerkin method. The stream function is obtained by a standard Poisson solver using continuous finite elements. There is a natural matching between these two finite element spaces, since the normal component of the velocity field is continuous across element boundaries. This allows for a correct upwinding gluing in the discontinuous Galerkin framework, while still maintaining total energy conservation with no numerical dissipation and total enstrophy stability. The method is efficient for inviscid or high Reynolds number flows. Optimal error estimates are proved and verified by numerical experiments.

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.