A fourth order finite difference method is presented for the 2D unsteady viscous incompressible Boussinesq equations in vorticity-stream function formulation. The method is especially suitable for moderate to large Reynolds number flows. The momentum equation is discretized by a compact fourth order scheme with the no-slip boundary condition enforced using a local vorticity boundary condition. Fourth order long-stencil discretizations are used for the temperature transport equation with one-sided extrapolation applied near the boundary. The time stepping scheme for both equations is classical fourth order Runge–Kutta. The method is highly efficient. The main computation consists of the solution of two Poisson-like equations at each Runge–Kutta time stage for which standard FFT based fast Poisson solvers are used. An example of Lorenz flow is presented, in which the full fourth order accuracy is checked. The numerical simulation of a strong shear flow induced by a temperature jump, is resolved by two perfectly matching resolutions. Additionally, we present benchmark quality simulations of a differentially-heated cavity problem. This flow was the focus of a special session at the first MIT conference on Computational Fluid and Solid Mechanics in June 2001.

A multilevel quality-guided phase unwrapping algorithm for real-time 3D shape measurement is presented. The quality map is generated from the gradient of the phase map. Multilevel thresholds are used to unwrap the phase level by level. Within the data points in each level, a fast scan-line algorithm is employed. The processing time of this algorithm is approximately 18.3 ms for an image size of 640480 pixels in an ordinary computer. We demonstrate that this algorithm can be implemented into our real-time 3D shape measurement system for real-time 3D reconstruction. Experiments show that this algorithm improves the previous scan-line phase unwrapping algorithm significantly although it reduces its processing speed slightly.

The modules of principal parts$P$^{$k$}($E$) of a locally free sheaf ε on a smooth scheme$X$is a sheaf of$O$_{$X$}-bimodules which is locally free as left and right$O$_{$X$}-module. We explicitly split the modules of principal parts$P$^{$k$}($O$($n$)) on the projective line in arbitrary characteristic, as left and right$O$_{p1}-modules. We get examples when the splitting-type as left module differs from the splitting-type as right module. We also give examples showing that the splitting-type of the principal parts changes with the characteristic of the base field.

Consider the image restoration from incomplete noisy frequency data with total variation and sparsity regularizing penalty terms. Firstly, we establish an unconstrained optimization model with dierent smooth approximations on the regularizing terms. Then, to weaken the amount of computations for cost functional with total variation term, the alternating iterative scheme is developed to obtain the exact solution through shrinkage thresholding in inner loop, while the nonlinear Euler equation is appropriately linearized at each iteration in exterior loop, yielding a linear system with diagonal coefficient matrix in frequency domain. Finally the linearized iteration is proven to be convergent in generalized sense for suitable regularizing parameters, and the error between the linearized iterative solution and the one gotten from the exact nonlinear Euler equation is rigorously estimated, revealing the essence of the proposed alternative iteration scheme. Numerical tests for different configurations show the validity of the proposed scheme, compared with some existing algorithms.

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