Singular matrix inversion in fluid dynamics computations

A fast algorithm for sparse matrix computations related to inversion

We have developed a fast algorithm for computing certain entries of the inverse of a sparse matrix. Such computations are critical to many applications, such as the calculation of nonequilibrium Green’s functions G and G for nano-devices. The FIND (Fast Inverse using Nested Dissection) algorithm is optimal in the big-O sense. However, in practice, FIND suffers from two problems due to the width...

Fast Cauchy-like and Singular Toeplitz-like Matrix Computations

computes the solution ~ x = T ?1 ~ b to a nonsingular Toeplitz or Toeplitz-like linear system T~ x = ~ b, a short displacement generator for the inverse T ?1 of T, and det T. We extend this algorithm to the similar computations with nn Cauchy (generalized Hilbert) and Cauchy-like matrices. Recursive triangular factorization of such a matrix can be computed by our algorithm at the cost of execut...

Recurrent implicit dynamics for online matrix inversion

A novel kind of recurrent implicit dynamics together with its electronic realization is proposed and exploited for real-time matrix inversion. Compared to conventional explicit neural dynamics, our proposed model in the form of implicit dynamics has the following advantages: (a) can coincide better with systems in practice; and (b) has higher abilities in representing dynamic systems. More impo...

High-Performance Java Codes for Computational Fluid Dynamics and Sparse Matrix Computations

Analysis of Some Singular Solutions in Fluid Dynamics

Studies on singular flows in which either the velocity fields or the vorticity fields change dramatically on small regions are of considerable interests in both the mathematical theory and applications. Important examples of such flows include supersonic shock waves, boundary layers, and motions of vortex sheets, whose studies pose many outstanding challenges in both theoretical and numerical a...