Our randomized preprocessing enables pivoting-free and orthogonalization-free solution of homogeneous linear systems of equations, which leads to significant acceleration of the known algorithms in the cases of both general and structured input matrices. E.g., in the case of Toeplitz inputs, we decrease the estimated solution time from quadratic to nearly linear, and our tests show dramatic dec...

We propose a fast structured selected inversion method for extracting the diagonal blocks of the inverse of a sparse symmetric matrix A, using the multifrontal method and rank structures. When A arises from the discretization of some PDEs and has a low-rank property (the intermediate dense matrices in the factorization have small off-diagonal numerical ranks), structured approximations of the d...

There has been much effort to improve the runtime of matrix multiplication. The trivial algorithm multiplies n×n matrices in O(n) time. Strassen (’69) surprised everyone by giving an O(n) time algorithm. This began a long line of improvements until in 1986, Coppersmith and Winograd achieved O(n). After 24 years of no progress, in 2010 Andrew Stothers, a graduate student in Edinburgh, improved t...

Our randomized preprocessing enables pivoting-free and orthogonalization-free solution of homogeneous linear systems of equations, which leads to significant acceleration of the known algorithms in the cases of both general and structured input matrices. E.g., in the case of Toeplitz inputs, we decrease the estimated solution time from quadratic to nearly linear, and our tests show dramatic dec...

Full waveform inversion (FWI) has been regarded as an effective tool to build the velocity model for the following pre-stack depth migration. Traditional inversion methods are built on Born approximation and are initial model dependent, while this problem can be avoided by introducing Transmission matrix (T-matrix), because the T-matrix includes all orders of scattering effects. The T-matrix ca...

Tracing the flow of electricity becomes an essential part in transmission fixed cost allocation since the existing methods based on transaction are incapable of providing fair and non-discriminatory transmission service pricing. Conventional power tracing based on matrix inversion confronts with the problem of singular matrix possibility as the solution will be indeterminate and undefined. This...

Three recurrent neural networks are presented for computing the pseudoinverses of rank-deficient matrices. The first recurrent neural network has the dynamical equation similar to the one proposed earlier for matrix inversion and is capable of Moore–Penrose inversion under the condition of zero initial states. The second recurrent neural network consists of an array of neurons corresponding to ...

Focusing light through scattering media has broad applications in optical imaging, manipulation and therapy. The contrast of the focus can be quantified by peak-to-background intensity ratio (PBR). Here, we theoretically and numerically show that by using a transmission matrix inversion method to achieve focusing, within a limited field of view and under a low noise condition in transmission ma...

An efficient pad assignment methodology to minimize voltage drop on a power distribution network is proposed. A combination of successive pad assignment (SPA) with incremental matrix inversion (IMI) determines both location and number of power supply pads to satisfy drop voltage constraint. The SPA creates an equivalent resistance matrix which preserves both pad candidates and power consumption...

using adaptive beam forming, " [J. Abstract Abstract: Harrison [J. Acoust. Soc. Am. 125, 3511– 3513 (2009)] presented a mathematical explanation for a sign-inversion induced to the passive fathometer response by minimum variance distortionless response (MVDR) beamforming. Here a concise mathematical formulation is offered, which decomposes the cross-spectral density matrix into coherent and inc...