Matrix Compression Algorithm Research Articles

Matrix compression along isogenic blocks

A matrix-compression algorithm is derived from a novel isogenicblock decomposition for square matrices. The resulting compression andinflation operations possess strong functorial and spectral-permanenceproperties. The basic observation that Hadamard entrywise functionalcalculus preserves isogenic blocks has already proved to be of paramountimportance for thresholding large correlation matrices. The proposedisogenic stratification of the set of complex matrices bears similarities tothe Schubert cell stratification of a homogeneous algebraic manifold. Anarray of potential applications to current investigations in computationalmatrix analysis is briefly mentioned, touching concepts such as symmetricstatistical models, hierarchical matrices and coherent matrix organizationinduced by partition trees.

A fast and oblivious matrix compression algorithm for Volterra integral operators

The numerical solution of dynamical systems with memory requires the efficient evaluation of Volterra integral operators in an evolutionary manner. After appropriate discretization, the basic problem can be represented as a matrix-vector product with a lower diagonal but densely populated matrix. For typical applications, like fractional diffusion or large-scale dynamical systems with delay, the memory cost for storing the matrix approximations and complete history of the data then becomes prohibitive for an accurate numerical approximation. For Volterra integral operators of convolution type, the fast and oblivious convolution quadrature method of Schädle, Lopez-Fernandez, and Lubich resolves this issue and allows to compute the discretized evaluation with N time steps in O(N log N) complexity and only requires O(log N) active memory to store a compressed version of the complete history of the data. We will show that this algorithm can be interpreted as an {{mathscr{H}}}-matrix approximation of the underlying integral operator. A further improvement can thus be achieved, in principle, by resorting to {{mathscr{H}}}^{2}-matrix compression techniques. Following this idea, we formulate a variant of the {{mathscr{H}}}^{2}-matrix-vector product for discretized Volterra integral operators that can be performed in an evolutionary and oblivious manner and requires only O(N) operations and O(log N) active memory. In addition to the acceleration, more general asymptotically smooth kernels can be treated and the algorithm does not require a priori knowledge of the number of time steps. The efficiency of the proposed method is demonstrated by application to some typical test problems.

A Truncated Singular Value Decomposition Enhanced Nested Complex Source Beam Method

This work presents a novel matrix compression algorithm to improve the computational efficiency of the nested complex source beam (NCSB) method. The algorithm is based on the application of the truncated singular value decomposition (TSVD) to the multilevel aggregation, translation, and disaggregation operations in NCSB. In our implementation, the aggregation/disaggregation matrices are solved by the truncated far-field matching, which is based on the directional far-field radiation property of the complex source beams (CSBs). Furthermore, the translation matrices are obtained according to the beam width of CSBs. Due to the high directivity of the radiation patterns of CSBs, all the far-field related interaction matrices are low-ranked. Therefore, TSVD can be employed and a new set of equivalent sources can be constructed by a linear combination of the original CSBs. It is proved that the radiation power of the new sources is proportional to the square of the corresponding singular values. This provides a theoretical guideline to drop the insignificant singular vectors in the calculation. In doing so, the efficiency of the original NCSB method can be much improved while a reasonably good accuracy is maintained. Several numerical tests are conducted to validate the proposed method.

The Application of Matrix Compression Algorithm in Network Education Platform for the User Security Authentication

Using the matrix compression algorithm in the network education platform for the user information security certification is a good way. The sensitive user information is transfer in an open channel and it can be authentication for using the matrix compression/decompression, matrix decomposition/reduction algorithm. The client conduct random capture and compression by the user information been divided into a number of rectangular block size, corresponding to generate a key and cipher text. The server take out the corresponding data from the data queue of receiving and to extract In accordance with the key rules, then to reduce of information in corresponding positions. Thus the user identity information can be authenticity and integrity verification.

Numerical simulation of mechanisms of sound conduction through the human head with a fast elasto‐acoustics integral equation solver.

The purpose of this paper is to describe our progress in development and applications of numerical simulation tools designed to (a) investigate mechanisms of energy transfer to cochlea through air‐ and bone‐conduction pathways and to (b) assess effectiveness of suitable noise protection devices. Our numerical simulations employ a recently implemented acoustoelastic integral equation solver, capable (through the use of nonlossy fast Fourier transform based matrix compression algorithm and parallelization on distributed‐memory systems) of accurate large‐scale numerical simulations with anatomically realistic models of the human head, discretized with several millions of tetrahedral elements. Recently added new solver features (including treatment of shear waves, described in terms of node‐based linear elements) enable us to analyze detailed aspects of acoustic/elastic wave interaction with the human head with significant accuracy and efficiency. In particular, they facilitate examination of the relative amounts of energy transferred to the middle and inner ears directly through the outer ear and indirectly via excitation of elastic waves in the skull and soft tissues. [This work is supported by AFOSR.]

On the Use of the Singular Value Decomposition in the Fast Multipole Method

The authors present a new matrix compression algorithm to improve the efficiency of the fast multipole method (FMM). The method is based on the application of the singular value decomposition (SVD) to the plane wave FMM aggregation matrices. These matrices are low-ranked, which is exploited to provide alternative sets of orthonormal singular basis functions, obtained as linear combinations of the original basis. By choosing only the most relevant singular functions, a much more compact representation is obtained to accurately handle the interactions between the FMM groups. The new formulation provides a reduction close to one order of magnitude both in computational cost and memory requirements, with a moderate impact on the accuracy of the solution.

Accurate solution of multi-region continuum biomolecule electrostatic problems using the linearized Poisson-Boltzmann equation with curved boundary elements.

We present a boundary-element method (BEM) implementation for accurately solving problems in biomolecular electrostatics using the linearized Poisson-Boltzmann equation. Motivating this implementation is the desire to create a solver capable of precisely describing the geometries and topologies prevalent in continuum models of biological molecules. This implementation is enabled by the synthesis of four technologies developed or implemented specifically for this work. First, molecular and accessible surfaces used to describe dielectric and ion-exclusion boundaries were discretized with curved boundary elements that faithfully reproduce molecular geometries. Second, we avoided explicitly forming the dense BEM matrices and instead solved the linear systems with a preconditioned iterative method (GMRES), using a matrix compression algorithm (FFTSVD) to accelerate matrix-vector multiplication. Third, robust numerical integration methods were employed to accurately evaluate singular and near-singular integrals over the curved boundary elements. Fourth, we present a general boundary-integral approach capable of modeling an arbitrary number of embedded homogeneous dielectric regions with differing dielectric constants, possible salt treatment, and point charges. A comparison of the presented BEM implementation and standard finite-difference techniques demonstrates that for certain classes of electrostatic calculations, such as determining absolute electrostatic solvation and rigid-binding free energies, the improved convergence properties of the BEM approach can have a significant impact on computed energetics. We also demonstrate that the improved accuracy offered by the curved-element BEM is important when more sophisticated techniques, such as nonrigid-binding models, are used to compute the relative electrostatic effects of molecular modifications. In addition, we show that electrostatic calculations requiring multiple solves using the same molecular geometry, such as charge optimization or component analysis, can be computed to high accuracy using the presented BEM approach, in compute times comparable to traditional finite-difference methods.

A Single-Level Low Rank IE-QR Algorithm for PEC Scattering Problems Using EFIE Formulation

This paper presents a single-level matrix compression algorithm, termed IE-QR, based on a low-rank approximation to speed up the electric field integral equation (EFIE) formulation. It is shown, with the number of groups chosen to be proportional to N/sup 1/2/, where N is the number of unknowns, the memory and CPU time for the resulting algorithm are both O(N/sup 1.5/). The unique features of the algorithm are: a. The IE-QR algorithm is based on the near-rank-deficiency property for well-separated groups. This near-rank-deficiency assumption holds true for many integral equation methods such as Laplacian, radiation, and scattering problems in electromagnetics (EM). The same algorithm can be adapted to other applications outside EM with few or no modifications; and, b. The rank estimation is achieved by a dual-rank process, which ranks the transmitting and receiving groups, respectively. Thus, the IE-QR algorithm can achieve matrix compression without assembling the entire system matrix. Also, a "geometric-neighboring" preconditioner is presented in this paper. This "geometric-neighboring" preconditioner when used in conjunction with GMRES is proven to be both efficient and effective for solving the compressed matrix equations.

A Low-Rank IE-QR Algorithm for Matrix Compression in Volume Integral Equations

A single-level matrix compression algorithm based on pivoted QR factorization with partial matrix assembling, which exploits the rank deficiency of matrix blocks for physically separated groups of basis functions, is presented for the volume integral equation solution of electromagnetic scattering from arbitrarily shaped dielectric bodies. For a system of N equations, an amount of work of the order O(N/sup 2/) has traditionally been required by the method of moments (MoM). The algorithm of the present paper reduces both computational complexity and storage requirement to O(N/sup 1.5/) with relatively less dependence on the integral equation kernel. Hence, the proposed algorithm is more practical for large-scale problems and can be implemented in a wide range of applications with few or no modifications.

Iterative Selection of Expansion Functions From an Overcomplete Dictionary of Wavelet Packets for Impedance Matrix Compression

The paper further develops the recently introduced idea of compressing the impedance matrix by an iterative selection of expansion functions. The improved algorithm uses an overcomplete dictionary comprising bases whose inherent properties match some features of the given scattering problem. Thus, the overcomplete dictionary offers a variety of expansion functions from which a solution-oriented spanning set for the unknown current is extracted. The number of iterations in the proposed algorithm is suppressed by selecting more than one expansion function at each iteration. This latter section is facilitated by a matching pursuit process. It is shown that the compression ratio of the resultant compressed impedance matrix is superior to the one achieved by the ordinary iterative matrix compression algorithm.

An improvement to Ziegler's sparse matrix compression algorithm

We propose a heuristic algorithm to compress a sparse matrix. An algorithm is also presented for recovering the original matrix. Using this compression algorithm, we can compress any sparse matrix to get a linear result and map this linear result directly into memory. After being compressed, the sparse matrix becomes a more compact linear array. Thus, we can reduce memory size for storing the matrix by using our compression algorithm and recover the original matrix without any difficulty.