Sparse Algebraic Equations Research Articles

Using small-scale quantum devices to solve algebraic equations

Solving algebraic equations over GF(2) is a problem which has a wide range of applications, including NP-Hard problems and problems related to cryptography. The existing mature algorithms are difficult to solve large-scale problems. Inspired by Schoning’s algorithm and its quantum version, we apply related methods to solve algebraic equations over GF (2). The new algorithm we proposed has a significant improvement of solving efficiency in large-scale and sparse algebraic equations. As a hybrid algorithm, the new algorithm can not only run on a classic computer alone, but also use small-scale quantum devices to assist acceleration. And the new algorithm can be seen as an example of solving a large-scale problem on a small-scale quantum device.

Improved Agreeing-Gluing Algorithm

Asymptotical complexity of solving a system of sparse algebraic equations over finite fields is studied here. An equation is called sparse if it depends on a bounded number of variables. Finding efficiently solutions to the system of such equations is an underlying hard problem in the cryptanalysis of modern ciphers. New deterministic Improved Agreeing-Gluing Algorithm is introduced. The expected running time of the algorithm on uniformly random instances of the problem is rigorously estimated. The estimate is at present the best theoretical bound on the complexity of solving average instances of the problem. In particular, this is a significant improvement over those in our earlier papers (Semaev, Des Codes Cryptogr 49:47–60, 2008; Semaev, SIAM J Comput 39:388–409 2009). In sparse Boolean equations a gap between the present worst case and the average time complexity of the problem has significantly increased. We formulate Average Time Complexity Conjecture. If proved that will have far-reaching consequences in the field of cryptanalysis and in computing in general.

APPLICATION OF THE FINITE STRIP METHOD TO MODELING OF VIBRATIONS OF COLLECTING ELECTRODES

The paper presents an application of the finite strip method to modeling of vibrations of the collecting electrodes, which are shells with large length (up to 16 m), width of 0.5 m and thickness of 0.002 m. The models and computer programs have been worked out and validated. Comparison of results obtained from numerical simulations and experimental measurements are presented and discussed. The equations of motion have been solved using methods for solution of sparse algebraic equations and Newmark method. The strip method has proved to be numerically effective. The programs enable us to carry out calculations for a system with several hundred thousands of degrees of freedom with time of analysis requiring thousand integration steps during less than 90 min on a PC computer. High numerical efficiency enables the geometrical parameters of the collecting electrodes to be selected in order to ensure large accelerations caused by a beater to be spread evenly over the surface of the electrodes. Conclusions concerning the influence of length of the collecting electrodes on the normal and tangentz accelerations are formulated.

Dynamic stiffness removal for direct numerical simulations

A systematic approach was developed to derive non-stiff reduced mechanisms for direct numerical simulations (DNS) with explicit integration solvers. The stiffness reduction was achieved through on-the-fly elimination of short time-scales induced by two features of fast chemical reactivity, namely quasi-steady-state (QSS) species and partial-equilibrium (PE) reactions. The sparse algebraic equations resulting from QSS and PE approximations were utilized such that the efficiency of the dynamic stiffness reduction is high compared with general methods of time-scale reduction based on Jacobian decomposition. Using the dimension reduction strategies developed in our previous work, a reduced mechanism with 52 species was first derived from a detailed mechanism with 561 species. The reduced mechanism was validated for ignition and extinction applications over the parameter range of equivalence ratio between 0.5 and 1.5, pressure between 10 and 50 atm, and initial temperature between 700 and 1600 K for ignition, and worst-case errors of approximately 30% were observed. The reduced mechanism with dynamic stiffness removal was then applied in homogeneous and 1-D ignition applications, as well as a 2-D direct numerical simulation of ignition with temperature inhomogeneities at constant volume with integration time-steps of 5–10 ns. The integration was numerically stable and good accuracy was achieved.

On solving sparse algebraic equations over finite fields

A system of algebraic equations over a finite field is called sparse if each equation depends on a small number of variables. In this paper new deterministic algorithms for solving such equations are presented. The mathematical expectation of their running time is estimated. These estimates are at present the best theoretical bounds on the complexity of solving average instances of the above problem. In characteristic 2 the estimates are significantly lower the worst case bounds provided by SAT solvers.

Solution of Hallen's integral equation using multiwavelets

An effective method based upon Alpert multiwavelets is proposed for the solution of Hallen's integral equation. The properties of Alpert multiwavelets are first given. These wavelets are utilized to reduce the solution of Hallen's integral equation to the solution of sparse algebraic equations. In order to save memory requirement and computation time, a threshold procedure is applied to obtain algebraic equations. Through numerical examples, performance of the present method is investigated concerning the convergence and the sparseness of resulted matrix equation.

Efficient Numerical Techniques for Solving Pocklington's Integral Equation Using Multiwavelets

A procedure is presented for solving the electric field integral equation for a thin wire. The method is based upon Alpert multiwavelets functions approximation. The properties of Alpert multiwavelets are first given. These wavelets are utilized to reduce the solution of Pocklington's integral equation to the solution of sparse algebraic equations. In order to save memory requirement and computation time, a threshold procedure is applied to obtain sparse algebraic equations. Through numerical examples, performance of the present method is investigated concerning the convergence and the sparseness of the resulted matrix equation.

Parallel solution of sparse algebraic equations

Two methods that have been studied for solving large, sparse sets of algebraic equations connected with power system problems, the multiple factoring method and the W-matrix method, are shown to be two independent methods of explaining equivalent computational procedures. The forward and backward substitution part of these methods are investigated using parallel processing techniques on commercially available computers. The results are presented from testing the proposed methods on two local memory machines, the Intel iPSC/1 and iPSC/860 hypercubes, and a shared memory machine, the Sequent SymmetryS81. With the iPSC/1, which is characterised by its slow communication rate and high communication overhead for a short message, the best speedup obtained is less than 2.5, and that was with only 8 of the 16 available processors in use. The iPSC/860, a more advanced model of the iPSC family, is even worse as far as these parallel methods are concerned. Much better results were obtained on the Sequent Symmetry where a speedup of 7.48 was obtained with 16 processors.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Time-Domain Analysis of Lumped-Distributed Networks

A new method for time-domain analysis of networks containing transmission lines and lumped linear/nonlinear elements is presented. A key feature of the method is a procedure for generating a system matrix in a manner that involves only sums of subnetwork (or element) terms (no products or quotients). Numerical integration algorithms are used to reduce the problem to a solution of sparse algebraic equations.