The null-field approach in series form—The direct and inverse problems

CANM, a program for numerical solution of a system of nonlinear equations using the continuous analog of Newton's method

Numerous scientific and engineering applications require numerically solving systems of equations. Classically solving a general set of polynomial equations requires iterative solvers, while linear equations may be solved either by direct matrix inversion or iteratively with judicious preconditioning. However, the convergence of iterative algorithms is highly variable and depends, in part, on the condition number. We present a direct method for solving general systems of polynomial equations based on quantum annealing, and we validate this method using a system of second-order polynomial equations solved on a commercially available quantum annealer. We then demonstrate applications for linear regression, and discuss in more detail the scaling behavior for general systems of linear equations with respect to problem size, condition number, and search precision. Finally, we define an iterative annealing process and demonstrate its efficacy in solving a linear system to a tolerance of 10−8.

The security of many recently proposed cryptosystems is based on the difficulty of solving large systems of quadratic multivariate polynomial equations. This problem is NP-hard over any field. When the number of equations m is the same as the number of unknowns n the best known algorithms are exhaustive search for small fields, and a Gröbner base algorithm for large fields. Gröbner base algorithms have large exponential complexity and cannot solve in practice systems with n ≥ 15. Kipnis and Shamir [9] have recently introduced a new algorithm called “relinearization”. The exact complexity of this algorithm is not known, but for sufficiently overdefined systems it was expected to run in polynomial time.In this paper we analyze the theoretical and practical aspects of relinearization. We ran a large number of experiments for various values of n and m, and analysed which systems of equations were actually solvable. We show that many of the equations generated by relinearization are linearly dependent, and thus relinearization is less efficient that one could expect. We then develop an improved algorithm called XL which is both simpler and more powerful than relinearization. For all 0 < ε ≤ 1/2, and m ≥ εn 2, XL and relinearization are expected to run in polynomial time of approximately \( n^{\mathcal{O}(1/\sqrt \varepsilon )} \). Moreover, we provide strong evidence that relinearization and XL can solve randomly generated systems of polynomial equations in subexponential time when m exceeds n by a number that increases slowly with n.KeywordsQuadratic EquationExhaustive SearchAsymptotic ComplexityExact ComplexityMultivariate EquationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

97/04852 Coal pulsating combustion model: Xu, W. Ranshao Kexue Yu Jishu., 1996, 2, (3), 260–269. (In Chinese)

98/03268 Reopening of pulverized coal injection in blast furnace after a pause

99/00738 On the combustion of heat pump with solar steam turbine

99/00035 ESR study of the macromolecular structure of copper-porphyrin-labeled coal

The solution of a system of nonlinear algebraic or transcendental equations is probably one of the most difficult problems in numerical analysis. In general, it is almost impossible to find the solution, unless an approximate value is already known. Linear equations occur more frequently in numerical calculations, simply because it is difficult and sometimes impossible to solve the corresponding nonlinear problem. Most of the equations governing the physical world are nonlinear, but very often for simplicity and tractability linear models are considered. Solution of boundary value problems for differential equations also lead to a nonlinear equation or a system of nonlinear equations. Solution of a system of nonlinear equations also arises in numerical analysis, while deriving certain formulae. As we have seen in Chapter 6, the Gaussian quadrature formulae can be obtained by solving a system of nonlinear equations.

We relate different approaches for proving the unsatisfiability of a system of real polynomial equations over Boolean variables. On the one hand, there are the static proof systems Sherali-Adams and sum-of-squares (a.k.a. Lasserre), which are based on linear and semi-definite programming relaxations. On the other hand, we consider polynomial calculus, which is a dynamic algebraic proof system that models Gröbner basis computations. Our first result is that sum-of-squares simulates polynomial calculus: any polynomial calculus refutation of degree d can be transformed into a sum-of-squares refutation of degree 2d and only polynomial increase in size. In contrast, our second result shows that this is not the case for Sherali-Adams: there are systems of polynomial equations that have polynomial calculus refutations of degree 3 and polynomial size, but require Sherali-Adams refutations of large degree and exponential size. A corollary of our first result is that the proof systems Positivstellensatz and Positivstellensatz Calculus, which have been separated over non-Boolean polynomials, simulate each other in the presence of Boolean axioms.

We prove that rational data of bounded input length are uniformly distributed (in the classical sense of H. Weyl, in [42]) with respect to the probability distribution of condition numbers of numerical analysis. We deal both with condition numbers of linear algebra and with condition numbers for systems of multivariate polynomial equations. For instance, we prove that for a randomly chosen n\times n rational matrix M of bit length O(n 4 log n) + log w , the condition number k(M) satisfies k(M) ≤ w n 5/2 with probability at least 1-2w -1 . Similar estimates are established for the condition number μ_ norm of M. Shub and S. Smale when applied to systems of multivariate homogeneous polynomial equations of bounded input length. Finally, we apply these techniques to estimate the probability distribution of the precision (number of bits of the denominator) required to write approximate zeros of systems of multivariate polynomial equations of bounded input length.

The subject of this work is the development of a nonlinear model of the interaction of an unsteady fluid flow with a structure and finding analytical solutions for the system of equations that correspond to the specified model. The convection effect of the fluid velocity field was already considered in the previous works of the authors of this paper. These studies are devoted to the water hammer without considering the "flow-structure" interaction. This work expands the possibilities of modeling and considers four equations instead of two equations of the theory of the water hammer (equations of conservation of mass and momentum), two of which relate to the motion of particles of a solid body (pipes or structures). The novelty of this work lies in the consideration of the model that describes the interaction of the flow with the structure, the convection in the velocity field, and the effect, together with the friction of the fluid against the solid wall, on the dynamics of the shock pulse propagation process both in the fluid and in the solid body. It should be noted that the solution of the nonlinear system of differential equations as a whole is carried out by an analytical method, which makes it possible to obtain an exact (rather than numerical) solution of the problem. Since the effects of various factors should be evaluated by comparison with the main components, dimensionless equations containing six parameters (dimensionless combinations) were obtained in this study. Two of these parameters were named after scientists – Darcy and Weisbach (steady friction) and Bruno (unsteady friction). Particular cases of the general (full) model were considered, and the effects of various factors on the dynamics of the interaction of the flow with the structure during the propagation of the shock pulse were determined. Research methods are purely theoretical. The concepts of a self-similar equation and a system of equations, balances of forces acting on particles of a fluid and a solid body, and a standard method of reducing a system of equations to a single equivalent equation are used. Conclusions. An extended model of the interaction between the unsteady fluid flow and the structure is proposed. The transition to a self-similar variable makes it possible to solve a nonlinear system of differential equations and obtain an analytical (exact) solution. The functions of longitudinal stress in a solid body, pressure disturbance, and velocity of motion of particles in a solid body (pipe) are linearly expressed by the velocity of shock pulse propagation in the fluid. It should also be noted that the results for the particular case of the linear model completely agree with the already known ones. The advantage of using a self-similar solution is that it is easy to obtain. The results of previous studies on the water hammer problem were qualitatively consistent. As the fluid viscosity increases, the shock pulse domain becomes more concentrative.

We present the inverse problem of successive determination of the two unknowns that are a coefficient characterizing the properties of a medium with weakly horizontal inhomogeneity and the kernel of an integral operator describing the memory of the medium. The direct initial-boundary value problem involves the zero data and the Neumann boundary condition. The trace of the Fourier image of a solution to the direct problem on the boundary of the medium serves as additional information. Studying inverse problems, we assume that the unknown coefficient is expanded into an asymptotic series in powers of a small parameter. Also, we construct some method for finding the coefficient that accounts for the memory of the environment to within an error of order $ O(\varepsilon^{2}) $ . At the first stage, we determine a solution to the direct problem in the zero approximation and the kernel of the integral operator, while the inverse problem reduces to an equivalent problem of solving the system of nonlinear Volterra integral equations of the second kind. At the second stage, we consider the kernel given and recover a solution to the direct problem in the first approximation and the unknown coefficient. In this case, the solution to the equivalent inverse problem agrees with a solution to the linear system of Volterra integral equations of the second kind. We prove some theorems on the unique local solvability of the inverse problems and present the results of numerical calculations of the kernel and the coefficient.

The construction of highresolution differenceschemes for hyperbolic equations has been addressedin numerous works (see [1, 2] and the referencestherein). Several methods are available for improvingthe accuracy of schemes in space. They can be basedon multipoint stencils; differential consequences ofthe original equations; compact approximations ofderivatives [3]; and combinations of grid functionsobtained on different grids, for example, the Richardson method [4, 5]. There are also approaches thatcombine these methods. For example, in the wellknown work [6] concerning gasdynamic computations, a fourthorder compact approximation and differential consequences of the original equation wereused to construct a fourthorder accurate scheme on atwopoint stencil for a thirdorder scalar nonlineardifferential equation. An analysis of the literature shows that [6] is apparently the first to construct a fourthorder accuratescheme on a twopoint stencil in space. Importantly,in the computation of highgradient flows on the basisof hyperbolic conservation laws [1, 2], twopoint compact schemes seem more promising than threepointones [3] because the former have the following important properties. First, twopoint stencil schemes preserve the order of accuracy in the transition from auniform to a nonuniform grid. Second, interpolationacross a discontinuity in compact twopoint schemescan be avoided by placing a grid node at the discontinuity point. The major elements of the technique of [6] for constructing difference schemes include the introductionof derivatives as additional unknown functions inorder to reduce highorder equations to systems offirstorder equations, the use of the integrointerpolation method, and the Simpson and Maclaurin quadrature rules. Note that in this technique replacing theSimpson and Maclaurin formulas by the trapezoidalrule yields secondorder accurate twopoint schemes.In [6, 7], a secondorder scheme on a twopointminimum stencil for hyperbolic equations wasobtained via the introduction of the derivative of theunknown function as an additional independent variable. In [8] the same technique was used to construct afourthorder scheme on a twopoint stencil for theheat equation. In the recent paper [9], higher orderaccurate schemes on a twopoint stencil were called,for brevity, bicompact. Below, we will follow this (inour view apt) terminology.In [10] the bicompact scheme of [6] for a thirdorder scalar differential equation was modified andextended to systems of secondorder nonlinear differential equations. This modified scheme was used in[11, 12] to solve inner and outer problems in viscousgas dynamics.In contrast to [6–12], for deriving bicompactschemes, we use, as auxiliary functions, primitives ofthe original unknown functions rather than their spatial derivatives. Importantly, the auxiliary functionsthus introduced are once more differentiable than theoriginal unknown functions. For example, if a gasdynamic variable is a piecewise continuous function,then its primitive is a piecewise smooth function. In this work, we construct unconditionally stablefourthorder accurate implicit bicompact schemes forlinear and nonlinear equations and systems of equations of the hyperbolic type written in conservativeform. Based on these schemes, the solution to a mixedCauchy problem [13] is determined by the runningcalculation method. The properties of these schemesare examined. Test computations confirm the accuracy and high quality of the resulting solutions.Below, we construct and analyze bicompactschemes for linear and nonlinear hyperbolic equationsand systems of equations wri tten in conservative form.

A theoretical analysis of steady and unsteady magnetohydrodynamic (MHD) flow of blood through a stenosed artery with variable viscosity and slip conditions is presented in this work. Slip conditions and variable viscosity dependent on hematocrit for both steady and unsteady blood flow models are taken into consideration in this study. The governing equations have been derived by treating blood as incompressible magnetohydrodynamic (MHD) Newtonian fluid. Galerkin Weighted Residual method, Newton Raphson method and fourth order Runge-Kutta method are used to solve the non-linear differential equations for velocity profiles of the blood flow, flow rate, shear stress and flow resistance for different values of hematocrit parameter, magnetic field parameter and slip velocity. The Galerkin Weighted Residual method is applied to transform the partial differential equations describing the flow into systems of non-linear equations and system of non-linear first order differential equations for the time independent and time dependent MHD flow models respectively. Newton Raphson’s method with the aid of maple software is used to solve the system of non-linear equations while the fourth order Runge-Kutta method is used to solve the system of non-linear first order differential equations. The research analyzed how the flow velocity, flow rate, shear stress and resistance to fluid flow respond to changes in some key parameters such as hematocrit parameter, magnetic field parameter and slip velocity. The results as presented graphically reveal that flow velocity and flow rate decrease while shear stress and resistance to flow increase when the hematocrit parameter increases. Also, flow velocity, flow rate and shear stress increase while resistance to flow decrease when the slip velocity increases. Finally, the increase in magnetic field parameter led to a decrease in flow velocity, flow rate and shear stress but increased the flow resistance. The flow velocity of the unsteady blood flow model is higher when compared to steady blood flow model.

The similarity solution for the steady two-dimensional flow of an incompressible viscous and electrically conducting fluid over a non-linearly semi-infinite stretching sheet in the presence of a chemical reaction and under the influence of a magnetic field gives a system of non-linear ordinary differential equations. These non-linear differential equations are analytically solved by applying a newly developed method, namely the Homotopy Analysis Method (HAM). The analytic solutions of the system of non-linear differential equations are constructed in the series form. The convergence of the obtained series solutions is carefully analyzed. Graphical results are presented to investigate the influence of the Schmidt number, magnetic parameter and chemical reaction parameter on the velocity and concentration fields. It is noted that the behavior of the HAM solution for concentration profiles is in good agreement with the numerical solution given in reference [A. Raptis, C. Perdikis, Int. J. Nonlinear Mech. 41, 527 (2006)].