Arbitrary High Accuracy Research Articles

The local discontinuous Galerkin method for the nonlinear quantum Zakharov system

In this paper, we design an energy conserving local discontinuous Galerkin (LDG) method to solve the quantum Zakharov system (QZS). We first give theoretical proof for the conservation of mass and energy and then focus our discussion on the priori estimates for the semi-discrete approximation of the QZS. With the proof for the L2 boundedness of the numerical solutions and the discrete Poincaré inequalities, we obtain the optimal error estimates. Next, we develop a decoupled and implicit linear fully-discrete scheme using the semi-implicit spectral deferred correction (SISDC) method for temporal discretization. Another strength of our approach is the arbitrary high accuracy which depends only on the degree of the approximating piecewise polynomials. Finally, several numerical examples are reported to validate it and the conservation properties. Especially, the second-order convergence rate of the QZS to its limiting model in the semi-classical limit is achieved as expected.

High-accuracy control systems with singular perturbations

In this paper, a linear quadratic problem for a continuous scalar control plant with stationary perturbations is considered. A class of perturbations is selected, in which arbitrary high accuracy of control by the criterion of steady-state response variance is ensured. An algorithm for synthesizing a stabilizing controller providing the desired control performance is proposed.

Completeness of the set of scattering amplitudes

Let $f\in L^2(S^2)$ be an arbitrary fixed function with small norm on the unit sphere $S^2$, and $D\subset \R^3$ be an arbitrary fixed bounded domain. Let $k>0$ and $\alpha\in S^2$ be fixed. It is proved that there exists a potential $q\in L^2(D)$ such that the corresponding scattering amplitude $A(\alpha')=A_q(\alpha')=A_q(\alpha',\alpha,k)$ approximates $f(\alpha')$ with arbitrary high accuracy: $\|f(\alpha')-A_q(\alpha')_{L^2(S^2)}\|\leq\ve$ where $\ve>0$ is an arbitrarily small fixed number. This means that the set $\{A_q(\alpha')\}_{\forall q\in L^2(D)}$ is complete in $L^2(S^2)$. The results can be used for constructing nanotechnologically "smart materials".

Computationally Efficient Approximations of the Joint Spectral Radius

The joint spectral radius of a set of matrices is a measure of the maximal asymptotic growth rate that can be obtained by forming long products of matrices taken from the set. This quantity appears in a number of application contexts but is notoriously difficult to compute and to approximate. We introduce in this paper a procedure for approximating the joint spectral radius of a finite set of matrices with arbitrary high accuracy. Our approximation procedure is polynomial in the size of the matrices once the number of matrices and the desired accuracy are fixed. For the special case of matrices with nonnegative entries we give elementary proofs of simple inequalities that we then use to obtain approximations of arbitrary high accuracy. From these inequalities it follows that the spectral radius of matrices with nonnegative entries is given by the simple expression $$ \rho(A_1, \ldots, A_m)= \lim_{k \rightarrow \infty} \rho^{1/k} ( A_1^{\otimes k} + \cdots + A_m^{\otimes k}), $$ where it is somewhat surprising to notice that the right-hand side does not directly involve any mixed product between the matrices. ($A^{\otimes k}$ denotes the kth Kronecker power of A.) For matrices with arbitrary entries (not necessarily nonnegative), we introduce an approximation procedure based on semidefinite liftings that can be implemented in a recursive way. For two matrices, even the first step of the procedure gives an approximation whose relative accuracy is at least ${1 / \sqrt{2}}$, that is, more than $70\%$. The subsequent steps improve the accuracy but also increase the dimension of the auxiliary problems from which the approximation can be found. Our approximation procedures provide approximations of relative accuracy $1-\epsilon$ in time polynomial in $n^{(\ln m) /\epsilon}$, where m is the number of matrices and n is their size. These bounds are close from optimality since we show that, unless P=NP, no approximation algorithm is possible that provides a relative accuracy of $1-\epsilon$ and runs in time polynomial in n and $1/\epsilon$. As a by-product of our results we prove that a widely used approximation of the joint spectral radius based on common quadratic Lyapunov functions (or on ellipsoid norms) has relative accuracy $1/\sqrt m$, where m is the number of matrices.

Combined Preisach–Mayergoyz-neural-network vector hysteresis model for electrical steel sheets

Three versions of a vector hysteresis model for electrical steel sheets are presented, based on the function approximation capabilities of feed-forward neural networks and the memory mechanism of vector hysteresis proposed by Mayergoyz. The first model handles arbitrary vector magnetization patterns, but requires a very extended data set for the training of the neural network. The second model is suitable for convex induction loci and allows a reduction of the required training set. The third model handles the features of the considered magnetization pattern in an alternative way and relaxes the convexity requirement. The choice of the specific model, its parameters, and the network training set depends on the types of magnetization patterns concerned. Arbitrary high accuracy can be reached by extending the complexity of the model and/or the size of the training set. Experimental results for the third model are presented and show the good accuracy of the approach. Standard neural network algorithms are used.

Universal variational expansion for high-precision bound-state calculations inthree-body systems. Applications to weakly bound, adiabatic and two-shell clustersystems

The results of high-precision variational calculations are reported for anumber of bound states in various Coulomb three-body systems, includinghelium and helium-muonic atoms, some adiabatic systems (H2+, D2+ andDT+ions) and muonic molecular ions ppμ, ddμ, ttμ and dtμ. The hyperfinesplittings for the double electron-excited states in the helium-muonic 3He2+ μ−e− and 4He2+ μ−e−atoms have also been determined. The results of this study are significantly moreaccurate than results known from earlier calculations for all considered systemsand states. The approach can be used to determine the bound-state spectra invarious three-body systems to arbitrary high accuracy. We also discuss a numberof complications which are usually detected in high-precision bound-statecalculations of few-body systems.

Fully Discrete Convergence Estimates for Vortex Methods in Bounded Domains

In this paper the authors study vortex method for 2-dimensionsal Euler equations of incompressible flow in bounded domains. To approximate the initial vortex field by a sum of vortex blobs with arbitrary high accuracy, this field is extended smoothly to a small neighborhood of the boundary. And the computation is carried out in the extended domain. To construct the velocity field from vorticity, a second-order isoparametric finite element method is applied, and to solve the characteristic equations, the explicit Euler’s scheme is considered. Optimal error bounds for this fully discrete scheme are obtained.

Modeling of continuous time systems using a discrete time representation

The problem of the representation of continuous time systems by difference equations is studied. A solution is proposed using a sample integration technique. The convolution integrals are approximated by digital filters resulting in an arbitrary high accuracy in a wide frequency band.

On the algebraic fitting of experimental thermodynamic data with special regard to employing of orthonormal polynomials

Following the Weierstrass approximation theorem the thermodynamic excess functions are representable with arbitrary high accuracy by means of polynomials of sufficient high degrees in the mole fraction x. So, algebraic fitting of experimental thermodynamic excess data can be based upon mathematical polynomial expressions without any loss of generality. With respect to the necessary scattering of experimental results, algebraic evaluation of those data can only be solved by employing the calculus of observations. The least square method is the only principle of fitting with full justification by statistical mathematics, and which can be applied directly for algebraic fitting of experimental data by means of a computer. The general linear problem of fitting is solved explicitly (i) by means of Gauss method of elimination, and (ii) by employing the property of “orthonormality” of polynomials. In the latter case the explicit form of the “orthonormal” polynomials depends strongly on the number of experimental data which has to be fitted. A convenient procedure is presented to generate polynomials which are orthonormal with respect to an actual set of experimental data. Computer-programs in PORTRAN-language are enclosed 1) to employ Gauss method of elimination, and 2) to generate discrete orthonormal polynomials.