Showing papers on "Auxiliary function published in 2010"

Auxiliary-function-based independent component analysis for super-Gaussian sources

Abstract: This paper presents new algorithms of independent component analysis (ICA) for super-Gaussian sources based on auxiliary function technique. The algorithms consist of two alternative updates: 1) update of demixing matrix and 2) update of weighted covariance matrix, which include no tuning parameters such as step size. The monotonic decrease of the objective function at each update is guaranteed. The experimental results show that the derived algorithms are robust to nonstationary data and outliers, and the convergence is faster than natural-gradient-based algorithm.

Free In-Plane Vibration Analysis of Rectangular Plates With Elastically Point-Supported Edges

Abstract: In comparison with the transverse vibrations of rectangular plates, far less attention has been paid to the in-plane vibrations even though they may play an equally important role in affecting the vibrations and power flows in a built-up structure. In this paper, a generalized Fourier method is presented for the in-plane vibration analysis of rectangular plates with any number of elastic point supports along the edges. Displacement constraints or rigid point supports can be considered as the special case when the stiffnesses of the supporting springs tend to infinity. In the current solution, each of the in-plane displacement components is expressed as a 2D Fourier series plus four auxiliary functions in the form of the product of a polynomial times a Fourier cosine series. These auxiliary functions are introduced to ensure and improve the convergence of the Fourier series solution by eliminating all the discontinuities potentially associated with the original displacements and their partial derivatives along the edges when they are periodically extended onto the entire x-y plane as mathematically implied by the Fourier series representation. This analytical solution is exact in the sense that it explicitly satisfies, to any specified accuracy, both the governing equations and the boundary conditions. Numerical examples are given about the in-plane modes of rectangular plates with different edge supports. It appears that these modal data are presented for the first time in literature, and may be used as a benchmark to evaluate other solution methodologies. Some subtleties are discussed about corner support arrangements.

Fourth-order accurate bicompact schemes for hyperbolic equations

Abstract: 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.

Time-dependent auxiliary density perturbation theory.

Abstract: The recently developed auxiliary density perturbation theory is extended to time-dependent perturbations. As its static counterpart, it is based on auxiliary density functional theory in which the Coulomb and exchange-correlation potentials are expressed through one auxiliary function density. As in the case of static perturbations a noniterative alternative to the corresponding coupled perturbed Kohn–Sham method is formulated. The new methodology is validated by local and gradient corrected dynamical polarizability calculations. Comparison with experiment indicates that for low frequencies reliable dynamical polarizabilities are obtained. Our discussion also shows that the computational performance of time-dependent auxiliary density perturbation theory is similar to the previously described static approach. In order to demonstrate the potential of this new methodology, dynamic polarizabilities of C60, C180, and C240 are calculated.

Construction of Analytical Solutions to Fractional Differential Equations Using Homotopy Analysis Method

Abstract: In this paper, we present an algorithm of the homotopy analysis method (HAM) to obtain symbolic approximate solutions for linear and nonlinear differential equations of fractional order. We show that the HAM is different from all analytical methods; it provides us with a simple way to adjust and control the convergence region of the series solution by introducing the auxiliary parameter ℏ, the auxiliary function �� �� , the initial guess �� ��(�� ) and the auxiliary linear operator . Three examples, the fractional oscillation equation, the fractional Riccati equation and the fractional Lane-Emden equation, are tested using the modified algorithm. The obtained results show that the Adomain decomposition method, Variational iteration method and homotopy perturbation method are special cases of homotopy analysis method. The modified algorithm can be widely implemented to solve both ordinary and partial differential equations of fractional order.

Parameter-dependent optimal stopping problems for one-dimensional diffusions

Abstract: We consider a class of optimal stopping problems for a regular one-dimensional diffusion whose payoff depends on a linear parameter. As shown in Bank and Follmer (2003) problems of this type may allow for a universal stopping signal that characterizes optimal stopping times for any given parameter via a level-crossing principle of some auxiliary process. For regular one-dimensional diffusions, we provide an explicit construction of this signal in terms of the Laplace transform of level passage times. Explicit solutions are available under certain concavity conditions on the reward function. In general, the construction of the signal at a given point boils down to finding the infimum of an auxiliary function of one real variable. Moreover, we show that monotonicity of the stopping signal corresponds to monotone and concave (in a suitably generalized sense) reward functions. As an application, we show how to extend the construction of Gittins indices of Karatzas (1984) from monotone reward functions to arbitrary functions.

Contour extraction device and program

Abstract: An object of the present invention is to provide the technology capable of appropriately balancing the preciseness of contour extraction and calculation cost. In order to achieve this object, an energy function setting section sets an energy function that is expressed by a weighted linear sum of a plurality of kinds of energy terms defined correspondingly to a state of an active curve and is formulated so as to have a smaller value as approaching a shape of the contour to be extracted, and an iterative computation processing section minimizes the energy function by an iterative computation. An end instruction section sets an auxiliary function formulated so as to monotonously increase in accordance with the number of iteration times of iterative computation and sets a judging function expressed by a linear sum of the auxiliary function and the energy function. Then, a point of time when a local minimum appears in the judging function in the course of the iterative computation is judged as the end timing of iterative computing.

Solvability Criteria for Some Set-Valued Inequality Systems

Abstract: Arising from studying some multivalued von Neumann model, three set-valued inequality systems are introduced, and two solvability questions are considered. By constructing some auxiliary functions and studying their minimax and saddle-point properties, solvability criteria composed of necessary and sufficient conditions regarding these inequality systems are obtained.

Damping feedback stabilization for time-dependent nonlinear control affine systems

Abstract: This paper studies the stabilization of time-dependent affine nonlinear control systems by using time-dependent damping feedback control. A time-dependent auxiliary function is first construct using a homotopy operator. Providing that the obtained function is a weak Jurdjevic-Quinn function, the time-varying auxiliary function is used to construct a damping feedback controller. Stability of the closed-loop system is then shown using a Lyapunov function obtained by reshaping the auxiliary function. An application example is presented to illustrate the proposed construction.

A global optimized solution of nonlinear programming with equality constraints

Abstract: This paper combines the penalty function method with an auxiliary function method for the global optimization problem without constraint.Firstly,we construct a penalty function of the nonlinear programming problem with equality constraints.The original problem can be transformed into an unconstrained problem.Then the auxiliary function method called GOM is used to solve the unconstrained penalty problem.Finally the global minimizer of the original nonlinear programming problem can be obtained.

A dynamic convexized method for the TSP

Abstract: This paper describes a dynamic convexized method for solving the symmetric traveling salesman problem (TSP). We construct an auxiliary function and design an algorithm based on this function. The possibility of sinking into a previous local minimizer can be reduced by adjusting the value of the parameter in the auxiliary function. We have verified the correctness of this approach both in theory and experiment. Computational tests show that the algorithm is effective.

The initial and Neumann boundary value problem for a class of parabolic Monge-Ampère equations

Abstract: It is proved that the classical solution to the initial and Neumann boundary value problem for parabolic-type Monge-Ampere equations is existent and unique.Using the comparison principle,the uniqueness of the classical solution is shown.By employing proper auxiliary functions and barrier functions,the priori estimations are obtained.The existence of the strict convex classical solution is obtained by the continuous method.

The Kinematics Design with the secant-homotopy Continuation Method

Abstract: In this paper, the traditional Newton-Raphson method is modified by the secant theory and the homotopy continuation technique to a secant-homotopy continuation formula and is applied to the kinematics design problems. By means of choosing the auxiliary function, we can solve the nonlinear equations and guarantee the solutions exactly without divergence rather than the traditional numerical methods such as the Newton-Raphson method and so on.

Efficient calculation of boys integrals

Abstract: �An efficient procedure has been presented for the calculation of Boys integrals by dividing the integral region [0, 1] into several parts. The obtained formula includes factorials, exponentials and a well known auxiliary function that converges very well. A computer program has been constructed in Maple symbolic programming language for comparing our results with literature. It is seen that the computational results of the presented algorithm agree well with the numerical results obtained by direct integration with Maple and results in the available literature.

Contour extraction device and program

Abstract: Disclosed is a technique capable of appropriately balancing the adequacy and calculation cost of contour extraction. An energy function setting unit sets an energy function which is expressed by the weighted linear sum of a plurality of kinds of energy terms that are defined correspondingly to the state of a dynamic curve, and which is formulated so that the closer to the shape of a contour to be extracted, the smaller the value becomes, and an iterative operation processing unit minimizes the energy function by means of iterative operation. A completion instruction unit sets an auxiliary function which is formulated so as to be monotonically increased in accordance with the number of iteration of the iterative operation, and sets a determination function which is expressed by the linear sum of the auxiliary function and the energy function. Then, the completion instruction unit determines the point at which the minimum value has appeared in the determination function in the process of the iterative operation as the completion timing of the iterative operation processing.

Notice of Retraction A hybrid population-based algorithm for constrained global programming

Abstract: In this paper, a hybrid population-based algorithm (HPBA) for constrained global programming is studied. Firstly , an auxiliary function is constructed, the properties of this kind of auxiliary function are analyzed. Some nice properties of the constructed auxiliary function have proven by some theorems. Finally, the detailed steps of HPBA and numerical experiments for constrained global programming are presented. The effectiveness of HPBA is proven by comparing other existed algorithms under the same benchmark functions.