Stabilization of geometrically nonlinear topology optimization by the Levenberg–Marquardt method

This paper deals with the design of compliant mechanisms in a continuum-based finite-element representation. Because the displacements of mechanisms are intrinsically large, the geometric nonlinearity is essential for designing such mechanisms. However, the consideration of the geometric nonlinearity may cause some instability in topology optimization. The problem is in the analysis part but not in the optimization part. To alleviate the analysis problem and eventually stabilize the optimization process, this paper proposes to apply the Levenberg–Marquardt method to the nonlinear analysis of compliant mechanisms.     

JavaFit: A platform independent program for interactive nonlinear least-squares fitting using the Levenberg—Marquardt method

The JavaFit program is a package for carrying out interactive nonlinear least-squares fitting to determine the parameters of physical models from experimental data. It has been conceived as a platform independent package aimed at the relatively modest computational needs of spectroscopists, where it is often necessary to determine physical parameters from a variety of spectral lineshape models. The program is platform independent, provided that a Java runtime module is available for the host platform. The program is also designed to read a wide variety of data in ASCII column formats produced on DOS, Macintosh and UNIX platforms.     

The local discontinuous Galerkin method for 2D nonlinear time-fractional advection–diffusion equations

This paper presents a numerical solution of time-fractional nonlinear advection–diffusion equations (TFADEs) based on the local discontinuous Galerkin method. The trapezoidal quadrature scheme (TQS) for the fractional order part of TFADEs is investigated. In TQS, the fractional derivative is replaced by the Volterra integral equation which is computed by the trapezoidal quadrature formula. Then the local discontinuous Galerkin method has been applied for space-discretization in this scheme. Additionally, the stability and convergence analysis of the proposed method has been discussed. Finally some test problems have been investigated to confirm the validity and convergence of the proposed method.

     

Gauss–Newton filtering incorporating Levenberg–Marquardt methods for tracking

This paper shows that the Levenberg–Marquardt algorithms (LMA) can be merged into the Gauss–Newton filters (GNF) to track difficult, non-linear trajectories, with improved convergence. The GNF discussed first in this paper is an iterative filter, with memory that was introduced by Norman Morrison (1969) [1]. To improve the computation demands of the GNF, we adapted the GNF to a recursive version. The original GNF uses back propagation of the predicted state to compute the Jacobian matrix over the filter memory length. The LMA are optimisation techniques widely used for data fitting (Marquardt, 1963 [2]). These optimisation techniques are iterative and guarantee local convergence.     

Laguerre wavelet method for solving Thomas–Fermi type equations

Engineering with Computers - An efficient numerical algorithm based on the Laguerre wavelets collocation technique for numerical solutions of a class of Thomas–Fermi boundary value problems,...     

The variational iteration method for solving systems of equations of Emden–Fowler type

In this paper, the variational iteration method (VIM) is used to study systems of linear and nonlinear equations of Emden–Fowler type arising in astrophysics. The VIM overcomes the singularity at the origin and the nonlinearity phenomenon. The Lagrange multipliers for all cases of the parameter α,α>0, are determined. The work is supported by examining specific systems of two or three Emden–Fowler equations where the convergence of the results is emphasized.     

Wilson wavelets-based approximation method for solving nonlinear Fredholm–Hammerstein integral equations

Some of mathematical physics models deal with nonlinear integral equations such as diffraction problems, scattering in quantum mechanics, conformal mapping and etc. In fact, analytically solving such nonlinear integral equations is usually difficult, therefore, it is necessary to propose proper numerical methods. In this paper, an efficient and accurate computational method based on the Wilson wavelets and collocation method is proposed to solve a class of nonlinear Fredholm–Hammerstein integral equations. In the proposed method, Kumar and Sloan scheme is used. Convergence of the Wilson expansion is investigated and also the error analysis of the proposed method is proved. Some numerical examples are provided to demonstrate the accuracy and efficiency of the method.     

Pressure projection stabilized finite element method for Navier–Stokes equations with nonlinear slip boundary conditions

In this paper, we consider the pressure projection stabilized finite element method for the Navier–Stokes equation with nonlinear slip boundary conditions whose variational formulation is the variational inequality problem of the second kind with Navier–Stokes operator. The H ¹ and L ² error estimates for the velocity and the L ² error estimate for the pressure are obtained. Finally, the numerical results are displayed to verify the theoretical analysis.     

The extended Pouzet–Runge–Kutta methods for nonlinear neutral delay-integro-differential equations

This paper deals with the extended Pouzet–Runge–Kutta methods for nonlinear neutral delay-integro-differential equations. Nonlinear stability and numerical implementation of the methods are investigated. It is proven under the suitable conditions that the extended Pouzet–Runge–Kutta methods are globally and asymptotically stable for problems of the class ${\mathbb{NRI}{(\alpha,\beta,\gamma,\nu)}}$ . Numerical examples further illustrate the theoretical results and the methods’ effectiveness.     

Parallel iteratively regularized Gauss–Newton method for systems of nonlinear ill-posed equations

We propose a parallel version of the iteratively regularized Gauss–Newton method for solving a system of ill-posed equations. Under certain widely used assumptions, the convergence rate of the parallel method is established. Numerical experiments show that the parallel iteratively regularized Gauss–Newton method is computationally convenient for dealing with underdetermined systems of nonlinear equations on parallel computers, especially when the number of unknowns is much larger than that of equations.     

Finite difference method for time–space linear and nonlinear fractional diffusion equations

ABSTRACT

In this paper a finite difference method is presented to solve time–space linear and nonlinear fractional diffusion equations. Specifically, the centred difference scheme is used to approximate the Riesz fractional derivative in space. A trapezoidal formula is used to solve a system of Volterra integral equations transformed from spatial discretization. Stability and convergence of the proposed scheme is discussed which shows second-order accuracy both in temporal and spatial directions. Finally, examples are presented to show the accuracy and effectiveness of the schemes.     

Levenberg–Marquardt Algorithm for Acousto-Electric Tomography based on the Complete Electrode Model

Journal of Mathematical Imaging and Vision - The inverse problem in acousto-electric tomography concerns the reconstruction of the electric conductivity in a body from knowledge of the power...     

The well-posedness for fractional nonlinear Schrödinger equations

In this paper we apply the tools of harmonic analysis to study the Cauchy problem for time fractional Schrödinger equations. The existence and a sharp decay estimate for solutions of the given problem in two different spaces are addressed. Some fundamental properties of operators appearing in the solution of the problem are also discussed.     

A high-order compact method for nonlinear Black–Scholes option pricing equations of American options

Due to transaction costs, illiquid markets, large investors or risks from an unprotected portfolio the assumptions in the classical Black–Scholes model become unrealistic and the model results in nonlinear, possibly degenerate, parabolic diffusion–convection equations. Since in general, a closed-form solution to the nonlinear Black–Scholes equation for American options does not exist (even in the linear case), these problems have to be solved numerically. We present from the literature different compact finite difference schemes to solve nonlinear Black–Scholes equations for American options with a nonlinear volatility function. As compact schemes cannot be directly applied to American type options, we use a fixed domain transformation proposed by ?ev?ovi? and show how the accuracy of the method can be increased to order four in space and time.     

An efficient split-step quasi-compact finite difference method for the nonlinear fractional Ginzburg–Landau equations

In this paper, we propose a split-step quasi-compact finite difference method to solve the nonlinear fractional Ginzburg–Landau equations both in one and two dimensions. The original equations are split into linear and nonlinear subproblems. The Riesz space fractional derivative is approximated by a fourth-order fractional quasi-compact method. Furthermore, an alternating direction implicit scheme is constructed for the two dimensional linear subproblem. The unconditional stability and convergence of the schemes are proved rigorously in the linear case. Numerical experiments are performed to confirm our theoretical findings and the efficiency of the proposed method.     

A stable Gaussian radial basis function method for solving nonlinear unsteady convection–diffusion–reaction equations

We investigate a novel method for the numerical solution of two-dimensional time-dependent convection–diffusion–reaction equations with nonhomogeneous boundary conditions. We first approximate the equation in space by a stable Gaussian radial basis function (RBF) method and obtain a matrix system of ODEs. The advantage of our method is that, by avoiding Kronecker products, this system can be solved using one of the standard methods for ODEs. For the linear case, we show that the matrix system of ODEs becomes a Sylvester-type equation, and for the nonlinear case we solve it using predictor–corrector schemes such as Adams–Bashforth and implicit–explicit (IMEX) methods. This work is based on the idea proposed in our previous paper (2016), in which we enhanced the expansion approach based on Hermite polynomials for evaluating Gaussian radial basis function interpolants. In the present paper the eigenfunction expansions are rebuilt based on Chebyshev polynomials which are more suitable in numerical computations. The accuracy, robustness and computational efficiency of the method are presented by numerically solving several problems.     

Relaxation method for unsteady convection–diffusion equations

We propose and implement a relaxation method for solving unsteady linear and nonlinear convection–diffusion equations with continuous or discontinuity-like initial conditions. The method transforms a convection–diffusion equation into a relaxation system, which contains a stiff source term. The resulting relaxation system is then solved by a third-order accurate implicit–explicit (IMEX) Runge–Kutta method in time and a fifth-order finite difference WENO scheme in space. Numerical results show that the method can be used to effectively solve convection–diffusion equations with both smooth structures and discontinuities.