Modifications of Steepest Descent Method and Conjugate Gradient Method Against Noise for Ill-posed Linear Systems

It is well known that the numerical algorithms of the steepest descent method (SDM), and the conjugate gradient method (CGM) are effective for solving well-posed linear systems. However, they are vulnerable to noisy disturbance for solving ill-posed linear systems. We propose the modifications of SDM and CGM, namely the modified steepest descent method (MSDM), and the modified conjugate gradient method (MCGM). The starting point is an invariant manifold defined in terms of a minimum functional and a fictitious time-like variable; however, in the final stage we can derive a purely iterative algorithm including an acceleration parameter. Through the Hopf bifurcation, this parameter indeed plays a major role to switch the situation of slow convergence to a new situation that the functional is stepwisely decreased very fast. Several numerical examples are examined and compared with exact solutions, revealing that the new algorithms of MSDM and MCGM have good computational efficiency and accuracy, even for the highly ill-conditioned linear equations system with a large noise being imposed on the given data

The Solution of SO

In many applications we need to solve an orthogonal transformation tensor Q∈SO(3) from a tensorial equation Q˙ = WQ under a given spin history W. In this paper, we address some interesting issues about this equation. A general solution of Q is obtained by transforming the governing equation into a new one in the space of ℝP3. Then, we develop a novel method to solve Q in terms of a single parameter, whose governing equation is a single nonlinear ordinary differential equation (ODE)

Solving an Inverse Sturm-Liouville Problem by a Lie-Group Method

Solving an inverse Sturm-Liouville problem requires a mathematical process to determine unknown function in the Sturm-Liouville operator from given data in addition to the boundary values. In this paper, we identify a Sturm-Liouville potential function by using the data of one eigenfunction and its corresponding eigenvalue, and identify a spatial-dependent unknown function of a Sturm-Liouville differential operator. The method we employ is to transform the inverse Sturm-Liouville problem into a parameter identification problem of a heat conduction equation. Then a Lie-group estimation method is developed to estimate the coefficients in a system of ordinary differential equations discretized from the heat conduction equation. Numerical tests confirm the accuracy and efficiency of present approach. Definite and random disturbances are also considered when comparing the present method with that by using a technique of numerical differentiation

Solving an inverse Sturm-Liouville problem by a Lie-group method

Solving an inverse Sturm-Liouville problem requires a mathematical process to determine unknown function in the Sturm-Liouville operator from given data in addition to the boundary values. In this paper, we identify a Sturm-Liouville potential function by using the data of one eigenfunction and its corresponding eigenvalue, and identify a spatial-dependent unknown function of a SturmLiouville differential operator. The method we employ is to transform the inverse Sturm-Liouville problem into a parameter identification problem of a heat conduction equation. Then a Lie-group estimation method is developed to estimate the coefficients in a system of ordinary differential equations discretized from the heat conduction equation. Numerical tests confirm the accuracy and efficiency of present approach. Definite and random disturbances are also considered when comparing the present method with that by using a technique of numerical differentiation

Solving Inverse Conductivity Problems In Doubly Connected Domains By the Homogenization Functions of Two Parameters

In the paper, we make the first attempt to derive a family of two-parameter homogenization functions in the doubly connected domain, which is then applied as the bases of trial solutions for the inverse conductivity problems. The expansion coefficients are obtained by imposing an extra boundary condition on the inner boundary, which results in a linear system for the interpolation of the solution in a weighted Sobolev space. Then, we retrieve the spatial- or temperature-dependent conductivity function by solving a linear system, which is obtained from the collocation method applied to the nonlinear elliptic equation after inserting the solution. Although the required data are quite economical, very accurate solutions of the space-dependent and temperature-dependent conductivity functions, the Robin coefficient function and also the source function are available. It is significant that the nonlinear inverse problems can be solved directly without iterations and solving nonlinear equations. The proposed method can achieve accurate results with high efficiency even for large noise being imposed on the input data

Point contact Andreev reflection spectroscopy of NdFeAsO_0.85

The newly discovered oxypnictide family of superconductors show very high critical temperatures of up to 55K. Whilst there is growing evidence that suggests a nodal order parameter, point contact Andreev reflection spectroscopy can provide crucial information such as the gap value and possibly the number of energy gaps involved. For the oxygen deficient NdFeAsO0.85 with a Tc of 45.5K, we show that there is clearly a gap value at 4.2K that is of the order of 7meV, consistent with previous studies on oxypnictides with lower Tc. Additionally, taking the spectra as a function of gold tip contact pressure reveals important changes in the spectra which may be indicative of more complex physics underlying this structure.Comment: 11 pages, 3 figures. New references included, extra discussion. This version is accepted in Superconductor Science and Technolog

A Third-Order p-Laplacian Boundary Value Problem Solved by an SL(3,ℝ) Lie-Group Shooting Method

The boundary layer problem for power-law fluid can be recast to a third-order p-Laplacian boundary value problem (BVP). In this paper, we transform the third-order p-Laplacian into a new system which exhibits a Lie-symmetry SL(3,ℝ). Then, the closure property of the Lie-group is used to derive a linear transformation between the boundary values at two ends of a spatial interval. Hence, we can iteratively solve the missing left boundary conditions, which are determined by matching the right boundary conditions through a finer tuning of r∈[0,1]. The present SL(3,ℝ) Lie-group shooting method is easily implemented and is efficient to tackle the multiple solutions of the third-order p-Laplacian. When the missing left boundary values can be determined accurately, we can apply the fourth-order Runge-Kutta (RK4) method to obtain a quite accurate numerical solution of the p-Laplacian