Interior-Point Gradient Method for Large-Scale Totally Nonnegative Least Squares Problems

We study an interior-point gradient method for solving a class of so-called totally nonnegative least-squares problems. At each iteration, the method decreases the residual norm along a diagonally-scaled negative gradient direction with a special scaling. We establish the global convergence of the method and present some numerical examples to compare the proposed method with a few similar methods including the affine scaling method.This author was supported in part by DOE/LANL Contract 03891-99-23This author was supported in part by NSF Grant DMS-0442065     

Use of the Minimum-Norm Search Direction in a Nonmonotone Version of the Gauss-Newton Method

In this work, a new stabilization scheme for the Gauss-Newton method is defined, where the minimum norm solution of the linear least-squares problem is normally taken as search direction and the standard Gauss-Newton equation is suitably modified only at a subsequence of the iterates. Moreover, the stepsize is computed by means of a nonmonotone line search technique. The global convergence of the proposed algorithm model is proved under standard assumptions and the superlinear rate of convergence is ensured for the zero-residual case. A specific implementation algorithm is described, where the use of the pure Gauss-Newton iteration is conditioned to the progress made in the minimization process by controlling the stepsize. The results of a computational experimentation performed on a set of standard test problems are reported.     

Determination of the orientation distribution function of anisotropic paramagnetic species from the angular dependence of the ESR spectra

A method for determination of the orientation distribution function of anisotropic paramagnetic species from the angular dependence of the ESR spectra based on simulation of the spectra was developed. The orientation distribution function is represented as a sum of orthonormal functions. The weight coefficients are determined by minimizing the deviations of the theoretical spectra from experimental ones. The method was numerically tested. The method is relatively insensitive to the presence of noise and foreign ESR signals not related to the sample orientation in the magnetic field.__________Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 12, pp. 2595–2601, December, 2004.     

部分线性变系数模型中估计的渐进正态性

作为部分线性模型与变系数模型的推广,部分线性变系数模型是一类应用非常广泛的模型,本文基于Profile最小二乘方法给出了模型中参数分量与非参数分量的估计,并在异方差情形下证明了这些估计的渐进正态性.     

Curve fitting and identification of physical spectra

A modification of the trust-region Gauss-Newton method for the identification of physical spectra is described and analysed. Local convergence results are presented.     

A simple Marquardt algorithm for the nonlinear least-squares problem

A new implementation of the Marquardt method for the nonlinear least-squares problem is presented. The algorithm is very simple but its performance with nine test functions is at least comparable with either Davidon-Fletcher-Powell's method or Moré's adaptive Marquardt method.     

The approximation of the Maxwell eigenvalue problem using a least-squares method

In this paper we consider an approximation to the Maxwell's eigenvalue problem based on a very weak formulation of two div-curl systems with complementary boundary conditions. We formulate each of these div-curl systems as a general variational problem with different test and trial spaces, i.e., the solution space is and components in the test spaces are in subspaces of , the Sobolev space of order one on the computational domain . A finite-element least-squares approximation to these variational problems is used as a basis for the approximation. Using the structure of the continuous eigenvalue problem, a discrete approximation to the eigenvalues is set up involving only the approximation to either of the div-curl systems. We give some theorems that guarantee the convergence of the eigenvalues to those of the continuous problem without the occurrence of spurious values. Finally, some results of numerical experiments are given.

     

A stochastic framework for recursive computation of spline functions: Part II,smoothing splines

Many of the optimal curve-fitting problems arising in approximation theory have the same structure as certain estimation problems involving random processes. We develop this structural correspondence for the problem of smoothing inaccurate data with splines and show that the smoothing spline is a sample function of a certain linear least-squares estimate. Estimation techniques are then used to derive a recursive algorithm for spline smoothing.     

Application of compressed sensing to in vivo 3D ¹⁹F CSI

This study shows how applying compressed sensing (CS) to (19)F chemical shift imaging (CSI) makes highly accurate and reproducible reconstructions from undersampled datasets possible. The missing background signal in (19)F CSI provides the required sparsity needed for application of CS. Simulations were performed to test the influence of different CS-related parameters on reconstruction quality. To test the proposed method on a realistic signal distribution, the simulation results were validated by ex vivo experiments. Additionally, undersampled in vivo 3D CSI mouse datasets were successfully reconstructed using CS. The study results suggest that CS can be used to accurately and reproducibly reconstruct undersampled (19)F spectroscopic datasets. Thus, the scanning time of in vivo(19)F CSI experiments can be significantly reduced while preserving the ability to distinguish between different (19)F markers. The gain in scan time provides high flexibility in adjusting measurement parameters. These features make this technique a useful tool for multiple biological and medical applications.