Linear Regularization Method Research Articles

A linear regularization method for a parameter identification problem in heat equation

Abstract Recently, Nair and Roy (2017) considered a linear regularization method for a parameter identification problem in an elliptic PDE. In this paper, we consider similar procedure for identifying the diffusion coefficient in the heat equation, modifying the Sobolev spaces involved appropriately. We derive error estimates under appropriate conditions and also consider the finite-dimensional realization of the method, which is essential for practical application. In the analysis of finite-dimensional realization, we give a procedure to obtain finite-dimensional subspaces of an infinite-dimensional Hilbert space L 2 ⁢ ( 0 , T ; H 1 ⁢ ( Ω ) ) {L^{2}(0,T;H^{1}(\Omega))} by doing double discretization, that is, discretization corresponding to both the space and time domain. Also, we analyze the parameter choice strategy and obtain an a posteriori parameter which is order optimal.

Performance of the CMS Zero Degree Calorimeters in the 2016 pPb run

Two neutral particle detectors, Zero Degree Calorimeters (ZDCs) at the LHC-CMS experiment, cover the |η| > 8.5 region. The ZDCs are Cherenkov calorimeters that use tungsten as the absorber and quartz clad quartz fibers as the active medium. They have a five element electromagnetic section followed by a hadronic section divided into four depth segments. For the 2016 pPb run, the ZDCs were calibrated using test beam data and the single spectator neutron peak at 2.56 TeV. Peaks corresponding to 1, 2 and 3 neutrons are visible in the ZDC total signal distribution. The effect of pileup is corrected by a Fourier deconvolution method. Using this, the spectator neutron number distribution can be unfolded by a linear regularization method. This information serves as a strong constraint to models of pPb collisions and has the potential to produce an unbiased measure of centrality in pPb collisions.

Parameter identification for continuous point emission source based on Tikhonov regularization method coupled with particle swarm optimization algorithm

In order to identify the parameters of hazardous gas emission source in atmosphere with less previous information and reliable probability estimation, a hybrid algorithm coupling Tikhonov regularization with particle swarm optimization (PSO) was proposed. When the source location is known, the source strength can be estimated successfully by common Tikhonov regularization method, but it is invalid when the information about both source strength and location is absent. Therefore, a hybrid method combining linear Tikhonov regularization and PSO algorithm was designed. With this method, the nonlinear inverse dispersion model was transformed to a linear form under some assumptions, and the source parameters including source strength and location were identified simultaneously by linear Tikhonov-PSO regularization method. The regularization parameters were selected by L-curve method. The estimation results with different regularization matrixes showed that the confidence interval with high-order regularization matrix is narrower than that with zero-order regularization matrix. But the estimation results of different source parameters are close to each other with different regularization matrixes. A nonlinear Tikhonov-PSO hybrid regularization was also designed with primary nonlinear dispersion model to estimate the source parameters. The comparison results of simulation and experiment case showed that the linear Tikhonov–PSO method with transformed linear inverse model has higher computation efficiency than nonlinear Tikhonov-PSO method. The confidence intervals from linear Tikhonov-PSO are more reasonable than that from nonlinear method. The estimation results from linear Tikhonov-PSO method are similar to that from single PSO algorithm, and a reasonable confidence interval with some probability levels can be additionally given by Tikhonov-PSO method. Therefore, the presented linear Tikhonov-PSO regularization method is a good potential method for hazardous emission source parameters identification.

Adaptive error estimation technique of the Trefftz method for solving the over-specified boundary value problem

In this paper, numerical solutions are investigated based on the Trefftz method for an over-specified boundary value problem contaminated with artificial noise. The main difficulty of the inverse problem is that divergent results occur when the boundary condition on over-specified boundary is contaminated by artificial random errors. The mechanism of the unreasonable result stems from its ill-posed influence matrix. The accompanied ill-posed problem is remedied by using the Tikhonov regularization technique and the linear regularization method, respectively. This remedy will regularize the influence matrix. The optimal parameter λ of the Tikhonov technique and the linear regularization method can be determined by adopting the adaptive error estimation technique. From this study, convergent numerical solutions of the Trefftz method adopting the optimal parameter can be obtained. To show the accuracy of the numerical solutions, we take the examples as numerical examination. The numerical examination verifies the validity of the adaptive error estimation technique. The comparison of the Tikhonov regularization technique and the linear regularization method was also discussed in the examples.

Influence of the fine structure of glass forming materials on the overlapping of the alpha and secondary relaxations

The dielectric behavior of a set of poly benzyl methacrylates with methyl, fluorine, chlorine and difluorine substitutes in the phenyl ring is reported. Attention is paid to the deconvolution of overlapping processes with the aim of obtaining a deeper insight into the influence of the fine chemical structure on the splitting of the α and β relaxations. Deconvolutions are carried out on the retardation time spectra calculated from the dielectric loss in the frequency domain using the linear regularization method. Some difficulties associated with the detection of the β relaxation of this set of polymers were resolved by means of the thermostimulated depolarization currents experiments.

Determining axial fuel-rod power-density profiles from in-core neutron flux measurements

A method is proposed for determining power-density profiles in nuclear reactor fuel rods from neutron flux measurements obtained by multiple in-core micropocket fission detectors (MPFD). A general formulation is presented that relates the power-density profiles in a reactor's fuel rods to the flux density at detector locations. Key to this formulation is the construction of an appropriate response function that relates the flux at any position in the core to the rate at which neutrons are born at different depths in the fuel rods. In this preliminary study simple, although not unrealistic, response functions for thermal neutrons and for fast neutrons are derived and used to illustrate the analysis methods. To find the unknown power-density profiles in the fuel rods, given the measured fluxes at various locations through the core, requires inversion of an ill-posed Fredholm integral equation. With numerical quadrature, this integral equation can be converted into a (generally, underdetermined) set of algebraic equations for the power densities at various axial positions in the fuel rods. To solve these underdetermined equations, the linear regularization method is employed. Results are presented for simulated data for a single fuel rod and for multiple rods in various configurations. This preliminary study indicates that in-core flux measurements can indeed be used to infer power-density profiles in reactor fuel rods.

Inexact Newton Regularization Using Conjugate Gradients as Inner Iteration

In our papers [Inverse Problems, 15 (1999), pp. 309--327] and [Numer. Math., 88 (2001), pp. 347--365] we proposed algorithm {\tt REGINN}, an inexact Newton iteration for the stable solution of nonlinear ill-posed problems. {\tt REGINN} consists of two components: the outer iteration, which is a Newton iteration stopped by the discrepancy principle, and an inner iteration, which computes the Newton correction by solving the linearized system. The convergence analysis presented in both papers covers virtually any linear regularization method as inner iteration, especially Landweber iteration, $\nu$-methods, and Tikhonov--Phillips regularization. In the present paper we prove convergence rates for {\tt REGINN} when the conjugate gradient method, which is nonlinear, serves as inner iteration. Thereby we add to a convergence analysis of {Hanke}, who had previously investigated {\tt REGINN} furnished with the conjugate gradient method [Numer. Funct. Anal. Optim., 18 (1997), pp. 971--993]. By numerical experiments we illustrate that the conjugate gradient method outperforms the $\nu$-method as inner iteration.

Determination of the linear mass power spectrum from the mass function of galaxy clusters

We develop a new method to determine the linear mass power spectrum using the mass function of galaxy clusters. We obtain the rms mass fluctuation a (M) using the expression for the mass function in the Press & Schechter, Sheth, Mo & Tormen and Jenkins et al. formalisms. We apply different techniques to recover the adimensional power spectrum Δ 2 (k) from a(M) namely the k e f f approximation, the singular value decomposition and the linear regularization method. The application of these techniques to the rCDM and ACDM GIF simulations shows a high efficiency in recovering the theoretical power spectrum over a wide range of scales. We compare our results with those derived from the power spectrum of the spatial distribution of the same sample of clusters in the simulations obtained by application of the classical Feldman, Kaiser & Peacock (FKP) method. We find that the mass function based method presented here can provide a very accurate estimate of the linear power spectrum, particularly for low values of k. This estimate is comparable to, or even better behaved than, the FKP solution. The principal advantage of our method is that it allows the determination of the linear mass power spectrum using the joint information of objects of a wide range of masses without dealing with specific assumptions on the bias relative to the underlying mass distribution.

ECHO MAPPING OF AGN BROAD-LINE REGIONS: FUNDAMENTAL ALGORITHMS

ABSTRACT We formulate and test a series of algorithms for echo mapping the emission-line regions near active galactic nuclei from measurements of correlated variability in their line and continuum light curves. The Linear Regularization Method (LRM) employs a direct inversion of evenly-spaced light curve data, with a regularization parameter that can be used to control the trade-off between noise and resolution. Matrix formulae express the formal solution as well as its variance and co-variance in terms of uncertainties in the measurements. Unlike the maximum entropy method (MEM), LRM applies to kernels with both positive and negative values, but the results are somewhat limited by ringing effects. A positivity constraint proves effective in controlling the ringing. MEM combines regularization and positivity in a natural way, but similar results are also found using positivity constraints with non-entropic regularization functions. Direct inversions of unevenly-sampled light curves require interpolating the noisy data. In this case better results are found by solving for both the continuum light curve and kernel function in a simultaneous fit to the data. Our conclusion is that while echo mapping currently gives ambiguous results, the algorithms are not the limiting factor. Progress depends on efforts to increase the accuracy and completeness of sampling of the observed light curves.