Optimization of the Neutron Spectrum Unfolding Algorithm Using Shifted Legendre Polynomials Based on Weighted Tikhonov Regularization

The well-known approach to solve the ill-posed problem is Tikhonov regularization scheme. But, the approximate solution of Tikhonov scheme may not contain all the details of the exact solution. To circumference this problem, weighted Tikhonov regularization has been introduced. In this article, we propose two a posteriori parameter choice rules to choose the regularization parameter for weighted Tikhonov regularization and establish the optimal rate of convergence $$O(\delta ^\frac{\alpha +1}{\alpha +2})$$ for the scheme based on these proposed rules. The numerical results are documented to demonstrate the significance of the theoretical results.

One of the most popular feature extraction algorithms for brain-computer interfaces (BCI) is common spatial patterns (CSPs). Despite its known efficiency and widespread use, CSP is also known to be very sensitive to noise and prone to overfitting. To address this issue, it has been recently proposed to regularize CSP. In this paper, we present a simple and unifying theoretical framework to design such a regularized CSP (RCSP). We then present a review of existing RCSP algorithms and describe how to cast them in this framework. We also propose four new RCSP algorithms. Finally, we compare the performances of 11 different RCSP (including the four new ones and the original CSP), on electroencephalography data from 17 subjects, from BCI competition datasets. Results showed that the best RCSP methods can outperform CSP by nearly 10% in median classification accuracy and lead to more neurophysiologically relevant spatial filters. They also enable us to perform efficient subject-to-subject transfer. Overall, the best RCSP algorithms were CSP with Tikhonov regularization and weighted Tikhonov regularization, both proposed in this paper.

Computation of control for linear approximately controllable system using weighted Tikhonov regularization

To overcome ill-posed problem of force identification in the Transfer Path Analysis (TPA) of a rocket engine, this paper proposes a novel TPA based on weighted Tikhonov regularization. Firstly, the upper bound of relative force identification error of the traditional TPA is derived. Secondly, weighting matrix and Bayesian theory are adopted to improve the accuracy of force identification, and then the theory of the improved TPA is built. Finally, a ground vibration testing of the rocket engine is performed to analyze its path contributions. The results show that the proposed TPA is more accurate than the traditional TPA.

Design of experiments for discrete ill-posed problems is a relatively new area of research. While there has been some limited work concerning the linear case, little has been done to study design criteria and numerical methods for ill-posed nonlinear problems. We present an algorithmic framework for nonlinear experimental design with an efficient numerical implementation. The data are modeled as indirect, noisy observations of the model collected via a set of plausible experiments. An inversion estimate based on these data is obtained by a weighted Tikhonov regularization whose weights control the contribution of the different experiments to the data misfit term. These weights are selected by minimization of an empirical estimate of the Bayes risk that is penalized to promote sparsity. This formulation entails a bilevel optimization problem that is solved using a simple descent method. We demonstrate the viability of our design with a problem in electromagnetic imaging based on direct current resistivity and magnetotelluric data.

Study on iterative regularization method and application to neutron spectrum unfolding of multi-sphere spectrometer measurement

The criteria for selecting a method for unfolding neutron spectra based on the information entropy theory

Regularized heuristic method for activation monitor neutron spectrum unfolding

Development and experimental validation of a fast neutron spectrometry system based on Superheated Drop Detectors (SDDs) operating under different external pressures

To develop a physics-based approach to improve the accuracy and robustness of the ill-conditioned problem of unfolding megavoltage bremsstrahlung spectra from transmission data. Spectra are specified using a rigorously-benchmarked functional form. Since ion chambers are the typical detector used in transmission measurements, the energy response of a Farmer chamber is calculated using the EGSnrc Monte Carlo code, and the effect of approximating the energy response on the accuracy of the unfolded spectra is studied. A proposal is introduced to enhance spectral sensitivity by combining transmission data measured with multiple detectors of different energy response and by combining data from multiple attenuating materials. Monte Carlo methods are developed to correct for nonideal exponential attenuation (e.g., scatter effects and secondary attenuation). The performance of the proposed methods is evaluated for a diverse set of validated clinical spectra (3.5-25 MV) using analytical transmission data with simulated experimental noise. The approximations commonly used in previous studies for the ion-chamber energy response lead to significant errors in the unfolded spectra. Of the configurations studied, the one with best spectral sensitivity is to measure four full transmission curves using separate low-Z and high-Z attenuators in conjunction with two detectors of different energy response (the authors propose a Farmer-type ion chamber, once with a low-Z, and once with a high-Z buildup cap material), then to feed the data simultaneously to the unfolding algorithm. Deviations from ideal exponential attenuation are as much as 1.5% for the smallest transmission signals, and the proposed methods properly correct for those deviations. The transmission data with enhanced spectral sensitivity, combined with the accurate and flexible spectral functional form, lead to robust unfolding without requiring a priori knowledge of the spectrum. Compared with the commonly-used methods, the accuracy is improved for the unfolded spectra and for the unfolded mean incident electron kinetic energy by at least factors of three and four, respectively. With simulated experimental noise and a lowest transmission of 1%, the unfolded energy fluence spectra agree with the original spectra with a normalized root-mean-square deviation, %Δ(ψ), of 2.3%. The unfolded mean incident electron kinetic energies agree, on average, with the original values within 1.4%. A lowest transmission of only 10% still allows unfolding with %Δ(ψ) of 3.3%. In the presence of realistic experimental noise, the proposed approach significantly improves the accuracy and robustness of the spectral unfolding problem for all therapy and MV imaging beams of clinical interest.

PurposeThe purpose of the paper is to extend the differential quadrature method (DQM) for solving time and space fractional non-linear partial differential equations on a semi-infinite domain.Design/methodology/approachThe proposed method is the combination of the Legendre polynomials and differential quadrature method. The authors derived and constructed the new operational matrices for the fractional derivatives, which are used for the solutions of non-linear time and space fractional partial differential equations.FindingsThe fractional derivative of Lagrange polynomial is a big hurdle in classical DQM. To overcome this problem, the authors represent the Lagrange polynomial in terms of shifted Legendre polynomial. They construct a transformation matrix which transforms the Lagrange polynomial into shifted Legendre polynomial of arbitrary order. Then, they obtain the new weighting coefficients matrices for space fractional derivatives by shifted Legendre polynomials and use these in conversion of a non-linear fractional partial differential equation into a system of fractional ordinary differential equations. Convergence analysis for the proposed method is also discussed.Originality/valueMany engineers can use the presented method for solving their time and space fractional non-linear partial differential equation models. To the best of the authors’ knowledge, the differential quadrature method has never been extended or implemented for non-linear time and space fractional partial differential equations.

This article is devoted to developing an accurate operational matrix (OM) method for the solution of a new category of nonlinear optimal control problems (OCPs) explained by fractal‐fractional dynamical systems. The fractal‐fractional differentiation is considered in the sense of Atangana‐Riemann‐Liouville. The method is based on the shifted Legendre polynomials (LPs). Through the way, a new OM of fractal‐fractional differentiation is extracted for the shifted LPs. The optimality conditions are converted to an algebraic system of equations by using the shifted LP expansions of the state and control variables and applying the method of constrained extrema. More precisely, these variables are approximated by the shifted LPs with undetermined coefficients. Then, these expansions are replaced in the performance index, and the Gauss‐Legendre integration method is used to approximate the performance index for extracting a nonlinear algebraic equation. In the sequel, the mentioned OM and the collocation method are used to generate some algebraic equations from the dynamical system. Finally, the method of the constrained extrema is used by coupling the algebraic constraints generated from the dynamical system and the initial condition with the algebraic equation elicited from the performance index by a set of unknown Lagrange multipliers. The accuracy of the method is studied by solving some numerical examples.

A novel MLEM stopping criterion for unfolding neutron fluence spectra in radiation therapy

An adaptive deviation-resistant neutron spectrum unfolding method based on transfer learning

A time–space spectral tau method for the time fractional cable equation and its inverse problem