Volume 29, Issue 4

In this paper, we propose the r - d class predictors which are general predictors of the best linear unbiased predictor (BLUP), the principal components regression (PCR) and the Liu predictors in the linear mixed models. Superiorities of the linear combination of the new predictors to each of these predictors are done in the sense of the mean square error matrix criterion. Finally, numerical examples and a simulation study are done to illustrate the findings.

Stability estimates of Hölder type for weak solutions to backward one-dimensional semi-linear parabolic equations with locally Lipschitz source are obtained. It is noticed that stability results for weak solutions to nonlinear inverse problems are very rare in the literature.

Parameter estimation in chaotic dynamical systems is an important and practical issue. Nevertheless, the high-dimensionality and the sensitive dependence on initial conditions typically makes the problem difficult to solve. In this paper, we propose an innovative parameter estimation approach, utilizing numerical differentiation for observation data preprocessing. Given plenty of noisy observations on a portion of dependent variables, numerical differentiation allows them and their derivatives to be accurately approximated. Substituting those approximations into the original system can effectively simplify the parameter estimation problem. As an example, we consider parameter estimation in the well-known Lorenz model given partial noisy observations. According to the Lorenz equations, the estimated parameters can be simply given by least squares regression using the approximated functions provided by data preprocessing. Numerical examples show the effectiveness and accuracy of our method. We also prove the uniqueness and stability of the solution.

In this manuscript, a purely data-driven statistical regularization method is proposed for extracting the information from big data with randomly distributed noise. Since the variance of the noise may be large, the method can be regarded as a general data preprocessing method in ill-posed problems, which is able to overcome the difficulty that the traditional regularization method is unable to solve, and has superior advantage in computing efficiency. The unique solvability of the method is proved, and a number of conditions are given to characterize the solution. The regularization parameter strategy is discussed, and the rigorous upper bound estimation of the confidence interval of the error in the L2L^{2} norm is established. Some numerical examples are provided to illustrate the appropriateness and effectiveness of the method.

We propose a data and knowledge driven approach for SPECT by combining a classical iterative algorithm of SPECT with a convolutional neural network. The classical iterative algorithm, such as ART and ML-EM, is employed to provide the model knowledge of SPECT. A modified U-net is then connected to exploit further features of reconstructed images and data sinograms of SPECT. We provide mathematical formulations for the architecture of the proposed networks. The networks are trained by supervised learning using the technique of mini-batch optimization. We apply the trained networks to the problems of simulated lung perfusion imaging and simulated myocardial perfusion imaging, and numerical results demonstrate their effectiveness of reconstructing source images from noisy data measurements.

The paper contributes to the clarification of the mechanism of one-dimensional pulsating detonation wave propagation for the transition regime with two-scale pulsations. For this purpose, a novel numerical algorithm has been developed for the numerical investigation of the gaseous pulsating detonation wave using the two-stage model of kinetics of chemical reactions in the shock-attached frame. The influence of grid resolution, approximation order and the type of rear boundary conditions on the solution has been studied for four main regimes of detonation wave propagation for this model. Comparison of dynamics of pulsations with results of other authors has been carried out.

We consider the Sturm–Liouville operator on quantum graphs with a loop with the standard matching conditions in the internal vertex and the jump conditions at the boundary vertex. Given the potential on the loop, we try to recover the potential on the boundary edge from the subspectrum. The uniqueness theorem and a constructive algorithm for the solution of this partial inverse problem are provided.

Limited-angle computed tomography (CT) reconstruction problem arises in some practical applications due to restrictions in the scanning environment or CT imaging device. Some artifacts will be presented in image reconstructed by conventional analytical algorithms. Although some regularization strategies have been proposed to suppress the artifacts, such as total variation (TV) minimization, there is still distortion in some edge portions of image. Guided image filtering (GIF) has the advantage of smoothing the image as well as preserving the edge. To further improve the image quality and protect the edge of image, we propose a coupling method, that combines ℓ0{\ell_{0}} gradient minimization and GIF. An intermediate result obtained by ℓ0{\ell_{0}} gradient minimization is regarded as a guidance image of GIF, then GIF is used to filter the result reconstructed by simultaneous algebraic reconstruction technique (SART) with nonnegative constraint. It should be stressed that the guidance image is dynamically updated as the iteration process, which can transfer the edge to the filtered image. Some simulation and real data experiments are used to evaluate the proposed method. Experimental results show that our method owns some advantages in suppressing the artifacts of limited angle CT and in preserving the edge of image.

This paper tries to examine the recovery of the time-dependent implied volatility coefficient from market prices of options for the time fractional Black–Scholes equation (TFBSM) with double barriers option. We apply the linearization technique and transform the direct problem into an inverse source problem. Resultantly, we get a Volterra integral equation for the unknown linear functional, which is then solved by the regularization method. We use L1{L_{1}}-forward difference implicit approximation for the forward problem. Numerical results using L1{L_{1}}-forward difference implicit approximation (L1{L_{1}}-FDIA) for the inverse problem are also discussed briefly.

We consider dynamic inverse problems for a dynamical system associated with a finite Jacobi matrix and for a system describing propagation of waves in a finite Krein–Stieltjes string. We offer three methods of recovering unknown parameters: entries of a Jacobi matrix in the first problem and point masses and distances between them in the second, from dynamic Dirichlet-to-Neumann operators. We also answer a question on a characterization of dynamic inverse data for these two problems.

We propose a method for increasing the speed and spectral efficiency of information transmission over soliton communication lines, based on algorithms for solving inverse and direct scattering problems, in the frame of a modern soliton orthogonal frequency division multiplexing (SOFDM) approach. The proposed method uses a simultaneous phase and frequency coding. This phase–frequency coding method retains all the technological advantages of the SOFDM method, and due to the additional frequency coding, a noticeable increase in the information transfer rate over the soliton optical lines is expected. Numerical modeling confirmed the proposed method for telecommunication applications’ prospects.