Volume 32, Issue 6

The Bell regression model (BRM), a member of the generalized linear models (GLMs), can be used when the dependent variable consists of overdispersed count data. The maximum likelihood estimator (MLE) is generally used to estimate unknown regression coefficients. The major drawback of the MLE is an inflated variance when multicollinearity problems occur. In this study, we proposed a new biased estimator to cope with the multicollinearity in the BRM. The simulation study is conducted to illustrate the performance of the proposed estimator over the MLE and Bell ridge estimator (BRE). Also, we give real data examples to approve the applicability of the proposed estimator in real data problems.

The characterization of the subsurface profile properties that are susceptible to liquefaction during earthquakes is of great importance in accident prevention. Field data can be used in a stochastic inverse problem to estimate the material properties using Markov chain Monte Carlo Methods (McMC) that incorporates prior knowledge about the unknown parameters as well as the data available. Here, we numerically study the Random Walk (RW) and Differential Evolution (DE) variations of the Metropolis algorithm in the context of seismic modeling considering the viscoelastic wave equation. The algorithms are tested with data from the 1987 Superstition Hills earthquake recorded at the Wildlife Site. The results show the effectiveness and accuracy of these algorithms, with emphasis to DE which reached convergence earlier compared to RW.

We consider an ensemble Kalman inversion scheme for inverse elastic scattering problems in which the unknown quantity is the shape of the scatterer. Assume that the scatterer is a piecewise constant function with known value inside inhomogeneities. The level set method is described as an implicit representation of the scatterer boundary, with Gaussian random fields serving as prior to provide information on the level set functions. The ensemble Kalman filter method is then employed based on the level set functions to reconstruct the shape of the scatterer. We demonstrate the effectiveness of the proposed method using several numerical examples.

Inverse problems are commonly used in many fields as they enable the estimation of parameters that cannot be experimentally measured. However, the complex nature of inverse problems requires a strong background in data and signal processing. Moreover, ill-posed problems yield parameters that have a strong linear dependence on the problem. The ill-posed nature of these problems lead to many errors in numerical computations that can make parameter identification nearly impossible. In this paper, a new data processing tool is proposed to maximize the sensitivity of the model to the parameters of interest, while reducing the correlation between them. The effectiveness of the toll is demonstrated through a given inverse problem example using Periodically Pulsed Photothermal Radiometry (PPTR).

In this paper, we formulate the modified iteratively regularized Landweber iteration method in Banach spaces to solve the inverse problems for which the forward operator may be smooth or non-smooth. We study the convergence analysis of the modified method for both the perturbed as well as unperturbed data by utilizing the Hölder stability estimates. In the presence of perturbed data, we terminate the method via a discrepancy principle and show that it is in fact a convergence regularization method that terminates within a few iterations. In the presence of unperturbed data, we show that the iterates converge to the exact solution. Additionally, we deduce the convergence rates in the presence of perturbed as well as unperturbed data. Finally, we discuss two inverse problems on which the method is applicable.

Sparse identification of nonlinear dynamics is a popular approach to system identification. In this approach system identification is reformulated as a sparse regression problem, and the use of a good sparse regression method is crucial. Sparse Bayesian learning based on collaborative neurodynamic optimization is a recent method that consistently produces high-quality solutions. In this article, we extensively assess how this method performs for ordinary differential equation identification. We find that it works very well compared with sparse regression algorithms currently used for this task in terms of the tradeoff between the approximation accuracy and the complexity of the identified system. We also propose a way to substantially reduce the computational complexity of this algorithm compared with its original implementation, thus making it even more practical.

At present, computationally intensive numerical weather prediction systems based on physics equations are widely used for short-term weather forecasting. In this paper, we investigate the potential of accelerating the Weather Research and Forecasting (WRF-ARW) model using machine learning techniques. Two main approaches are considered. First, we assess the viability of complete replacing the numerical weather model with deep learning models, capable of predicting the full range forecast directly from basic initial data. Second, we consider a “super-resolution” technique involving low-resolution WRF computation and a machine learning based downscaling using coarse-grid forecast for conditioning. The process of downscaling is intrinsically an ill-posed problem. In both categories, several prominent and promising machine learning methods are evaluated and compared on real data from a variety of sources. for the Moscow region Namely, in addition to the ground truth WRF forecasts that were utilized for training, we compare the model predictions against ERA5 reanalysis and measurements from local weather stations. We show that deep learning approaches can be successfully applied to accelerate a numerical model and even produce more realistic forecasts in other aspects. As a practical outcome, this study offers empirically validated guidance for the selection and application of deep learning methods to accelerate the computation of detailed short-term atmospheric forecasts tailored to specific needs.

In this paper, we consider the simplified Levenberg–Marquardt method for nonlinear ill-posed inverse problems in Hilbert spaces for obtaining stable approximations of solutions to the ill-posed nonlinear equations of the form F ⁢ ( u ) = y {F(u)=y} , where F : 𝒟 ⁢ ( F ) ⊂ 𝖴 → 𝖸 {F:\mathcal{D}(F)\subset\mathsf{U}\to\mathsf{Y}} is a nonlinear operator between Hilbert spaces 𝖴 {\mathsf{U}} and 𝖸 {\mathsf{Y}} . The method is defined as follows: u n + 1 δ = u n δ - ( T 0 ∗ ⁢ T 0 + α n ⁢ I ) - 1 ⁢ T 0 ∗ ⁢ ( F ⁢ ( u n δ ) - y δ ) , u_{n+1}^{\delta}=u_{n}^{\delta}-(T_{0}^{\ast}T_{0}+\alpha_{n}I)^{-1}T_{0}^{% \ast}(F(u_{n}^{\delta})-y^{\delta}), where T 0 = F ′ ⁢ ( u 0 ) {T_{0}=F^{\prime}(u_{0})} and T 0 ∗ = F ′ ⁢ ( u 0 ) ∗ {T_{0}^{\ast}=F^{\prime}(u_{0})^{\ast}} . Here F ′ ⁢ ( u 0 ) {F^{\prime}(u_{0})} denotes the Frèchet derivative of F at an initial guess u 0 ∈ 𝒟 ⁢ ( F ) {u_{0}\in\mathcal{D}(F)} for the exact solution u † {u^{\dagger}} , F ′ ⁢ ( u 0 ) ∗ {F^{\prime}(u_{0})^{\ast}} is the adjoint of F ′ ⁢ ( u 0 ) {F^{\prime}(u_{0})} and { α n } {\{\alpha_{n}\}} is an a priori chosen sequence of non-negative real numbers satisfying suitable properties. We use Morozov-type stopping rule to terminate the iterations. Under suitable non-linearity conditions on operator F , we show convergence of the method and also obtain a convergence rate result under a Hölder-type source condition on the element u 0 - u † {u_{0}-u^{\dagger}} . Furthermore, we derive convergence of the method for the case when no source conditions are used and the study concludes with numerical examples which validate the theoretical conclusions.

We study the inverse spectral problems of recovering Sturm–Liouville differential operator with two constant delays a 1 {a_{1}} and a 2 {a_{2}} greater than one third of the interval. It has been proved that the operator can be recovered uniquely from four spectra under the condition 2 ⁢ a 1 + a 2 2 ≥ π {2a_{1}+\frac{a_{2}}{2}\geq\pi} , while it is not possible otherwise.

The main goal of this paper is to investigate a newly proposed hybrid and hybrid inclusion problem consisting of fractional differential problems involving two different fractional derivatives of order μ, Caputo and Liouville–Riemann operators, with multi-order mixed Riemann–Liouville integro-derivative conditions. Although α is between one and two, we need three boundary value conditions to find the integral equation. The study investigates the results of existence for hybrid, hybrid inclusion, and non-hybrid inclusion problems by employing several analytical approaches, including Dhage’s technique, α - ψ {\alpha-\psi} -contractive mappings, fixed points, and endpoints of the product operators. To further illustrate our findings, we present three examples.

The probe and singular sources methods are two well-known classical direct reconstruction methods in inverse obstacle problems governed by partial differential equations. In this paper, by considering an inverse obstacle problem governed by the Laplace equation in a bounded domain as a prototype case, an integrated theory of the probe and singular sources methods is proposed. The theory consists of three parts: (i) introducing the singular sources method combined with the notion of the probe method; (ii) finding a third indicator function whose two ways decomposition yields the indicator functions in the probe and singular sources methods; (iii) finding the completely integrated version of the probe and singular sources methods.

In this paper, a space-time spectral method for solving an inverse problem in the Korteweg–de Vries equation is considered. Optimal order of convergence of the semi-discrete method is obtained in L 2 {L^{2}} -norm. The discrete schemes of the method are based on the modified Fourier pseudospectral method in spatial direction and the Legendre-tau method in temporal direction. The nonlinear term is computed via the fast Fourier transform and fast Legendre transform. The method is implemented by the explicit-implicit iterative method. Numerical results are given to show the accuracy and capability of this space-time spectral method.