REGULARIZATION METHODS FOR NONLINEAR ILL-POSED PROBLEMS WITH ACCRETIVE OPERATORS

This article is devoted to the regularization of nonlinear ill-posed problems with accretive operators in Banach spaces. The data involved are assumed to be known approximately. The authors concentrate their discussion on the convergence rates of regular solutions.

Arnoldi Projection Fractional Tikhonov for Large Scale Ill-Posed Problems

It is well known that Tikhonov regularization in standard form may determine approximate solutions that are too smooth for ill-posed problems,so fractional Tikhonov methods have been introduced to remedy this shortcoming.And Tikhonov regularization for large-scale linear ill-posed problems is commonly implemented by determining apartial Arnoldi decomposition of the given matrix.In this paper,we propose a new method to compute an approximate solution of large scale linear discrete ill-posed problems which applies projection fractional Tikhonov regularization in Krylov subspace via Arnoldi process.The projection fractional Tikhonov regularization combines the fractional matrices and orthogonal projection operators.A suitable value of the regularization parameter is determined by the discrepancy principle.Numerical examples with application to image restoration are carried out to examine that the performance of the method.

A Newton-type method for nonlinear ill-posed problems with A-smooth regularization

In this paper we present a regularized Newton-type method for ill-posed problems, by using the A-smooth regularization to solve the linearized ill-posed equations. For noisy data a proper a posteriori stopping rule is used that yields convergence of the Newton iteration to a solution, as the noise level goes to zero, under certain smoothness conditions on the nonlinear operator. Some appropriate assumptions on the closedness and smoothness of the starting value and the solution are shown to lead to optimal convergence rates.

A mixed Newton-Tikhonov method for nonlinear ill-posed problems

Newton type methods are one kind of the efficient methods to solve nonlinear ill-posed problems, which have attracted extensive attention. However, computational cost of Newton type methods is high because practical problems are complicated. We propose a mixed Newton-Tikhonov method, i.e., one step Newton-Tikhonov method with several other steps of simplified Newton-Tikhonov method. Convergence and stability of this method are proved under some conditions. Numerical experiments show that the proposed method has obvious advantages over the classical Newton method in terms of computational costs.

Nonlinear implicit iterative method for solving nonlinear ill-posed problems

In the paper, we extend the implicit iterative method for linear ill-posed operator equations to solve nonlinear ill-posed problems. We show that under some conditions the error sequence of solutions of the nonlinear implicit iterative method is monotonically decreasing and, with this monotonicity, prove convergence of the new method for both the exact and perturbed equations.

Fast Multilevel Methods for Solving Ill-posed Problems

1 Introduetion Many industrial and engineering applieations require numerieally solving ill-posed Problems. Regularization methods are employed to find approximate solutions of these problems.The choice of regularization

On the Regularization Method for Solving Ill-Posed Problems with Unbounded Operators

Let be a linear, closed, and densely defined unbounded operator, where X and Y are Hilbert spaces. Assume that A is not boundedly invertible. Suppose the equation Au=f is solvable, and instead of knowing exactly f only know its approximation satisfies the condition: In this paper, we are interested a regularization method to solve the approximation solution of this equation. This approximation is a unique global minimizer of the functional , for any , defined as follows: . We also study the stability of this method when the regularization parameter is selected a priori and a posteriori. At the same time, we give an application of this method to the weak derivative operator equation in Hilbert space.

A UNIFIED TREATMENT OF REGULARIZATION METHODS FOR LINEAR ILL-POSED PROBLEMS

By presenting a general framework, some regularization methods for solving linear ill-posed problems are considered in a unified manner. Applications to some specific approaches are illustrated.

MULTILEVEL ITERATION METHODS FOR SOLVING LINEAR ILL-POSED PROBLEMS

In this paper we develop multilevel iteration methods for solving linear systems resulting from the Galerkin method and Tikhonov regularization for ill-posed problems. The algorithm and its convergence analysis are presented in an abstract framework.

Optimal Shape Factor and Fictitious Radius in the MQ-RBF:Solving Ill-Posed Laplacian Problems

To solve the Laplacian problems,we adopt a meshless method with the multiquadric radial basis function(MQRBF)as a basis whose center is distributed inside a circle with a fictitious radius.A maximal projection technique is developed to identify the optimal shape factor and fictitious radius by minimizing a merit function.A sample function is interpolated by theMQ-RBF to provide a trial coefficient vector to compute the merit function.We can quickly determine the optimal values of the parameters within a preferred rage using the golden section search algorithm.The novel method provides the optimal values of parameters and,hence,an optimal MQ-RBF;the performance of the method is validated in numerical examples.Moreover,nonharmonic problems are transformed to the Poisson equation endowed with a homogeneous boundary condition;this can overcome the problem of these problems being ill-posed.The optimal MQ-RBF is extremely accurate.We further propose a novel optimal polynomial method to solve the nonharmonic problems,which achieves high precision up to an order of 10^(−11).

A FINITE DIMENSIONAL METHOD FOR SOLVINGNONLINEAR ILL-POSED PROBLEMS

We propose a finite dimensional method to compute the solution of nonlinear ill-posed problems approximately and show that under certain conditions, the convergence can be guaranteed. Moreover, we obtain the rate of convergence of our method provided that the true solution satisfies suitable smoothness condition. Finally, we present two examples from the parameter estimation problems of differential equations and illustrate the applicability of our method.

An Iterative Multigrid Regularization Method for Toeplitz Discrete Ill-Posed Problems

Iterative regularization multigrid methods have been successful applied to signal/image deblurring problems.When zero-Dirichlet boundary conditions are imposed the deblurring matrix has a Toeplitz structure and it is potentially full.A crucial task of a multilevel strategy is to preserve the Toeplitz structure at the coarse levels which can be exploited to obtain fast computations.The smoother has to be an iterative regularization method.The grid transfer operator should preserve the regularization property of the smoother.This paper improves the iterative multigrid method proposed in[11]introducing a wavelet soft-thresholding denoising post-smoother.Such postsmoother avoids the noise amplification that is the cause of the semi-convergence of iterative regularization methods and reduces ringing effects.The resulting iterative multigrid regularization method stabilizes the iterations so that and imprecise(over)estimate of the stopping iteration does not have a deleterious effect on the computed solution.Numerical examples of signal and image deblurring problems confirm the effectiveness of the proposed method.

A Comparative Study of Metaheuristic Optimization Algorithms for Solving Real-World Engineering Design Problems

Real-world engineering design problems with complex objective functions under some constraints are relatively difficult problems to solve.Such design problems are widely experienced in many engineering fields,such as industry,automotive,construction,machinery,and interdisciplinary research.However,there are established optimization techniques that have shown effectiveness in addressing these types of issues.This research paper gives a comparative study of the implementation of seventeen new metaheuristic methods in order to optimize twelve distinct engineering design issues.The algorithms used in the study are listed as:transient search optimization(TSO),equilibrium optimizer(EO),grey wolf optimizer(GWO),moth-flame optimization(MFO),whale optimization algorithm(WOA),slimemould algorithm(SMA),harris hawks optimization(HHO),chimp optimization algorithm(COA),coot optimization algorithm(COOT),multi-verse optimization(MVO),arithmetic optimization algorithm(AOA),aquila optimizer(AO),sine cosine algorithm(SCA),smell agent optimization(SAO),and seagull optimization algorithm(SOA),pelican optimization algorithm(POA),and coati optimization algorithm(CA).As far as we know,there is no comparative analysis of recent and popular methods against the concrete conditions of real-world engineering problems.Hence,a remarkable research guideline is presented in the study for researchersworking in the fields of engineering and artificial intelligence,especiallywhen applying the optimization methods that have emerged recently.Future research can rely on this work for a literature search on comparisons of metaheuristic optimization methods in real-world problems under similar conditions.

An Effective Meshless Approach for Inverse Cauchy Problems in 2D and 3D Electroelastic Piezoelectric Structures

In the past decade,notable progress has been achieved in the development of the generalized finite difference method(GFDM).The underlying principle of GFDM involves dividing the domain into multiple sub-domains.Within each sub-domain,explicit formulas for the necessary partial derivatives of the partial differential equations(PDEs)can be obtained through the application of Taylor series expansion and moving-least square approximation methods.Consequently,the method generates a sparse coefficient matrix,exhibiting a banded structure,making it highly advantageous for large-scale engineering computations.In this study,we present the application of the GFDM to numerically solve inverse Cauchy problems in two-and three-dimensional piezoelectric structures.Through our preliminary numerical experiments,we demonstrate that the proposed GFDMapproach shows great promise for accurately simulating coupled electroelastic equations in inverse problems,even with 3%errors added to the input data.

Highly Accurate Golden Section Search Algorithms and Fictitious Time Integration Method for Solving Nonlinear Eigenvalue Problems

This study sets up two new merit functions,which are minimized for the detection of real eigenvalue and complex eigenvalue to address nonlinear eigenvalue problems.For each eigen-parameter the vector variable is solved from a nonhomogeneous linear system obtained by reducing the number of eigen-equation one less,where one of the nonzero components of the eigenvector is normalized to the unit and moves the column containing that component to the right-hand side as a nonzero input vector.1D and 2D golden section search algorithms are employed to minimize the merit functions to locate real and complex eigenvalues.Simultaneously,the real and complex eigenvectors can be computed very accurately.A simpler approach to the nonlinear eigenvalue problems is proposed,which implements a normalization condition for the uniqueness of the eigenvector into the eigenequation directly.The real eigenvalues can be computed by the fictitious time integration method(FTIM),which saves computational costs compared to the one-dimensional golden section search algorithm(1D GSSA).The simpler method is also combined with the Newton iterationmethod,which is convergent very fast.All the proposed methods are easily programmed to compute the eigenvalue and eigenvector with high accuracy and efficiency.

Oscillation properties of eigenfunctions for Sturm-Liouville problems with interface conditions via Prufer transformation

A class of Sturm-Liouville problems with discontinuity is studied in this paper.The oscillation properties of eigenfunctions for Sturm-Liouville problems with interface conditions are obtained.The main method used in this paper is based on Prufer transformation,which is different from the classical ones.Moreover,we give two examples to verify our main results.

Association of preschool children behavior and emotional problems with the parenting behavior of both parents

BACKGROUND Parental behaviors are key in shaping children’s psychological and behavioral development,crucial for early identification and prevention of mental health issues,reducing psychological trauma in childhood.AIM To investigate the relationship between parenting behaviors and behavioral and emotional issues in preschool children.METHODS From October 2017 to May 2018,7 kindergartens in Ma’anshan City were selected to conduct a parent self-filled questionnaire-Health Development Survey of Preschool Children.Children’s Strength and Difficulties Questionnaire(Parent Version)was applied to measures the children’s behavioral and emotional performance.Parenting behavior was evaluated using the Parental Behavior Inventory.Binomial logistic regression model was used to analyze the association between the detection rate of preschool children’s behavior and emotional problems and their parenting behaviors.RESULTS High level of parental support/participation was negatively correlated with conduct problems,abnormal hyperactivity,abnormal total difficulty scores and abnormal prosocial behavior problems.High level of maternal support/participation was negatively correlated with abnormal emotional symptoms and abnormal peer interaction in children.High level of parental hostility/coercion was positively correlated with abnormal emotional symptoms,abnormal conduct problems,abnormal hyperactivity,abnormal peer interaction,and abnormal total difficulty scores in children(all P<0.05).Moreover,paternal parenting behaviors had similarly effects on behavior and emotional problems of preschool children compared with maternal parenting behaviors(all P>0.05),after calculating ratio of odds ratio values.CONCLUSION Our study found that parenting behaviors are associated with behavioral and emotional issues in preschool children.Overall,the more supportive or involved the parents are,the fewer behavioral and emotional problems the children experience;conversely,the more hostile or controlling the parents are,the more behavioral and emotional problems the children face.Moreover,the impact of fathers’parenting behaviors on preschool children’s behavior and emotions is no less significant than that of mothers’parenting behaviors.

Quafu-Qcover:Explore combinatorial optimization problems on cloud-based quantum computers

We introduce Quafu-Qcover,an open-source cloud-based software package developed for solving combinatorial optimization problems using quantum simulators and hardware backends.Quafu-Qcover provides a standardized and comprehensive workflow that utilizes the quantum approximate optimization algorithm(QAOA).It facilitates the automatic conversion of the original problem into a quadratic unconstrained binary optimization(QUBO)model and its corresponding Ising model,which can be subsequently transformed into a weight graph.The core of Qcover relies on a graph decomposition-based classical algorithm,which efficiently derives the optimal parameters for the shallow QAOA circuit.Quafu-Qcover incorporates a dedicated compiler capable of translating QAOA circuits into physical quantum circuits that can be executed on Quafu cloud quantum computers.Compared to a general-purpose compiler,our compiler demonstrates the ability to generate shorter circuit depths,while also exhibiting superior speed performance.Additionally,the Qcover compiler has the capability to dynamically create a library of qubits coupling substructures in real-time,utilizing the most recent calibration data from the superconducting quantum devices.This ensures that computational tasks can be assigned to connected physical qubits with the highest fidelity.The Quafu-Qcover allows us to retrieve quantum computing sampling results using a task ID at any time,enabling asynchronous processing.Moreover,it incorporates modules for results preprocessing and visualization,facilitating an intuitive display of solutions for combinatorial optimization problems.We hope that Quafu-Qcover can serve as an instructive illustration for how to explore application problems on the Quafu cloud quantum computers.

Dirac method for nonlinear and non-homogenous boundary value problems of plates

The boundary value problem plays a crucial role in the analytical investigation of continuum dynamics. In this paper, an analytical method based on the Dirac operator to solve the nonlinear and non-homogeneous boundary value problem of rectangular plates is proposed. The key concept behind this method is to transform the nonlinear or non-homogeneous part on the boundary into a lateral force within the governing function by the Dirac operator, which linearizes and homogenizes the original boundary, allowing one to employ the modal superposition method for obtaining solutions to reconstructive governing equations. Once projected into the modal space, the harmonic balance method(HBM) is utilized to solve coupled ordinary differential equations(ODEs)of truncated systems with nonlinearity. To validate the convergence and accuracy of the proposed Dirac method, the results of typical examples, involving nonlinearly restricted boundaries, moment excitation, and displacement excitation, are compared with those of the differential quadrature element method(DQEM). The results demonstrate that when dealing with nonlinear boundaries, the Dirac method exhibits more excellent accuracy and convergence compared with the DQEM. However, when facing displacement excitation, there exist some discrepancies between the proposed approach and simulations;nevertheless, the proposed method still accurately predicts resonant frequencies while being uniquely capable of handling nonuniform displacement excitations. Overall, this methodology offers a convenient way for addressing nonlinear and non-homogenous plate boundaries.

Data-Driven Learning Control Algorithms for Unachievable Tracking Problems

For unachievable tracking problems, where the system output cannot precisely track a given reference, achieving the best possible approximation for the reference trajectory becomes the objective. This study aims to investigate solutions using the Ptype learning control scheme. Initially, we demonstrate the necessity of gradient information for achieving the best approximation.Subsequently, we propose an input-output-driven learning gain design to handle the imprecise gradients of a class of uncertain systems. However, it is discovered that the desired performance may not be attainable when faced with incomplete information.To address this issue, an extended iterative learning control scheme is introduced. In this scheme, the tracking errors are modified through output data sampling, which incorporates lowmemory footprints and offers flexibility in learning gain design.The input sequence is shown to converge towards the desired input, resulting in an output that is closest to the given reference in the least square sense. Numerical simulations are provided to validate the theoretical findings.