Tikhonov Regularization Method Research Articles

A Fractional Tikhonov Regularization Method for Identifying a Time-Independent Source in the Fractional Rayleigh–Stokes Equation

The aim of this paper is to identify a time-independent source term in the Rayleigh–Stokes equation with a fractional derivative where additional data are considered at a fixed time point. This inverse problem is proved to be ill-posed in the sense of Hadamard. By using a fractional Tikhonov regularization method, we construct a regularized solution. Then, according to a priori and a posteriori regularization parameter selection rules, we prove the convergence estimates of the regularization method. Finally, we provide some numerical examples to prove the effectiveness of the proposed method.

Stability and regularization for ill-posed Cauchy problem of a stochastic parabolic differential equation

Abstract In this paper, we investigate an ill-posed Cauchy problem involving a stochastic parabolic equation. We first establish a Carleman estimate for this equation. Leveraging this estimate, we derive the conditional stability and convergence rate of the Tikhonov regularization method for the aforementioned ill-posed Cauchy problem. To complement our theoretical analysis, we employ kernel-based learning theory to implement the completed Tikhonov regularization method for several numerical examples.

A two-parameter Tikhonov regularization for a fractional sideways problem with two interior temperature measurements

This paper deals with a fractional sideways problem of determining the surface temperature of a heat body from two interior temperature measurements. Mathematically, it is formulated as a problem for the one-dimensional heat equation with Caputo fractional time derivative of order α∈(0,1], where the data are given at two interior points, namely x=x1 and x=x2, and the solution is determined for x∈(0,L),0<x1<x2≤L. The problem is challenging since it is severely ill-posed for x∉[x1,x2]. For the ill-posed problem, we apply the Tikhonov regularization method in Hilbert scales to construct stable approximation problems. Using the two-parameter Tikhonov regularization, we obtain the order optimal convergence estimates in Hilbert scales by using both a priori and a posteriori parameter choice strategies. Numerical experiments are presented to show the validity of the proposed method.

Scientific problems of ensuring technological sovereignty in the field of artificial intelligence technologies

The article discusses the scientific problems of ensuring technological sovereignty in the field of artificial intelligence technologies. One of the key scientific problems is computer modeling of human cognitive functions with finite resources, which provide the ability to process and analyze finite–dimensional and infinite-dimensional data along with finite-dimensional ones. A well–known solution to this problem is the A.N. Tikhonov regularization method, due to the high complexity and cost of implementation of which, less complex and expensive empirically constructed foreign INS are used to build AI solutions, but without guarantees of error-free results. The consequence of this is a rather low level of implementation of AI solutions in industry and in the field of information technology and a practical lack of data on the economic effect. It is proposed to consider the economic efficiency of domestic AI solutions as the main criterion for the need for their development.

IDENTIFICATION OF AN ARBITRARY MOVING AXISYMMETRIC LOAD ACTING ON A CYLINDRICAL SHELL

Various types of external non-stationary loads can act on various structural elements and cylindrical shells of finite length in particular distributed and concentrated, stationary and moving load. When using various external load identification methods, the type of external load is usually known. In prac- tice this is not always the case. The purpose of the study is to develop a method for identifying an arbitrary axisymmetric load acting on an elastic cy- lindrical shell of finite length, which can be used to identify a moving load. To model the non-stationary load of a cylindrical shell, a system of differ- ential equations of the refined Timoshenko theory for medium thickness shells was used. The solution to this system of differential equations is ob- tained by expanding the unknown functions into Fourier series and applying the integral Laplace transform. The solution of the inverse problem was obtained using the theory of integral equations and the Tikhonov regularization method. As a result of the study, a solution of the inverse problem of solid mechanics to identify an arbitrary axisymmetric non-stationary load was obtained. A numerical experiment was carried out to use the developed method to identify a moving non-stationary load acting on a simply supported cylindrical shell of medium thickness. The simulation results indicate a fairly accurate identification of both the change in time and the distribution along the shell of a non-stationary axisymmetric moving load. A method has been developed for identifying an external non-stationary load randomly distributed along a cylindrical shell, which makes it possible to identify a moving load without preliminary information about the type of this load. The developed method allows us to reproduce moving non-stationary loads, which are often encountered in practice, and expand it to other types of structural elements.

Generalized Nash Equilibrium Seeking for Noncooperative Game With Different Monotonicities by Adaptive Neurodynamic Algorithm.

This article proposes a novel adaptive neurodynamic algorithm (ANA) to seek generalized Nash equilibrium (GNE) of the noncooperative constrained game with different monotone conditions. In the ANA, the adaptive penalty term, which acts as trajectory-dependent penalty parameters, evolves based on the degree of constraints violation until the trajectory enters the action set of noncooperative game. It is shown that the trajectory of the ANA enters the action set in finite time benefited from the adaptive penalty term. Moreover, it is proven that the trajectory exponentially (or polynomially) converges to the unique GNE when the pseudo-gradient of cost function in noncooperative game satisfies strong (or "generalized" strong) monotonicity. To the best of our knowledge, this is the first time to study the polynomial convergence of GNE seeking algorithm. Furthermore, when the pseudo-gradient mentioned above satisfies monotonicity in general, based on Tikhonov regularization method, a new ANA for finding its ε -generalized Nash equilibrium ( ε -GNE) is proposed, and the related exponential convergence of the algorithm is established. Finally, the river basin pollution game and 5G base station location game are given as examples to showcase the algorithm's effectiveness.

Particle swarm optimization method for parameter selecting in Tikhonov regularization method for solving inverse problems

In this article the Tikhonov method used as a regularization technique for solving the inverse problem liner operator equation of the first kind with noisy and noise-free data. We also provide the Tikhonov method's essential analysis for tackling inverse problems. Following that, we go over how to choose the Tikhonov regularization parameter by implementing the Particle Swarm Optimization (PSO) technique. The effectiveness of combining these two approaches particle swarm optimization and Tikhonov is demonstrated to be highly practicable.

A hybrid approach for power quality event identification in power systems: Elasticnet Regression decomposition and optimized probabilistic neural networks

The transformation of traditional grid networks towards smart-grid and microgrid concepts raises many critical issues, and quality in the power supply is one of the prominent ones that needs further research. Developing and applying power quality (PQ) recognition methods with efficient and reliable analysis are essential to the fast-growing issues related to modern smart power distribution systems. In this regard, a hybrid algorithm is proposed for PQ events detection and classification using Elasticnet Regression-based Variational Mode Decomposition (ER-VMD) and Salp Swarm Algorithm optimized Probabilistic Neural Network (SSA-PNN). The Elasticnet Regression (ER) process is suggested to modify the conventional VMD approach instead of the Tikhonov Regularization (TR) method to enhance performance and obtain better band-limited intrinsic mode functions. This idea results in robust and effective reconstruction features and helps to obtain accurate classification using the classifier. In the classification stage, a Salp Swarm Algorithm (SSA) based PNN is used for the PQ event, considering the relevant features obtained from ER-VMD. The system parameters often influence PNN performance, and SSA is used to determine the ideal values to improve the PNN's capacity for more accurate classification. The numerical values of the accuracy percentage, percentage of sensitivity, and percentage of specificity in the case of real-time data are found as 98.58, 100, and 98.46, respectively. The acquired comparison findings demonstrate the effectiveness and robustness of the proposed technique in terms of rapid learning speed, smaller computational complexity, robust performance for anti-noise conditions, and accurate identification and categorization.

Use of electrical impedance tomography for lung volume reconstruction

The article presents a study of the application of electro-impedance tomography (EIT) in diagnosing lung capacity using the Tikhonov regularization method. The possibility of reconstructing the lungs to monitor the degree of air filling was investigated. The experiment included a series of tests using a torso phantom designed to simulate different states of the lungs - from fully inflated to fully deflated. Lung-filling states were manipulated in controlled scenarios to test nine main experimental conditions reflecting different lung-filling states. In addition, the quality of reconstruction was checked using various types of reference backgrounds. The results show significant differences in lung volume reconstructions depending on the lung filling state. The most successful reconstructions, which were obtained using the 'No phantom' background, provided the most explicit visualization of the lungs, reassuring the method's reliability. The experiments confirm the potential of EIT to distinguish between different lung states and reconstruct the degree of lung filling. The study also underscores the need to optimize the reference background to increase the precision of the images, especially for the left lung.

Stochastic force identification for uncertain structures based on matrix equilibration and improved Tikhonov regularization method

Accurate identification and estimation of stochastic forces applied to in-service engineering structures play a vital role in structural safety assessments. This study devised an effective force power spectral density (PSD) identification method to address the challenge of identifying multipoint stationary stochastic forces in uncertain structures. Initially, a probability model was employed to characterize structural uncertainties. Subsequently, an integral relationship was established between the probability density function (PDF) of the random structural parameters and that of the stochastic force PSD. By employing a point-selection technique based on the generalized F-discrepancy and a smoothing method, the uncertainty problem was transformed into a finite number of stochastic force PSD identification problems for deterministic structures. Simultaneously, based on the inverse pseudo-excitation method, a matrix equilibration approach and an improved Tikhonov regularization method were used to address the problem of large identification errors near structural natural frequencies. In comparison to the traditional weighting matrix method, the proposed method further reduces the condition number of frequency response function matrices, thereby enhancing the accuracy of force PSD identification. Finally, numerical examples were presented to validate the effectiveness of the proposed method in solving the stochastic force identification problem.

Noncoercive convex sweeping processes with velocity constraints

In this paper, we investigate noncoercive monotone convex sweeping processes with velocity constraints, a topic not previously investigated. Using some fundamental results on Bochner integration, the Tikhonov regularization method, a solution existence result for coercive convex sweeping processes with velocity constraints and a useful fact on measurable single-valued mappings, we prove the solution existence and properties of the solution set of noncoercive convex sweeping processes with velocity constraints under suitable conditions. These results represent a significant contribution, addressing a specific aspect of an open question posed in our recent work [Adly et al., Convex and nonconvex sweeping processes with velocity constraints: well-posedness and insights. Appl Math Optim. 2023;88(Paper No. 45)]. Additionally, we resolve two other open questions from the same paper concerning the behaviour of the regularized trajectories, relying on the Dominated Convergence Theorem for Bochner integration.

Iterated fractional Tikhonov method for recovering the source term and initial data simultaneously in a two-dimensional diffusion equation

In this paper, an inverse problem of a two-dimensional diffusion equation is considered. The purpose here includes recovering not only the source term but also the initial value from given observations at two fixed times t=T1 and t=T2. The uniqueness of the solutions for the inverse problem is given. Instead of the classical Tikhonov regularization strategy, we propose an iterated fractional Tikhonov regularization method to solve this problem, an a priori and an a posterior parameter selection rules and corresponding convergence rates are derived. For verification of the theoretical estimates, several numerical examples are constructed and compared with the standard iterated Tikhonov regularization method.

Parameter Choice Strategy that Computes Regularization Parameter before Computing the Regularized Solution

The modeling of many problems of practical interest leads to nonlinear ill-posed equations (for example, the parameter identification problem (see the Numerical section)). In this article, we introduce a new source condition (SC) and a new parameter choice strategy (PCS) for the Tikhonov regularization (TR) method for nonlinear ill-posed problems. The new PCS is introduced using a new SC to compute the regularization parameter (RP) before computing the regularized solution. The theoretical results are verified using a numerical example.

On the numerical solution of weakly singular integral equations of the first kind using a regularized projection method

This study investigates a numerical method based on the Jacobi–Gauss quadrature for solving Fredholm integral equations of the first kind with a weakly singular kernel by combining the Tikhonov regularization and projection methods. This numerical method reduces the solution of the weakly singular integral equations of the first kind to the solution of a linear system of algebraic equations. The theoretical analysis of the proposed technique is provided. Finally, several tests are presented to show the validity and efficiency of this approach.

The Application of Piecewise Regularization Reconstruction to the Calibration of Strain Beams.

Standard beams are mainly used for the calibration of strain sensors using their load reconstruction models. However, as an ill-posed inverse problem, the solution to these models often fails to converge, especially when dealing with dynamic loads of different frequencies. To overcome this problem, a piecewise Tikhonov regularization method (PTR) is proposed to reconstruct dynamic loads. The transfer function matrix is built both using the denoised excitations and the corresponding responses. After singular value decomposition (SVD), the singular values are divided into submatrices of different sizes by utilizing a piecewise function. The regularization parameters are solved by optimizing the piecewise submatrices. The experimental result shows that the MREs of the PTR method are 6.20% at 70 Hz and 5.86% at 80 Hz. The traditional Tikhonov regularization method based on GCV exhibits MREs of 28.44% and 29.61% at frequencies of 70 Hz and 80 Hz, respectively, whereas the L-curve-based approach demonstrates MREs of 29.98% and 18.42% at the same frequencies. Furthermore, the PREs of the PTR method are 3.54% at 70 Hz and 3.73% at 80 Hz. The traditional Tikhonov regularization method based on GCV exhibits PREs of 27.01% and 26.88% at frequencies of 70 Hz and 80 Hz, respectively, whereas the L-curve-based approach demonstrates PREs of 29.50% and 15.56% at the same frequencies. All in all, the method proposed in this paper can be extensively applied to load reconstruction across different frequencies.

A dynamic light scattering inversion method based on regularization matrix reconstruction for flowing aerosol measurement.

Tikhonov regularization, or truncated singular value decomposition (TSVD), is usually used for dynamic light scattering (DLS) inversion of particles in suspension. The Tikhonov regularization method uses a regularization matrix to modify all singular values in the kernel matrix. The modification of large singular values cannot effectively reduce the variance of the estimated values but may introduce bias in the solution, resulting in poor disturbance resistance in the inversion results. The TSVD method, on the other hand, truncates all small singular values, which leads to the loss of particle size information during the inversion process. The shortcomings of the two methods mentioned above do not have a significant impact on the inversion of high signal-to-noise ratio data. However, compared to the classical DLS particle size inversion for non-flowing suspended particles, the DLS inversion of flowing aerosols is more significantly affected by noise, and the extraction of particle size information is more difficult due to the effect of flow velocity, resulting in worse inversion results with increasing aerosol flow velocity for both methods. To improve the accuracy of the particle size distribution (PSD) of flowing aerosols, we introduced a kernel matrix into the regularization matrix, and based on the principles of the two methods, the spectral information of the kernel matrix was utilized to make the modification of small singular values by the regularization matrix. Correspondingly, weak or no modification was made according to the values of large singular values to reduce the introduction of bias. The inversion results of simulated and measured data indicate that the reconstruction of the regularization matrix improves the anti-disturbance performance and avoids the loss of particle size information during the regularization inversion process, thereby significantly improving the PSD accuracy, which is affected by the dual effects of flow velocity and noise in the DLS measurement of flowing particles. The peak error and distribution error of the inversion results by reconstructing the regularization matrix are lower than those of Tikhonov regularization.

SIMULTANEOUS INVERSION OF THE SOURCE TERM AND INITIAL VALUE OF THE TIME FRACTIONAL DIFFUSION EQUATION

In this paper, the problem we investigate is to simultaneously identify the source term and initial value of the time fractional diffusion equation. This problem is ill-posed, i.e., the solution (if exists) does not depend on the measurable data. We give the conditional stability result under the a-priori bound assumption for the exact solution. The modified Tikhonov regularization method is used to solve this problem, and under the a-priori and the a-posteriori selection rule for the regularization parameter, the convergence error estimations for this method are obtained. Finally, numerical example is given to prove the effectiveness of this regularization method.

Wavelet denoising based on comprehensive index optimization and improved L2 regularization for load identification

Load identification, as an inverse problem, is still one of the challenges in the field of vibration engineering. In this paper, a comprehensive load identification method combining entropy weight Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) noise reduction evaluation algorithm and improved L2 regularization method is proposed to improve the load identification accuracy from the following two perspectives. Firstly, the entropy weight TOPSIS method is used to evaluate the wavelet denoising scheme comprehensively, and the optimal denoising scheme is found to reduce the noise interference in the process of inverse problem solving. Secondly, we introduce a new regularization filter function, and search for the regularization parameters by Generalized Cross-Validation (GCV) criterion and one-dimensional optimization algorithm to improve the ill-posedness of the inverse problem. Finally, the simulation and experimental verification of single-source load identification are carried out with the scaled crankcase model, while the accuracy is compared with the Tikhonov regularization method. The results show that the integrated algorithm proposed in this paper can improve the accuracy of load identification while overcoming the inadequacy of inverse problem. Unlike previous studies, the comprehensive evaluation method of denoising effect introduced in this paper provides a way to judge the denoising effect in practical engineering, and the experimental model is closer to practical engineering structures, which shows potential engineering application value.

A new second-order dynamical method for solving linear inverse problems in Hilbert spaces

A new second-order dynamic method (SODM) is proposed for solving ill-posed linear inverse problems in Hilbert spaces. The SODM can be viewed as a combination of Tikhonov regularization and second-order asymptotical regularization methods. As a result, a double-regularization-parameter strategy is adopted. The regularization properties of SODM are demonstrated under both a priori and a posteriori stopping rules. In the context of time discretization, we propose several iterative schemes with different choices of damping parameters. A truncated discrepancy principle is employed as the stop criterion. Finally, numerical experiments are performed to show the efficiency of the SODM: on the whole, compared with the classical Tikhonov method and the first-order dynamical-system method, the SODM leads to more-accurate approximate solutions while requiring fewer-iterative numbers.

A hybrid surface shape control method for optimizing thermal deformation of FEL reflection mirror

The surface shape errors induced by thermal deformation of the free electron lasers (FEL) beamline reflection mirror directly affect the quality of FEL beam. In order to reduce the thermal deformation and meet the stringent requirements of surface shape error of the FEL reflection mirror, this paper proposes a surface shape control method that combines mirror bending and active thermal control (ATC) technique, aiming to minimize surface shape errors and enhance the overall performance of the FEL beamline. A bending mechanism has been designed to achieve a bending sensitivity of 60 nrad RMS/N using only one bending actuator. It does not obstruct the FEL beam with extremely small incident angle. This mechanism can compensate for thermal deformation by eliminating the cylindrical surface component of the thermal deformation. Additionally, a novel ATC algorithm is proposed based on the Tikhonov regularization method. When combined with the bending mechanism, it proves effective in controlling the surface shape of the reflection mirror. The simulation results indicate that this surface control method can effectively reduce the surface shape errors of the reflection mirror. Taking the 3 nm wavelength FEL as an example, the RMS value of surface slope error of the reflection mirror is substantially reduced from 17.1974 μrad (with heat load) to 0.0913 μrad after bending. Based on the bending result, the ATC technique further reduces the slope error to 0.0303 μrad RMS. For 1 nm and 2 nm wavelength FELs, the mirror slope errors ultimately reach 0.0147 μrad RMS and 0.0295 μrad RMS respectively. This surface shape control method ensures that the reflection mirror slope errors meet the beamline requirement, which is 0.05 μrad RMS, thereby guaranteeing the beam quality.