Tikhonov Regularization Method Research Articles

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.

A Second-Order Generalized Total Variation with Improved Alternating Direction Method of Multipliers Algorithm for Electrical Impedance Tomography Reconstruction

The issue of Electrical Impedance Tomography (EIT) is a well-known inverse problem that presents challenging characteristics. In order to address the difficulties associated with ill-conditioned inverses, regularization methods are typically employed. One commonly used approach is total variation (TV) regularization, which has shown effectiveness in EIT. In order to meet the requirements of real-time tracking, it is essential to acquire fast and reliable algorithms for image reconstruction. Therefore, we present a modified second-order generalized regularization algorithm that enables more-accurate reconstruction of organ boundaries and internal structures, to reduce EIT artifacts, and to overcome the inability of the conventional Tikhonov regularization method in solving the step effect of the medium boundary. The proposed algorithm uses the improved alternating direction method of multipliers (ADMM) to tackle this optimization issue and adopts the second-order generalized total variation (SOGTV) function with strong boundary-preserving features as the regularization generalization function. The experiments are based on simulation data and the physical model of the circular water tank that we developed. The results showed that SOGTV regularization can improve image realism compared with some classic regularization.

FPGA-Based Hardware Implementation of a Stable Inverse Source Problem Algorithm in a Non-Homogeneous Circular Region

Objective: This work presents an implementation of a stable algorithm that recovers sources located at the boundary separating two homogeneous media in field-programmable gate arrays. Two loop unrolling architectures were developed and analyzed for this purpose. This inverse source problem is ill-posed due to numerical instability, i.e., small errors in the measurement can produce significant changes in the source location. Methodology: To handle the numerical instability when recovering these sources, the Tikhonov regularization method in combination with the Fourier series truncation method are applied in the stable algorithm. This stable algorithm is implemented in two different architectures developed in this work: The first architecture (Mode 1) allows for different operating speeds, which is an advantage depending on whether we work with fast or slow signals. The second one (Mode 2) reduces resource consumption by exploiting the characteristics of the source identification algorithm, which is an advantage for multichannel problems such as inverse electrocardiography or electroencephalography. Results: The architectures were tested on four devices of the 7 Series of Xilinx: Spartan-7 xc7s100fgga484, Virtex-7 xc7v585tffg1157, Kintex-7 xc7k70tfbg484, and Artix-7 xc7a35tcpg236. The two hardware implementations of the stable algorithm were validated using synthetic examples implemented in MATLAB, which shows the advantages of each architecture. Contributions: We developed two efficient architectures based on a loop unrolling design for source identification problems. These are effective strategies to divide and assign tasks to the configurable hardware, and they appear as an appropriate technique for implementing the algorithm. The first one is simple and allows for different operating speeds. The second one uses a control system based on multiplexors that reduce resource consumption and complexity of the design and can be used for multichannel problems. From the numerical test, we found the regularization parameters. The synthetic examples developed here can be considered for similar problems and can be extended to concentric spheres.

Chebyshev polynomials to Volterra-Fredholm integral equations of the first kind

Numerous methods have been studied and discussed for solving ill-posed Volterra integral equations and ill-posed Fredholm integral equations, but rarely for both simultaneously. In this study, we focus on numerically solving the ill-posed Volterra-Fredholm integral equation of the first kind by replacing it with its perturbed counterpart. We employ Chebyshev polynomials of the first kind to solve the perturbed equation. Our findings suggest that this technical approach is superior to the regularization method of Tikhonov. It is simpler, less cumbersome, and this simplicity is demonstrated through various examples.

Identifying the Heat Source in Radially Symmetry and Axis-Symmetry Problems

This paper solves the inverse source problem of heat conduction in which the source term only varies with time. The application of the discrete regularization method, a kind of effective radial symmetry and axisymmetric heat conduction problem source identification that does not involve the grid integral numerical method, is put forward. Taking the fundamental solution as the fundamental function, the classical Tikhonov regularization method combined with the L-curve criterion is used to select the appropriate regularization parameters, so the problem is transformed into a class of ill-conditioned linear algebraic equations to solve with an optimal solution. Several numerical examples of inverse source problems are given. Simultaneously, a few numerical examples of inverse source problems are given, and the effectiveness and superiority of the method is shown by the results.

Surface charge inversion algorithms based on integral equation method

AbstractAccurate surface charge inversion can guide the research on surface modification of insulators in GIS/GIL. The current inversion algorithms have disadvantages of high computational cost and low accuracy. Based on that, the integral equation method (IEM) is proposed to calculate the transformation matrix. Compared with the traditional analytical method (AM), IEM has a simple calculation process. The calculation speed of IEM is much faster than that of AM. To suppress the numerical divergence in IEM, the Tikhonov regularisation method is introduced and Tikhonov‐IEM is proposed. For square insulators, compared to IEM, the peak‐mean square error (PMSE) is reduced by about 40 percent. However, Tikhonov‐IEM is not suitable for basin insulators. Therefore, the least square method (LSM) is introduced and the LSM‐IEM is proposed. For basin insulators, compared to IEM, the PMSE is reduced by about 30 percent. Finally, the accuracy of the algorithms is verified by physical tests.

Estimation of pollutant source using source-receptor response matrix from maximum-length sequence pulse technique

Accurate estimation of pollutant sources in built environments is crucial for monitoring air quality and preventing harm to health and property. This study presented an improved method for estimating the dynamical pollutant source strength using the concentration signal monitored at the receptor point. We employed a built environment system subjected to a pollutant pulse signal in accordance with the maximum-length sequence (m-sequence) to acquire response concentration. Subsequently, the source-receptor response matrix was inversely identified based on the known pulse signal and the response concentration. Then, the unknown source release rate was inversely estimated by the monitored concentration signal and previously identified response matrix. In addition, the performances of different inverse algorithms and m-sequence orders were also analyzed. Here both simulation and application to a real experiment of a wind duct were implemented, and the results demonstrated the higher-order m-sequence method exhibited better noise resistance compared to the traditional unit pulse method. In both simulation and experiment, the m-sequence pulse method reduced the root mean square error (RMSE) by 39.9 % and 70.7 %, respectively, compared to the unit pulse method. In experiment, the simultaneous multiplicative algebraic reconstruction technique (SMART) combined with the Tikhonov regularization (TR) method showed relatively stable performance.

Two Regularization Methods for Identifying the Spatial Source Term Problem for a Space-Time Fractional Diffusion-Wave Equation

In this paper, we delve into the challenge of identifying an unknown source in a space-time fractional diffusion-wave equation. Through an analysis of the exact solution, it becomes evident that the problem is ill-posed. To address this, we employ both the Tikhonov regularization method and the Quasi-boundary regularization method, aiming to restore the stability of the solution. By adhering to both a priori and a posteriori regularization parameter choice rules, we derive error estimates that quantify the discrepancies between the regularization solutions and the exact solution. Finally, we present numerical examples to illustrate the effectiveness and stability of the proposed methods.

A new improved fractional Tikhonov regularization method for moving force identification

To address the problem of poor recognition accuracy of integer-order Tikhonov regularization method (Tik) in identifying bridge moving loads, a bridge moving load recognition method based on Improved fractional Tikhonov (IF-Tik) regularization method is proposed in this paper. The bridge moving load identification model is established according to the theory of time domain method, and the process of driving a two-axle vehicle on the bridge is simulated. The moving load is represented as the differential form of the kernel function of bending moment response and acceleration response. The differential equation is transformed into a linear system of equations by the discretization method, and solved by the improved fractional-order Tikhonov regularization method. The results show that the IF-Tik method has certain advantages over the F-Tik and Tik methods in terms of recognition accuracy and noise immunity, and the IF-Tik method has a higher accuracy of load recognition, and the recognition error is only 15% of that of the F-Tik method; the recognition results are less affected by the moment response, and the proposed method has good robustness, and is more suitable for the bridge moving load identification.

Sensitivity and reliability analysis to MSBAS regularization for the estimation of surface deformation over a mine

To systematically and thoroughly analyze the sensitivity and reliability of the MSBAS regularization for the estimation of surface deformation over a mine in combination with an application example, this study processed 101 Sentinel-1A/B SAR images, constructed and solved the 2D deformation models using SVD and Tikhonov regularization methods with different orders and parameters, and estimated the vertical and east-west surface deformation time series in a mine of China. Then, this study collected the leveling-monitoring vertical surface deformation data on three leveling points, and compared and analyzed the sensitivity and reliability of the MSBAS regularization methods for estimating vertical surface deformation. The results indicate that different regularization orders and parameters can lead to thousands of times differences in condition numbers and significant differences in ill-posed degree of the deformation models. The zero-order Tikhonov regularized deformation model with regularization parameter of 0.1 has the minimum condition number and the equation is not ill-posed. The first-order Tikhonov regularized deformation model with regularization parameter of 0.001 has the maximum condition number and the equation is seriously ill-posed. As a result, the estimates of 2D surface deformation models with different parameters and orders are also different in terms of numerical values and accuracy. Compared with the leveling-monitoring data, the first- and second-order MSBAS regularization methods with parameter 0.1 have the minimum fluctuation and the maximum correlation coefficients between the estimated values and the leveling-monitoring values, and are also closest to the leveling-monitoring results with the highest accuracy.

On a Backward Problem for the Rayleigh–Stokes Equation with a Fractional Derivative

The Rayleigh–Stokes equation with a fractional derivative is widely used in many fields. In this paper, we consider the inverse initial value problem of the Rayleigh–Stokes equation. Since the problem is ill-posed, we adopt the Tikhonov regularization method to solve this problem. In addition, this paper not only analyzes the ill-posedness of the problem but also gives the conditional stability estimate. Finally, the convergence estimates are proved under two regularization parameter selection rules.

“Spectral Method” for Determining a Kernel of the Fredholm Integral Equation of the First Kind of Convolution Type and Suppressing the Gibbs Effect

A set of one-dimensional (as well as one two-dimensional) Fredholm integral equations (IEs) of the first kind of convolution type is solved. The task for solving these equations is ill-posed (first of all, unstable); therefore, the Wiener parametric filtering method (WPFM) and the Tikhonov regularization method (TRM) are used to solve them. The variant is considered when a kernel of the integral equation (IE) is unknown or known inaccurately, which generates a significant error in the solution of IE. The so-called “spectral method” is being developed to determine the kernel of an integral equation based on the Fourier spectrum, which leads to a decrease of the error in solving the IE and image improvement. Moreover, the authors also propose a method for diffusing the solution edges to suppress the possible Gibbs effect (ringing-type distortions). As applications, the problems for processing distorted (smeared, defocused, noisy, and with the Gibbs effect) images are considered. Numerical examples are given to illustrate the use of the “spectral method” to enhance the accuracy and stability of processing distorted images through their mathematical and computer processing.