Search Results (80)

This paper investigates the problem of identifying a time-dependent source term in distributed-order time-space fractional diffusion equations (FDEs) based on boundary observation data. Firstly, the existence, uniqueness, and regularity of the solution to the direct problem are proved. Using the regularity of the solution and a Gronwall inequality with a weakly singular kernel, the uniqueness and stability estimates of the solution to the inverse problem are obtained. Subsequently, the inverse source problem is transformed into a minimization problem of a functional using the Tikhonov regularization method, and an approximate solution is obtained by the conjugate gradient method. Numerical experiments confirm that the method provides both accurate and robust results. Full article

Inverse Laplace transforms (ILTs) are fundamental to a wide range of scientific and engineering applications—from diffusion NMR spectroscopy to medical imaging—yet their numerical inversion remains severely ill-posed, particularly in the presence of noise or sparse data. The primary objective of this study is to develop robust and efficient numerical methods that improve the stability and accuracy of ILT reconstructions under challenging conditions. In this work, we introduce a novel family of Kaczmarz-based ILT solvers that embed advanced regularization directly into the iterative projection framework. We propose three algorithmic variants—Tikhonov–Kaczmarz, total variation (TV)–Kaczmarz, and Wasserstein–Kaczmarz—each incorporating a distinct penalty to stabilize solutions and mitigate noise amplification. The Wasserstein–Kaczmarz method, in particular, leverages optimal transport theory to impose geometric priors, yielding enhanced robustness for multi-modal or highly overlapping distributions. We benchmark these methods against established ILT solvers—including CONTIN, maximum entropy (MaxEnt), TRAIn, ITAMeD, and PALMA—using synthetic single- and multi-modal diffusion distributions contaminated with 1% controlled noise. Quantitative evaluation via mean squared error (MSE), Wasserstein distance, total variation, peak signal-to-noise ratio (PSNR), and runtime demonstrates that Wasserstein–Kaczmarz attains an optimal balance of speed (0.53 s per inversion) and accuracy (MSE = $4.7\times {10}^{-8}$), while TRAIn achieves the highest fidelity (MSE = $1.5\times {10}^{-8}$) at a modest computational cost. These results elucidate the inherent trade-offs between computational efficiency and reconstruction precision and establish regularized Kaczmarz solvers as versatile, high-performance tools for ill-posed inverse problems. Full article

This paper investigates the inverse problem of identifying a time-dependent source term in multi-term time–space fractional diffusion Equations (TSFDE). First, we rigorously establish the existence and uniqueness of strong solutions for the associated direct problem under homogeneous Dirichlet boundary conditions. A novel implicit finite difference scheme incorporating matrix transfer technique is developed for solving the initial-boundary value problem numerically. Regarding the inverse problem, we prove the solution uniqueness and stability estimates based on interior measurement data. The source identification problem is reformulated as a variational problem using the Tikhonov regularization method, and an approximate solution to the inverse problem is obtained with the aid of the optimal perturbation algorithm. Extensive numerical simulations involving six test cases in both 1D and 2D configurations demonstrate the high effectiveness and satisfactory stability of the proposed methodology. Full article

The number of hologram points in near-field acoustical holography (NAH) for a vibro-acoustic system plays a vital role in conditioning the transfer function between the source and measuring points. The requirement for many overdetermined hologram points for extended sources to obtain high accuracy poses a problem for the practical applications of NAH. Furthermore, overdetermination does not generally ensure enhanced accuracy, stability, and convergence, owing to the problem of rank deficiency. To achieve satisfactory reconstruction accuracy with underdetermined hologram data, the best practice for choosing hologram points and regularization methods is determined by comparing cross-linked sets of data-sorting and regularization methods. Three typical data selection and treatment methods are compared: iterative discarding of the most dependent data, monitoring singular value changes during the data reduction process, and zero padding in the patch holography technique. To test the regularization method for inverse conditioning, which is used together with the data selection method, the Tikhonov method, Bayesian regularization, and the data compression method are compared. The inverse equivalent source method is chosen as the holography method, and a numerical test is conducted with a point-excited thin plate. The simulation results show that selecting hologram points using the effective independence method, combined with regularization via compressed sensing, significantly reduces the reconstruction error and enhances the modal assurance criterion value. The experimental results also support the proposed best practice for inverting underdetermined hologram data by integrating the NAH data selection and regularization techniques. Full article

In this paper, we propose and develop a stationary Stokes Inverse Model (SIM) to estimate the stress distributions that are difficult to measure directly in flows. We estimate the driving stresses from the velocities by solving the inverse problem governed by Stokes equations under iterative Tikhonov (IT) regularization. We investigate the heuristic L-curve criterion to determine the proper regularization parameter. The solution existence and uniqueness for the Stokes inverse problem have been analyzed. We also conducted convergence analysis and error estimation for perturbed data, providing a fast and stable convergence. The finite element method is applied to the numerical approach. Following the theoretical investigation and formulation, we validate the model and demonstrate that the velocity data closely match the velocity fields that were reconstructed using the computed stress distributions. In particular, the proposed SIM can be used to reliably derive the stress distributions for the flows governed by the Stokes equations with small Reynolds number. Additionally, the model is robust to a certain number of perturbations, which enables the precise and effective estimation of the stress distributions. The proposed stationary SIM may be widely applicable in the estimation of stresses from experimental velocity fields in engineering and biological applications. Full article

The Cauchy problem for the Laplace equation in an annular bounded region consists of finding a harmonic function from the Dirichlet and Neumann data known on the exterior boundary. This work considers a fractional boundary condition instead of the Dirichlet condition in a circular annular region. We found the solution to the fractional boundary problem using circular harmonics. Then, the Tikhonov regularization is used to handle the numerical instability of the fractional Cauchy problem. The regularization parameter was chosen using the L-curve method, Morozov’s discrepancy principle, and the Tikhonov criterion. From numerical tests, we found that the series expansion of the solution to the Cauchy problem can be truncated in $N=20$, $N=25$, or $N=30$ for smooth functions. For other functions, such as absolute value and the jump function, we have to choose other values of N. Thus, we found a stable method for finding the solution to the problem studied. To illustrate the proposed method, we elaborate on synthetic examples and MATLAB 2021 programs to implement it. The numerical results show the feasibility of the proposed stable algorithm. In almost all cases, the L-curve method gives better results than the Tikhonov Criterion and Morozov’s discrepancy principle. In all cases, the regularization using the L-curve method gives better results than without regularization. Full article

Least-squares reverse time migration (LSRTM) is an advanced seismic imaging technique that reconstructs subsurface models by minimizing the residuals between simulated and observed data. Mathematically, the LSRTM inversion of the sub-surface reflectivity is a large-scale, highly ill-posed sparse inverse problem, where conventional inversion methods typically lead to poor imaging quality. In this study, we propose a regularized LSRTM method based on the flexible Krylov subspace inversion framework. Through the strategy of the Krylov subspace projection, a basis set for the projection solution is generated, and then the inversion of a large ill-posed problem is expressed as the small matrix optimization problem. With flexible preconditioning, the proposed method could solve the sparse regularization LSRTM, like with the Tikhonov regularization style. Sparse penalization solution is implemented by decomposing it into a set of Tikhonov penalization problems with iterative reweighted norm, and then the flexible Golub–Kahan process is employed to solve the regularization problem in a low-dimensional subspace, thereby finally obtaining a sparse projection solution. Numerical tests on the Valley model and the Salt model validate that the LSRTM based on Krylov subspace method can effectively address the sparse inversion problem of subsurface reflectivity and produce higher-quality imaging results. Full article

Recovering the relaxation spectrum, a fundamental rheological characteristic of polymers, from experiment data requires special identification methods since it is a difficult ill-posed inverse problem. Recently, a new approach relating the identification index directly with a completely unknown real relaxation spectrum has been proposed. The integral square error of the relaxation spectrum model was applied. This paper concerns regularization aspects of the linear-quadratic optimization task that arise from applying Tikhonov regularization to relaxation spectra direct identification problem. An influence of the regularization parameter on the norms of the optimal relaxation spectra models and on the fit of the related relaxation modulus model to the experimental data was investigated. The trade-off between the integral square norms of the spectra models and the mean square error of the relaxation modulus model, parameterized by varying regularization parameter, motivated the definition of two new multiplicative indices for choosing the appropriate regularization parameter. Two new problems of the regularization parameter optimal selection were formulated and solved. The first and second order optimality conditions were derived and expressed in the matrix-vector form and, alternatively, in finite series terms. A complete identification algorithm is presented. The usefulness of the new regularization parameter selection rules is demonstrated by three examples concerning the Kohlrausch–Williams–Watts spectrum with short relaxation times and uni- and double-mode Gauss-like spectra with middle and short relaxation times. Full article

Dynamic light scattering (DLS) is a highly efficient approach for extracting particle size distributions (PSDs) from autocorrelation functions (ACFs) to measure nanoparticle particles. However, it is a technical challenge to get an exact inversion of the PSD in DLS. Generally, Tikhonov regularization is widely used to address this issue; it uses the L₂ norm for both the data fitting term (DFT) and the regularization constraint term. However, the L₂ norm’s DFT has poor robustness, and its regularization term lacks sparsity, making the solution susceptible to noise and a reduction in accuracy. To solve this problem, the L_p,q norm restrictive model is formulated to examine the impact of various norms in the DFT and regularization term on the inversion results. On this basis, combined with the robustness of DFT and the sparsity of regularization terms, an L_1,∞-constrained Tikhonov regularization model was constructed. This model improves the inversion accuracy of PSD and offers a better noise-resistance performance. Simulation tests reveal that the L_1,∞ model has strong noise resistance, exceptional inversion precision, and excellent bimodal resolution. The inversion outcomes for the 33 nm unimodal particles, the 55 nm unimodal, and the 33 nm/203 nm bimodal experimental particles show that L_1,∞ reduces peak errors by at most 6.06%, 5.46%, and 12.12%/3.94% compared to L_2,2, L_1,2, and L_2,∞ models, respectively. These simulations are validated by experimental data. Full article

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. Full article

Relaxation time and frequency spectra are not directly available by measurement. To determine them, an ill-posed inverse problem must be solved based on relaxation stress or oscillatory shear relaxation data. Therefore, the quality of spectra models has only been assessed indirectly by examining the fit of the experiment data to the relaxation modulus or dynamic moduli models. As the measures of data fitting, the mean sum of the moduli square errors were usually used, the minimization of which was an essential step of the identification algorithms. The aim of this paper was to determine a relaxation spectrum model that best approximates the real unknown spectrum in a direct manner. It was assumed that discrete-time noise-corrupted measurements of a relaxation modulus obtained in the stress relaxation experiment are available for identification. A modified relaxation frequency spectrum was defined as a quotient of the real relaxation spectrum and relaxation frequency and expanded into a series of linearly independent exponential functions that are known to constitute a basis of the space of square-integrable functions. The spectrum model, given by a finite series of these basis functions, was assumed. An integral-square error between the real unknown modified spectrum and the spectrum model was taken as a measure of the model quality. This index was proved to be expressed in terms of the measurable relaxation modulus at uniquely defined sampling instants. Next, an empirical identification index was introduced in which the values of the real relaxation modulus are replaced by their noisy measurements. The identification consists of determining the spectrum model that minimizes this empirical index. Tikhonov regularization was applied to guarantee model smoothness and noise robustness. A simple analytical formula was derived to calculate the optimal model parameters and expressed in terms of the singular value decomposition. A complete identification algorithm was developed. The analysis of the model smoothness and model accuracy for noisy measurements was carried out. The equivalence of the direct identification of the relaxation frequency and time spectra has been demonstrated when the time spectrum is modeled by a series of functions given by the product of the relaxation frequency and its exponential function. The direct identification concept can be applied to both viscoelastic fluids and solids; however, some limitations to its applicability have been pointed out. Numerical studies have shown that the proposed identification algorithm can be successfully used to identify Gaussian-like and Kohlrausch–Williams–Watt relaxation spectra. The applicability of this approach to determining other commonly used classes of relaxation spectra was also examined. Full article

The uneven distribution of global navigation satellite system (GNSS) continuous stations in the Yellow River Basin, combined with the sparse distribution of GNSS continuous stations in some regions and the weak far-field load signals, poses challenges in using GNSS vertical displacement data to invert terrestrial water storage changes (TWSCs). To achieve the inversion of water reserves in the Yellow River Basin using unevenly distributed GNSS continuous station data, in this study, we employed the Tikhonov regularization method to invert the terrestrial water storage (TWS) in the Yellow River Basin using vertical displacement data from network engineering and the Crustal Movement Observation Network of China (CMONOC) GNSS continuous stations from 2011 to 2022. In addition, we applied an inverse distance weighting smoothing factor, which was designed to account for the GNSS station distribution density, to smooth the inversion results. Consequently, a gridded product of the TWS in the Yellow River Basin with a spatial resolution of 0.5 degrees on a daily scale was obtained. To validate the effectiveness of the proposed method, a correlation analysis was conducted between the inversion results and the daily TWS from the Global Land Data Assimilation System (GLDAS), yielding a correlation coefficient of 0.68, indicating a strong correlation, which verifies the effectiveness of the method proposed in this paper. Based on the inversion results, we analyzed the spatial–temporal distribution trends and patterns in the Yellow River Basin and found that the average TWS decreased at a rate of 0.027 mm/d from 2011 to 2017, and then increased at a rate of 0.010 mm/d from 2017 to 2022. The TWS decreased from the lower-middle to lower reaches, while it increased from the upper-middle to upper reaches. Furthermore, an attribution analysis of the terrestrial water storage changes in the Yellow River Basin was conducted, and the correlation coefficients between the monthly average water storage changes inverted from the results and the monthly average precipitation, evapotranspiration, and surface temperature (AvgSurfT) from the GLDAS were 0.63, −0.65, and −0.69, respectively. This indicates that precipitation, evapotranspiration, and surface temperature were significant factors affecting the TWSCs in the Yellow River Basin. Full article

A body may have a structural, thermal, electromagnetic or optical role. In wave propagation, many models are described for transmission problems, whose solutions are defined in two or more domains. In this paper, we consider an inverse source hyperbolic problem on disconnected intervals, using solution point constraints. Applying a transform method, we reduce the inverse problems to direct ones, which are studied for well-posedness in special weighted Sobolev spaces. This means that the inverse problem is said to be well posed in the sense of Tikhonov (or conditionally well posed). The main aim of this study is to develop a finite difference method for solution of the transformed hyperbolic problems with a non-local differential operator and initial conditions. Numerical test examples are also analyzed. Full article

We are interested in the determination of the local nonlinear magnetic material behaviour in electrical steel sheets due to cutting and punching effects. For this purpose, the inverse problem has to be solved, where the objective function, which penalises the difference between the measured and the simulated magnetic flux density, has to be minimised under a constraint defined according to the corresponding partial differential equation model. We use the adjoint method to efficiently obtain the gradients of the objective function with respect to the material parameters. The optimisation algorithm is low-memory Broyden–Fletcher–Goldfarb–Shanno (BFGS), the forward and adjoint formulations are solved using the finite element (FE) method and the ill-posedness is handled via Tikhonov regularisation, in combination with the discrepancy principle. Realistic numerical case studies show promising results. Full article

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. Full article