Search Results (86)

In marine seismic exploration, ghost waves distort reflection waveforms and narrow the frequency band of seismic records. Traditional deghosting methods are susceptible to practical limitations from sea surface fluctuations and velocity variations. This paper proposes a τ-p domain deghosting method based on the Hybrid Least Squares Residual (HyBR LSMR) algorithm. We first establish a linear forward model in the τ-p domain that describes the relationship between the total wavefield and upgoing wavefield, transforming deghosting into a linear inverse problem. The method then employs the hybrid LSMR algorithm with Tikhonov regularization to address the inherent ill-posedness. A key innovation is the integration of the Generalized Cross Validation (GCV) criterion to adaptively determine regularization parameters and iteration stopping points, effectively avoiding the semi-convergence phenomenon and enhancing solution stability. Applications to both synthetic and field data demonstrate that the proposed method effectively suppresses ghost waves under various acquisition conditions, significantly improves the signal-to-noise ratio and resolution, broadens the effective frequency band, and maintains good computational efficiency, providing a reliable solution for high-precision seismic data processing in complex marine environments. Full article

In the present work, we aim to study the inverse problem of recovering an unknown spatial source term in a multi-term time-fractional diffusion equation involving the fractional Laplacian. The forward problem is first analyzed in appropriate fractional Sobolev spaces, establishing the existence, uniqueness, and regularity of solutions. Exploiting the spectral representation of the solution and properties of multinomial Mittag–Leffler functions, we prove uniqueness and derive a stability estimate for the spatial source term from finaltime observations. The inverse problem is then formulated as a Tikhonov regularized optimization problem, for which existence, uniqueness, and strong convergence of the regularized minimizer are rigorously established. On the computational side, we propose an efficient reconstruction algorithm based on the conjugate gradient method, with temporal discretization via an L¹-type scheme for Caputo derivatives and spatial discretization using a Galerkin approach adapted to the nonlocal fractional Laplacian. Numerical experiments confirm the accuracy and robustness of the proposed method in reconstructing the unknown source term. Full article

In high-temperature testing scenarios that rely on contact, fine-wire thermocouples demonstrate commendable dynamic performance. Nonetheless, their thermal inertia leads to notable dynamic nonlinear inaccuracies, including response delays and amplitude reduction. To mitigate these challenges, a novel dynamic error correction approach is introduced, which combines a Continuous Restricted Boltzmann Machine, Deep Belief Network, and Physics-Informed Neural Network (CDBN-PINN). The unique heat transfer properties of the thermocouple’s bimetallic structure are represented through an Inverse Heat Conduction Equation (IHCP). An analysis is conducted to explore the connection between the analytical solution’s ill-posed nature and the thermocouple’s dynamic errors. The transient temperature response’s nonlinear characteristics are captured using CRBM-DBN. To maintain physical validity and minimize noise amplification, filtered kernel regularization is applied as a constraint within the PINN framework. This approach was tested and confirmed through laser pulse calibration on thermocouples with butt-welded and ball-welded configurations of 0.25 mm and 0.38 mm. Findings reveal that the proposed method achieved a peak relative error of merely 0.83%, superior to Tikhonov regularization by −2.2%, Wiener deconvolution by 20.40%, FBPINNs by 1.40%, and the ablation technique by 2.05%. In detonation tests, the corrected temperature peak reached 1045.7 °C, with the relative error decreasing from 77.7% to 5.1%. Additionally, this method improves response times, with the rise time in laser calibration enhanced by up to 31 ms and in explosion testing by 26 ms. By merging physical constraints with data-driven methodologies, this technique successfully corrected dynamic errors even with limited sample sizes. Full article

Due to the difficulty of directly measuring external loads on jacket platform structures and the challenges in accurately expressing them through analytical formulas, this study proposes a load inversion method based on local strain measurement data to obtain the time–history curves of structural loads. The method establishes a mapping relationship between unknown loads and measured strains based on the quasi-static superposition principle. An Improved Sine Cosine Algorithm, combined with an Opposition-Based Learning, is introduced to optimize the placement of strain sensors. The unknown loads are solved using a least squares approach integrated with Tikhonov regularization. The method was validated through indoor loading experiments under eight conditions, where the inverted load time–history curves accurately reflected the periodic characteristics of the applied loads, achieving a maximum Mean Absolute Relative Error (MARE) of 6.91%, demonstrating high stability and accuracy. The further application of the method to an in-service jacket platform in a marine environment yielded inverted wind and wave loads with a maximum MARE of 11.63% compared to loads calculated from measured wind and wave data, validating the method’s practical applicability and robustness. This approach offers a more accurate load basis for the safety assessment and residual life prediction of jacket platform structures. Full article

Civil infrastructure, such as bridges and buildings, is susceptible to damage from unforeseen low-speed impacts during service. Impact force identification from dynamic response measurements is essential for structural health monitoring and structural design. Force identification is an ill-posed inverse problem, and the regularization technique is widely used to solve this problem using a full transfer matrix. However, existing regularization techniques are not suitable for large-scale practical structures due to the high computational cost for the inverse calculation of a high-dimensional transfer matrix, and impact excitation locations are often unknown in practice. To address these challenges, a novel two-step truncated transfer matrix-based impact force identification method is proposed in this study. In the first step, a sparse regularization-based technique is developed to determine unknown force locations using modal superposition. In the second step, the full transfer matrix is truncated by time windows corresponding to short durations of impact excitations, and a Tikhonov regularization-based technique is adopted to reconstruct the time history of impact forces. The proposed method is verified numerically on a simply supported beam and experimentally on a 10 m steel–concrete composite bridge deck. The results show that the proposed method could determine the impact locations and reconstruct the time history of impact forces accurately. Compared with existing Tikhonov and sparse regularization methods, the proposed method demonstrates superior accuracy and computational efficiency for impact force identification. The robustness of the proposed method to noise level and the number of modes and sensors is investigated. Experimental studies for both single-force and multiple-force localization and identification are conducted. The results indicate that the proposed method is efficient and accurate in identifying impact forces. Full article

In this work, we proposed a dynamic inverse solution with spatio-temporal constraints of the nonlinear heat diffusion problem in 1D and 2D based on a regularized Gauss–Newton and Krylov subspace with a preconditioner. The preconditioner is computed by approximating the Jacobian of the nonlinear system at each Gauss–Newton iteration. The proposed approach is used for estimation of the initial value from measurements of the last value by considering spatial and spatio-temporal constraints. The system is compared to a dynamic Tikhonov inverse solution and generalized minimal residual method (GMRES) with and without a preconditioner. The system is evaluated under noise conditions in order to verify the robustness of the proposed approach. It can be seen that the proposed spatio-temporal regularized Gauss–Newton method with GMRES and a preconditioner shows better estimation results than the other methods for both spatial and spatio-temporal constraints. Full article

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