Search Results (71)

To address the ill-posedness of the Hessian matrix in monocular visual-inertial SLAM (Simultaneous Localization and Mapping) caused by unobservable depth of feature points, which leads to convergence difficulties and reduced robustness, this paper proposes a Pre-Dog-Leg feature optimization method based on an adaptive preconditioner. First, we propose a multi-candidate initialization method with robust characteristics. This method effectively circumvents erroneous depth initialization by introducing multiple depth assumptions and geometric consistency constraints. Second, we address the pathology of the Hessian matrix of the feature points by constructing a hybrid SPAI-Jacobi adaptive preconditioner. This preconditioner is capable of identifying matrix pathology and dynamically enabling preconditioning as a strategy. Finally, we construct a hybrid adaptive preconditioner for the traditional Dog-Leg numerical optimization method. To address the issue of degraded convergence performance when solving pathological problems, we map the pathological optimization problem from the original parameter space to a well-conditioned preconditioned space. The optimization equivalence is maintained by variable recovery. The experiments on the EuRoC dataset show that the method reduces the number of Hessian matrix conditionals by a factor of 7.9, effectively suppresses outliers, and significantly improves the overall convergence time. From the analysis of trajectory error, the absolute trajectory error is reduced by up to 16.48% relative to RVIO2 on the MH_01 sequence, 20.83% relative to VINS-mono on the MH_02 sequence, and up to 14.73% relative to VINS-mono and 34.0% relative to OpenVINS on the highly dynamic MH_05 sequence, indicating that the algorithm achieves higher localization accuracy and stronger system robustness. Full article

This paper addresses the ill-posed problem in calculating subgroup parameters for resonance self-shielding within nuclear reactor physics. The conventional Pade approximation method often yields negative subgroup cross-sections lacking physical meaning due to its treatment of overdetermined nonlinear systems, making the subgroup transport equations unsolvable. To overcome this, an optimized Pade approximation method is proposed: a resonance factor criterion is used to select energy groups requiring calculation; a systematic procedure dynamically traverses background cross-section combinations starting from a minimal subgroup number, incrementally increasing it until solutions meeting accuracy constraints with positive parameters are found; and, given the insufficiency of background points, a high-resolution resonance integral table is constructed, particularly for ranges exhibiting significant cross-section variations. Numerical validation confirms the method eliminates negative parameters, ensures physical validity, and significantly improves accuracy across benchmark cases including typical fuel pins, burnt pellets, and Gd-bearing lattices. This approach effectively resolves the ill-posedness of the traditional method, offering a more robust and precise subgroup resonance treatment for high-fidelity core neutronics. Full article

In 2071, the Hydraulic community will commemorate the second centenary of the Baré de Saint-Venant equations, also known as the Shallow Water Equations (SWE). These equations are fundamental to the study of open-channel flow. As non-linear partial differential equations, their solutions were largely unattainable until the development of computers and numerical methods. Following 1960, various numerical schemes emerged, with Preissmann’s scheme becoming the most widely employed in many software applications. In the 1990s, some researchers identified a significant limitation in existing software and codes: the inability to simulate transcritical flow. At that time, Preissmann’s scheme was the dominant method employed in hydraulics tools, leading the research community to conclude that this scheme could not handle transcritical flow due to suspected instability. In response to this concern, several researchers suggested modifications to Preissmann’s scheme to enable the simulation of transcritical flow. This paper will demonstrate that these accusations against the Preissmann scheme are unfounded and that the proposed improvements are unnecessary. The observed instability is not due to the numerical method itself, but rather a mathematical instability inherent to the SWE, which can lead to ill-posed conditions if a specific derived condition is not met. In the context of a friction slope formula based on Manning or Chézy types, the condition for ill-posedness of the 1D shallow water equations simplifies to the Vedernikov number condition, which is necessary for roll waves to develop in uniform flow. This derived condition is also relevant for the formation of roll waves in unsteady flow when the 1D shallow water equations become ill-posed. Full article

This paper investigates the well-posedness of the inverse problem for the time-fractional Burgers equation, which aims to reconstruct initial conditions from terminal observations. Such equations are crucial for the modeling of hydrodynamic phenomena with memory effects. The inverse problem involves inferring initial conditions from terminal observation data, and such problems are typically ill-posed. A framework based on optimal control theory is proposed, addressing the ill-posedness via $H^1$ regularization. Three substantial results are achieved: (1) a rigorous mathematical framework transforming the ill-posed inverse problem into a well-posed optimization problem with proven existence of solutions; (2) theoretical guarantee of solution uniqueness when the regularization parameter is $\alpha >0$ and the stability is of order O($\delta$) with respect to observation noise ($\delta$); and (3) the discovery of a “super-stability” phenomenon in numerical experiments, where the actual stability index (0.046) significantly outperforms theoretical expectations (1.0). Finally, the theoretical framework is validated through comprehensive numerical experiments, demonstrating the accuracy and practical effectiveness of the proposed optimal control approach for the reconstruction of hydrodynamic initial conditions. Full article

This paper identifies a source term depending on spatial variable in a heat equation from just part of the boundary conditions. The measurement data are specified at an internal moment of time. The ill-posedness of the problem is higher than most of the previous source identification problems. This is because the problem becomes a noncharacteristic Cauchy problem for the heat equation if the source term is given, which is known as severely ill-posed. The method of fundamental solutions (MFS) in conjunction with the classical Tikhonov regularization method is proposed to reconstruct a stable approximation. The fundamental solutions for the heat equation are spherically symmetric in spatial variable and satisfy the equation automatically, and thus only the boundary conditions need to be satisfied. This characteristic allows the discretization to be performed only on boundary-like geometry and improve the computational efficiency. In this paper, several numerical examples are listed to show the feasibility and effectiveness of the suggested method. Full article

Conventional methods for solving inverse wave problems struggle with ill-posedness, significant computational demands, and discretization errors. In this study, we propose an innovative framework for solving inverse problems in wave equations by using deep learning techniques with spacetime radial basis functions (RBFs). The proposed method capitalizes on the pattern recognition strength of deep neural networks (DNNs) and the precision of spacetime RBFs in capturing spatiotemporal dynamics. By utilizing initial conditions, boundary data, and radial distances to construct spacetime RBFs, this approach circumvents the need for wave equation discretization. Notably, the model maintains accuracy even with incomplete or noisy boundary data, illustrating its robustness and offering significant advancements over traditional techniques in solving wave equations. Full article

Correctly fixing the integer ambiguity of GNSS is the key to realizing the application of GNSS high-precision positioning. When solving the float solution of ambiguity based on the double-difference model epoch by epoch, the common method for resolving the integer ambiguity needs to solve the coordinate parameter information, due to the influence of limited GNSS phase data observations. This type of method will lead to an increase in the ill-posedness of the double-difference solution equation, so that the fixed success rate of the integer ambiguity is not high. Therefore, a new integer ambiguity resolution method based on eliminating coordinate parameters and ant colony algorithm is proposed in this paper. The method eliminates the coordinate parameters in the observation equation using QR decomposition transformation, and only estimates the ambiguity parameters using the Kalman filter. On the basis that the Kalman filter will obtain the float solution of ambiguity, the decorrelation processing is carried out based on continuous Cholesky decomposition, and the optimal solution of integer ambiguity is searched using the ant colony algorithm. Two sets of static and dynamic GPS experimental data are used to verify the method and compared with conventional least squares and LAMBDA methods. The results show that the new method has good decorrelation effect, which can correctly and effectively realize the integer ambiguity resolution. Full article

The problem of blind image deblurring remains a challenging inverse problem, due to the ill-posed nature of estimating unknown blur kernels and latent images within the Maximum A Posteriori (MAP) framework. To address this challenge, traditional methods often rely on sparse regularization priors to mitigate the uncertainty inherent in the problem. In this paper, we propose a novel blind deblurring model based on the MAP framework that leverages Composite-Gradient Feature (CGF) variations in edge regions after image blurring. This prior term is specifically designed to exploit the high sparsity of sharp edge regions in clear images, thereby effectively alleviating the ill-posedness of the problem. Unlike existing methods that focus on local gradient information, our approach focuses on the aggregation of edge regions, enabling better detection of both sharp and smoothed edges in blurred images. In the blur kernel estimation process, we enhance the accuracy of the kernel by assigning effective edge information from the blurred image to the smoothed intermediate latent image, preserving critical structural details lost during the blurring process. To further improve the edge-preserving restoration, we introduce an adaptive regularizer that outperforms traditional total variation regularization by better maintaining edge integrity in both clear and blurred images. The proposed variational model is efficiently implemented using alternating iterative techniques. Extensive numerical experiments and comparisons with state-of-the-art methods demonstrate the superior performance of our approach, highlighting its effectiveness and real-world applicability in diverse image-restoration tasks. Full article

The third-order pseudoparabolic equations represent models of filtration, the movement of moisture and salts in soils, heat and mass transfer, etc. Such non-classical equations are often referred to as Sobolev-type equations. We consider an inverse problem for identifying an unknown time-dependent boundary condition in a two-dimensional linear pseudoparabolic equation from integral-type measured output data. Using the integral measurements, we reduce the two-dimensional inverse problem to a one-dimensional problem. Then, we apply appropriate substitution to overcome the non-local nature of the problem. The inverse ill-posed problem is reformulated as a direct well-posed problem. The well-posedness of the direct and inverse problems is established. We develop a computational approach for recovering the solution and unknown boundary function. The results from numerical experiments are presented and discussed. Full article

Sliding bearings are widely used in wind turbine gearboxes, and the accurate identification of coupling interface loads is critical for ensuring the reliability and performance of these systems. However, the space–time coupling nature of these loads makes them difficult to calculate and measure directly. An improved method utilizing the POD decomposition algorithm and polynomial selection technology is proposed in this paper to identify the sliding bearing coupling interface loads. By using the POD decomposition algorithm, the sliding bearing coupling interface loads can be decomposed into the form of a series of independent oil film time history and spatial distribution functions. Then, it can be converted into space–time independent sub-coupled interface load identification in which oil film time history can be transformed into the recognition of a certain order modal load and the corresponding oil film spatial distribution function can be fitted with a set of Chebyshev orthogonal polynomial. To address the ill-posedness caused by the weak correlation between the modal matrix and polynomial options during the identification process, this paper introduces polynomial structure selection technology. Firstly, displacement responses are collected, and a series of modal loads are identified using conventional concentrated load identification methods. Then, the polynomial structure selection technology is applied to select the effective modal shape matrix, using a specific mode load as the oil film time history function. The load ratios of other mode loads to this reference mode load are compared, and the effective Chebyshev orthogonal polynomials are selected based on the error reduction ratio. Finally, multiplying the identified oil film time histories by the corresponding oil film spatial distribution functions yields the coupling interface load. The results of the numerical examples verify the improved method’s rationality and effectiveness. Full article

Similarity to reality is a necessary property of models in earth sciences. Similarity information can thus possess a large potential in advancing geophysical modeling and data assimilation. We present a formalism for utilizing similarity within the existing theoretical data assimilation framework. Two examples illustrate the usefulness of utilizing similarity in data assimilation. The first, theoretical example shows changes in the accuracy of the amplitude estimate in the presence of a phase error in a sine function, where correcting the phase error prior to the assimilation reduces the degree of ill-posedness of the assimilation problem. This signifies the importance of accounting for the phase error in order to reduce the error in the amplitude estimate of the sine function. The second, real-world example illustrates that timing errors in simulated flow degrade the data assimilation performance, and that the flow gradient-informed shifting of rainfall time series improved the assimilation results with less adjusting model states. This demonstrates the benefit of utilizing streamflow gradients in shifting rainfall time series in a way to improve streamflow timing—vital information for flood early warning and preparedness planning. Finally, we discuss the implications, potential issues, and future challenges associated with utilizing similarity in hydrologic data assimilation. Full article

In clinical settings, computed tomography (CT), magnetic resonance imaging (MRI), or positron emission tomography (PET) are commonly employed in brain imaging to assist clinicians in determining the type of stroke in patients. However, these modalities are associated with potential hazards or limitations. In contrast, microwave imaging emerges as a promising technique, offering advantages such as non-ionizing radiation, low cost, lightweight, and portability. The primary challenges faced by microwave tomography include the severe ill-posedness of the electromagnetic inverse scattering problem and the time-consuming nature and unsatisfactory resolution of iterative quantitative algorithms. This paper proposes a learning electric field enhancement imaging method (LEFEIM) to achieve quantitative brain imaging based on a microwave tomography system. LEFEIM comprises two cascaded networks. The first, based on a convolutional neural network, utilizes the electric field from the receiving antenna to predict the electric field distribution within the imaging domain. The second network employs the electric field distribution as input to learn the dielectric constant distribution, thereby realizing quantitative brain imaging. Compared to the Born Iterative Method (BIM), LEFEIM significantly improves imaging time, while enhancing imaging quality and goodness-of-fit to a certain extent. Simultaneously, LEFEIM exhibits anti-noise capabilities. Full article

We examine a time-fractional pseudo-hyperbolic equation involving positive operators. We explore the determination of initial velocity and perturbation. It is demonstrated that these initial inverse problems are ill posed. Additionally, we prove that under certain conditions, the inverse problems exhibit well-posedness properties. Our focus is on developing a theoretical framework for these initial inverse problems associated with time-fractional pseudo-hyperbolic equations, laying the groundwork for future studies on numerical algorithms to solve these problems. This investigation is crucial for understanding the fundamental behavior of the equations under various initial conditions and perturbations. By establishing a rigorous theoretical framework, we pave the way for future research to focus on practical numerical methods and simulations. Our results provide a deeper insight into the mathematical structure of time-fractional pseudo-hyperbolic equations, ensuring that future computational approaches are built on a solid theoretical foundation. Full article

This article is devoted to solving full-wave electromagnetic inverse scattering problems (EM-ISPs), which determine the geometrical and physical properties of scatterers from the knowledge of scattered fields. Due to the intrinsic ill-posedness and nonlinearity of EM-ISPs, traditional non-iterative and iterative methods struggle to meet the requirements of high accuracy and real-time reconstruction. To overcome these issues, we propose a two-step contrast source learning approach, cascading convolutional neural networks (CNNs) into the inversion framework, to tackle 2D full-wave EM-ISPs. In the first step, a contrast source network based on the CNNs architecture takes the determined part of the contrast source as input and then outputs an estimate of the total contrast source. Then, the recovered total contrast source is directly converted into the initial contrast. In the second step, the rough initial contrast obtained beforehand is input into the U-Net for refinement. Consequently, the EM-ISPs can be quickly solved with much higher accuracy, even for high-contrast objects, almost achieving real-time imaging. Numerical examples have demonstrated that the proposed two-step contrast source learning approach is able to improve accuracy and robustness even for high-contrast scatterers. The proposed approach offers a promising avenue for advancing EM-ISPs by integrating strengths from both traditional and deep learning-based approaches, to achieve real-time quantitative microwave imaging for high-contrast objects. Full article

The Rational Function Model (RFM) is composed of numerous highly correlated Rational Polynomial Coefficients (RPCs), establishing a mathematical relationship between two-dimensional images and three-dimensional spatial coordinates. Due to the existence of ill-posedness and overparameterization, the estimated RPCs are sensitive to any slight perturbations in the observation data, particularly when handling a limited number of Ground Control Points (GCPs). Recently, Principal Component Analysis (PCA) has demonstrated significant performance improvements in the RFM optimization problem. In the PCA-based RFM, each Principal Component (PC) is a linear combination of all variables in the design matrix. However, some original variables are noise related and have very small or almost zero contributions to the construction of PCs, which leads to the overparameterization problem and makes the RPC estimation process ill posed. To address this problem, in this paper, we propose an Adaptive Sparse Principal Component Analysis-based RFM method (ASPCA-RFM) for RPC estimation. In this method, the Elastic Net sparsity constraint is introduced to ensure that each PC contains only a small number of original variables, which automatically eliminates unnecessary variables during PC computation. Since the optimal regularization parameters of the Elastic Net vary significantly in different scenarios, an adaptive regularization parameter approach is proposed to dynamically adjust the regularization parameters according to the explained variance of PCs and degrees of freedom. By adopting the proposed method, the noise and error in the design matrix can be reduced, and the ill-posedness and overparameterization of the RPC estimation can be significantly mitigated. Additionally, we conduct extensive experiments to validate the effectiveness of our method. Compared to existing state-of-the-art methods, the proposed method yields markedly improved or competitive performance. Full article