First-order Tikhonov Regularization Research Articles

Tomography of fast ion distribution function under neutral beam injection and ion cyclotron resonance heating on EAST

In magnetic confinement fusion devices, velocity-space tomography of fast-ion velocity distribution function is crucial for investigating fast-ion distribution and transport. In the neutral beam injection (NBI) and ion cyclotron resonance heating (ICRF) synergistic heating experiments in Experimental Advanced Superconducting Tokamak (EAST), high-energy particles with energy exceeding the particle energy in NBI are observed. Simulations of synergistic effect on fast-ion velocity distribution function given by TRANSP also show the existence of particles with energy higher than the particle energy in NBI. To investigate the behaviors of fast ion distribution and calculate the velocity distribution functions under different heating conditions, the first-order Tikhonov regularization tomographic inversion method with higher inversion accuracy is introduced by comparing various regularization techniques. The limitations of the dual-view fast-ion D<sub>α</sub> (FIDA) diagnostic measurements in velocity space are addressed by incorporating prior information such as null measurement and the known peaks and effectively mitigate the occurrence of artifacts. This method is first employed in the case of NBI heating. The NBI peak is successfully reconstructed at the expected location in velocity space, which shows significant improvement in the inversion results. In order to further validate the synergistic effect of NBI-ICRF heating and study the mechanism of fast ion distribution under synergistic heating, the combination of FIDA and neutron emission spectrometer (NES) is applied to the first-order Tikhonov regularization tomographic inversion method for enhancing the coverage of velocity space, through which the issue of artifacts in the inversion results is significantly improved, and thus the precision of the obtained fast-ion velocity distribution functions is enhanced. Based on the benefit described above, the method of combining NES diagnosis and FIDA diagnosis is used to obtain fast-ion velocity distribution functions in the NBI and ICRF synergistic heating discharge. The synergistic heating effect is manifested in the fast-ion velocity distribution. The availability of this inversion method in reconstructing fast-ion velocity distribution functions during high-performance operation of NBI-ICRF synergistic heating in the EAST experiment is confirmed. In the next-step EAST research, high performance discharge will demand more efficiency NBI and ICRF synergistic heating, the present work builds the stage for investigating the underlying mechanism of synergistic heating and the intricate behaviors associated with fast ion distribution and transport.

Comparison of non-parametric T2 relaxometry methods for myelin water quantification

Multi-component T2 relaxometry allows probing tissue microstructure by assessing compartment-specific T2 relaxation times and water fractions, including the myelin water fraction. Non-negative least squares (NNLS) with zero-order Tikhonov regularization is the conventional method for estimating smooth T2 distributions. Despite the improved estimation provided by this method compared to non-regularized NNLS, the solution is still sensitive to the underlying noise and the regularization weight. This is especially relevant for clinically achievable signal-to-noise ratios. In the literature of inverse problems, various well-established approaches to promote smooth solutions, including first-order and second-order Tikhonov regularization, and different criteria for estimating the regularization weight have been proposed, such as L-curve, Generalized Cross-Validation, and Chi-square residual fitting. However, quantitative comparisons between the available reconstruction methods for computing the T2 distribution, and between different approaches for selecting the optimal regularization weight, are lacking. In this study, we implemented and evaluated ten reconstruction algorithms, resulting from the individual combinations of three penalty terms with three criteria to estimate the regularization weight, plus non-regularized NNLS. Their performance was evaluated both in simulated data and real brain MRI data acquired from healthy volunteers through a scan-rescan repeatability analysis. Our findings demonstrate the need for regularization. As a result of this work, we provide a list of recommendations for selecting the optimal reconstruction algorithms based on the acquired data. Moreover, the implemented methods were packaged in a freely distributed toolbox to promote reproducible research, and to facilitate further research and the use of this promising quantitative technique in clinical practice.

Fast 3D magnetic inversion of a surface relief in the space domain

We have developed a fast 3D regularized magnetic inversion algorithm for depth-to-basement estimation based on an efficient way to compute the total-field anomaly produced by an arbitrary interface separating nonmagnetic sediments from a magnetic basement. We approximate the basement layer by a grid of 3D vertical prisms juxtaposed in the horizontal directions, in which the prisms’ tops represent the depths to the magnetic basement. To compute the total-field anomaly produced by the basement relief, the 3D integral of the total-field anomaly of a prism is simplified by a 1D integral along the prism thickness, which in turn is multiplied by the horizontal area of the prism. The 1D integral is calculated numerically using the Gauss-Legendre quadrature produced by dipoles located along the vertical axis passing through the prism center. This new magnetic forward modeling overcomes one of the main drawbacks of the nonlinear inverse problem for estimating the basement depths from magnetic data: the intense computational cost to calculate the total-field anomaly of prisms. The new sensitivity matrix is simpler and computationally faster than the one using classic magnetic forward modeling based on the 3D integrals of a set of prisms that parameterize the earth’s subsurface. To speed up the inversion at each iteration, we used the Gauss-Newton approximation for the Hessian matrix keeping the main diagonal only and adding the first-order Tikhonov regularization function. The large sparseness of the Hessian matrix allows us to construct and solve a linear system iteratively that is faster and demands less memory than the classic nonlinear inversion with prism-based modeling using 3D integrals. We successfully inverted the total-field anomaly of a simulated smoothing basement relief with a constant magnetization vector. Tests on field data from a portion of the Pará-Maranhão Basin, Brazil, retrieved a first depth-to-basement estimate that was geologically plausible.

A Boltzmann machine for high-resolution prestack seismic inversion

Seismic inversion is one popular approach that aims at predicting some indicative properties to support the geologic interpretation process. Existing inversion techniques indicate weaknesses when dealing with complex geologic area, where the uncertainties arise from the guiding model, which are provided by the interpreters. We have developed a prestack seismic inversion algorithm using a machine-learning algorithm called the Boltzmann machine. Unlike common inversion approaches, this stochastic neural network does not require a starting model at the beginning of the process to guide the solution; however, low-frequency models are required to convert the inversion-derived reflectivity terms to the absolute elastic P- and S-impedance as well as density. Our algorithm incorporates a single-layer Hopfield neural network whose neurons can be treated as the desired reflectivity terms. The optimization process seeks the global minimum solution by combining the network with a stochastic model update from the mean-field annealing algorithm. Also, we use a Z-shaped sample sorting scheme and the first-order Tikhonov regularization to improve the lateral continuity of the results and to stabilize the inversion process. The algorithm is applied to a field 2D data set to invert for high-resolution indicative P- and S-impedance sections to better capture some features away from the reservoir zone. The resulting models are strongly supported by the well results and reveal some realistic features that are not clearly displayed in the model-based deterministic inversion result. In combination with well-log analyses, the new features appear to be a good prospect for hydrocarbon.

The generalized finite difference method for an inverse boundary value problem in three-dimensional thermo-elasticity

In this study, a new framework for the efficient and accurate solutions for an inverse problem associated with three-dimensional (3D) coupled thermo-elasticity equation is presented. The ill-conditioned problem is solved here with the generalized finite difference method (GFDM) together with the first-order Tikhonov regularization technique and the L-curve criterion. The GFDM uses the Taylor series expansions and the moving least squares approximation to derive explicit formulae for the required partial derivatives of unknown variables. The method is truly meshless that can be applied for solving problems merely defined over irregular clouds of points. In addition, for problems involving complex geometries, a new distance criterion for adaptive selection of nodes in the GFDM simulations is proposed. Preliminary numerical experiments show that the regularized GFDM proposed in this study are very promising for accurate and efficient inverse thermo-elasticity simulations, even with a comparatively large level of noise added into the input data.

Inverse time-dependent source problems for the heat equation with nonlocal boundary conditions

In this paper, we consider inverse problems of finding the time-dependent source function for the population model with population density nonlocal boundary conditions and an integral over-determination measurement. These problems arise in mathematical biology and have never been investigated in the literature in the forms proposed, although related studies do exist. The unique solvability of the inverse problems are rigorously proved using generalized Fourier series and the theory of Volterra integral equations. Continuous dependence on smooth input data also holds but, as in reality noisy errors are random and non-smooth, the inverse problems are still practically ill-posed. The degree of ill-posedness is characterised by the numerical differentiation of a noisy function. In the numerical process, the boundary element method together with either a smoothing spline regularization or the first-order Tikhonov regularization are employed with various choices of regularization parameter. One is based on the discrepancy principle and another one is the generalized cross-validation criterion. Numerical results for some benchmark test examples are presented and discussed in order to illustrate the accuracy and stability of the numerical inversion.

Investigation of the Tikhonov Regularization Method in Regional Gravity Field Modeling by Poisson Wavelets Radial Basis Functions

The application of Tikhonov regularization method for dealing with the ill-conditioned problems in the regional gravity field modeling by Poisson wavelets is studied. In particular, the choices of the regularization matrices as well as the approaches for estimating the regularization parameters are investigated in details. The numerical results show that the regularized solutions derived from the first-order regularization are better than the ones obtained from zero-order regularization. For cross validation, the optimal regularization parameters are estimated from L-curve, variance component estimation (VCE) and minimum standard deviation (MSTD) approach, respectively, and the results show that the derived regularization parameters from different methods are consistent with each other. Together with the first-order Tikhonov regularization and VCE method, the optimal network of Poisson wavelets is derived, based on which the local gravimetric geoid is computed. The accuracy of the corresponding gravimetric geoid reaches 1.1 cm in Netherlands, which validates the reliability of using Tikhonov regularization method in tackling the ill-conditioned problem for regional gravity field modeling.

High-resolution regional gravity field recovery from Poisson wavelets using heterogeneous observational techniques

We adopt Poisson wavelets for regional gravity field recovery using data acquired from various observational techniques; the method combines data of different spatial resolutions and coverage, and various spectral contents and noise levels. For managing the ill-conditioned system, the performances of the zero- and first-order Tikhonov regularization approaches are investigated. Moreover, a direct approach is proposed to properly combine Global Positioning System (GPS)/leveling data with the gravimetric quasi-geoid/geoid, where GPS/leveling data are treated as an additional observation group to form a new functional model. In this manner, the quasi-geoid/geoid that fits the local leveling system can be computed in one step, and no post-processing (e.g., corrector surface or least squares collocation) procedures are needed. As a case study, we model a new reference surface over Hong Kong. The results show solutions with first-order regularization are better than those obtained from zero-order regularization, which indicates the former may be more preferable for regional gravity field modeling. The numerical results also demonstrate the gravimetric quasi-geoid/geoid and GPS/leveling data can be combined properly using this direct approach, where no systematic errors exist between these two data sets. A comparison with 61 independent GPS/leveling points shows the accuracy of the new geoid, HKGEOID-2016, is around 1.1 cm. Further evaluation demonstrates the new geoid has improved significantly compared to the original model, HKGEOID-2000, and the standard deviation for the differences between the observed and computed geoidal heights at all GPS/leveling points is reduced from 2.4 to 0.6 cm. Finally, we conclude HKGEOID-2016 can be substituted for HKGEOID-2000 for engineering purposes and geophysical investigations in Hong Kong.Graphical abstract.

Regularized Laplace–Fourier-Domain Full Waveform Inversion Using a Weighted l 2 Objective Function

Full waveform inversion (FWI) can be applied to obtain an accurate velocity model that contains important geophysical and geological information. FWI suffers from the local minimum problem when the starting model is not sufficiently close to the true model. Therefore, an accurate macroscale velocity model is essential for successful FWI, and Laplace–Fourier-domain FWI is appropriate for obtaining such a velocity model. However, conventional Laplace–Fourier-domain FWI remains an ill-posed and ill-conditioned problem, meaning that small errors in the data can result in large differences in the inverted model. This approach also suffers from certain limitations related to the logarithmic objective function. To overcome the limitations of conventional Laplace–Fourier-domain FWI, we introduce a weighted l 2 objective function, instead of the logarithmic objective function, as the data-domain objective function, and we also introduce two different model-domain regularizations: first-order Tikhonov regularization and prior model regularization. The weighting matrix for the data-domain objective function is constructed to suitably enhance the far-offset information. Tikhonov regularization smoothes the gradient, and prior model regularization allows reliable prior information to be taken into account. Two hyperparameters are obtained through trial and error and used to control the trade-off and achieve an appropriate balance between the data-domain and model-domain gradients. The application of the proposed regularizations facilitates finding a unique solution via FWI, and the weighted l 2 objective function ensures a more reasonable residual, thereby improving the stability of the gradient calculation. Numerical tests performed using the Marmousi synthetic dataset show that the use of the weighted l 2 objective function and the model-domain regularizations significantly improves the Laplace–Fourier-domain FWI. Because the Laplace–Fourier-domain FWI is improved, the frequency-domain FWI, in which the Laplace–Fourier-domain FWI result is used as the starting model, yields inversion result much closer to the true velocity.

Development of regularization methods on simulated ground-penetrating radar signals to predict thin asphalt overlay thickness

The range resolution of ground-penetrating radar (GPR) signal is important in thin asphalt overlay thickness estimation. In this paper, regularized deconvolution is utilized to analyze simulated GPR signals to increase their range resolution. Four types of regularization methods, including Tikhonov regularization and total variation, were applied on noisy GPR signals; and their performance was evaluated in terms of accuracy in estimating distance of close impulses. The L-curve method was used to choose the appropriate regularization parameter. The total variation regularization method and zeroth-order Tikhonov regularization outperform first-order and second-order Tikhonov regularization in terms of average asphalt layer thickness estimation error and the standard deviation of the error. An example of the field GPR data is provided to validate the proposed algorithm. The study shows that the algorithm based on regularization is a simple and effective approach to increase the GPR signal range resolution with presence of noise in the case of thin asphalt overlay thickness prediction.

Inversion methods for fast-ion velocity-space tomography in fusion plasmas

Velocity-space tomography has been used to infer 2D fast-ion velocity distribution functions. Here we compare the performance of five different tomographic inversion methods: truncated singular value decomposition, maximum entropy, minimum Fisher information and zeroth- and first-order Tikhonov regularization. The inversion methods are applied to fast-ion measurements taken just before and just after a sawtooth crash in the ASDEX Upgrade tokamak as well as to synthetic measurements from different test distributions. We find that the methods regularizing by penalizing steep gradients or maximizing entropy perform best. We assess the uncertainty of the calculated inversions taking into account photon noise, uncertainties in the forward model as well as uncertainties introduced by the regularization which allows us to distinguish regions of high and low confidence in the tomographies. In high confidence regions, all methods agree that ions with pitch values close to zero, as well as ions with large pitch values, are ejected from the plasma center by the sawtooth crash, and that this ejection depletes the ion population with large pitch values more strongly.

Blood perfusion estimation in heterogeneous tissue using BEM based algorithm

Abstract The estimation of space-dependent perfusion coefficient in homogeneous and non-homogeneous tissue has been investigated. While initial and Dirichlet boundary conditions are known, additional heat-flux measurement data is needed to render a unique solution. A numerical approach based on Boundary Element Method (BEM) combined with two different optimization routines and first-order Tikhonov regularization using L-Curve method has been developed. Efficiency of the algorithm, effect of initial guess, noise, perfusion distribution and non-homogeneous tissue on retrieving the perfusion coefficient has been studied on two test examples using exact as well as noisy data. Results show very good agreement with the true perfusion function under exact and low-noisy data, using Levenberg–Marquardt (LM) method combined with first-order regularization process. If the true perfusion function first-derivative is large, the function can be successfully retrieved only in the region near the boundary measurement, which is especially noticeable for a non-monotonic function in non-homogeneous tissue. This study represents the base for further research on the field of successful non-invasive blood perfusion determination in non-homogeneous tissue.

Regularized seismic full waveform inversion with prior model information

Full waveform inversion (FWI) delivers high-resolution quantitative images and is a promising technique to obtain macroscale physical property model of the subsurface. In most geophysical applications, prior information, such as that collected in wells, is available and should be used to increase the image reliability. For this, we propose to introduce three terms in the definition of the FWI misfit function: the data misfit itself, the first-order Tikhonov regularization term acting as a smoothing operator, and a prior model norm term. This last term is the way to smoothly introduce prior information into the FWI workflow. On a selected target of the Marmousi synthetic example, significant improvement was obtained when using the prior model term for noise-free and noisy synthetic data. The prior model term may significantly reduce the inversion sensitivity to incorrect initial conditions. The limited range of spatial wavenumber sampling by the acquisition may be compensated with the prior model information, for multiple-free and multiple-contaminated data. Prior and initial models play different roles in the inversion scheme. The starting model is used for wave propagation and therefore drives the data-misfit gradient, whereas the prior model is never explicitly used for solving the wave equation and only drives the optimization step as an additional constraint to minimize the total objective function. Thus, the prior model is not required to follow kinematic properties as precisely as the initial model, except in zones of poor illumination. In addition, we investigate the influence of a simple dynamic decreasing weighting of the prior model term. Once the cycle-skipping problem has been solved, the impact of the prior model term is gradually reduced within the misfit function to be driven by seismic-data only.

Source geometry estimation using the mass excess criterion to constrain 3-D radial inversion of gravity data

SUMMARY We present a gravity-inversion method for estimating the geometry of an isolated 3-D source, assuming prior knowledge about its top and density contrast. The subsurface region containing the geological sources is discretized into an ensemble of 3-D vertical prisms juxtaposed in the vertical direction of a right-handed coordinate system. The prisms’ thicknesses and density contrasts are known, but their horizontal cross-sections are described by unknown polygons. The horizontal coordinates of the polygon vertices approximately represent the edges of horizontal depth slices of the 3-D geological source. The polygon vertices of each prism are describedbypolarcoordinateswithanunknownoriginwithintheprism.Ourmethodestimates the horizontal Cartesian coordinates of the unknown origin and the radii associated with the vertices of each polygon for a fixed number of equally spaced central angles from 0 o to 360 o . By estimating these parameters from gravity data, we retrieve a set of vertically stacked prisms with polygonal horizontal sections that represents a set of juxtaposed horizontal depth slices of the estimated source. This set, therefore, approximates the 3-D source’s geometry. To obtain stable estimates we impose constraints on the source shape. The judicious use of first-order Tikhonov regularization on either all or a few parameters allows estimating both vertical and inclined sources whoseshapes can beisometricoranisometric. Theestimated solution, despite being stable and fitting the data, will depend on the maximum depth assumed for the set of juxtaposed 3-D prisms. To reduce the class of possible solutions compatible with the gravity anomaly and the constraints, we use a criterion based on the relationship between the datamisfit measure and the estimated total-anomalous mass computed along successive inversions, using different tentative maximum depths for the set of assumed juxtaposed 3-D prisms. In applying this criterion, we plotted the curve of the estimated total-anomalous mass mt versus data-misfit measure s for the range of different tentative maximum depths. The tentative value forthe maximum depth producing thesmallest value of data-misfit measure inthemt ×s curve is the best estimate of the true (or minimum) depth to the bottom of the source, depending on whether the true source produces a gravity anomaly that is able (or not) to resolve the depth to the source bottom. This criterion was theoretically deduced from Gauss’ theorem. Tests with synthetic data shows that the correct depth-to-bottom estimate of the source is obtained if the minimum of s on the mt × s curve is well defined; otherwise this criterion provides just a lower bound estimate of the source’s depth to the bottom. These synthetic results show that the method efficiently recovers source geometries dipping at different angles. Test on real data from the Matsitama intrusive complex (Botswana) retrieved a dipping intrusion with variable dips and strikes and with bottom depth of 8.0 ± 0.5 km.

State space regularization in the nonstationary inverse problem for diffuse optical tomography

In this paper, we present a regularization method in the nonstationary inverse problem for diffuse optical tomography (DOT). The regularization is based on a choosing time evolution process such that in a stationary state it has a covariance function which corresponds to a process with similar smoothness properties as the first-order smoothness Tikhonov regularization. The proposed method is computationally more lightweight than the method where the regularization is augmented as a measurement. The method was tested in the case of the inverse problem of DOT. A solid phantom with optical properties similar to tissue was made, incorporating two moving parts that simulate two different physiological processes: a localized change in absorption and a surrounding rotating two-part shell which simulates slow oscillations in the tissue background physiology. A sequence of measurements of the phantom was made and the reconstruction of the image sequence was computed using this method. It allows the recovery of the full time series of images from relatively slow measurements with one source active at a time. In practice, this allows instruments with a larger dynamic range to be applied to the imaging of functional phenomena using DOT.

Inverse temperature-dependent perfusion coefficient reconstruction

The identification of the temperature-dependent perfusion coefficient in the one-dimensional transient bio-heat conduction equation is investigated. If Neumann boundary conditions are prescribed, then the additional measurement sufficient to render a unique solution is a temperature measurement on a part of the boundary. A numerical approach based on a Crank–Nicolson-type finite-difference scheme combined with the first-order Tikhonov regularization method is developed. Numerical results are presented and discussed.

Discretization Error Analysis and Adaptive Meshing Algorithms for Fluorescence Diffuse Optical Tomography: Part I

For imaging problems in which numerical solutions need to be computed for both the inverse and the underlying forward problems, discretization can be a major factor that determines the accuracy of imaging. In this work, we analyze the effect of discretization on the accuracy of fluorescence diffuse optical tomography. We model the forward problem by a pair of diffusion equations at the excitation and emission wavelengths and consider a finite element discretization method for the numerical solution of the forward problem. For the inverse problem, we use an optimization framework which allows incorporation of a priori information in the form of zeroth- and first-order Tikhonov regularization terms. Next, we convert the inverse problem into a variational problem and use Galerkin projection to discretize the inverse problem. Following the discretization, we analyze the error in reconstructed images due to the discretization of the forward and inverse problems and present two theorems which point out the factors that may lead to high error such as the mutual dependence of the forward and inverse problems, the number of sources and detectors, their configuration and their positions with respect to fluorophore concentration, and the formulation of the inverse problem. Finally, we demonstrate the results and implications of our error analysis by numerical experiments. In the second part of the paper, we apply our results to design novel adaptive discretization algorithms.

Total wavefield-based inversion in PML-truncated domains

We discuss a total wavefield-based inversion approach for the reconstruction of the material profile of heterogeneous semi-infinite domains, directly in the time domain, based on scant surficial measurements of the domain's response to prescribed wave illumination. Of particular interest is the ability to recover the in-depth profile of moduli/wave velocities typically associated with small-scale geotechnical site characterization applications. We address four key issues associated with the wavefield-based inversion: a) to limit the semi-infinite extent of the physical domain, a perfectly-matched-layer (PML) is introduced at truncation interfaces to render a finite computational domain; b) to account for the introduction of the PML, while retaining the second-order temporal character of the hyperbolic problem, we discuss a new mixed unsplit-field formulation for the coupled PML-regular-domain problem; c) to tackle the inversion, we adopt a PDE-constrained optimization framework that formally leads to a classic (time-dependent) KKT (Karush-Kuhn-Tucker) system; and d) to alleviate solution multiplicity, we discuss a first-order Tikhonov regularization scheme endowed with a regularization factor continuation algorithm. We report on one- and two-dimensional experiments that lead efficiently to the reconstruction of heterogeneous profiles involving both horizontal and inclined layers, as well as of inclusions within layered systems.

A Computational Investigation of Microwave Breast Imaging Using Deformable Reflector

In recent years, active microwave breast imaging is increasingly being viewed as a promising complementary imaging modality for cancer detection. In this paper, we present a novel deformable reflector microwave tomography technique for noninvasive characterization of the breast tissue. In contrast to conventional multitransceiver designs, the proposed technique utilizes a continuously deformable reflector with metallic coating to acquire field measurements for imaging. Computational feasibility of the proposed technique to image heterogeneous dielectric tissue property is evaluated using simplified 2-D breast models. The robustness of the deformable reflector-based tomography technique in imaging the spatial distribution of the tissue dielectric property in the presence of measurement noise is investigated using first-order Tikhonov regularization. Preliminary results obtained for the 2-D breast models appear promising and indicate further investigation of the new microwave tomography technique for breast imaging.

A general solution to the identification of moving vehicle forces on a bridge

AbstractBridge weigh‐in‐motion systems measure bridge strain caused by the passing of a truck to estimate static axle weights. For this calculation, they commonly use a static algorithm that takes the bridge influence line as reference. Such a technique relies on adequate filtering to remove bridge dynamics and noise. However, filtering can lead to the loss of a significant component of the underlying signal in bridges where the vibration does not have time to complete sufficient number of cycles and in cases of closely spaced axles traveling at high vehicle speeds. In order to overcome these limitations and also to provide additional information on the dynamics of the applied forces, this paper presents an algorithm based on first‐order Tikhonov regularization and dynamic programming. First, strain measurements are simulated using an elaborate three‐dimensional vehicle and orthotropic bridge interaction system. Then, strain is contaminated with noise and input into the moving force identification algorithm. The procedure to implement the algorithm and to derive the applied forces from the simulated strain record is described. Vehicle axle forces are shown to be accurately predicted for smooth and rough road profiles and a range of speeds. Copyright © 2008 John Wiley & Sons, Ltd.