Hybrid ℓ1/ℓ2 minimization with applications to tomography

Geophysics ◽  
1997 ◽  
Vol 62 (4) ◽  
pp. 1183-1195 ◽  
Author(s):  
Kenneth P. Bube ◽  
Robert T. Langan
Keyword(s):  
Least Squares ◽  
Computational Cost ◽  
Nonlinear Problems ◽  
Full Rank ◽  
Approximate Solutions ◽  
Seismic Inversion ◽  
Smooth Transition ◽  
Data Points ◽  
Linear Problems ◽  
Data Error

Least squares or [Formula: see text] solutions of seismic inversion and tomography problems tend to be very sensitive to data points with large errors. The [Formula: see text] minimization for 1 ≤ p < 2 gives more robust solutions, but usually with higher computational cost. Iteratively reweighted least squares (IRLS) gives efficient approximate solutions to these [Formula: see text] problems. We apply IRLS to a hybrid [Formula: see text] minimization problem that behaves like an [Formula: see text] fit for small residuals and like an [Formula: see text] fit for large residuals. The smooth transition from [Formula: see text] to [Formula: see text] behavior is controlled by a parameter that we choose using an estimate of the standard deviation of the data error. For linear problems of full rank, the hybrid objective function has a unique minimum, and IRLS can be proven to converge to it. We obtain a robust efficient method. For nonlinear problems, a version of the Gauss‐Newton algorithm can be applied. Synthetic crosswell tomography examples and a field‐data VSP tomography example demonstrate the improvement of the hybrid method over least squares when there are outliers in the data.

scholarly journals Anderson acceleration for seismic inversion

Geophysics ◽  
2021 ◽  
Vol 86 (1) ◽  
pp. R99-R108
Author(s):  
Yunan Yang
Keyword(s):  
Fixed Point ◽  
Least Squares ◽  
Optimization Problems ◽  
Computational Cost ◽  
Seismic Inversion ◽  
Steepest Descent ◽  
Memory Storage ◽  
Steepest Descent Method ◽  
Time Migration ◽  
Anderson Acceleration

State-of-the-art seismic imaging techniques treat inversion tasks such as full-waveform inversion (FWI) and least-squares reverse time migration (LSRTM) as partial differential equation-constrained optimization problems. Due to the large-scale nature, gradient-based optimization algorithms are preferred in practice to update the model iteratively. Higher-order methods converge in fewer iterations but often require higher computational costs, more line-search steps, and bigger memory storage. A balance among these aspects has to be considered. We have conducted an evaluation using Anderson acceleration (AA), a popular strategy to speed up the convergence of fixed-point iterations, to accelerate the steepest-descent algorithm, which we innovatively treat as a fixed-point iteration. Independent of the unknown parameter dimensionality, the computational cost of implementing the method can be reduced to an extremely low dimensional least-squares problem. The cost can be further reduced by a low-rank update. We determine the theoretical connections and the differences between AA and other well-known optimization methods such as L-BFGS and the restarted generalized minimal residual method and compare their computational cost and memory requirements. Numerical examples of FWI and LSRTM applied to the Marmousi benchmark demonstrate the acceleration effects of AA. Compared with the steepest-descent method, AA can achieve faster convergence and can provide competitive results with some quasi-Newton methods, making it an attractive optimization strategy for seismic inversion.

scholarly journals Highway Speed Prediction Using Gated Recurrent Unit Neural Networks

2021 ◽  
Vol 11 (7) ◽  
pp. 3059
Author(s):  
Myeong-Hun Jeong ◽  
Tae-Young Lee ◽  
Seung-Bae Jeon ◽  
Minkyo Youm
Keyword(s):  
Neural Network ◽  
Intelligent Transportation Systems ◽  
Moving Average ◽  
Computational Cost ◽  
Nonlinear Problems ◽  
Transportation Systems ◽  
Mobility Data ◽  
Challenges And Opportunities ◽  
Moving Average Model ◽  
Gated Recurrent Unit

Movement analytics and mobility insights play a crucial role in urban planning and transportation management. The plethora of mobility data sources, such as GPS trajectories, poses new challenges and opportunities for understanding and predicting movement patterns. In this study, we predict highway speed using a gated recurrent unit (GRU) neural network. Based on statistical models, previous approaches suffer from the inherited features of traffic data, such as nonlinear problems. The proposed method predicts highway speed based on the GRU method after training on digital tachograph data (DTG). The DTG data were recorded in one month, giving approximately 300 million records. These data included the velocity and locations of vehicles on the highway. Experimental results demonstrate that the GRU-based deep learning approach outperformed the state-of-the-art alternatives, the autoregressive integrated moving average model, and the long short-term neural network (LSTM) model, in terms of prediction accuracy. Further, the computational cost of the GRU model was lower than that of the LSTM. The proposed method can be applied to traffic prediction and intelligent transportation systems.

EXPRESS: Baseline Correction Based on a Search Algorithm from Artificial Intelligence

2020 ◽  
pp. 000370282097751
Author(s):  
Xin Wang ◽  
Xia Chen
Keyword(s):  
Artificial Intelligence ◽  
Objective Function ◽  
Least Squares ◽  
Least Squares Method ◽  
Search Algorithm ◽  
Absolute Error ◽  
Search Problem ◽  
Baseline Correction ◽  
Current Spectrum ◽  
Data Points

Many spectra have a polynomial-like baseline. Iterative polynomial fitting (IPF) is one of the most popular methods for baseline correction of these spectra. However, the baseline estimated by IPF may have substantially error when the spectrum contains significantly strong peaks or have strong peaks located at the endpoints. First, IPF uses temporary baseline estimated from the current spectrum to identify peak data points. If the current spectrum contains strong peaks, then the temporary baseline substantially deviates from the true baseline. Some good baseline data points of the spectrum might be mistakenly identified as peak data points and are artificially re-assigned with a low value. Second, if a strong peak is located at the endpoint of the spectrum, then the endpoint region of the estimated baseline might have significant error due to overfitting. This study proposes a search algorithm-based baseline correction method (SA) that aims to compress sample the raw spectrum to a dataset with small number of data points and then convert the peak removal process into solving a search problem in artificial intelligence (AI) to minimize an objective function by deleting peak data points. First, the raw spectrum is smoothened out by the moving average method to reduce noise and then divided into dozens of unequally spaced sections on the basis of Chebyshev nodes. Finally, the minimal points of each section are collected to form a dataset for peak removal through search algorithm. SA selects the mean absolute error (MAE) as the objective function because of its sensitivity to overfitting and rapid calculation. The baseline correction performance of SA is compared with those of three baseline correction methods: Lieber and Mahadevan–Jansen method, adaptive iteratively reweighted penalized least squares method, and improved asymmetric least squares method. Simulated and real FTIR and Raman spectra with polynomial-like baselines are employed in the experiments. Results show that for these spectra, the baseline estimated by SA has fewer error than those by the three other methods.

Analytic approximate solutions for a class of variable order fractional differential equations using the polynomial least squares method

2017 ◽  
Vol 20 (4) ◽  
Author(s):  
Constantin Bota ◽  
Bogdan Căruntu
Keyword(s):  
Differential Equations ◽  
Least Squares ◽  
Fractional Differential Equations ◽  
Least Squares Method ◽  
Approximate Solutions ◽  
Variable Order ◽  
Polynomial Solutions ◽  
Fractional Differential ◽  
Approximate Polynomial

AbstractIn this paper a new way to compute analytic approximate polynomial solutions for a class of nonlinear variable order fractional differential equations is proposed, based on the Polynomial Least Squares Method (PLSM). In order to emphasize the accuracy and the efficiency of the method several examples are included.

From acoustic to elastic inverse extended Born modeling: a first insight in the marine environment

Geophysics ◽  
2021 ◽  
pp. 1-73
Author(s):  
Milad Farshad ◽  
Hervé Chauris
Keyword(s):  
Least Squares ◽  
Marine Environment ◽  
Computational Cost ◽  
Physical Parameters ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
S Wave ◽  
Time Migration ◽  
Physical Density ◽  
Acoustic Approximation

Elastic least-squares reverse time migration is the state-of-the-art linear imaging technique to retrieve high-resolution quantitative subsurface images. A successful application requires many migration/modeling cycles. To accelerate the convergence rate, various pseudoinverse Born operators have been proposed, providing quantitative results within a single iteration, while having roughly the same computational cost as reverse time migration. However, these are based on the acoustic approximation, leading to possible inaccurate amplitude predictions as well as the ignorance of S-wave effects. To solve this problem, we extend the pseudoinverse Born operator from acoustic to elastic media to account for the elastic amplitudes of PP reflections and provide an estimate of physical density, P- and S-wave impedance models. We restrict the extension to marine environment, with the recording of pressure waves at the receiver positions. Firstly, we replace the acoustic Green's functions by their elastic version, without modifying the structure of the original pseudoinverse Born operator. We then apply a Radon transform to the results of the first step to calculate the angle-dependent response. Finally, we simultaneously invert for the physical parameters using a weighted least-squares method. Through numerical experiments, we first illustrate the consequences of acoustic approximation on elastic data, leading to inaccurate parameter inversion as well as to artificial reflector inclusion. Then we demonstrate that our method can simultaneously invert for elastic parameters in the presence of complex uncorrelated structures, inaccurate background models, and Gaussian noisy data.

NURBS Surface Reconstruction From a Large Set of Image and World Data Points

1993 ◽  
Author(s):  
Vassilios E. Theodoracatos ◽  
Vasudeva Bobba
Keyword(s):  
Least Squares ◽  
Vision System ◽  
Physical Object ◽  
Free Form ◽  
Approximation Technique ◽  
Global Coordinate System ◽  
Large Set ◽  
Affine Invariant ◽  
Nurbs Surface ◽  
Data Points

Abstract In this paper an approach is presented for the generation of a NURBS (Non-Uniform Rational B-splines) surface from a large set of 3D data points. The main advantage of NURBS surface representation is the ability to analytically describe both, precise quadratic primitives and free-form curves and surfaces. An existing three dimensional laser-based vision system is used to obtain the spatial point coordinates of an object surface with respect to a global coordinate system. The least-squares approximation technique is applied in both the image and world space of the digitized physical object to calculate the homogeneous vector and the control net of the NURBS surface. A new non-uniform knot vectorization process is developed based on five data parametrization techniques including four existing techniques, viz., uniform, chord length, centripetal, and affine invariant angle and a new technique based on surface area developed in this study. Least-squares error distribution and surface interrogation are used to evaluate the quality of surface fairness for a minimum number of NURBS control points.

Outlier removal method for the refinement of optical measured displacement field based on critical factor least squares and subdomain division

2021 ◽  
Author(s):  
Bo Wang ◽  
Chen Sun ◽  
Keming Zhang ◽  
Jubing Chen
Keyword(s):  
Least Squares ◽  
Displacement Field ◽  
Critical Factor ◽  
Least Square ◽  
Outlier Removal ◽  
Full Field ◽  
Removal Method ◽  
Strain Estimation ◽  
Data Points ◽  
Measured Displacement

Abstract As a representative type of outlier, the abnormal data in displacement measurement often inevitably occurred in full-field optical metrology and significantly affected the further evaluation, especially when calculating the strain field by differencing the displacement. In this study, an outlier removal method is proposed which can recognize and remove the abnormal data in optically measured displacement field. A iterative critical factor least squares algorithm (CFLS) is developed which distinguishes the distance between the data points and the least square plane to identify the outliers. A successive boundary point algorithm is proposed to divide the measurement domain to improve the applicability and effectiveness of the CFLS algorithm. The feasibility and precision of the proposed method are discussed in detail through simulations and experiments. Results show that the outliers are reliably recognized and the precision of the strain estimation is highly improved by using these methods.

Autoregressive processes with infinite variance

1977 ◽  
Vol 14 (02) ◽  
pp. 411-415 ◽  
Author(s):  
E. J. Hannan ◽  
Marek Kanter
Keyword(s):  
Least Squares ◽  
Autoregressive Model ◽  
Infinite Variance ◽  
Autoregressive Processes ◽  
Stable Law ◽  
Least Squares Estimators ◽  
Data Points

The least squares estimators β i(N), j = 1, …, p, from N data points, of the autoregressive constants for a stationary autoregressive model are considered when the disturbances have a distribution attracted to a stable law of index α &lt; 2. It is shown that N1/δ(β i(N) – β) converges almost surely to zero for any δ &gt; α. Some comments are made on alternative definitions of the βi (N).

scholarly journals On the Evaluation of Rhamnolipid Biosurfactant Adsorption Performance on Amberlite XAD-2 Using Machine Learning Techniques

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Fengqin Chen ◽  
Jinbo Huang ◽  
Xianjun Wu ◽  
Xiaoli Wu ◽  
Arash Arabmarkadeh
Keyword(s):  
Least Squares ◽  
Machine Learning Techniques ◽  
Lower Percentage ◽  
Support Vector ◽  
Mean Deviation ◽  
Group Management ◽  
Learning Techniques ◽  
Data Points ◽  
Artificial Neural Network Ann ◽  
Rhamnolipid Biosurfactant

Biosurfactants are a series of organic compounds that are composed of two parts, hydrophobic and hydrophilic, and since they have properties such as less toxicity and biodegradation, they are widely used in the food industry. Important applications include healthy products, oil recycling, and biological refining. In this research, to calculate the curves of rhamnolipid adsorption compared to Amberlite XAD-2, the least-squares vector machine algorithm has been used. Then, the obtained model is formed by 204 adsorption data points. Various graphical and statistical approaches are applied to ensure the correctness of the model output. The findings of this study are compared with studies that have used artificial neural network (ANN) and data group management method (GMDH) models. The model used in this study has a lower percentage of absolute mean deviation than ANN and GMDH models, which is estimated to be 1.71%.The least-squares support vector machine (LSSVM) is very valuable for investigating the breakthrough curve of rhamnolipid, and it can also be used to help chemists working on biosurfactants. Moreover, our graphical interface program can assist everyone to determine easily the curves of rhamnolipid adsorption on Amberlite XAD-2.

Sign in / Sign up

Export Citation Format

Share Document