Least-squares Optimization Method Research Articles

Determination of a diffusion coefficient function for long rooms using a least square optimization approach

Over the last decades, considerable attention has been devoted to improving the diffusion equation model within the field of room acoustics. It has been found that this model is subject to certain limitations, with one constraint regarding the diffusion coefficient. The diffusion coefficient, a parameter integrated into the model, is a value that scales the sound intensity to the gradient of the sound energy density. Currently, this value is often treated as a constant, but research has shown that it is spatially dependent. This paper explores a method to optimise the diffusion coefficient for long rooms such as corridors and tunnels by comparing the results of the diffusion equation model with those computed with the radiosity model and using the least-square optimization method, minimising the difference between the results of both methods. The parameters of a diffusion coefficient function are optimised, targeting a function that can be applied to a range of elongated spaces.

Convergent least-squares optimization methods for variational data assimilation

Data assimilation combines prior (or background) information with observations to estimate the initial state of a dynamical system over a given time-window. A common application is in numerical weather prediction where a previous forecast and atmospheric observations are used to obtain the initial conditions for a numerical weather forecast. In four-dimensional variational data assimilation (4D-Var), the problem is formulated as a nonlinear least-squares problem, usually solved using a variant of the classical Gauss-Newton (GN) method. However, we show that GN may not converge if poorly initialized. In particular, we show that this may occur when there is greater uncertainty in the background information compared to the observations, or when a long time-window is used in 4D-Var allowing more observations. The difficulties GN encounters may lead to inaccurate initial state conditions for subsequent forecasts. To overcome this, we apply two convergent GN variants (line search and regularization) to the long time-window 4D-Var problem and investigate the cases where they locate a more accurate estimate compared to GN within a given budget of computational time and cost. We show that these methods are able to improve the estimate of the initial state, which may lead to a more accurate forecast. Highlights Poor initialization of Gauss-Newton method may result in failure to converge. Safeguarded Gauss-Newton improves initial state estimate within limited time/cost. Results using twin experiments with long time-window and chaotic Lorenz models. Apply state of the art least-squares convergence theory to data assimilation. Improvements to initial state estimate may lead to a more accurate forecast.

Analytical solutions considering face advance and time-dependent behavior for back-analysis of convergence measurements in deep circular tunnels under isotropic initial stress state

A methodology is presented for the back-analysis of convergence measurements in deep tunnels to determine the constitutive parameters of the surrounding rock mass. Since increasing deformations and stresses with time are due to both the face advance and the time-dependent behavior of the ground, the two effects must be considered during the excavations. To that end, an analytical solution assuming an unlined circular tunnel excavated in a homogeneous isotropic ground under an initial isotropic stress field and assuming a fractional viscoelastic plastic behavior is developed. A second closed-form solution is also derived assuming an instantaneous excavation. Additionally, combining the developed analytical solution that takes into account the progressive face advance and an empirical approach, convergences are back-analyzed based on a least-squares optimization method to calibrate the constitutive parameters of the ground. The presented methodology aims to characterize the long-term behavior of tunnels and offers the advantage of being directly applicable during the excavation phase as soon as convergence measurements are available. Finally, the method is illustrated by two case studies related to the Fréjus road tunnel and the Saint-Martin-la-Porte access gallery (SMP2).

Thermal-hydraulic characterization of a 50-kW medicinal-aimed aqueous homogeneous reactor (MAHR)

This paper analyses the thermal–hydraulic performance of a proposed 50-kW aqueous homogeneous reactor (AHR) aimed at producing Mo-99. The reactor thermal–hydraulic characteristics are analyzed using ANSYS-FLUENT software and Relap code, and the results are validated through experiments conducted by a prototypic cylindrical sector of the reactor vessel. Boundary conditions in the CFD simulation are set up based on Relap modelling, while CFD outputs are validated by experimental results achieved using the laboratory model. Then, various parameters, including coolant mass flow rate, first cycle pipe diameter, coolant outlet temperature, average fuel solution temperature, and the elevation difference between the reactor and heat exchanger, are optimized using the least-squares optimization method. Results demonstrate that the heat removal system provides sufficient cooling capacity to ensure stable operation of the proposed AHR core by preventing fuel solution overheating and, consequently, boiling (void formation) in the active fuel solution.

Research on the differential coefficient least-squares optimization method of reverse time migration in acoustic-reflected S-wave imaging logging

Research on the differential coefficient least-squares optimization method of reverse time migration in acoustic-reflected S-wave imaging logging

Angle-dependent 4D seismic time-strain inversion for estimating subsurface thickness and velocity changes

In time-lapse (also called 4D) seismic analysis, the time shift of a certain seismic reflection event is caused by changes in the seismic velocity and the depth of the event. An interpretation of 4D time shifts is normally simplified by neglecting displacement changes (strains) or assuming linear relations between thickness and velocity strains. Here, we go beyond these assumptions and propose a least-squares optimization method to simultaneously estimate the thickness and velocity strains in vertical transverse isotropic media from angle-dependent 4D seismic time strains. Through examples from synthetic and field data, we show that 4D thickness strains, velocity strains, and anisotropic parameter changes can be estimated simultaneously without prior knowledge about the geomechanics of the survey area. Our time-strain inversion method can be applied to any other 4D seismic data set in which angle-stack images are available. We see that our method has high potential in many other applications, because thickness and velocity strains are the fundamental components of most physical properties used in 4D seismic and geomechanics applications.

Interpreting gas sorption isotherms in glassy polymers using a Bayesian framework: A view on parameter uncertainty propagation into mixture sorption predictions

Investigation of mixed-gas sorption is necessary for robust design and optimization of membrane-based processes. While sorption models for glassy polymers are well-established, they often deviate from observed mixture data. In doing so, however, parameters of these models are typically estimated via traditional least-squares optimization methods, and so parametric uncertainty is often ignored in mixture sorption predictions. As an alternative, we use Bayesian Inference (BI) to estimate probability distributions for sorption models' parameters and thus provide statistically meaningful mixture sorption forecasts that reflect parameter uncertainty rigorously. To this end, we exploit molecular sorption simulations, combining Grand Canonical Monte Carlo (GCMC) and Equilibrium Molecular Dynamics (EMD), to focus on two different glassy polymer systems (i.e., single- and mixed-gas sorption of CO2/CH4 in a fluorinated polyimide and CH4/H2 in a polymer of intrinsic microporosity) and three popular mixture sorption models (the Dual-Mode Sorption model, Non-Equilibrium Thermodynamics for Glassy Polymers model, and the Ideal Adsorbed Solution Theory) to demonstrate the benefits of this technique for uncertainty quantification and propagation in membrane applications. We show that observed sorption data at typical working pressures (e.g., 0−25atm) are often insufficient to accurately estimate model parameters, and consequently, these models often fail to represent observed mixture data adequately. Furthermore, we show that sorption data (e.g., high-pressure data), able to capture intrinsic isotherm nonlinearities of sorption models, are critical to considerably improve parameter inference from collective model-data fits and thus accurately predict mixture sorption.

Efficient reinforcement learning with least-squares soft Bellman residual for robotic grasping

Grasping control of intelligent robots has to deal with the difficulties of model uncertainties and nonlinearities. In this paper, we propose the Kernel-based Least-Squares Soft Bellman residual Actor–Critic (KLSAC) algorithm for robotic grasping. In the proposed approach, a novel linear temporal-difference learning algorithm using the least-squares soft Bellman residual (LS2BR) method is designed for policy evaluation. In addition, KLSAC adopts a sparse-kernel feature representation method based on approximate linear dependency (ALD) analysis to construct features for continuous state–action space. Compared with typical deep reinforcement learning algorithms, KLSAC has two main advantages: firstly, the critic module has the capacity for rapid convergence by computing the fixed point of the linear soft Bellman equation via the least-squares optimization method. Secondly, the kernel-based features construction approach only requires predefining the basic kernel function and can improve the generalization ability of KLSAC. The simulation studies on robotic grasping control were conducted in the V-REP simulator. The results demonstrate that compared with other typical RL algorithms (e.g., SAC and BMPO), the proposed KLSAC algorithm can achieve better performance in terms of sample efficiency and asymptotic convergence property. Furthermore, experimental results on a real UR5 robot validated that KLSAC performed well in the real world.

Integer Ambiguity Parameter Identification for Fast Satellite Positioning and Navigation Based on LAMBDA-GWO with Tikhonov Regularization

Satellite positioning is one of the main navigation technologies in unmanned aerial vehicles (UAVs), the accuracy of which has an important impact on the safety, stability, and flexibility of UAVs. The parameters of integer ambiguity are important factors affecting the accuracy of satellite positioning. However, the accuracy of the integer ambiguity cannot be guaranteed when only a few epoch data can be obtained in the fast positioning such that the identification matrix of the integer ambiguity parameters is seriously ill-conditioned and the information of position deviation is enlarged. In this paper, an error checking and correcting strategy is proposed, where a Least-square Ambiguity Decorrelation Adjustment-Grey Wolf Optimization (LAMBDA-GWO) Method combined with the Tikhonov regularization method is developed to improve the accuracy of integer ambiguity for fast satellite positioning. More specifically, the LAMBDA-GWO is first used to search the integer ambiguity parameters. To reduce the ill-condition of the integer ambiguity parameter identification matrix, the Tikhonov regularization method is introduced to regularize the identification matrix such that a reliable integer ambiguity floating-point solution can be obtained. Furthermore, the correctness of the integer ambiguity is checked according to the prior accuracy information of the initial coordinates and the Total Electron Content (TEC), and the part that fails the test is corrected by the Grey Wolf Optimization (GWO) Method. Finally, experimental studies based on a 522 m baseline and a 975 m baseline show that the identification success rates of the proposed method are both above 99%, which is 12% and 23% higher than that of traditional LAMBDA, respectively.

Electrically Evoked Compound Action Potential Studies Based on Finite Element and Neuron Models

Electrically evoked compound action potential (ECAP) represents a synchronous response of a group of nerve fibers activated by electrical stimulation. ECAP has been an important physiological signal for the cochlear implant (CI) used in various clinical studies. A 3-D CI model based on a patient’s medical images was created to simulate ECAP using the finite element method (FEM) and neuron models in this study. ECAPs obtained by alternating polarity approach were recorded by several sensing electrodes with different current levels. The least-square optimization method was used to calibrate the ECAP simulation results and they were compared with a patient’s clinical ECAP measurements. Results showed that calibrated simulations of ECAPs were consistent with the ECAP measurements. This study demonstrated that simulated ECAP based on CI models could be used as a reference for patient-specific CI studies for CI objective programming.

Two-Dimensional Beam Pattern Synthesis for Phased Arrays With Arbitrary Element Geometry via Magnitude Least Squares Optimization

In this paper, an optimization procedure for synthesizing shaped beams with an arbitrary geometry phased array system with real-time capability is presented. The algorithm involves a combination of an arbitrary extension of the Woodward-Lawson synthesis procedure with the magnitude least-squares optimization method. This simple combination significantly reduces the optimization time, enabling real-time beam shaping in response to changing requirements on beam shape or evolving geometry in distributed arrays. The technique places no restrictions on element positioning in all three spatial dimensions and is demonstrated to accurately reproduce the desired beam shape in both angular dimensions. The algorithm is demonstrated for multiple beam cases with a large, randomly generated array in computational simulation. The applicability of the algorithm to practical phased array hardware is also demonstrated using full-wave electromagnetic simulations and measurements of a realized arbitrary phased array system using a near-field scanner in an anechoic chamber.

Virtual structured-light three-dimensional point cloud compression with geometric reshaping

Converting three-dimensional (3D) point cloud data to two-dimensional (2D) image based on virtual structured-light system is a popular method in 3D data compression because the image format is easier to process by the exiting storing and transmitting methods. When this method is used in the 3D point cloud compression, the quantization error is introduced during the coordinate mapping. To solve this problem, a virtual structured-light 3D point cloud compression algorithm based on geometric reshaping is presented. And a least-square system parameter optimization method is proposed to further improve the data decoding accuracy. In the proposed method, the 3D spatial coordinates are reshaped to a 2D matrix first, and then the 2D matrix is stored as a “Holoimage” by using the parameter optimized virtual structured-light system. The quantization error introduced by coordinate mapping between X and Y coordinates of the 3D point cloud and the 2D image pixel coordinates is suppressed, so the decoding accuracy is improved. In addition, the geometric information of the 3D point cloud is hidden synchronously when the 3D point cloud is compressed, which is of great significance in the copyright protection of the point cloud data. Experiments verify the effectiveness of the proposed algorithm. The decoding root mean square error (RMSE) of the proposed method is decreased by 79.86% on average compared with the traditional one under the same compression ratio, and the compression ratio of the proposed method is 1.96 times bigger than the traditional one under the similar decoding accuracy.

Investigating key cell types and molecules dynamics in PyMT mice model of breast cancer through a mathematical model.

The most common kind of cancer among women is breast cancer. Understanding the tumor microenvironment and the interactions between individual cells and cytokines assists us in arriving at more effective treatments. Here, we develop a data-driven mathematical model to investigate the dynamics of key cell types and cytokines involved in breast cancer development. We use time-course gene expression profiles of a mouse model to estimate the relative abundance of cells and cytokines. We then employ a least-squares optimization method to evaluate the model’s parameters based on the mice data. The resulting dynamics of the cells and cytokines obtained from the optimal set of parameters exhibit a decent agreement between the data and predictions. We perform a sensitivity analysis to identify the crucial parameters of the model and then perform a local bifurcation on them. The results reveal a strong connection between adipocytes, IL6, and the cancer population, suggesting them as potential targets for therapies.

The use of asymmetric time constraints in 4-D ERT inversion

Time-lapse resistivity surveys are commonly used to monitor temporal changes in the subsurface. In certain cases, it is known from other information that the resistivity will only decrease or increase with time. The 4-D resistivity smoothness-constrained inversion method reduces artifacts due to noise by including a temporal roughness filter constraint that ensures the temporal changes vary in a smooth manner. A least-squares optimization method is used to find a solution by attempting to locate the minimum of an objective function that consists of the data misfit and model (spatial and temporal) roughness. In some cases, the 4-D time-lapse inverse models show an increase in the resistivity with time in parts of the subsurface where it is only expected to decrease (or vice versa). We compare two methods, the barrier function and transformation methods, that attempt to minimize or eliminate these artifacts. We incorporate the barrier function constraint into the 4-D inverse method by using a modified difference matrix as a temporal roughness filter. The barrier function constraint includes an additional term that increases the objective function value greatly if the model values cross the allowed thresholds. This greatly minimizes the artifacts but does not completely eliminate them. It has the advantage that there are minimal changes in the objective function in regions of model space that are not close to the imposed thresholds. The method of transformations changes the model parameter such that the additional positivity or negativity constraints are implicitly included in the transformed model parameter. It has the advantage that it can completely eliminate the artifacts. However, it modifies the entire objective function which could be a disadvantage in some cases. We also explore a combination of the two methods, using the barrier function method to generate an initial model that minimizes the artifacts followed by the transformation method. This hybrid technique completely removes the residual artifacts left by the barrier method, and produces an inverse model which is closer to the true model for a synthetic data set. We also describe a post-inversion modification of the L-curve method to determine the optimum model that takes into account the non-linear nature of the inverse problem and the forward modelling method. The technique gave an estimate of the noise level for a field data set and produced a model which is consistent with independent hydrological measurements at the test site.

SAR Coregistration by Robust Selection of Extended Targets and Iterative Outlier Cancellation

This letter extends the constrained least-squares (CLS) optimization method developed to coregister multitemporal synthetic aperture radar (SAR) images affected by a joint rotation effect and range/azimuth shifts enforcing the absence of zooming effects. To take advantage of the structural information extracted from the scene, the method starts with a detection stage that identifies extended targets/areas in the images. The selected tie-points allow the CLS problem to be reformulated to find its (initial) solution based on a robust subset of image blocks. Then, the mean square error (MSE) of each equation evaluated from the initial solution allows to implement an iterative cancellation procedure to further skim the CLS equation set. The effectiveness of the proposed procedure is validated on real SAR data in comparison with the standard CLS.

Soil radioactivity-depth profiles from regularized inversion of borehole gamma spectrometry data

An in situ borehole gamma logging method using a LaBr3 gamma detector has been developed to characterize a137Cs contaminated site. The activity-depth distribution of 137Cs was derived by inversion of the in situ measurement data using two different least squares methods, (i) Least squares optimization (LSO) and (ii) Tikhonov regularization. The regularization parameter (λ) of the Tikhonov regularization method was estimated using three different methods i.e. the L-curve, Generalized Cross Validation (GCV) and a prior information based method (PIBM). The considered inversion method variants were first validated for a137Cs contaminated pipe, and in most of the cases, the calculated activity of 137Cs was found to be within the acceptable range. The calculated 137Cs activity-depth profiles from in situ measurements were also in good agreement with the ones obtained from soil sample analysis, with an R2 ranging from 0.76 to 0.82. The GCV method for estimating λ appeared to perform better than the two other methods in terms of R2 and root mean squared error (RMSE). The L-curve method resulted in higher RMSE than the other Tikhonov regularization methods. Instability was observed in the activity concentration depth profile obtained from the LSO method. Therefore, we recommend the Tikhonov regularization with GCV for estimating λ for estimating the activity concentration-depth profile. The site studied showed 137Cs activity concentrations above the exemption limit down to depths of 0.50–0.90 m.

An inverse source identification by nonlinear optimization in a two-dimensional hyperbolic problem

This study deals with the identification of source function from final time state observation in a two-dimensional hyperbolic problem. The solution to the direct problem is obtained by the weak solution approach and finite element method. In the part of the inverse problem, the trust-region method and Levenberg–Marquardt method, which are nonlinear least-squares optimization methods, are used for the identification of source function. The findings are presented with numerical examples.

Physics-driven deep-learning inversion with application to transient electromagnetics

Machine learning, and specifically deep-learning (DL) techniques applied to geophysical inverse problems, is an attractive subject, which has promising potential and, at the same time, presents some challenges in practical implementation. Some obstacles relate to scarce knowledge of the searched geologic structures, a problem that can limit the interpretability and generalizability of the trained DL networks when applied to independent scenarios in real applications. Commonly used (physics-driven) least-squares optimization methods are very efficient local optimization techniques but require good starting models close to the correct solution to avoid local minima. We have developed a hybrid workflow that combines both approaches in a coupled physics-driven/DL inversion scheme. We exploit the benefits and characteristics of both inversion techniques to converge to solutions that typically outperform individual inversion results and bring the solution closer to the global minimum of a nonconvex inverse problem. The completely data-driven and self-feeding procedure relies on a coupling mechanism between the two inversion schemes taking the form of penalty functions applied to the model term. Predictions from the DL network are used to constrain the least-squares inversion, whereas the feedback loop from inversion to the DL scheme consists of the network retraining with partial results obtained from inversion. The self-feeding process tends to converge to a common agreeable solution, which is the result of two independent schemes with different mathematical formalisms and different objective functions on the data and model misfit. We determine that the hybrid procedure is converging to robust and high-resolution resistivity models when applied to the inversion of the synthetic and field transient electromagnetic data. Finally, we speculate that the procedure may be adopted to recast the way we solve inverse problems in several different disciplines.

A Robust Wavenumber-Domain Superdirective Beamforming for Endfire Arrays

Among various array configurations, superdirective beamforming based on linear differential arrays has attracted much attention. It is well known that its directivity performance degrades at higher frequencies due to the limitation of finite difference approximation, and its robustness deteriorates due to the mismatch of practical systems. In this paper, we develop a wavenumber-domain superdirective beamforming for endfire arrays, an alternative way of phase mode beamforming that cannot be applied directly to linear arrays. Here, “wavenumber-domain” means the spatial derivatives are calculated in the wavenumber domain through multiplication operations. An important feature of proposed method is that it provides a novel way to obtain the higher order derivatives from the reconstructed sound field utilizing all sensors of array, which is valid when the Nyquist criterion is satisfied, resulting in a broadband frequency-independent beamformer. On the other hand, since the proposed $N\rm {th}$ -order superdirective beamformer has no upper limit of number of sensors, its robustness at lower frequencies can be enhanced through increasing the number of sensors and using an optimum truncation parameter which plays a role to remove small singular values sensitive to the mismatch of array. Although it is demonstrated that the proposed method and the weighted least-squares optimization method lead to almost identical performances, the main advantage of the proposed method is that an optimum truncation parameter determined by the experiment is adopted to increase the robustness of beamformers, where the high sensitivity of the higher eigenbeams to random errors is exploited.

Modelling and Identification of the Hysteretic Dynamics of Inerters

This paper deals with an experimental study on the modeling and identification of the hysterical behavior of inerters. Unlike existing methods that can only consider a constant inertance to capture a static model of the device, we develop three different dynamic models for a ball-screw type inerter. To eliminate the effects of the measurement noise, an empirical mode decomposition (EMD) method is proposed. Then, three dynamic models—the Dahl, LuGre and Bouc–Wen model—are used in order to model the friction behavior of the device. Using the least-square optimization method, the parameters of the models are estimated. The results of the tuned models are compared together within different frequencies. The good agreement between predicted and measured data shows that LuGre and Bouc–Wen models can be effective for modelling the hysteretic behavior of friction inside the inerter mechanism. It is also shown that the Bouc–Wen model has better correlation with the experimental results in all test frequencies and amplitudes.