Adaptive Least-squares Research Articles

Adaptive least-squares methods for convection-dominated diffusion-reaction problems

This paper studies adaptive least-squares finite element methods for convection-dominated diffusion-reaction problems. The least-squares methods are based on the first-order system of the primal and dual variables with various ways of imposing outflow boundary conditions. The coercivity of the homogeneous least-squares functionals are established, and the a priori error estimates of the least-squares methods are obtained in a norm that incorporates the streamline derivative. All methods have the same convergence rate provided that meshes in the layer regions are fine enough. To increase computational accuracy and reduce computational cost, adaptive least-squares methods are implemented and numerical results are presented for some test problems.

An adaptive least-squares algorithm for the elliptic Monge–Ampère equation

An adaptive least-squares algorithm for the elliptic Monge–Ampère equation

Adaptive least-squares finite element methods: Guaranteed upper bounds and convergence in L2 norm of the dual variable

We consider adaptive least-squares finite element methods. First, we develop a guaranteed upper bound for the dual error in the L2 norm, and this can be used as a stopping criterion for the adaptive procedures. Secondly, based on the a posteriori error estimates for the dual variable, we develop an error indicator that identifies the local area to refine, and establish the convergence of the adaptive procedures based on the Dörfler's marking strategy. Our convergence analysis is valid for the entire range of the bulk parameter 0<Θ≤1 and it shows the effect of bulk parameter and reduction factor of elements on the convergence rate. Confirming numerical experiments are provided.

Adaptive vertical-deformation field estimation and current tectonic deformation significance analysis in Shanxi rift valley

The Shanxi rift valley is a continental rift system that is characterized by mantle uplift, crustal thinning, and tectonic deformation. A continuous, vertical crustal-deformation field can elucidate the constraints for understanding the mechanisms currently driving the deformation of the rift. Accordingly, we acquired and processed data from 250 continuous and mobile GPS stations located in the Shanxi rift valley and its surrounding region. Accounting for the influence of fault activity, we established the vertical crustal-deformation field of the Shanxi rift valley as a continuous function of space, using an adaptive least-squares collocation method. The main research findings are stated as follows: 1) the adaptive least-squares model yielded a reliable interpolation prediction results with adequate robustness, even for relatively sparse actual observation data. 2) The current general deformation pattern of the Shanxi rift valley exhibits an uplift of the mountainous regions on both sides and subsidence of the central basin. The average uplift rate of the mountain area is 2–3 mm/a, and the subsidence rate of the basin is not uniform and is positively correlated with the spatial distribution of the Cenozoic sediment thickness. However, in certain areas, a high subsidence rate of 10–30 mm/a is associated with human activities such as groundwater exploitation. 3) In summary, the current vertical crustal deformation occurring in the Shanxi rift valley correlates with the pattern over a time scale of millions of years. Overall, it is controlled by regional geological structure pattern, and is influenced by nonstructural factors in the shallow crust, exhibiting both complex and orderly characteristics in its spatial distribution.

Improved inference in financial factor models

Conditional heteroskedasticity of the error terms is a common occurrence in financial factor models, such as the CAPM and Fama–French factor models. This feature necessitates the use of heteroskedasticity consistent (HC) standard errors to make valid inference for regression coefficients. In this paper, we show that using weighted least squares (WLS) or adaptive least squares (ALS) to estimate model parameters generally leads to smaller HC standard errors compared to ordinary least squares (OLS), which translates into improved inference in the form of shorter confidence intervals and more powerful hypothesis tests. In an extensive empirical analysis based on historical stock returns and commonly used factors, we find that conditional heteroskedasticity is pronounced and that WLS and ALS can dramatically shorten confidence intervals compared to OLS, especially during times of financial turmoil.

Image downsampling expedited adaptive least-squares (IDEAL) fitting improves intravoxel incoherent motion (IVIM) analysis in the human kidney.

To improve the reliability of intravoxel incoherent motion (IVIM) model parameter estimation for the DWI in the kidney using a novel image downsampling expedited adaptive least-squares (IDEAL) approach. The robustness of IDEAL was investigated using simulated DW-MRI data corrupted with different levels of Rician noise. Subsequently, the performance of the proposed method was tested by fitting bi- and triexponential IVIM model to in vivo renal DWI data acquired on a clinical 3 Tesla MRI scanner and compared to conventional approaches (fixed D* and segmented fitting). The numerical simulations demonstrated that the IDEAL algorithm provides robust estimates of the IVIM parameters in the presence of noise (SNR of 20) as indicated by relatively low absolute percentage bias (maximal sMdPB <20%) and normalized RMSE (maximal RMSE <28%). The analysis of the in vivo data showed that the IDEAL-based IVIM parameter maps were less noisy and more visually appealing than those obtained using the fixed D* and segmented methods. Further, coefficients of variation for nearly all IVIM parameters were significantly reduced in cortex and medulla for IDEAL-based biexponential (coefficients of variation: 4%-50%) and triexponential (coefficients of variation: 7.5%-75%) IVIM modelling compared to the segmented (coefficients of variation: 4%-120%) and fixed D* (coefficients of variation: 17%-174%) methods, reflecting greater accuracy of this method. The proposed fitting algorithm yields more robust IVIM parameter estimates and is less susceptible to poor SNR than the conventional fitting approaches. Thus, the IDEAL approach has the potential to improve the reliability of renal DW-MRI analysis for clinical applications.

In vivo pH mapping with omega plot-based quantitative chemical exchange saturation transfer MRI.

Chemical exchange saturation transfer (CEST) MRI is promising for detecting dilute metabolites and microenvironment properties, which has been increasingly adopted in imaging disorders such as acute stroke and cancer. However, in vivo CEST MRI quantification remains challenging because routine asymmetry analysis (MTRasym ) or Lorentzian decoupling measures a combined effect of the labile proton concentration and its exchange rate. Therefore, our study aimed to quantify amide proton concentration and exchange rate independently in a cardiac arrest-induced global ischemia rat model. The amide proton CEST (APT) effect was decoupled from tissue water, macromolecular magnetization transfer, nuclear Overhauser enhancement, guanidinium, and amine protons using the image downsampling expedited adaptive least-squares (IDEAL) fitting algorithm on Z-spectra obtained under multiple RF saturation power levels, before and after global ischemia. Omega plot analysis was applied to determine amide proton concentration and exchange rate simultaneously. Global ischemia induces a significant APT signal drop from intact tissue. Using the modified omega plot analysis, we found that the amide proton exchange rate decreased from 29.6 ± 5.6 to 12.1 ± 1.3s-1 (P < 0.001), whereas the amide proton concentration showed little change (0.241 ± 0.035% vs. 0.202 ± 0.034%, P=0.074) following global ischemia. Our study determined the labile proton concentration and exchange rate underlying the in vivo APT MRI. The significant change in the exchange rate, but not the concentration of amide proton demonstrated that the pH effect dominates the APT contrast during tissue ischemia.

How to prove optimal convergence rates for adaptive least-squares finite element methods∗

Abstract The convergence analysis with rates for adaptive least-squares finite element methods (ALSFEMs) combines arguments from the a posteriori analysis of conforming and mixed finite element schemes. This paper provides an overview of the key arguments for the verification of the axioms of adaptivity for an ALSFEM for the solution of a linear model problem. The formulation at hand allows for the simultaneous analysis of first-order systems of the Poisson model problem, the Stokes equations, and the linear elasticity equations. Following [Carstensen and Park, SIAM J. Numer. Anal. 53 (1), 2015], the adaptive algorithm is driven by an alternative residual-based error estimator with exact solve and includes a separate marking strategy for quasi-optimal data resolution of the right-hand side. This presentation covers conforming discretisations for an arbitrary polynomial degree and mixed homogeneous boundary conditions.

User-Independent Hand Gesture Recognition Classification Models Using Sensor Fusion.

Recently, it has been proven that targeting motor impairments as early as possible while using wearable mechatronic devices for assisted therapy can improve rehabilitation outcomes. However, despite the advanced progress on control methods for wearable mechatronic devices, the need for a more natural interface that allows for better control remains. To address this issue, electromyography (EMG)-based gesture recognition systems have been studied as a potential solution for human–machine interface applications. Recent studies have focused on developing user-independent gesture recognition interfaces to reduce calibration times for new users. Unfortunately, given the stochastic nature of EMG signals, the performance of these interfaces is negatively impacted. To address this issue, this work presents a user-independent gesture classification method based on a sensor fusion technique that combines EMG data and inertial measurement unit (IMU) data. The Myo Armband was used to measure muscle activity and motion data from healthy subjects. Participants were asked to perform seven types of gestures in four different arm positions while using the Myo on their dominant limb. Data obtained from 22 participants were used to classify the gestures using three different classification methods. Overall, average classification accuracies in the range of 67.5–84.6% were obtained, with the Adaptive Least-Squares Support Vector Machine model obtaining accuracies as high as 92.9%. These results suggest that by using the proposed sensor fusion approach, it is possible to achieve a more natural interface that allows better control of wearable mechatronic devices during robot assisted therapies.

On the Determination of the Output Power in Mono/Multicrystalline Photovoltaic Cells

In the present work, two artificial intelligence-based models were proposed to determine the output power of two types of photovoltaic cells including multicrystalline (multi-) and monocrystalline (mono-). Adaptive neuro-fuzzy inference system (ANFIS) and Least-squares support vector machine (LSSVM) are applied for the output power calculations. The estimation results are very close to the actual data based on graphical and statistical analysis. The coefficients of determination ( R 2 ) of monocrystalline cell output power for LSSVM and ANFIS models are as 0.997 and 0.962, respectively. Additionally, multicells have R 2 values of 0.999 and 0.995 for LSSVM and ANFIS, respectively. The acceptable values for R 2 and various error parameters prove the accuracy of suggested models. The visualization of these comparisons clarifies the accuracy of suggested models. Additionally, the proposed models are compared with previously published machine learning methods. The accurate performance of proposed models in comparison with others showed that our models can be helpful tools for the estimation of output power. Moreover, a sensitivity analysis for the effects of inputs parameters on output power has been employed. The sensitivity output shows that light intensity has more on output power. The outcomes of this study provide interesting tools which have potential to apply in different parts of renewable energy industries.

An adaptive least-squares finite element method for Giesekus viscoelastic flow problems

In this study, a least-squares (LS) finite element method with an adaptive mesh approach is investigated for Giesekus viscoelastic flow problems. We consider the weighted LS method on uniform and adaptive meshes for the Newton linearized viscoelastic problem, where adaptive grids are automatically generated by the least-squares solutions. We use a residual-type a-posteriori error estimator to adjust weights in the LS functional and compare the convergence behaviour of adaptive meshes generated using different grading functions. Numerical results demonstrate that the adaptive LS method shows at least the first-order convergence rate when equal-order linear interpolation functions are used for all variables, which agrees with the theoretical estimate. In addition, adaptive grids generated using the velocity outperform those based on the a-posteriori error estimator, yielding better numerical results.

Collective marking for arbitrary order adaptive least-squares finite element methods with optimal rates

The collective marking strategy with alternative refinement-indicators in adaptive mesh-refining of least-squares finite element methods (LSFEMs) has recently been shown to lead to optimal convergence rates in Carstensen (2020). The proofs utilize explicit identities for the lowest-order Raviart–Thomas and the Crouzeix–Raviart finite elements. This paper generalizes those results to arbitrary polynomial degree and mixed boundary conditions with some novel arguments. The analysis is outlined for the Poisson equation in 3D with mixed boundary conditions.

FPGA Implementation of RLSE Algorithm for Multichannel Brain Imaging

This paper describes the implementation of a computationally efficient embedded system on an Field Programmable Gate Array (FPGA) platform for real-time brain activity estimation with multiple channels. The brain signals from multiple channels are considered as output of independent linear systems with unknown parameters representing the brain activity in corresponding channels. Multiple adaptive Recursive Least-Squares Estimation (RLSE) cores are implemented in FPGA to independently estimate the brain activity in each channel concurrently. The proposed RLSE-FPGA system provides dedicated (no time or resource sharing) and parallel processing environment. The universal asynchronous receiver transmitter core is also developed to communicate the measured and estimated parameters supported by storage facility programmed as shared memory. The computational precision is guaranteed by deploying a 32-bit floating point core for all the variables. The validation carried out by real Functional Near-Infrared Spectroscopy dataset and comparative analysis with the previously reported result, demonstrates the effectiveness of the proposed system. The computational cost endorses the effectiveness of concurrent processing of multiple channelsꞌ data in a sample before the arrival of the next sample. The proposed methodology has potential in real-time medical, military and industrial applications.

Research on Zero-crossing Point Prediction Technology of CFI Based on A New Adaptive LS Algorithm

Fast identification of fault current parameters and accurate prediction of current waveform zero-crossing are the key technologies of controlled fault interruption(CFI). Aiming at the problem that the least squares(LS) algorithm has truncation error, this paper propose a new adaptive least square(LS) algorithm using the change rate of the fault current, avoiding the approximation of the attenuated DC decaying component. Based on the fault current simulated by MATLAB, the results show that the new algorithm has higher accuracy when the noise is not less than 40dB and the data window is more than 10ms. For the actual simulated fault current data by PSCAD, the proposed algorithm also had high accuracy, which can meet the requirements of CFI.

A short note on plain convergence of adaptive least-squares finite element methods

We show that adaptive least-squares finite element methods driven by the canonical least-squares functional converge under weak conditions on PDE operator, mesh-refinement, and marking strategy. Contrary to prior works, our plain convergence does neither rely on sufficiently fine initial meshes nor on severe restrictions on marking parameters. Finally, we prove that convergence is still valid if a contractive iterative solver is used to obtain the approximate solutions (e.g., the preconditioned conjugate gradient method with optimal preconditioner). The results apply within a fairly abstract framework which covers a variety of model problems.

An Adaptive Online Blade Health Monitoring Method: From Raw Data to Parameters Identification

Blade tip timing (BTT) methods have been increasingly implemented for blade health monitoring (BHM). However, there are two drawbacks of current signal analysis methods preventing them from applying to online monitoring: first, current online monitoring requires the manual judgment of the resonance region, which is time-consuming. Second, existing BTT resonance signal analysis methods are not suitable for online monitoring. The spectral-analysis-based method presents spectral aliasing, while the computational complexity of the sparse-based method is usually high. In this article, we propose an adaptive online BHM method that includes two steps: automatic resonance region recognition and parameters identification in resonance area. For the former step, we demonstrate different methods for synchronous and asynchronous resonances, use the cross correlation to judge the occurrence of synchronous vibration, and use the linear estimation to determine the appearance of asynchronous vibration. For the latter step, an iterative adaptive least-squares periodogram is adopted for its tradeoff between spectral aliasing and computational complexity. The effectiveness of the above steps is first verified using different simulation data separately. Then, the laboratory data are used to test the effectiveness of the whole method. Finally, the online monitoring function of the proposed method is verified by the engineering data with both synchronous and asynchronous resonances.

Adaptive least-squares finite element methods for linear transport equations based on an H(div) flux reformulation

In this paper, we study the least-squares finite element methods (LSFEM) for the linear hyperbolic transport equations. The linear transport equation naturally allows discontinuous solutions and discontinuous inflow conditions, while the normal component of the flux across the mesh faces needs to be continuous. Traditional LSFEMs using continuous finite element approximations will introduce unnecessary extra error for discontinuous solutions and boundary conditions. In order to separate the continuity requirements, a new flux variable is introduced. With this reformulation, the continuities of the flux and the solution can be handled separately in natural H(div;Ω)×L2(Ω) conforming finite element spaces. Several variants of the methods are developed to handle the inflow boundary condition strongly or weakly.With the reformulation, the new LSFEMs can handle discontinuous solutions and boundary conditions much better than the traditional LSFEMs with continuous polynomial approximations. With least-squares functionals as a posteriori error estimators, the adaptive methods can naturally identify error sources including singularity and non-matching discontinuity. For discontinuity aligned mesh, no extra error is introduced. If an RT0×P0 pair is used to approximate the flux and the solution, the new adaptive LSFEMs can approximate discontinuous solutions with almost no overshooting even when the mesh is not aligned with discontinuity.Existence and uniqueness of the solutions and a priori and a posteriori error estimates are established for the proposed methods. Extensive numerical tests are performed to show the effectiveness of the methods developed in the paper.

Adaptive weighted least-squares polynomial chaos expansion with basis adaptivity and sequential adaptive sampling

An efficient framework to obtain stochastic models of responses with polynomial chaos expansion (PCE) using an adaptive least-squares approach is presented in this paper. PCE is a high accuracy spectral expansion technique for uncertainty quantification; however, it is hugely affected by the curse of dimensionality with the increase in stochastic dimensions. To alleviate this effect, the basis polynomials are added in an adaptive manner, unlike selecting basis polynomials from a large predefined set in the traditional approach. Also, a refinement strategy is proposed to cull the unnecessary PCE terms based on their contribution to the variance of the response. Furthermore, a sequential optimal sampling is utilized that is capable of adding new samples based on the most recent basis polynomials and also reutilizes the old set of samples. The additional highlights of the algorithm include the implementation of weighted least-squares to reduce the effect of outliers and Kullback–Leibler Divergence to check the convergence of PCE. The algorithm has been implemented to analytical benchmark problems and a composite laminate problem. The substantial computational savings of the proposed framework compared to traditional PCE approaches and a large number of random simulations to achieve similar accuracy were demonstrated by the results.

Solar vector solving method based on photovoltaic array and its application in attitude estimation of navigation system

Solar-powered unmanned aerial vehicles usually fly at high altitudes, and they are mainly powered by the photocells covering the body of unmanned aerial vehicles. Considering that the solar vector cannot be affected by the disturbing magnetic field and harmful acceleration, a unit solar vector solving method based on photovoltaic array is proposed in this paper. The photocells with different installation angles are selected to form the photovoltaic array. The solar vector is solved by the least-squares method on the basis of normalization by using the output currents of the photovoltaic array. For eliminating the influence of faults and reflected light on the solving of the solar vector, an adaptive least-squares unit solar vector solving method is proposed. In addition, a solar vector measuring device is designed in order to verify the effectiveness of the proposed methods. By employing the structural advantages of the device, the current generated by the reflected light of the sky can be solved according to the currents generated by all photocells of this device. Thus, the solved current generated by the reflected light of the sky is more accurate. Moreover, strapdown inertial navigation system/solar vector/global positioning system integrated navigation Kalman filtering algorithm is proposed, in which the adaptive least-squares unit solar vector solving method is applied to the measurement update of the filter. The effectiveness of the methods proposed in this paper is illustrated by some numerical and physical simulations.

Adaptive Least-Squares Collocation Algorithm Considering Distance Scale Factor for GPS Crustal Velocity Field Fitting and Estimation

High-precision, high-reliability, and high-density GPS crustal velocity are extremely important requirements for geodynamic analysis. The least-squares collocation algorithm (LSC) has unique advantages over crustal movement models to overcome observation errors in GPS data and the sparseness and poor geometric distribution in GPS observations. However, traditional LSC algorithms often encounter negative covariance statistics, and thus, calculating statistical Gaussian covariance function based on the selected distance interval leads to inaccurate estimation of the correlation between the random signals. An unreliable Gaussian statistical covariance function also leads to inconsistency in observation noise and signal variance. In this study, we present an improved LSC algorithm that takes into account the combination of distance scale factor and adaptive adjustment to overcome these problems. The rationality and practicability of the new algorithm was verified by using GPS observations. Results show that the new algorithm introduces the distance scale factor, which effectively weakens the influence of systematic errors by improving the function model. The new algorithm can better reflect the characteristics of GPS crustal movement, which can provide valuable basic data for use in the analysis of regional tectonic dynamics using GPS observations.