Least-squares Minimization Research Articles

RKHSMetaMod: An R Package to Estimate the Hoeffding Decomposition of a Complex Model by Solving RKHS Ridge Group Sparse Optimization Problem

In this paper, we propose an R package, called [RKHSMetaMod](https://CRAN.R-project.org/package=RKHSMetaMod), that implements a procedure for estimating a meta-model of a complex model. The meta-model approximates the Hoeffding decomposition of the complex model and allows us to perform sensitivity analysis on it. It belongs to a reproducing kernel Hilbert space that is constructed as a direct sum of Hilbert spaces. The estimator of the meta-model is the solution of a penalized empirical least-squares minimization with the sum of the Hilbert norm and the empirical $L^2$-norm. This procedure, called RKHS ridge group sparse, allows both to select and estimate the terms in the Hoeffding decomposition, and therefore, to select and estimate the Sobol indices that are non-zero. The [RKHSMetaMod](https://CRAN.R-project.org/package=RKHSMetaMod) package provides an interface from the R statistical computing environment to the C++ libraries Eigen and GSL. In order to speed up the execution time and optimize the storage memory, except for a function that is written in R, all of the functions of this package are written using the efficient C++ libraries through [RcppEigen](https://CRAN.R-project.org/package=RcppEigen) and [RcppGSL](https://CRAN.R-project.org/package=RcppGSL) packages. These functions are then interfaced in the R environment in order to propose a user-friendly package.

A Smart Molecule Showing Spin Crossover Responsive Aggregation-Induced Emission

A Smart Molecule Showing Spin Crossover Responsive Aggregation-Induced Emission

Statistical Analysis of the Measurement Noise in Dynamic Impedance Spectra.

Different strategies can be used to acquire dynamic impedance spectra during a cyclic voltammetry experiment. The spectra are then analyzed by fitting them with a model using a weighted non‐linear least‐squares minimization algorithm. The choice of the weighting factors is not trivial and influences the value of the extracted parameters. At variance with the classic electrochemical impedance spectroscopy, dynamic impedance measurements are performed under non‐stationary conditions, making them typically more prone to errors arising from the voltage and current analog‐to‐digital conversion. Under the assumption that the noise in the voltage and current signals have a constant variance along the measurement and that it is uncorrelated between distinct samples, we calculate an expression for the expected variance of the error of the resulting immittances, which considers the specific procedure used to extract the spectra under the time‐varying nature of the measurements. The calculated variance of the error is then used as a rigorous way to evaluate the weighting factors of the least‐squares minimization, assuming that the fitted model is ideally exact and that there are no systematic errors. By exploring two classical electrochemical systems and fitting the measured spectra with a transfer function measurement model, namely with the Padé approximants, we show that the variance evaluated with our method captures the frequency dependence of the resulting residuals and can be used for reliably performing the complex non‐linear least‐squares fitting procedure.

Experimental and numerical multi-scale study of spruce wood degradation under inert atmosphere

The aim of the present work was to study the thermal degradation of spruce wood throughout a multiscale approach. First, thermogravimetric experiments under inert atmosphere with argon were performed with spruce samples reduced into powder. Three heating rates varying between 10 and 30 °C.min−1 were imposed to study the thermal degradation. A reactional mechanism consisting of three steps was defined and the associated kinetic parameters were determined by least-squares minimization using the Levenberg-Marquardt algorithm. The results showed efficient predictions in agreement with experimental data of mass and mass loss rate whatever the heating rate. At material scale, experiments were performed in an atmosphere controlled chamber placed in front of a cone calorimeter in vertical configuration. Spruce samples were exposed to three heat fluxes (38, 49 and 59 kW m−2) under inert atmosphere obtained with a constant flowrate of argon. Mass loss and in-depth temperature measured using twelve embedded thin wire thermocouples were recorded. At this scale, the thermal degradation was modeled using a one-dimensional pyrolysis model with GPYRO. The predicted in-depth temperatures are in good agreement despite differences explained by a possible overestimation of thermal properties such as thermal conductivity and heat capacity of char. The predictions showed similar mass loss and charring rates beside the experiments.

Full-volume three-component intraventricular vector flow mapping by triplane color Doppler

Objective. Intraventricular vector flow mapping (iVFM) is a velocimetric technique for retrieving two-dimensional velocity vector fields of blood flow in the left ventricular cavity. This method is based on conventional color Doppler imaging, which makes iVFM compatible with the clinical setting. We have generalized the iVFM for a three-dimensional reconstruction (3D-iVFM). Approach. 3D-iVFM is able to recover three-component velocity vector fields in a full intraventricular volume by using a clinical echocardiographic triplane mode. The 3D-iVFM problem was written in the spherical (radial, polar, azimuthal) coordinate system associated to the six half-planes produced by the triplane mode. As with the 2D version, the method is based on the mass conservation, and free-slip boundary conditions on the endocardial wall. These mechanical constraints were imposed in a least-squares minimization problem that was solved through the method of Lagrange multipliers. We validated 3D-iVFM in silico in a patient-specific CFD (computational fluid dynamics) model of cardiac flow and tested its clinical feasibility in vivo in patients and in one volunteer. Main results. The radial and polar components of the velocity were recovered satisfactorily in the CFD setup (correlation coefficients, = 0.99 and 0.78). The azimuthal components were estimated with larger errors ( = 0.57) as only six samples were available in this direction. In both in silico and in vivo investigations, the dynamics of the intraventricular vortex that forms during diastole was deciphered by 3D-iVFM. In particular, the CFD results showed that the mean vorticity can be estimated accurately by 3D-iVFM. Significance. Our results tend to indicate that 3D-iVFM could provide full-volume echocardiographic information on left intraventricular hemodynamics from the clinical modality of triplane color Doppler.

Eclipse timing modelling of three post-common envelope binaries: hybrid solutions

ABSTRACT We report 90 new observations of three post-common envelope binaries at primary eclipse spanning between 2018 December and 2022 February. We combine recent primary eclipse timing observations with previously published values to search for substellar circumbinary components consistent with timing variations from a linear ephemeris. We used a least-squares minimization fitting algorithm weighted by a Hill orbit stability function, followed by Bayesian inference, to determine best-fitting orbital parameters and associated uncertainties. For HS2231+2441, we find that the timing data are consistent with a constant period and that there is no evidence to suggest orbiting components. For HS0705 + 6700, we find both one- and two-component solutions stable for at least 10 Myr. For HW Vir, we find three- and four-component solutions that fit the timing data reasonably well, but are unstable on short time-scales, and therefore highly improbable. Conversely, solutions calculated using a Bayesian orbit stability prior result in a poor fit. The stable solutions significantly deviate from the ensemble timing data in both systems. We speculate that the observed timing variations for these systems, and very possibly other sdB binaries, may result from a combination of substellar component perturbations and an Applegate–Lanza mechanism.

AEROS: AdaptivE RObust Least-Squares for Graph-Based SLAM.

In robot localisation and mapping, outliers are unavoidable when loop-closure measurements are taken into account. A single false-positive loop-closure can have a very negative impact on SLAM problems causing an inferior trajectory to be produced or even for the optimisation to fail entirely. To address this issue, popular existing approaches define a hard switch for each loop-closure constraint. This paper presents AEROS, a novel approach to adaptively solve a robust least-squares minimisation problem by adding just a single extra latent parameter. It can be used in the back-end component of the SLAM system to enable generalised robust cost minimisation by simultaneously estimating the continuous latent parameter along with the set of sensor poses in a single joint optimisation. This leads to a very closely curve fitting on the distribution of the residuals, thereby reducing the effect of outliers. Additionally, we formulate the robust optimisation problem using standard Gaussian factors so that it can be solved by direct application of popular incremental estimation approaches such as iSAM. Experimental results on publicly available synthetic datasets and real LiDAR-SLAM datasets collected from the 2D and 3D LiDAR systems show the competitiveness of our approach with the state-of-the-art techniques and its superiority on real world scenarios.

Improving receiver function imaging with high-resolution Radon transform

SUMMARY The receiver function (RF) method is one of the most widely used passive seismic approaches for imaging subsurface structures from shallow sedimentary basins to the deep Earth’s interior. RF is typically computed with a deconvolution operation that enables separating structural response from source–time function embedded in teleseismic wavefields. However, seismic recordings of temporary arrays are often contaminated with strong random noise and even amplitude bias due to poor installation conditions. Both factors can cause instability to deconvolution and severely degrade the accuracy, thereby the imaging quality, of the RF method. This underscores the importance of effective denoising algorithms in RF processing. In this study, we explore the application of high-resolution Radon transform (RT) to improve the conventional RF imaging workflow. Contrary to the commonly implemented post-processing (i.e. after deconvolution) noise suppression schemes, we introduce the RT to data pre-processing (i.e. before deconvolution). This method seeks a sparse representation of teleseismic wavefields in the τ–p domain by iteratively solving a least-squares minimization problem with the conjugate gradient algorithm. Synthetic test with a 2-D step-Moho model shows that non-linear phase arrivals including incoherent noise and diffraction energy are effectively removed in resulting RFs, with the signal-to-noise ratio increased by as much as ∼8 dB. Real data experiments using the Hi-CLIMB network in the Tibetan Plateau demonstrate the superior performance of the proposed workflow in regularizing the wavefield and improving the coherence of converted phases across the recording array. Consequently, imaging results of common conversion point stacking using a single and a group of teleseismic events both recover subtle converted phases from the Moho and potential lithospheric discontinuities that are otherwise obscured by noise arising from the conventional processing workflow. This study highlights the necessity of wavefield regularization in the RF method and calls for improved data processing techniques in array-based seismic imaging.

Vertical Cracks Excited in Lock-in Vibrothermography Experiments: Identification of Open and Inhomogeneous Heat Fluxes.

Lock-in vibrothermography has proven to be very useful to characterizing kissing cracks producing ideal, homogeneous, and compact heat sources. Here, we approach real situations by addressing the characterization of non-compact (strip-shaped) heat sources produced by open cracks and inhomogeneous fluxes. We propose combining lock-in vibrothermography data at several modulation frequencies in order to gather penetration and precision data. The approach consists in inverting surface temperature amplitude and phase data by means of a least-squares minimization algorithm without previous knowledge of the geometry of the heat source, only assuming knowledge of the vertical plane where it is confined. We propose a methodology to solve this ill-posed inverse problem by including in the objective function penalty terms based on the expected properties of the solution. These terms are described in a comprehensive and intuitive manner. Inversions of synthetic data show that the geometry of non-compact heat sources is identified correctly and that the contours are rounded due to the penalization. Inhomogeneous smoothly varying fluxes are also qualitatively retrieved, but steep variations of the flux are hard to recover. These findings are confirmed by inversions of experimental data taken on calibrated samples. The proposed methodology is capable of identifying heat sources generated in lock-in vibrothermography experiments.

Data-Driven Modeling of Linear Dynamical Systems with Quadratic Output in the AAA Framework

We extend the Adaptive Antoulas-Anderson (AAA) algorithm to develop a data-driven modeling framework for linear systems with quadratic output (LQO). Such systems are characterized by two transfer functions: one corresponding to the linear part of the output and another one to the quadratic part. We first establish the joint barycentric representations and the interpolation theory for the two transfer functions of LQO systems. This analysis leads to the proposed AAA-LQO algorithm. We show that by interpolating the transfer function values on a subset of samples together with imposing a least-squares minimization on the rest, we construct reliable data-driven LQO models. Two numerical test cases illustrate the efficiency of the proposed method.

Reverse physically motivated frameworks for investigation of strain energy function in rubber-like elasticity

This paper introduces three innovative frameworks to obtain the strain energy function for rubber-like materials through an inverse approach. To this end, we inspire from the statistical background of physically motivated models to represent the strain energy function in terms of variables that correspond to the kinetics of a polymeric chain. We employ the micro-sphere concept to cast different structure-based models with unknown core functions or kernels and reveal them directly from experimental data. Accordingly, we use the basis spline (B-Spline) curve to represent these core functions and derive the macroscopic stress tensor based on the unknown control points associated with the B-Spline curves. Following this, we define a loss function to tie experimental data to these unknown control points, followed by a least-square error minimization technique that ultimately leads to a linear system of equations. In this way, we circumvent the nonlinear optimization in current structure-based models used for material parameter fitting that comes with the solution’s uncertainty. We also minimize simplifying hypotheses and bypass the unnecessary assumptions that have been influenced the accuracy of physical-based methods. Furthermore, one can integrate these functions with a high accuracy scheme over the micro-sphere, resulting in an integration-free model with competitive computational complexity to closed-form models. Finally, we validate the robustness of each framework in a challenging benchmark for three different polymeric materials. Considering the same dataset for the calibration of all models( which is uniaxial, pure shear, and equi-biaxial tests), the proposed frameworks outperform other well-known physical-based models in the literature in this challenging benchmark to the best of our knowledge ( we also show that the same performance is attainable by using only uniaxial and equi-biaxial tests ). Moreover, in the case of a limited dataset, we show that it is possible to apply prior knowledge regarding a material into one of the frameworks (RFN) to enhance the accuracy of predictions.

Optimization of semi-empirical mass formula co-efficients using least square minimization and variational Monte-Carlo approaches

In this paper, both least squares minimization (LSM) and variational Monte-Carlo techniques have been implemented to determine the co-efficients of semi-empirical mass formula (SEMF). First, the experimental binding energies (BEs) are determined for all the available nuclei from atomic mass evaluation (AME2016) data. Then, LSM technique is implemented in Gnumeric worksheet to minimize relative mean squared error (RMSE) to obtain the SEMF co-efficients by considering only the first three co-efficients which are deduced from liquid drop model. The mean percentage error (MPE) value, between obtained BEs from the optimized co-efficients and the experimental BEs, has been determined. Then, to emphasize the relevance of empirical terms, they have been introduced successively one after other and the procedure has been repeated. A reduction in MPE-value has been observed after each iteration. This same procedure has also been employed using Monte-Carlo approach to obtain SEMF co-efficients by minimizing RMSE-value as in variational principle.

Quality Control of Nuclear Magnetic Resonance Measurements Using Pseudo-T2 Distributions

Summary Nuclear magnetic resonance (NMR) measurements are extremely valuable in the assessment of fluid-flow properties of rocks. However, inverted NMR transverse-relaxation time (T2) distributions are often biased by positivity constraints and the implementation of regularization (stabilization) methods. In some cases, it is difficult to determine whether the estimated pore-size distributions are reliable or byproducts of the inversion. On-site quality control of NMR inversion results is thus essential to avoid erroneous analyses based on nonphysical interpretations across conventional or unconventional rocks. It is also important to adjust the measurement acquisition process in real time in response to variable signal/noise conditions in the borehole environment. We introduce and compare three quality-control approaches; two of them are based on two different signal processing practices, namely the semilogarithmic derivative and the Hilbert transform, while the third is based on a nonlinear optimization technique. Validation of the inverted T2 distribution is performed by applying either method to the echo train decay of proton magnetization to estimate a pseudo-T2 distribution that is not affected by inversion artifacts. These three methods are data-driven processes for on-site, continuous quality control of borehole NMR measurements. In the first method, a pseudo-T2 distribution is constructed by taking the derivative of the echo train decay with respect to the natural logarithm of the relaxation time. For the second method, the Hilbert transform is applied to the raw proton magnetization decay to approximate the T2 distribution. The latter method consists of an approximation of the derivative of the time decay of magnetization using least-squares minimization. NMR laboratory measurements were performed on a wide variety of rocks, the associated T2 distributions were estimated using linear inversion, and the three quality-control methods were applied for comparison. The reliability of our quality-control procedures was verified by benchmarking them against the inverted T2 distributions. Furthermore, these quality-control methods, when applied continuously to borehole NMR measurements, enable the optimization of the measurements in real time to mitigate the effects of biasing noise. By verifying the location of the dominant mode of the T2 distribution during NMR measurement acquisition, the operator can determine whether (a) more stacking of proton magnetization decays is needed, or (b) longer- or shorter-time sampling is needed to attain a high-quality estimation of the pore-size distribution before performing the T2 inversion. Our work provides the basis of effective quality-control methods for NMR measurements for the petrophysical interpretation of rocks with complex pore-size distributions, especially in unconventional rocks, where noise present in the measurements (notably at early acquisition times), can mask the useful signal originating from pore-size distributions and fluids. The fast and reliable quality control of estimated T2 distributions is not affected by inversion artifacts, relying only on unfiltered, raw data.

Activities Based on Empirical Data to Determine Coefficients in Semi-Empirical Mass Formula

In this paper, we propose activities based on empirical data to arrive at Coulomb, asymmetry, and pairing energy coefficients in the semi-empirical mass formula (SEMF). The Coloumb energy co-efficient is obtained using mirror nuclei data. Asymmetry energy co-efficient is determined from mass parabolas. The concept of neutron separation energies is used to arrive at the pairing energy co-efficient and its dependence with regards to the power of mass number A. Using these three obtained coefficients in SEMF for binding energies (BE), a plot of BE/A vs A−1/3 has resulted in Volume and surface energy coefficients using linear regression. It has found that the five coefficients obtained using these activities are closely matching with those obtained using Least Square Minimization (LSM) and Variational Monte Carlo (VMC) techniques. The mean percentage error (MPE), calculated between obtained binding energies from the optimized coefficients and the experimental binding energies, using the three approaches are almost equal. These activities enrich the understanding of students by laying importance on the physics involved, as they confirm that alternative strategies from experimental data are able to match those obtained using mathematical optimization procedures such as LSM and VMC.

Mathematical Analysis of Two Approaches for Optimal Parameter Estimates to Modeling Time Dependent Properties of Viscoelastic Materials

Mathematical models for phenomena in the physical sciences are typically parameter-dependent, and the estimation of parameters that optimally model the trends suggested by experimental observation depends on how model-observation discrepancies are quantified. Commonly used parameter estimation techniques based on least-squares minimization of the model-observation discrepancies assume that the discrepancies are quantified with the L2-norm applied to a discrepancy function. While techniques based on such an assumption work well for many applications, other applications are better suited for least-squared minimization approaches that are based on other norm or inner-product induced topologies. Motivated by an application in the material sciences, the new alternative least-squares approach is defined and an insightful analytical comparison with a baseline least-squares approach is provided.

A Universal, Analog, In-Memory Computing Primitive for Linear Algebra Using Memristors

The increasing demand for data-intensive computing applications, such as artificial intelligence (AI) and more specifically machine learning (ML), raises the need for novel computing hardware architectures capable of massive parallelism in performing core algebraic operations. Among the new paradigms, in-memory computing (IMC) with analogue devices is attracting significant interest for its large-scale integration potential, together with unrivaled speed and energy performance. Here, we present a fully-analogue, universal primitive capable of executing linear algebra operations such as regression, generalized least-square minimization and linear system solution with and without preconditioning. We study the impact of the main circuit parameters on accuracy and bandwidth with analytical closed-form expressions and SPICE simulations. Scaling challenges due to parasitic resistance/capacitance and their impact on key parameters such as bandwidth and accuracy are discussed. Finally, a comparison with existing solvers belonging to the same IMC framework is made to assess advantages and disadvantages of the proposed circuit.

A new logistic growth model applied to COVID-19 fatality data

Background:Recent work showed that the temporal growth of the novel coronavirus disease (COVID-19) follows a sub-exponential power-law scaling whenever effective control interventions are in place. Taking this into consideration, we present a new phenomenological logistic model that is well-suited for such power-law epidemic growth. Methods:We empirically develop the logistic growth model using simple scaling arguments, known boundary conditions and a comparison with available data from four countries, Belgium, China, Denmark and Germany, where (arguably) effective containment measures were put in place during the first wave of the pandemic. A non-linear least-squares minimization algorithm is used to map the parameter space and make optimal predictions. Results:Unlike other logistic growth models, our presented model is shown to consistently make accurate predictions of peak heights, peak locations and cumulative saturation values for incomplete epidemic growth curves. We further show that the power-law growth model also works reasonably well when containment and lock down strategies are not as stringent as they were during the first wave of infections in 2020. On the basis of this agreement, the model was used to forecast COVID-19 fatalities for the third wave in South Africa, which was in progress during the time of this work. Conclusion:We anticipate that our presented model will be useful for a similar forecasting of COVID-19 induced infections/deaths in other regions as well as other cases of infectious disease outbreaks, particularly when power-law scaling is observed.

A Fully-Automatic Gap Filling Approach for Motion Capture Trajectories

Missing marker information is a common problem in Motion Capture (MoCap) systems. Commercial MoCap software provides several methods for reconstructing incomplete marker trajectories; however, these methods still rely on manual intervention. Current alternatives proposed in the literature still present drawbacks that prevent their widespread adoption. The lack of fully automated and universal solutions for gap filling is still a reality. We propose an automatic frame-wise gap filling routine that simultaneously explores restrictions between markers’ distance and markers’ dynamics in a least-squares minimization problem. This algorithm constitutes the main contribution of our work by simultaneously overcoming several limitations of previous methods that include not requiring manual intervention, prior training or training data; not requiring information about the skeleton or a dedicated calibration trial and by being able to reconstruct all gaps, even if these are located in the initial and final frames of a trajectory. We tested our approach in a set of artificially generated gaps, using the full body marker set, and compared the results with three methods available in commercial MoCap software: spline, pattern and rigid body fill. Our method achieved the best overall performance, presenting lower reconstruction errors in all tested conditions.

Identifying an unknown heat source term in the third-order pseudo-parabolic equation from nonlocal integral observation

The aim of this paper is to find the time-dependent heat source term numerically in the third-order pseudo-parabolic equation with initial and boundary conditions supplemented by nonlocal integral observation. This is a very interesting and challenging nonlinear inverse source problem with important applications in various fields of mechanics and physics. Unique solvability theorem of this inverse problem is supplied. However, since the problem is still ill-posed (small errors in the integral input data cause large errors in the output force) the solution needs to be regularized. Therefore, to obtain a stable solution, a regularized objective function is minimized to retrieve the unknown heat source coefficient. The third-order pseudo-parabolic problem is discretized using the finite difference method (FDM) and recast as nonlinear least-squares minimization of the Tikhonov regularization function with simple bounds on the unknown force coefficient. Numerically, this is effectively solved using the MATLAB subroutine lsqnonlin. Both analytical and perturbed (noisy) data are inverted. Numerical results for three benchmark test examples are presented and discussed. In addition, the von Neumann stability analysis is also discussed.

Numerical and experimental flight verifications of a calibration matrix approach for load monitoring and temperature reconstruction and compensation

Mechanical and aeronautical structures could experience unexpected loads during operations, potentially reducing their operability. A load monitoring system enables one to recover the actual load spectra of the component and track its aging continuously. However, complex loading can be difficult to be identified, as for complex aerodynamic loads due to operational maneuvers in aeronautical structures, especially when environmental conditions vary.This work proposes a method for recovering the full strain and displacement fields in the entire structure, leveraging on the Calibration Matrix approach and an equivalent set of lumped forces, including the thermal variation among the unknown parameters. A least-squares minimization of an error functional defined as a comparison between measured and numerical strain is exploited for load and temperature reconstruction, automatically compensating thermal effects on the strain measurements without requiring any training data sets. By assuming a linear relationship between strain and the unknowns through a calibration matrix, the minimization can be performed analytically offline, resulting in a computationally very efficient algorithm that can be operated in real-time at a reduced computational burden.Though the mathematical formulation is general for any arbitrary component geometry and loading condition, the method is tested with a full-scale Unmanned Aerial Vehicle (UAV) undergoing different operational maneuvers, both numerically and with actual flight tests. The results confirm the robustness of the method to real environmental disturbances, correctly reconstructing the strain and displacement fields due to aerodynamic loads through triangular equivalent forces and automatically adapting to different load scenarios and temperature changes.