Least-squares Objective Function Research Articles

CSEM Optimization Using the Correspondence Principle

Traditionally, 3D modeling of marine controlled-source electromagnetic (CSEM) data (in the frequency domain) involves high-memory demand, requiring solving a large linear system for each frequency. To address this problem, we propose to solve Maxwell’s equations in a fictitious dielectric medium with time-domain finite-difference methods, with the support of the correspondence principle. As an advantage of this approach, we highlight the possibility of its implementation for execution with GPU accelerators, in addition to multi-frequency data modeling with a single simulation. Furthermore, we explore using the correspondence principle to the inversion of CSEM data by calculating the gradient of the least-squares objective function employing the adjoint-state method to establish the relationship between adjoint fields in a conductive medium and their counterparts in the fictitious dielectric medium, similar to the approach used in forward modeling. We validate this method through 2D inversions of three synthetic CSEM datasets, computed for a simple model consisting of two resistors in a conductive medium, a model adapted from a CSEM modeling and inversion package, and the last one based on a reference model of turbidite reservoirs on the Brazilian continental margin. We also evaluate the differences between the results of inversions using the steepest descent method and our proposed momentum method, comparing them with the limited-memory BFGS (Broyden–Fletcher–Goldfarb–Shanno) algorithm (L-BFGS-B). In all experiments, we use smoothing by model reparameterization as a strategy for regularizing and stabilizing the iterations throughout the inversions. The results indicate that, although it requires more iterations, our modified momentum method produces the best models, which are consistent with results from the L-BFGS-B algorithm and require less storage per iteration.

Phase retrieval for radar waveform design

Abstract The ability of a radar to discriminate in both range and Doppler velocity is completely characterized by the ambiguity function (AF) of its transmit waveform. Mathematically, it is obtained by correlating the waveform with its Doppler-shifted and delayed replicas. We consider the inverse problem of designing a radar transmit waveform that satisfies the specified AF magnitude. This process may be viewed as a signal reconstruction with some variation of phase retrieval methods. We provide a trust-region algorithm that minimizes a smoothed non-convex least-squares objective function to iteratively recover the underlying signal-of-interest for either time- or band-limited support. The method first approximates the signal using an iterative spectral algorithm and then refines the attained initialization based on a sequence of gradient iterations. Our theoretical analysis shows that unique signal reconstruction is possible using signal samples no more than thrice the number of signal frequencies or time samples. Numerical experiments demonstrate that our method recovers both time- and band-limited signals from sparsely and randomly sampled, noisy and noiseless AFs.

Steady-state flow under surface recharge through unconfined aquifer with depth-decaying hydraulic conductivity: Analytical results

This paper presents some analytical results related to groundwater flow through an unconfined horizontal aquifer induced by areal recharge on the top and a decaying hydraulic conductivity. Under steady-state conditions, and in the context of Boussinesq equation, exact analytical solutions describing drainage into one or two penetrating channels at the aquifer boundaries have been derived in closed forms. The analytical expressions have been used to investigate the existence of water divide inside the unconfined aquifer. Based on the derived analytical solutions, a closed-form analytical formula is developed for calculating groundwater transit time within Dupuit-type flow system. The analytical transit time expression is simply expressed in terms of common mathematical functions. Analytical results are verified against a numerical model and show excellent agreements. The analytical transit time is validated against a numerical one obtained from a two-dimensional model representing a rectangular unconfined aquifer. Based on the explicit closed-form solution, a fully analytical inverse problem to estimate the decay parameter is proposed using a least-square objective function. The solution of the inverse problem simply corresponds to the solution of a nonlinear algebraic equation obtained from the definition of the objective function and the form of the analytical solution.

Two-dimensional magnetotelluric inversion using unstructured triangular mesh implemented in Julia

This study presents a 2-D magnetotelluric inversion code tailored for unstructured triangular meshes, developed using Julia, a high-level, high-performance programming language designed for scientific and numerical computation. The forward modeling engine utilizes a node-based finite element method to solve electromagnetic fields throughout the modeling domain. The inversion process employs the Gauss-Newton optimization algorithm, iteratively updating the model via the minimization of a regularized least-squares objective function. We verified the accuracy of the forward modeling using two reference models and performed a synthetic inversion experiment to validate our inversion method and stress the necessity of accurately handling topography-influenced data. Through its application to constructing an electrical resistivity model beneath the northwestern end of Sumatra Island, Indonesia, we demonstrate the practicality of our code for inverting field datasets.

Dynamic Optimization for Petrophysical Property Estimation in Unsteady-State Coreflooding Using Pyomo

Summary This paper deals with a mathematical modeling and optimization-based approach for estimating relative permeability and capillary pressure from average water saturation data collected during unsteady-state waterflooding experiments. Assuming the Lomeland-Ebeltoft-Thomas (LET) model for the variation of the relative permeability with saturation, the appropriate governing equations, boundary, and initial conditions were solved within the Pyomo framework. Using interior point optimization (IPOPT) with a least-squares objective function, the six parameters of the LET model that ensure the history matching between the measured and calculated average saturation are determined. Additionally, we inferred the capillary pressure function and performed a Sobol sensitivity analysis on the LET model parameters. The results showcase the reliability and robustness of our proposed approach, as it estimates the crucial parameters driving the variation of oil-water flow relative permeability across several cases and effectively predicts the capillary pressure trend. The proposed approach can be seen as an alternative to experimental and numerical simulation-based techniques for predicting relative permeability and capillary pressure curves.

On robust estimation of the Gauss–Markov model with a singular covariance matrix

Although the robust parameter estimation has been intensively investigated in statistics and measurement science, no one considers the robust estimation within the Gauss–Markov model (GMM) with a singular covariance matrix. In this contribution, a method of finding the robust solutions in the singular linear model has been suggested based on the generalization of the unified least-squares objective function. The equivalent cofactor matrix in the robust objective function is established by the bifactor strategy based on the IGG III function. The robustness of the proposed robust algorithms is demonstrated with the examples of geodetic coordinates transformation and the free network deformation analysis.

Simultaneous estimation and group identification for network vector autoregressive model with heterogeneous nodes

Individuals or companies in a large social or financial network often display rather heterogeneous behaviors for various reasons. In this work, we propose a network vector autoregressive model with a latent group structure to model heterogeneous dynamic patterns observed from network nodes, for which group-wise network effects and time-invariant fixed-effects can be naturally incorporated. In our framework, the model parameters and network node memberships can be simultaneously estimated by minimizing a least-squares type objective function. In particular, our theoretical investigation allows the number of latent groups G to be over-specified when achieving the estimation consistency of the model parameters and group memberships, which significantly improves the robustness of the proposed approach. When G is correctly specified, valid statistical inference can be made for model parameters based on the asymptotic normality of the estimators. A data-driven criterion is developed to consistently identify the true group number for practical use. Extensive simulation studies and two real data examples are used to demonstrate the effectiveness of the proposed methodology.

A novel recursive least-squares adaptive method for streaming tensor-train decomposition with incomplete observations

Tensor tracking which is referred to as online (adaptive) decomposition of streaming tensors has recently gained much attention in the signal processing community due to the fact that many modern applications generate a huge number of multidimensional data streams over time. In this paper, we propose an effective tensor tracking method via the tensor-train format for decomposing high-order incomplete streaming tensors. On the arrival of new data, the proposed algorithm minimizes a weighted least-squares objective function accounting for both missing values and time-variation constraints on the underlying tensor-train cores, thanks to the recursive least-squares filtering technique and the block coordinate descent framework. Our algorithm is fully capable of tensor tracking from noisy, incomplete, and high-dimensional observations in both static and time-varying environments. Its tracking ability is validated with several experiments on both synthetic and real data.

Modeling and Parameter Tuning for Continuous Catalytic Reforming of Naphtha in an Industrial Reactor System

A two-dimensional mathematical model was developed to simulate naphtha reforming in a series of three industrial continuous catalytic regeneration (CCR) reactors. Discretization of the resulting partial differential equations (PDEs) in the vertical direction and a coordinate transformation in the radial direction were performed to make the model solvable using Aspen Custom Modeler. A sensitivity-based parameter subset selection method was employed to identify the most influential parameters within the model. Tuning of 8 out of 180 parameters was used to ensure that model predictions match experimental data from one steady-state run. The updated parameter values improved the model fit to the data, reducing the weighted least-squares objective function for parameter estimation by 73%. The proposed model was used to predict reactor temperatures, catalyst coke weight fraction at the exit of the third reactor, and benzene flowrate from the outlet of the third reactor. The simulation results demonstrated a good agreement between the simulated values and the industrial measurements. Finally, the reactor model was utilized to explore the effects of changes in inlet temperatures and inlet level of catalyst deactivation, providing valuable insights for identifying desirable operational conditions that will improve the overall efficiency of the CCR process.

A trust-region-based phase envelope construction algorithm

Pressure-temperature (PT) and pressure-composition (Px) phase diagrams play an important role in various chemical and petroleum engineering applications. Lots of algorithms based on the Newton-Raphson method have been proposed to numerically construct PT/Px phase envelopes. No attempt is made to improve the computational efficiency of such Newton-Raphson solution method. This work develops a new trust-region-based algorithm to replace the Newton-Raphson method to provide more robust and efficient phase envelope construction calculations. We first convert the phase envelope construction problem into a minimization problem by defining a least-squares objective function. We then apply the trust-region optimization method with an exact subproblem solver to minimize the objective function. We follow the general strategy proposed by Michelsen (1980) to develop the revised phase envelope construction algorithm. We demonstrate the good performance of the trust-region-based algorithm by comparing it with the conventional Newton's method. The computational results indicate that the new trust-region-based algorithm is more robust and cost-effective than the Newton-based algorithm.

Multivariate group-sparse mode decomposition

In signal decomposition, a complex and nonstationary signal is decomposed into a set of basic functions, usually called intrinsic mode functions (IMFs). In multivariate signal decomposition, it is assumed that analogous IMFs across all channels share common bandwidths (BWs). In this paper, a generic extension of the group-sparse mode decomposition (GSMD) for multivariate (multichannel) signal decomposition is developed. The univariate GSMD is based on the assumption that the spectral energy of the measured signal is concentrated in disjoint groups, and each group represents the spectral energy of one of the IMFs. This concept is generalized in the proposed algorithm to the multivariate measurements. Accordingly, the penalized least-squares objective function used for deriving GSMD is adapted in this paper to handle the multivariate measurements, where the weightedℓ0-norm is utilized as the penalty term, and the weighting parameters and the regularization parameter are calculated using techniques based on energy detection over short windows. In addition, two thresholding strategies were applied by the proposed algorithm to overcome the mode-mixing phenomenon. The derived algorithm is referred to as generalized GSMD (gGSMD). The gGSMD is further developed to handle the case when the analogous IMFs have disparate BWs across channels, and the resulting algorithm is referred to as multivariate GSMD (MvGSMD). The MvGSMD enjoys the following desired properties: robustness against noise, fast convergence, mode alignment, automatically estimating the number of IMFs, producing orthogonal IMFs, and efficiently adjusting the BW of each individual IMF even when analogous IMFs across channels have disparate BWs. All these properties were validated using numerical results on simulated data and real EEG and physiological data. The simulation results also show that the proposed algorithm outperforms many state-of-the-art multivariate signal decomposition algorithms.

On the Error of Li-ion Battery Parameter Estimation Subject to System Uncertainties

Emerging lithium-ion battery systems require high-fidelity electrochemical models for advanced control, diagnostics, and design. Accordingly, battery parameter estimation is an active research domain where novel algorithms are being developed to calibrate complex models from input-output data. Amidst these efforts, little focus has been placed on the fundamental mechanisms governing estimation accuracy, spurring the question, why is an estimate accurate or inaccurate? In response, we derive a generalized estimation error equation under the commonly adopted least-squares objective function, which reveals that the error can be represented as a combination of system uncertainties (i.e., in model, measurement, and parameter) and uncertainty-propagating sensitivity structures in the data. We then relate the error equation to conventional error analysis criteria, such as the Fisher information matrix, Cramér-Rao bound, and parameter sensitivity, to assess the benefits and limitations of each. The error equation is validated through several uni- and bivariate estimations of lithium-ion battery electrochemical parameters using experimental data. These results are also analyzed with the error equation to study the error compositions and parameter identifiability under different data. Finally, we show that adding target parameters to the estimation without increasing the amount of data intrinsically reduces the robustness of the results to system uncertainties.

Estimation of a Structural Break Point in Linear Regression Models

This study proposes a point estimator of the break location for a one-time structural break in linear regression models. If the break magnitude is small, the least-squares estimator of the break date has two modes at the ends of the finite sample period, regardless of the true break location. To solve this problem, I suggest an alternative estimator based on a modification of the least-squares objective function. The modified objective function incorporates estimation uncertainty that varies across potential break dates. The new break point estimator is consistent and has a unimodal finite sample distribution under small break magnitudes. A limit distribution is provided under an in-fill asymptotic framework. Monte Carlo simulation results suggest that the new estimator outperforms the least-squares estimator. I apply the method to estimate the break date in U.S. and U.K. stock return prediction models.

Solution of an acoustic transmission inverse problem by extended inversion

Study of a simple single-trace transmission example shows how an extended source formulation of full-waveform inversion can produce an optimization problem without spurious local minima (‘cycle skipping’), hence efficiently solvable via Newton-like local optimization methods. The data consist of a single trace extracted from a causal pressure field, propagating in a homogeneous fluid according to linear acoustics, and recorded at a given distance from a transient point energy source. The source intensity (‘wavelet’) is presumed quasi-impulsive, with zero energy for time lags greater than a specified maximum lag. The inverse problem is: from the recorded trace, recover both the sound velocity or slowness and source wavelet with specified support, so that the data is fit with prescribed RMS relative error. The least-squares objective function has multiple large residual minimizers. The extended inverse problem permits source energy to spread in time, and replaces the maximum lag constraint by a weighted quadratic penalty. A companion paper shows that for proper choice of weight operator, any stationary point of the extended objective produces a good approximation of the global minimizer of the least squares objective, with slowness error bounded by a multiple of the maximum lag and the assumed noise level. This paper summarizes the theory developed in the companion paper and presents numerical experiments demonstrating the accuracy of the predictions in concrete instances. We also show how to dynamically adjust the penalty scale during iterative optimization to improve the accuracy of the slowness estimate.

Retrieving electrical and structural carbonate formation properties from measurements on crushed rock using a multidata inversion approach

SUMMARY Quantification understanding, and prediction of physical rock properties rely so far predominantly on laboratory analyses of cores and plugs. Based on such data, petrophysical models are found that relate both microstructural properties and environmental conditions to geophysically accessible quantities. When considering reactive rock–fluid–gas systems, for example in geothermal energy, enhanced oil recovery or carbon dioxide sequestration, especially with carbonatic rock matrix, this approach is costly and time-consuming at best, or impossible to implement at worst. This is based on the two following reasons: First, porosity, permeability and accessible internal surface area in solid rock plugs are often so low that experimental time duration of many months or even years would be required to achieve chemical equilibrium. Secondly, plugs are single specimens of their — generally heterogeneous — original rock formation, which strongly questions the representativeness of single-plug data. To overcome these shortcomings, we present a new methodology based on the combination of systematic crushing, multimethod laboratory measurements and model-based computational evaluation with solving an inverse problem. As a first step, a large amount of undisturbed rock is intentionally crushed and divided in several particle size classes. Then, petrophysical laboratory measurements are carried out on all particle size classes. The resulting data set is finally inverted for the intended properties of the undisturbed rock. This inverse problem entails a three-level forward model, which parametrizes the undisturbed rock properties, particle characteristics and particle packings, but can also be freely adapted to other tasks by any suitable model representation. The three-level model is designed to enforce the petrophysical correlation of all properties at all levels while using a minimal set of model parameters, thus keeping the inverse problem overdetermined. For the inversion, we have developed a publicly available software tool (AnyPetro) based on a Gauss–Newton inversion scheme to minimize a damped least-squares objective function. To demonstrate and validate the proposed methodology, we present a study using five rock types — four carbonates and one sandstone as a reference. Laboratory measurements of complex electrical conductivity (from spectral induced polarization), specific surface (from nitrogen adsorption) and intraparticle porosity (from mercury intrusion) have been carried out on eight particle size classes and on plugs of each rock for comparison. Supportive and complementary analyses include, for example particle geometry, nuclear magnetic resonance, scanning electron microscopy, computer tomography, uniaxial compression strength and mineralogical composition. We show that our new methodology is highly capable of robustly recovering the complex electrical conductivity, specific surface and porosity of the undisturbed rocks from the measured data. The resulting sets of model parameters are petrophysically reasonable and verifiable. The presented methodology can further be applied to the use of drill cuttings as sample material, which is often the only available rock material from deep wells. Our findings also represent a methodological advance for laboratory experiments on reactive systems and both the interpretation and prediction of petrophysical rock properties in such systems.

Spectral-element method with an optimal mass matrix for seismic wave modelling

The spectral-element method (SEM), which combines the flexibility of the finite element method (FEM) with the accuracy of spectral method, has been successfully applied to simulate seismic wavefields in geological models on different scales. One kind of SEMs that adopts orthogonal Legendre polynomials is widely used in seismology community. In the SEM with orthogonal Legendre polynomials, the Gauss-Lobatto-Legendre (GLL) quadrature rule is employed to calculate the integrals involved in the SEM leading to a diagonal mass matrix. However, the GLL quadrature rule can exactly approximate only integrals with a polynomial degree below 2N-1 (N is the interpolation order in space) and cannot exactly calculate those of polynomials with degree 2N involved in the mass matrix. Therefore, the error of the mass matrix originating from inexact numerical integration may reduce the accuracy of the SEM. To improve the SEM accuracy, we construct a least-squares objective function in terms of numerical and exact integrals to increase the accuracy of the GLL quadrature rule. Then, we utilise the conjugate gradient method to solve the objective function and obtain a set of optimal quadrature weights. The optimal mass matrix can be obtained simultaneously by utilising the GLL quadrature rule with optimal integration weights. The improvement in the numerical accuracy of the SEM with an optimal mass matrix (OSEM) is demonstrated by theoretical analysis and numerical examples.

High-Precision SLAM Based on the Tight Coupling of Dual Lidar Inertial Odometry for Multi-Scene Applications

Simultaneous Localization and Mapping (SLAM) is an essential feature in many applications of mobile vehicles. To solve the problem of poor positioning accuracy, single use of mapping scene, and unclear structural characteristics in indoor and outdoor SLAM, a new framework of tight coupling of dual lidar inertial odometry is proposed in this paper. Firstly, through external calibration and an adaptive timestamp synchronization algorithm, the horizontal and vertical lidar data are fused, which compensates for the narrow vertical field of view (FOV) of the lidar and makes the characteristics of vertical direction more complete in the mapping process. Secondly, the dual lidar data is tightly coupled with an Inertial Measurement Unit (IMU) to eliminate the motion distortion of the dual lidar odometry. Then, the value of the lidar odometry after correcting distortion and the pre-integrated value of IMU are used as constraints to establish a non-linear least-squares objective function. Joint optimization is then performed to obtain the best value of the IMU state values, which will be used to predict the state of IMU at the next time step. Finally, experimental results are presented to verify the effectiveness of the proposed method.

ECG denoising based on successive local filtering

A new algorithm for denoising ECG data contaminated by wideband noise is proposed in this paper. In the proposed algorithm, a clean ECG data is modeled as a combination of different components. The components have the characteristics that they are disjoint in the time domain, their spectral coefficients overlap in the frequency domain, and they have different bandwidths. Based on this model, a successive local filtering approach is suggested in this paper to remove wideband noise from a recorded ECG data. In the proposed algorithm, a segmentation procedure is first developed to segment the recorded ECG data such that each segment approximately contains one dominant component. The denoised ECG signal is then constructed by successively denoising the constructed segments using ideal filters. The ideal filters are designed in the frequency domain by minimizing a penalized least-squares objective function, where the weighted ℓ0-norm is utilized as the penalty term to encourage the on-off group-sparsity of the ideal filters. In the proposed algorithm, the BW of each ideal filter is automatically adjusted to the BW of the dominant component in the analyzed segment. Simulation results on simulated and real ECG data show that the proposed algorithm can be successfully utilized to denoise ECG data contaminated by wideband noise. In addition, the proposed algorithm is also shown to produce significantly improved results compared to some existing ECG denoising techniques.

Optimized lattice-based metamaterials for elastostatic cloaking

An optimization-based approach is proposed to design elastostatic cloaking devices in two-dimensional (2D) lattices. Given an elastic lattice with a defect, i.e. a circular or elliptical hole, a small region (cloak) around the hole is designed to hide the effect of the hole on the elastostatic response of the lattice. Inspired by the direct lattice transformation approach to elastostatic cloaking in 2D lattices, the lattice nodal positions in the design region are obtained using a coordinate transformation of the reference (undisturbed) lattice nodes. Subsequently, additional connectivity (i.e. a ground structure) is defined in the design region and the stiffness properties of these elements are optimized to mimic the global stiffness characteristics of the reference lattice. A weighted least-squares objective function is proposed, where the weights have a physical interpretation—they are the design-dependent coefficients of the design lattice stiffness matrix. The formulation leads to a convex objective function that does not require a solution to an additional adjoint system. Optimization-based cloaks are designed considering uniaxial tension in multiple directions and are shown to exhibit approximate elastostatic cloaking, not only when subjected to the boundary conditions they were designed for but also for uniaxial tension in directions not used in design and for shear loading.

Detection method of inner lead wires of transformer bushing based on parameter estimation of ultrasonic echo model

In order to solve the problem of low accuracy of the ultrasonic detecting method for the lead wires of the transformer bushing, this paper used the method of parameter estimation of the ultrasonic echo model to improve the accuracy of the ultrasonic flight time. First, this paper proposed an ultrasonic Gaussian model that suitable for detecting inner lead wires of transformer bushings, constructs a least-squares objective function, and then uses a particle swarm optimization algorithm to search for the optimal solution of the objective function globally, and proposes an algorithm for parameter estimation of the echo model. Finally, it is proved that the detection method proposed in this paper has the advantage of anti-noise interference through experiment, and the detection accuracy is improved. So, this method can detect the state of the lead wires in the transformer bushing.