Nonlinear Least Squares Optimization Research Articles

Improved extrapolation of steady turbulent aerodynamics using a non-linear POD-based reduced order model

AbstractA reduced-order modelling (ROM) approach for predicting steady, turbulent aerodynamic flows based on computational fluid dynamics (CFD) and proper orthogonal decomposition (POD) is presented. Model-order reduction is achieved by parameter space sampling, solution space representation via POD and restriction of a CFD solver to the POD subspace. Solving the governing equations of fluid dynamics is replaced by solving a non-linear least-squares optimisation problem. The method will be referred to as LSQ-ROM method. Two approaches of extracting POD basis information from CFD snapshot data are discussed: POD of the full state vector (global POD) and POD of each of the partial states separately (variable-by-variable POD). The method at hand is demonstrated for a 2D aerofoil (NACA 64A010) as well as for a complete industrial aircraft configuration (NASA Common Research Model) in the transonic flow regime by computing ROMs of the compressible Reynolds-averaged Navier-Stokes equations, pursuing both the global and the variable-by-variable POD approach. The LSQ-ROM approach is tried for extrapolatory flow conditions. Results are juxtaposed with those obtained by POD-based extrapolation using Kriging and the radial basis functions spline method. As a reference, the full-order CFD solutions are considered. For the industrial aircraft configuration, the cost of computing the reduced-order solution is shown to be two orders of magnitude lower than that of computing the reference CFD solution.

Closed-Loop Identification of Second-Order Plus Time Delay (SOPTD) Model of Multivariable Systems by Optimization Method

The closed-loop identification of second-order plus time delay (SOPTD) transfer function models of multivariable systems is presented based on optimization method using the combined step-up and step-down responses. The need for combined step up and step down changes in the set point is brought out for the convergence of the model parameters. A standard nonlinear least-squares optimization method is used to obtain the parameters of the SOPTD model transfer function matrix by minimizing the sum of squared errors between the closed-loop responses of the model and the actual process responses. A simple method is proposed to obtain the initial guess values of SOPTD transfer function model parameters from the main and interaction responses of the actual process. This method was applied to two-input two-output (TITO) second-order plus time delay (SOPTD), higher order and 3 × 3 SOPTD transfer function models of multivariable systems. The proposed method considerably reduces the computational time for the optimiza...

Tension Force and Structural Parameter Identification of Bridge Cables

Vibration measurement is one of the most widely used methods for tension evaluation and condition assessment of stay cables in cable-stayed bridges. In the existing practice, the tension force of a cable is identified from the measured modal frequencies with the use of the taut string theory or empirical formulae, by assuming pre-determined structural parameters (geometric and material parameters) and boundary conditions of the cable. As a result, an inaccurate estimation of the cable tension may be obtained when there is an error in the pre-determined structural parameters and boundary conditions. Moreover, the commonly used empirical formulae are not applicable in the case when the cable is intermediately attached with dampers. In the present study, a method enabling simultaneous identification of cable tension and other structural parameters from the measured modal frequencies is developed. A precise finite element model (FEM) accounting for cable flexural rigidity, sag-extensibility, spatial variability of dynamic tension, boundary conditions, lumped masses and intermediate supports and/or dampers is first formulated as the reference model in parameter identification so that the modeling error is minimized. Then the measured multiple modal frequencies are used together with the FEM to figure out a nonlinear least-square optimization scheme which helps eliminate measurement error and allows for simultaneous identification of the cable tension and other structural parameters. Application of the proposed method to the Dongting Lake Bridge cables from in-situ ambient vibration measurements illustrates high identification accuracy and fidelity of the proposed method.

Spatial interpolation using nonlinear mathematical programming models for estimation of missing precipitation records

New mathematical programming models are proposed, developed and evaluated in this study for estimating missing precipitation data. These models use nonlinear and mixed integer nonlinear mathematical programming (MINLP) formulations with binary variables. They overcome the limitations associated with spatial interpolation methods relevant to the arbitrary selection of weighting parameters, the number of control points within a neighbourhood, and the size of the neighbourhood itself. The formulations are solved using genetic algorithms. Daily precipitation data obtained from 15 rain gauging stations in a temperate climatic region are used to test and derive conclusions about the efficacy of these methods. The developed methods are compared with some naïve approaches, multiple linear regression, nonlinear least-square optimization, kriging, and global and local trend surface and thin-plate spline models. The results suggest that the proposed new mathematical programming formulations are superior to those obtained from all the other spatial interpolation methods tested in this study. Editor D. Koutsoyiannis; Associate editor S. Grimaldi Citation Teegavarapu, R.S.V., 2012. Spatial interpolation using nonlinear mathematical programming models for estimation of missing precipitation records. Hydrological Sciences Journal, 57 (3), 383–406.

Spatial-Feature Identification and Parameter Estimation From Pressure-Transient Tests

This article, written by Senior Technology Editor Dennis Denney, contains highlights of paper SPE 142996, ’Grid-Based Inversion Methods for Spatial-Feature Identification and Parameter Estimation From Pressure-Transient Tests,’ by K.L. Morton, SPE, and R.J. Booth, Schlumberger; M. Onur, SPE, Istanbul Technical University; and F.J. Kuchuk, SPE, Schlumberger, prepared for the 2011 SPE Europec/EAGE Annual Conference and Exhibition, Vienna, Austria, 23-26 May. The paper has not been peer reviewed. Three grid-based inversion methods for estimating formation parameters and for spatial geological-feature identification based on pressure-transient-test (PTT) data from multiple-well locations were investigated. The first two methods use efficient adjoint schemes to determine the gradient of the objective functions. The third method uses ensemble Kalman filtering (EnKF) for data assimilation. With these methods, the existence and location of many subseismic features, such as strong spatial permeability variations, faults, fractures, and pinchouts, may be determined by use of exploration and production data. Introduction PTTs are used to determine the productivity of a well and the properties of the formation (reservoir) on the basis of downhole and/or surface pressure and flow-rate measurements. The main steps for interpretation are: Model identification—a possible set of reservoir models is found that may fit the data. Model-parameter estimation—model parameters are adjusted until the model behavior matches observed data. Model verification—consistency of the final model is verified by measuring the mismatch between the real system and the model or by comparing with other data. By use of conventional interpretation methods (i.e., semilog methods such as Horner or Miller-Dyes-Hutchinson, and/or type-curve matching of measured pressure and/or pressure derivative), data including reservoir pressure, an effective average permeability of the formation, skin factor, and wellbore storage can be estimated from the PTT data. In such interpretation, some sort of prior modeling is necessary to constrain the nonlinear parameter estimation because a model with many nonphysical reservoir parameters may match the observed PTT data. This prior knowledge may be available at small scale from logs and cores and at a larger scale from seismic surveys and outcrop analogies. Recently, nonlinear least-squares optimization has been applied to PTT data by use of numerical models with a similarly limited number of parameters—often models that are divided into a small number of regions, within which the reservoir parameters are assumed to be constant.

An inexact Gauss-Newton method for inversion of basal sliding and rheology parameters in a nonlinear Stokes ice sheet model

AbstractWe propose an infinite-dimensional adjoint-based inexact Gauss-Newton method for the solution of inverse problems governed by Stokes models of ice sheet flow with nonlinear rheology and sliding law. The method is applied to infer the basal sliding coefficient and the rheological exponent parameter fields from surface velocities. The inverse problem is formulated as a nonlinear least-squares optimization problem whose cost functional is the misfit between surface velocity observations and model predictions. A Tikhonov regularization term is added to the cost functional to render the problem well-posed and account for observational error. Our findings show that the inexact Newton method is significantly more efficient than the nonlinear conjugate gradient method and that the number of Stokes solutions required to solve the inverse problem is insensitive to the number of inversion parameters. The results also show that the reconstructions of the basal sliding coefficient converge to the exact sliding coefficient as the observation error (here, the noise added to synthetic observations) decreases, and that a nonlinear rheology makes the reconstruction of the basal sliding coefficient more difficult. For the inversion of the rheology exponent field, we find that horizontally constant or smoothly varying parameter fields can be reconstructed satisfactorily from noisy observations.

Parameter identification of multibody systems based on second order sensitivity analysis

An identification procedure for multibody systems is presented to determine optimal values for unknown or estimated model parameters from experimental test data. Based on a non-linear least-square optimization procedure, the Levenberg–Marquardt trust region method is developed to estimate the unknown parameters, in which the second order sensitivity analysis with the hybrid method is applied to improve the convergence and stability. Finally, numerical examples of slider-crank mechanism validate the accuracy and efficiency of the method presented.

Batch and column experiments to support heavy metals (Cu, Zn and Mn) in alluvial sediments

Laboratory batch sorption and column transport experiments were performed on heterogeneous alluvial soils with a wide range of physical characteristics from wells that are located in the region between Mogan Lake and Eymir Lake, Golbs1, Ankara. The mean values for the K(subscript d) of Cu were found to be highest in clay (32550.350 L/kg) and lowest in loamy sand (18170.76 L/kg). The minimum and maximum sorption capacity values (mean values) for Zn were found to be in clay (10985.148 mg/kg) and in silty loam (8597.14 mg/kg) units, respectively. Similarly, the minimum and maximum values for Mn were found in loamy sand (4908.695 mg/kg) and clay (7587.39 1 mg/kg) units. The non-linear least-squares optimization code ”CXTFIT” was used to determine transport parameter values by curve-fitting. The results of the column experiments demonstrate dispersivity values within the range of 0.024-1.13 cm for soil samples.

Multiple Faults Isolation for Hybrid Systems with Unknown Fault Pattern

This work is concerned with multiple faults isolation for hybrid systems based on Global Analytical Redundancy Relationships (GARRs) approach. GARRs are derived from the Hybrid Bond Graph (HBG) model of a hybrid system with a specified causality assignment procedure. In this article, multiple faults are considered in a complex hybrid system and these faults can develop during a mode when the faults are not detectable. Once a fault is detected, a fault candidates set is generated from mode dependent-fault signature matrix (MD-FSM) tables and a set of fault pattern hypothesis is created from the fault candidates set for further refinement. Fault isolation is carried out using a multiple nonlinear least square optimization (MNLSO) algorithm. The developed technique can deal with multiple faults with unknown pattern. The fault could be of incipient or abrupt nature. The simulation results show the effectiveness of the proposed method.

Nonlinear piezoelectricity in electroelastic energy harvesters: Modeling and experimental identification

We propose and experimentally validate a first-principles based model for the nonlinear piezoelectric response of an electroelastic energy harvester. The analysis herein highlights the importance of modeling inherent piezoelectric nonlinearities that are not limited to higher order elastic effects but also include nonlinear coupling to a power harvesting circuit. Furthermore, a nonlinear damping mechanism is shown to accurately restrict the amplitude and bandwidth of the frequency response. The linear piezoelectric modeling framework widely accepted for theoretical investigations is demonstrated to be a weak presumption for near-resonant excitation amplitudes as low as 0.5 g in a prefabricated bimorph whose oscillation amplitudes remain geometrically linear for the full range of experimental tests performed (never exceeding 0.25% of the cantilever overhang length). Nonlinear coefficients are identified via a nonlinear least-squares optimization algorithm that utilizes an approximate analytic solution obtained by the method of harmonic balance. For lead zirconate titanate (PZT-5H), we obtained a fourth order elastic tensor component of c1111p=−3.6673×1017 N/m2 and a fourth order electroelastic tensor value of e3111=1.7212×108 m/V.

Cooperative AUV Navigation using a Single Maneuvering Surface Craft

In this paper we describe the experimental implementation of an online algorithm for cooperative localization of submerged autonomous underwater vehicles (AUVs) supported by an autonomous surface craft. Maintaining accurate localization of an AUV is difficult because electronic signals, such as GPS, are highly attenuated by water. The usual solution to the problem is to utilize expensive navigation sensors to slow the rate of dead-reckoning divergence. We investigate an alternative approach that utilizes the position information of a surface vehicle to bound the error and uncertainty of the on-board position estimates of a low-cost AUV. This approach uses the Woods Hole Oceanographic Institution (WHOI) acoustic modem to exchange vehicle location estimates while simultaneously estimating inter-vehicle range. A study of the system observability is presented so as to motivate both the choice of filtering approach and surface vehicle path planning. The first contribution of this paper is to the presentation of an experiment in which an extended Kalman filter (EKF) implementation of the concept ran online on-board an OceanServer Iver2 AUV while supported by an autonomous surface vehicle moving adaptively. The second contribution of this paper is to provide a quantitative performance comparison of three estimators: particle filtering (PF), non-linear least-squares optimization (NLS), and the EKF for a mission using three autonomous surface craft (two operating in the AUV role). Our results indicate that the PF and NLS estimators outperform the EKF, with NLS providing the best performance.

Two nonlinear optimization methods for black box identification compared

In this paper, two nonlinear optimization methods for the identification of nonlinear systems are compared. Both methods estimate the parameters of e.g. a polynomial nonlinear state-space model by means of a nonlinear least-squares optimization of the same cost function. While the first method does not estimate the states explicitly, the second method estimates both states and parameters adding an extra constraint equation. Both methods are introduced and their similarities and differences are discussed utilizing simulation data. The unconstrained method appears to be faster and more memory efficient, but the constrained method has a significant advantage as well: it is robust for unstable systems of which bounded input–output data can be measured (e.g. a system captured in a stabilizing feedback loop). Both methods have successfully been applied on real-life measurement data.

Parameter sensitivity analysis of a nonlinear least-squares optimization-based anelastic full waveform inversion method

In a recent article, we described a seismic inversion method for determining the crustal velocity and attenuation of basins in earthquake-prone regions. We formulated the problem as a constrained nonlinear least-squares optimization problem in which the constraints are the equations that describe the forward wave propagation. Here, we conduct a parametric study to investigate the influence of parameters such as the form of the regularization function, receiver density, preconditioning, noise level of the data, and the multilevel continuation technique on the cost and quality of the inversion. We use the same 2D Los Angeles example as in our earlier study.

Characterization of wavelet coefficients for ultrasonic signals

A wavelet transform has been widely used to investigate the characteristics of wave signals for a decade. However, only qualitative investigation of the spectrogram was made rather than a quantitative interpretation. On the other hand, an analytical closed-form representation of the wavelet transformed wave signal can be used as a basis function in estimating parameters using nonlinear least-squares optimization. We derived a quantitative closed-form equation directly from the analytical continuous wavelet transformation of a pulse with a Gaussian spectrum. A fundamental three-dimensional shape of a wavelet in the spectrogram was obtained, and the analytical form was compared quantitatively with numerical results.

Non-linear reduced order models for steady aerodynamics

Abstract A reduced order modelling approach for predicting steady aerodynamic flows and loads data based on Computational Fluid Dynamics (CFD) and global Proper Orthogonal Decomposition (POD), that is, POD for multiple different variables of interest simultaneously, is presented. A suitable data transformation for obtaining problemadapted global basis modes is introduced. Model order reduction is achieved by parameter space sampling, reduced solution space representation via global POD and restriction of a CFD flow solver to the reduced POD subspace. Solving the governing equations of fluid dynamics is replaced by solving a non-linear least-squares optimization problem. Methods for obtaining feasible starting solutions for the optimization procedure are discussed. The method is demonstrated by computing reduced-order solutions to the compressible Euler equations for the NACA 0012 airfoil based on two different snapshot sets; one in the subsonic and one in the transonic flow regime, where shocks occur. Results are compared with those obtained by POD-based interpolation using Kriging and the Thin Plate Spline method (TPS).

Robust mobile robot localization in highly non-static environments

In this paper, we propose a robust pose tracking method for mobile robot localization with an incomplete map in a highly non-static environment. This algorithm will work with a simple map that does not include complete information about the non-static environment. With only an initial incomplete map, a mobile robot cannot estimate its pose because of the inconsistency between the real observations from the environment and the predicted observations on the incomplete map. The proposed localization algorithm uses the approach of sampling from a non-corrupted window, which allows the mobile robot to estimate its pose more robustly in a non-static environment even when subjected to severe corruption of observations. The algorithm sequence involves identifying the corruption by comparing the real observations with the corresponding predicted observations of all particles, sampling particles from a non-corrupted window that consists of multiple non-corrupted sets, and filtering sensor measurements to provide weights to particles in the corrupted sets. After localization, the estimated path may still contain some errors due to long-term corruption. These errors can be corrected using nonlinear constrained least-squares optimization. The incomplete map is then updated using both the corrected path and the stored sensor information. The performance of the proposed algorithm was verified via simulations and experiments in various highly non-static environments. Our localization algorithm can increase the success rate of tracking its pose to more than 95% compared to estimates made without its use. After that, the initial incomplete map is updated based on the localization result.

Myocardial material parameter estimation from in‐vivo myocardial strain measurements

The aim of this study was to develop and test a new method to estimate myocardial material parameters from 3‐D strains measured in individual canine hearts. The method can also be applied with non‐invasive imaging modalities.Diastolic myocardial 3‐D finite deformations were obtained in open‐chest anesthetized dog hearts by tracking the positions of implanted bead sets in the left ventricular free wall using biplane x‐ray cineradiography. Anatomically detailed finite element models of individual canine hearts were fitted to MR and CT images of perfusion‐fixed hearts.Resting myocardial material parameters were estimated by minimizing the differences between ventricular deformations predicted by the model and those measured in the intact heart using a Levenberg Marquardt nonlinear least‐squares optimization scheme. Artificially generated strains were used to confirm convergence to correct parameters in less than 20 iterations. Noisy artificial data with a realistic measurement error of 0.01 cm resulted in normally distributed parameter solutions, with respective means within 3% of true values. The means of the strains predicted by a model at marker locations with the noisy parameters had an average error of −1.2% ± 4.1% of the true strains. The best estimate of the material properties optimized by the real anatomic model and data are validated by subsequent comparison to experimental and published data.

Parameter Reduction in Estimated Model Sets for Robust Control

This paper proposes two techniques for reducing the number of uncertain parameters in order to simplify robust controller design and to reduce conservatism inherent in robust controllers. The system is assumed to have a known structure with parametric uncertainties that represent plant dynamics variation. An original set of parameters is estimated by nonlinear least-squares (NLS) optimization using noisy frequency response functions. Utilizing the property of asymptotic normality for NLS estimates, the original parameter set can be reparameterized by an affine function of the smaller number of uncorrelated parameters. The correlation among uncertain parameters is detected by the principal component analysis in one technique and optimization with a bilinear matrix inequality in the other. Numerical examples illustrate the usefulness of the proposed techniques.

Analysis of Fluorescence Lifetime Imaging Microscopy (FLIM) Data

A novel Fluorescence Lifetime Imaging Microscopy (FLIM) deconvolution method based on the linear expansion of fluorescence decays on a set of orthonormal Laguerre functions was recently proposed. The Laguerre deconvolution method applies linear least-square estimation to estimate the expansion coefficients of all pixel decays simultaneously, performing at least two orders of magnitude faster than the other algorithms. In the original Laguerre FLIM deconvolution implementation, however, the Laguerre parameter α is selected using a heuristic approach, making it unsuitable for online applications. In this study, we present a fully automated implementation of the Laguerre FLIM deconvolution, whereby the Laguerre parameter α is treated as a free parameter within a nonlinear least-squares optimization scheme. The performance of this method has been successfully validated on simulated data, and experimental FLIM images of standard fluorescent dyes and endogenous tissue fluorescence. The main advantage of the proposed method is that it does not require any user intervention for tuning up the deconvolution process. Thus, we believe this method will facilitate the translation of FLIM to online applications, including real-time clinical diagnosis.

Time-Dependent Deformation of Dredged Harbor Mud Used as Backfilling Material

Dredged harbor mud is being increasingly used as backfilling and construction material in harbor development or reorganization to gain financial and operative advantages compared to disposal or ex situ treatment of the often contaminated material. However, the suitability of such soft organic sediments as construction material is unclear. This study presents geotechnical parameters for describing the one-dimensional consolidation based on large-scale long-term and standard oedometer tests using organic harbor mud from the Osthafen (East Harbor) reorganization project in Bremerhaven, Germany. Determined compression indices Cc are in the range of peat, whereas the water permeabilities resemble those of clay. The coefficients of consolidation cv vary between 10−9 and 10−8m2∕s and large coefficients of secondary compression Cα between 0.065 and 0.14 are observed. Differences in parameters from large-scale and standard tests can be explained by volume and time effects likely linked to biogeochemical degredation of organic carbon and the resulting methane degassing. A curve fit based on the Levenberg–Marquardt nonlinear least-squares optimization method confirmed the applicability of the Terzaghi theory in combination with a creep term.