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SMU – an open-source MATLAB package for structural model updating
Published in Hiroshi Yokota, Dan M. Frangopol, Bridge Maintenance, Safety, Management, Life-Cycle Sustainability and Innovations, 2021
Yu Otsuki, Dan Li, Xinjun Dong, Yang Wang
where x∈Rnx is the vector variable; nx is the length of the vector variable; nr is the length of the residual vector. In order to use lsqnonlin, the objective functions in Eq. (3), Eq. (5), and Eq. (6) need to be rewritten in the format in Eq. (7). The current form of each objective function is readily expressed as a sum of squares of residual functions of the optimization variables. Two optimization algorithms available in lsqnonlin are Levenberg-Marquardt (L-M) algorithm (Moré 1978) and trust-region-reflective (TRR) algorithm (Coleman and Li 1996). L-M algorithm is a combination of steepest decent and Gauss-Newton algorithms, and is specially designed to solve nonlinear least-squares problems. TRR algorithm works by determining the step size for each iteration from the solution of quadratic trust-region subproblems.
Fitting models to data
Published in Victor A. Bloomfield, Using R for Numerical Analysis in Science and Engineering, 2018
Fitting to nonlinear models is done in base R with the nls() function, which uses a Gauss–Newton algorithm. The Gauss–Newton method assumes that the least squares function is locally quadratic, and finds the minimum of the quadratic. However, this approach can fail if the starting guess is too far from the true minimum. Therefore, the more commonly used method in the scientific literature for nonlinear least-squares minimization is the Levenberg–Marquardt (LM) method. The LM method combines two minimization methods: gradient descent (steepest descent) and Gauss–Newton. The gradient descent method reduces the sum of squared deviations by updating the unknown parameters in the direction of the steepest gradient of the least squares objective function. The LM method favors the gradient descent method when the sum of squared deviations is large, and favors the Gauss–Newton approach as the optimal value is approached.
Study of Toxic Compounds in River Bottoms at Metropolitan Areas
Published in John M. Bell, Proceedings of the 43rd Industrial Waste Conference May 10, 11, 12, 1988, 1989
Anthony F. Gaudy, Anand Ekambaram, Alan F. Rozich
Bounded Non-linear Least Squares Method. One method is to fit the data using a non-linear least squares technique. We and other investigators have discussed the difficulties with such methods.7–11 Our experience has been that, by providing reasonable boundaries for the numerical values for the biokinetic constants, it is possible to arrive at consistent and reasonably representative values of the constants, which exhibit considerable predictive power when used in engineering equations to predict system behavior. The method has been fully described in the recent literature and the boundary conditions for phenol as substrate were those we have previously reported.7
A hybrid optimization framework for road traffic accident data
Published in International Journal of Crashworthiness, 2021
Bulbula Kumeda, Zhang Fengli, Ghanim M. Alwan, Forster Owusu, Sadiq Hussain
The optimization model was built depending on the correlation of the collected experimental data. The steps of model building are 1. Refining of experimental data, 2. Limiting of objectives and decision (effective) variables, 3. Formulating of the model by statistical deterministic algorithm, and 4. Validation of the model (by comparison between the simulated results from a model with real experimental data within boundary conditions). The software tool (Statistica Version 10) was implemented to obtain a reliable mathematical structure. In this work, the advanced nonlinear least squares model estimation (Levenberg Marquardt method) was implemented with the aid of the software tool (Statistica Version 10). The Levenberg-Marquardt (LM) algorithm is an enhancement of the classic Gauss-Newton technique for solving nonlinear least-squares regression problems. Statistica data miner is an advanced analytics software platform that includes data mining, machine learning, data management, statistics, and text analytics and data visualization processes as well as a variety of, classification, clustering, exploratory techniques, and predictive modeling. Statistica data miner is the most effective, complete system, and user-friendly tool for the whole data mining procedure - from querying to producing the final reports.
An approach to modelling and simulation of epidemics of diseases from pathogens spreading over water distribution systems
Published in Canadian Water Resources Journal / Revue canadienne des ressources hydriques, 2019
Bartłomiej Fajdek, Radosław Pytlak, Marcin Stachura, Tomasz Tarnawski
The proposed approach is aimed at providing analytical tools which could help predict the possible outcome of an epidemic in terms of the number of infected people. A model of an epidemic evolution is useful provided that its parameters reflect the considered epidemic. Estimating epidemic model parameters rarely can be achieved by applying analytical formulae (Murray 2002; Murray 2001); in general one has to use numerical methods. Therefore, our computing environment has the functionality of calibrating epidemic models. We consider two types of calibrating tools. The first uses dynamic optimization numerical methods and is suitable for calibration problems in which only one reference (empirical) trajectory is available. It is a Gauss–Newton method for a nonlinear least squares problem adapted to calibration problems described by ordinary differential equations. If there are several reference trajectories (as is in the considered case study) a more suitable method is that based on multiobjective optimization techniques and genetic algorithms. Both these calibration techniques are discussed in some detail in paper due to the important role they play in building models aimed at forecasting epidemic extents.
A dynamical systems model of intrauterine fetal growth
Published in Mathematical and Computer Modelling of Dynamical Systems, 2018
Mohammad T. Freigoun, Daniel E. Rivera, Penghong Guo, Emily E. Hohman, Alison D. Gernand, Danielle Symons Downs, Jennifer S. Savage
In this section, the specific relative weights ( in the diagonal matrix in equation (32)) are presented for each participant. In addition, the specific initialization points (initial guesses) are also established in this section. It must be noted that given the limited amount of estimation data and the non-convexity of the optimization problem, the non-linear least squares solver becomes increasingly sensitive to relative weights and proper initialization as multiple local minima are expected. To avoid undesired solutions, solver features such as multistart can be used [46].