Several methods exists to find this derivative, including the direct—or classical—method, as well as the popular adjoint method and adjoint stiffness method. The classical method is the most natural in that it finds Du(a)(δa) by solving the variational problem from Theorem 1.2.2 for the direction δa. This leads to the need of solving a forward problem once for each derivative direction δa which becomes quickly intractable for large problems. A much more efficient alternative is the adjoint method which requires only the solution of a single additional forward problem. Recent uses of the adjoint method can be found in [18, 19, 20, 21, 22, 23, 24, 25], and a survey article of first and second-order adjoint methods is given by Tortorelli and Michaleris [26]. The adjoint stiffness method builds on the adjoint equation but linearizes out the parameter a. The derivative can then be found directly with a single matrix equations, versus the adjoint method which has to evaluate Du(a)(δa) for each direction δa. The drawback of the adjoint stiffness method is that by its nature it is only applicable to linear **parameters** a. We are dealing with nonlinearly appearing **parameters**, and as such we can’t make use of the adjoint stiffness method. Instead we will utilize the adjoint method and develop expressions for the **nonlinear** parameter case.

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on an evolving surface Γ(t), for m ≥ 1. We reformulate the equation as a local problem on the semi-infinite cylinder Γ(t) × [0, ∞), regularise the porous medium nonlinearity and truncate the cylinder. Then we pass to the limit first in the trun- cation parameter and then in the nonlinearity. The **identification** of limits is done using the theory of subdifferentials of convex functionals. In order to facilitate all of this, we begin by studying (in the setting of closed Riemannian manifolds and Sobolev spaces) the fractional Laplace–Beltrami operator which can be seen as the Dirichlet-to-Neumann map of a harmonic extension problem. A truncated harmonic extension problem will also be examined and convergence results of the solution to the (untruncated) harmonic extension will be given (these results are used in passing to the limit in the truncation described above). This theory is of course independent of the fractional porous medium equation and will be of use generally in the study of fractional elliptic and parabolic problems on manifolds. We will also consider some related extension problems on evolving hypersurfaces which will provide us with the language to formulate and solve the fractional porous medium equation (amongst others) on evolving hypersurfaces.

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DC motor has been widely used in the engineering field due to its simple structure, outstanding control performance and low cost. In high accuracy servo control system, high control performance of DC motor is needed. The traditional model of DC motor is a 2-order linear one, which ignores the dead **nonlinear** zone of the motor. Unfortunately, the dead zone caused by the **nonlinear** friction would bring great effect to servo systems. Therefore, it is vital to model the dead zone of DC motor accurately in order to improve the performance of servo system. Concerning the **nonlinear** friction of motor, Armstrong-helouvry B. et al. [1] have already conducted thorough research and proposed a **nonlinear** friction model. This friction model relates to the speed and time, and the motion goes four areas from the static friction to the coulomb friction. Moreover, this friction model structure is quite complex, and 7 **parameters** need to be identified. In order to simplify applications and reflect the real **nonlinear** friction of the motor accurately, a simplified friction model was proposed by Cong S. et al. [2], which is expressed as

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The real signals with multiple **parameters** were measured by interferometric fiber sensors. We obtained 7501 samples from each measurement. In the derived model, 𝜆 𝑚𝑖𝑛 = 1400 nm and ∆𝜆 = 0.04 nm. The first step for **parameters** **identification** is to determine parameter N. The signal was then transformed to frequency domain, where spectrum components that have high amplitude values were taken and calculated to estimate 𝑓 𝑖 . This results in the estimation of 𝑁 = 50.Then we built a 10 layers DNN model to estimate 𝐴 𝑖 , 𝑓 𝑖 and 𝜃 𝑖 . Some key hyper-**parameters** of DNN is listed below in table 1, where after 196 epochs of Pre-training, the loss value of second step was around 0.00001.

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Abstract: In this paper new robust **nonlinear** model construction algorithms for a large class of linear-in-the-**parameters** models are introduced to enhance model robustness, including three algorithms using combined A- or D-optimality or PRESS statistic (Predicted REsidual Sum of Squares) with regularised orthogonal least squares algorithm respectively. A common characteristic of these algorithms is that the inherent computation efficiency associated with the orthogonalisation scheme in orthogonal least squares or regularised orthogonal least squares has been extended such that the new algorithms are computationally efficient. A numerical example is included to demonstrate effectiveness of the algorithms. Copyright

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The primary diﬃculty in the **identification** of Hammerstein **nonlinear** systems (a static memoryless **nonlinear** system in series with a dynamic linear system) is that the output of the **nonlinear** system (input to the linear system) is unknown. By employing the theory of aﬃne projection, we propose a gradient-based adaptive Hammerstein algorithm with variable step-size which estimates the Hammerstein **nonlinear** system **parameters**. The adaptive Hammerstein **nonlinear** system parameter estimation algorithm proposed is accomplished without linearizing the systems nonlinearity. To reduce the eﬀects of eigenvalue spread as a result of the Hammerstein system nonlinearity, a new criterion that provides a measure of how close the Hammerstein filter is to optimum performance was used to update the step-size. Experimental results are presented to validate our proposed variable step-size adaptive Hammerstein algorithm given a real life system and a hypothetical case.

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In this study, we considered the Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZK-BBM) equation. We employed the new approach of generalized ( G ′ / G ) -expansion method for the exact solution to this equation and constructed some new solutions which are not found in the previous literature. The method offers solutions with free **parameters** that might be imperative to explain some intricate physical phenomena. This study shows that the new generalized ( G ′ / G ) -expansion method is quite efficient and practically well suited to be used in finding exact solutions of NLEEs. Also, we observe that the new generalized ( G ′ / G ) -expansion method is straightforward and can be applied to many other **nonlinear** evolution equations.

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Choose the th regressor’s covariance matrix as . The important algorithmic **parameters** that need to be chosen appropriately are the maximum repeating times and the termination criterion . To further simplify control, we may simply let the loop repeat times. Then we only needs to set an appropriate value for . We have applied this repeated weighted optimization algorithm as a generic global optimizer in several difficult optimization applications [32], and analysis and empirical results given in [32] have shown that this guided random search algorithm is effective. The need to determine the diagonal covariance matrices of every candidate regressors represents additional computational complexity of the proposed generalized kernel modeling approach, in comparison with the standard kernel method. However, the standard kernel approach would typically require cross validation for specifying the common single kernel variance, and this may involve additional validation data set and can also be computationally expensive. The proposed method does not require cross validation to tune kernel **parameters**, which is an important practical advantage. B. LROLS Algorithm With LOO Test Score for Subset Model Selection

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Inspired by the well-known analysis of variance (ANOVA) expansions [28], [29], a new class of fixed grid WNs is intro- duced in the present study for **nonlinear** system **identification**. In the new WNs, the model structure of a high-dimensional system is initially expressed as a superimposition of a number of functions with fewer variables. By expanding each func- tion using truncated wavelet decompositions, the multivariate **nonlinear** networks can then be converted into linear-in-the-pa- rameter problems, which can be solved using least-squares type methods. The new WNs are, therefore, in structure different from either the existing WNs [18], [24]–[26], [30]–[32] or wavelet mutiresolution models [33], [34]. A wavelet multires- olution model is in structure similar to a fixed grid WN. The former, however, forms a wavelet multiresolution decompo- sition similar to an ordinary multiresolution analysis (MAR), which involves not only a wavelet, but also another function, the associated scaling function, where some additional require- ments should be satisfied. An efficient model term detection approach based on a forward orthogonal least squares (OLS) algorithm, along with the error reduction ratio (ERR) crite- rion [35]–[37] is applied to solve the linear-in-the-**parameters** problem in the present study.

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In [9] the **identification** method is based on the best linear approximation technique using class of Gaussian (-like) signals. In [10], the authors show that there are many local minima, the estimation must to be repeated several times with different starting values to increase the chances of finding a model corresponding to a good local minimum. In [11], an approach based on the standard SVM for regression was presented. The quite poor results obtained in that work highlighted some of the limitations of the method. In particular, only a NFIR model structure was taken into account, which did not perform well since the considered system has a long impulse response. Another problem was given by the high computational time and memory usage, which made it difficult to work with a large amount of data. Several SVM-like approaches (e.g. [12]-[13]), based on the least squares SVM (LSSVM), are characterized by a very high number of **parameters**.

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For each method, accuracy (measured by the difference with an exact solution), computational time and storage are quantities defining the global efficiency of the method. Here, only the computational time (CPU time) is considered. In the following numerical tests, the simulation **parameters** have been tuned in order to require a computational time of one, five or ten minutes, on the same computer, with similarly optimized FORTRAN programs.

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conditioned spectral analysis which yield the underlying linear FRF can corrupt or bias the estimates of the **nonlinear** term **parameters**. The first objective of the current paper has been to suggest some simple strategies for improving the parameter estimates for the **nonlinear** terms. The basic ideas are (a) to bound the frequencies over which the parameter spectra are averaged to exclude high frequency fluctuations, (b) to improve the coefficient estimates by using weighting averages and (c) to estimate the **parameters** in isolation from each other in order to avoid accumulation of errors due to recursion. The overall novelty of this work is possibly in question as previous studies may simply have implicitly adopted these strategies; however, the current authors feel there is some benefit in presenting a summary. Numerical simulations have been carried out in order to study the influence of these strategies on the parameter estimates and some benefit has been demonstrated. The actual origin of the artefacts in the coefficient spectra as a result of the conditioning process and estimation of the linear FRF matrix has not been discussed in any detail and further work is in progress on this matter.

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The concepts related to limit cycles and polynomials are introduced in Section 2. Then we present in Section 3 a two-level approach based on Macaulay matrix to solve for the limit cycle. In Section 4, we review the concept of immersion and its novel use to extend our framework to non-polynomial limit cycle **identification**. Section 5 de- monstrates the entire procedure with examples. Finally, Section 6 concludes this paper.

In this paper Numerical analysis of composite leaf spring is carried out to find out effect of nonlinearities. For Linear and **Nonlinear** Finite element analysis ANSYS software is used. Experimental set up is developed in order to find out load-displacement characteristics of Composite Leaf spring. Load- displacement characteristics are studied in order to find out nonlinearities present in composite leaf spring. The present work deals with the numerical analysis of composite leaf spring with **nonlinear** **parameters**. In this paper Material and Geometric nonlinearities are considered.

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So, for example, in [2], to realize a **nonlinear** PID-regulator, the 2-layer artificial neural network of direct distribution containing 8 neurons of the hidden layer with sigmoid activation functions and one neuron of the output layer with linear activation function (Figure-1) is used.

For future research, we are looking into telecommunication sector, the nano– technology and also optical solitons. In 1973 Hasegawa and Tappert had proposed that soliton pulses could be used in optical communications through the balance of non- linearity and dispersion. They showed that these solitons would propagate according to the **nonlinear** Schrodinger equation (NLS), which had been solved by the inverse scattering method a year earlier by Zakharov and Shabat. At that time there was no capability to produce the fibers with the proper characteristics for doing this and the dispersive properties of optical fibers were not known. Also, the system required a laser which could produce very small wavelengths, which also was unavailable. It wasn’t until seven years later, when Mollenauer, Stollen and Gordon at AT&T Bell Laboratories had experimentally demonstrated the propagation of solitons in optical fibers. The original communications systems employed pulse trains with widths of about one nanosecond. However, there was still some distortion due to fiber loss. This was corrected by placing repeaters every several of tens of kilometers. As the width of the available pulses was decreased, the spacing of the repeaters was increased. In the mid 1980’s it was proposed that by sending in an additional pump wave along the fiber, the dispersion of a soliton could be halted through a process known as Raman scattering. In 1988 Mollenauer and his group had shown that this could be done by propagating a soliton over 6000 km without the need for repeaters.

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We study the spatial homogenisation of parabolic linear stochastic **PDEs** exhibiting a two-scale structure both at the level of the linear operator and at the level of the Gaussian driving noise. We show that in some cases, in particular when the forcing is given by space-time white noise, it may happen that the homogenised SPDE is not what one would expect from existing results for **PDEs** with more regular forcing terms.

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The 3 rd International Conference on Numerical Analysis in Engineering, Pula u Batam, Indonesia ..[r]